National University of Ireland Maynooth [606273]

National University of Ireland Maynooth
Evanescent Wave ReductionUsing a Segmented Wavemaker
in a Two Dimensional Wave Tank
Iain Keaney
A thesis submitted in partial fulllment for the degree of
Doctor of Philosophy
in the
Faculty of Science and Engineering
Electronic Engineering Department
Supervisor: Prof. John Ringwood
Head of Department: Dr. Ronan Farrell
October 2015

Declaration of Authorship
I, Iain Keaney, declare that this thesis titled `Evanescent Wave Reduction
Using a Segmented Wavemaker in a Two Dimensional Wave Tank' and the work
presented in it are my own. I conrm that:
This work was done wholly or mainly while in candidat: [anonimizat]
Where any part of this thesis has previously been submitted for a degree or
any other qualication at this University or any other institution, this has
been clearly stated.
Where I have consulted the published work of others, this is always clearly
attributed.
Where I have quoted from the work of others, the source is always given.
With the exception of such quotations, this thesis is entirely my own work.
I have acknowledged all main sources of help.
Where the thesis is based on work done by myself jointly with others, I have
made clear exactly what was done by others and what I have contributed
myself.
Signed:
Date:
i

Abstract
Evanescent waves are created by the wavemaking process during tank test-
ing. They have long been a nuisance for engineers as they contaminate the wave
eld in the tank and result in additional inertial force experienced by a wave-
maker. Evanescent waves are created by the mismatch between the motion of the
wavemaker and the motion of the
uid particles in a progressive wave. To avoid
contamination of test results, often a considerable distance must be left between
the wavemaker and the test area. This space requirement may be costly for small
research groups or companies who wish to have a facility to perform some basic
proof-of-concept tests in-house, but are restricted for space. The initial aim of
this project was to develop a wavemaker which minimised this space requirement
over a large range of frequencies. The exploration into the behaviour of evanes-
cent waves from the point of view of the fundamentals of hydrodynamics has been
very enlightening. It became clear with the discovery of an interference pattern
between the evanescent waves, that this pattern can be optimised to eectively
cancel out the evanescent wave eld. This interference pattern arises from a phase
shift ofradians experienced by some of the evanescent waves, with respect to
the others. The signicance of this in hydrodynamics is that it explains the exis-
tence of negative added mass. The application for this knowledge far out reaches
the topic of reducing the distortion in a wave tank. The ability to minimise the
added mass of a wavemaker has a great deal of potential in both active absorbing
wavemakers and wave energy conversion. For active absorbing wavemakers, the
minimisation of added mass may be useful in the absorption of unwanted waves
which can be particularly troublesome at high frequencies.
The concept of designing the geometry of a wavemaker to simply match the
motion of the
uid particles has long been proposed; however, the diculty with
designing such a wavemaker is that the ideal geometry is frequency dependent.
Hence, a design that eliminates the evanescent waves at one particular frequency
will not be able to do so for other frequencies. An investigation into the design of
a segmented wavemaker is presented here, as its geometry can easily be adjusted
to suit dierent frequencies.
The wavemaker theory for the multi-body problem of the segmented wave-
maker is developed, and a new aspect of wavemaker theory that predicts a phase
ii

shift ofradians in some of the evanescent waves is presented for the rst time.
A hypothesis is put forward, and then investigated, proposing that this phase
shift can be exploited to create an interference pattern that can eectively cancel
out the evanescent waves. The hydrodynamics of the segmented wavemaker were
constrained using the Newton-Euler equations of motion with Eliminated Con-
straints (NE-EC). This approach facilitated a comparison between wavemakers
with multiple degrees of freedom and traditional wavemakers with a single degree
of freedom.
The lengths and strokes of each segment in the wavemaker are optimised to
reduce the distortion caused by the evanescent waves using two approaches. Ap-
proach one follows the traditional ideas and optimises the lengths and strokes of
the segments to best approximate the motion of the
uid particles in a progres-
sive wave. Approach two optimises the lengths and strokes of the segments in
order to minimise the distance between the wavemaker and the testable area in
the tank. Approach two exploits the phase shift in the evanescent waves by nd-
ing the optimal interference pattern that eectively cancels out the evanescent
waves. A comparison between both approaches shows that eectively eliminating
the distortion caused by the evanescent waves is much more achievable by opti-
mising the interference pattern between the evanescent waves, rather than trying
to approximate a progressive wave.
The results for the segmented wavemakers optimised using approach two pre-
dicted that the distortion can be eectively eliminated for a wide range of frequen-
cies using a segment wavemaker consisting of three
aps. A sensitivity analysis
indicates that the performance of the wavemaker is somewhat eected by errors
in the segments strokes, but the overall performance is still better than what has
been developed to date.
iii

Acknowledgements
I would like to thank my supervisor, Prof. John Ringwood, and co-supervisor,
Dr. Ronan Costello, for their time and eorts on this project. The journey my
education has taken me on has certainly presented its share of challenges and I am
forever indebted to my parents for their constant support and encouragement over
the years, and to my partner, Anita, for her endless patience and understanding.
I would like to express my gratitude for the support of the rest of my family,
Corinna, Michael and Ciaran. I would also like to thank Harvey Appelbe for
the opportunities he's given me and for his support of my work. Finally, I am
grateful to Tue Vu, Francesco Fusco, Giorgio Bacelli, Boris Teillant, Tom Kelly,
Josh Davidson and the rest of my colleagues at COER for their friendship and
assistance over the years.
iv

Contents
1 Introduction 1
1.1 Contributions of the thesis . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Thesis layout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2 Wave eld in a tank 6
2.1 Potential
ow theory . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2 Fluid-structure interaction: Wavemaker theory . . . . . . . . . . . 10
2.2.1 Boundary value problem . . . . . . . . . . . . . . . . . . . 12
2.2.2 The general solution to the wavemaker problem . . . . . . 16
2.3 Unique solutions to the dispersion relation . . . . . . . . . . . . . 18
2.4 Evanescent waves . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.4.1 The eect of evanescent waves on the wave eld . . . . . . 21
2.4.2 Phase of evanescent waves . . . . . . . . . . . . . . . . . . 23
2.5 Validation of code . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3 Designs of wavemakers: A literature survey 32
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.2 History of wavemaker theory . . . . . . . . . . . . . . . . . . . . . 33
3.3 Vertical paddle wavemakers . . . . . . . . . . . . . . . . . . . . . 34
3.3.1 Design parameters . . . . . . . . . . . . . . . . . . . . . . 35
3.3.2 Consideration of evanescent waves . . . . . . . . . . . . . . 36
3.3.3 Design curves . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.3.4 Rolling seal design . . . . . . . . . . . . . . . . . . . . . . 38
3.3.5 Double articulated paddles . . . . . . . . . . . . . . . . . . 39
3.4 Double-
ap wavemaker . . . . . . . . . . . . . . . . . . . . . . . . 40
3.4.1 Eect on the evanescent wave eld . . . . . . . . . . . . . 41
3.5 Plunger wavemaker . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.5.1 Performance . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.5.2 Exponentially shaped plunger wavemaker . . . . . . . . . . 45
3.6 Flexible wavemakers . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.6.1 Evaluation of the evanescent wave eld . . . . . . . . . . . 47
v

CONTENTS
3.6.2 Construction of a
exible design . . . . . . . . . . . . . . . 48
3.7 Pneumatic wavemakers . . . . . . . . . . . . . . . . . . . . . . . . 48
3.8 Inclined
ap wavemaker . . . . . . . . . . . . . . . . . . . . . . . 49
3.9 Duck wavemaker . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.10 Porous wavemakers . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.11 Sloped Piston . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.12 Spiral wavemaker . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.13 Numerical wave tanks . . . . . . . . . . . . . . . . . . . . . . . . 53
3.13.1 Spinning dipole wavemaker . . . . . . . . . . . . . . . . . . 53
3.13.2 Boundary element method . . . . . . . . . . . . . . . . . . 54
3.13.3 Boundary collocation method . . . . . . . . . . . . . . . . 55
3.13.4 Conformal mapping . . . . . . . . . . . . . . . . . . . . . . 55
3.13.5 Computational
uid dynamics . . . . . . . . . . . . . . . . 56
3.14 Second order theories . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.15 Control systems and wave basins . . . . . . . . . . . . . . . . . . 58
3.15.1 First and second-order control . . . . . . . . . . . . . . . . 58
3.15.2 Comparison of force vs. position control . . . . . . . . . . 59
3.15.3 Active absorbers in NWTs . . . . . . . . . . . . . . . . . . 59
3.15.4 Directional wavemaking in wave basins . . . . . . . . . . . 60
3.16 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4 Segmented wavemaker design: A multi-body problem 61
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.2 Dening a segmented wavemaker . . . . . . . . . . . . . . . . . . 63
4.3 Hydrodynamics of the segmented wavemaker . . . . . . . . . . . . 64
4.4 Radiation impedance . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.4.1 Radiation damping . . . . . . . . . . . . . . . . . . . . . . 69
4.4.2 Added mass . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.5 Validation of code . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
4.6 Constrained dynamics of a segmented wavemaker . . . . . . . . . 80
4.6.1 Modes of motion of a wavemaker . . . . . . . . . . . . . . 81
4.6.2 Newton-Euler equations of motion with eliminated con-
straints . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
4.6.3 Constrained hydrodynamics . . . . . . . . . . . . . . . . . 85
4.7 Control parameter . . . . . . . . . . . . . . . . . . . . . . . . . . 92
4.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
5 Optimisation of segment length 94
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
5.2 Setting up the optimisation problem . . . . . . . . . . . . . . . . 97
vi

CONTENTS
5.2.1 Optimisation algorithms . . . . . . . . . . . . . . . . . . . 98
5.2.2 Wavemaker constraints . . . . . . . . . . . . . . . . . . . . 99
5.2.3 Dierential evolution algorithm . . . . . . . . . . . . . . . 100
5.2.4 Finding a global minimum . . . . . . . . . . . . . . . . . . 101
5.3 Approach 1: Matching the progressive wave kinematics . . . . . . 101
5.3.1 Tuning the scale factor and crossover probability . . . . . . 102
5.4 Approach 2: Minimising the wave eld distortion . . . . . . . . . 104
5.4.1 Objective function: position of 1% distortion away from
the wavemaker . . . . . . . . . . . . . . . . . . . . . . . . 104
5.4.2 Tuning the scale factor and the crossover probability . . . 105
5.5 Optimisation results: kinematics matching . . . . . . . . . . . . . 107
5.6 Optimisation results: minimisation of distortion . . . . . . . . . . 111
5.7 Comparison of results . . . . . . . . . . . . . . . . . . . . . . . . . 115
5.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
6 Optimisation of segment stroke 117
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
6.2 Tuning the optimisation parameters . . . . . . . . . . . . . . . . . 119
6.3 Case (1): Optimisation of strokes for segments of equal lengths . . 122
6.4 Case (2): Optimisation of strokes for optimised segment lengths . 125
6.5 Analysis of the evanescent wave eld (sample results) . . . . . . . 130
6.6 Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 135
6.6.1 Sensitivity of the distortion level . . . . . . . . . . . . . . 136
6.6.2 Sensitivity analysis: progressive wave height . . . . . . . . 148
6.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
7 Conclusion 162
7.1 Suggestions for future work . . . . . . . . . . . . . . . . . . . . . 165
7.2 Closing remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
Appendix A Results of complete optimised wavemakers 166
vii

Dedicated to my parents,
Michael and Mary Keaney
and to my wife,
Dr. Anita Keaney,
the world will be a better place because of your work.
viii

Chapter 1
Chapter 1
Introduction
The objective of this thesis is to investigate the nature of evanescent waves cre-
ated by a wavemaker in a wave tank. Evanescent waves arise from a mismatch
between the kinematics of a wavemaker and that of the
uid particles in the
progressive wave. This results in the existence of added mass and additional
inertial force, experienced by the wavemaker, due to the
uid it displaces. At
high frequencies the inertial force on a wavemaker due to added mass can be
very large in comparison to the force required to radiate a wave away from the
wavemaker, leading to diculties in absorbing high frequency waves. This issue
is also relevant to the topic of wave energy conversion.
The existence of negative added mass has long been known [1], yet to the
best of the authors knowledge no adequate explanation has been provided as to
why it occurs. Presented in this thesis is a clear explanation as to how negative
added mass is a result of evanescent waves undergoing a phase shift of radians,
relative to the motion of the wavemaker. It is demonstrated that this phase
shift is dependent on the geometry of the wavemaker. Furthermore, the thesis
discusses how only some of the evanescent waves experience a phase shift and
how this phase dierence leads to an interference pattern. It is proposed and
subsequently demonstrated that, by optimising the design of the wavemaker's
geometry, the phase dierence between the evanescent waves can be optimised to
create a destructive interference pattern, ultimately minimising the strength of
the evanescent wave eld. This knowledge can have a signicant impact on the
absorption of waves by wavemakers or wave energy converters. The presented
investigation helps to deepen our understanding of hydrodynamics by providing
an explanation as to why negative added mass exists and how it can be utilised
by optimising the wavemaker's geometry.
The work presented in this thesis is primarily focused around reducing the
distortion caused to the wave eld by evanescent waves. Evanescent waves con-
taminate the region in the wave tank near the wavemaker making test results
1

Chapter 1
unreliable. This leads to the forces experienced by a device tested in a wave tank
being dierent from that intended, leading to unreliable predictions of the power
output of the device if deployed at sea. The distortion due to the evanescent
waves decreases with distance away from the wavemaker. As a rule of thumb, to
avoid the accuracy of test results being compromised, devices are generally tested
a distance of two to three times the still water depth away from the wavemaker.
At the far-end of the tank, space must be provided for absorbing waves to pre-
vent re
ections back towards the test area. Often an active absorbing wavemaker
can be the most space ecient way of preventing re
ected waves travelling back
upstream to the test area. However, active absorbers create the same evanescent
waves as wavemakers do, hence, the same provisions must be provided to avoid
contamination. For large testing facilities this space allowance may not be a prob-
lem, but with the rising interest in wave energy, many smaller research groups
and companies simply cannot aord to house such large wave tanks. Thus, re-
ducing the evanescent wave distortion can be useful in allowing smaller facilities
to house their own wave tanks, ultimately speeding up the development of wave
energy converters.
Tank testing is an invaluable tool for testing how devices will perform and
survive in the ocean environment. Installing and maintaining devices in the
ocean is incredibly expensive, in some cases costing up to e100,000 per day for
installation. Hence, when a device is being installed in the ocean it is vital that
we know how it's going to behave and that it will be able to survive the abuse of
storms. Tank testing allows us to predict how devices will behave in the ocean.
An extensive knowledge of how a device behaves in various sea conditions can be
developed by performing numerous tank tests on scaled models for a fraction of
the cost. This has allowed companies to not only test the performance of their
device, but also optimise the installation process, helping to drive the cost of
installation and operation down. Furthermore, by proving that a device works
well under tank testing conditions, companies can nd it easier to secure grants
or investment to assist with the nancial burden of going to sea.
The motivation for this project originated in ITT Dublin, with a need to build
a wave tank in a very space restricted area. The space required to be able to take
meaningful measurements of the wave eld without them being contaminated
by the evanescent waves was out of the question, so focus shifted to designing
a wavemaker that would permit testing in shorter wave tanks. The tank was
limited to being 0 :6 m deep and 2 :5 m long, which meant that tests could not
be performed within a distance of 1 :2 m from either the wavemaker or the active
absorber, hence, useful tests were impossible. It should be noted that the use for
such a short tank would be very limited, but it could certainly play a role in the
2

Chapter 1
very early concept development of o-shore devices in the same manner as the
AMOEBA tank discussed by [2]. At the same time Omey Labs Ltd., a company
who develops low-end wave tanks for rapid and frequent early stage testing of
wave energy converters, were concerned that some clients may not be able to
house a large wave tank and were interested in nding space saving solutions.
A collaboration was established to design a wavemaker that would reduce the
distortion in the wave tank, for which a segmented wavemaker that consisted of
a number of paddles stacked on top of each other was chosen. The segmented
wavemaker design proposed in [3] and illustrated in Figure 4.1, was picked as it
oers a greater level of control over the wavemaker's geometry.
1.1 Contributions of the thesis
Previous to this thesis the literature surrounding segmented wavemakers [3,
4, 5, 6] was limited to two segment
ap wavemakers and did not provide a de-
tailed analysis into their ability to reduce distortion. To-date there has been no
attempts, to the author's knowledge, to understand how adding more segments
to the wavemaker will aect its performance, nor have attempts to optimise the
geometry of the wavemaker in order to minimise the distortion directly been pub-
lished. Clark and Menken [5] did use a linear regression method to nd the strokes
of the wavemaker that would best approximate the progressive wave. Hyun [4]
also presented an elementary investigation into how the lengths of the segments
aect the inertial pressure on the wavemaker. Neither [4] or [5] focused on min-
imising the distortion of the wave eld caused by the evanescent waves. As well as
studying the performance of the segmented wavemaker, this thesis presents a new
aspect of the well known linear wavemaker theory which predicts a phase shift in
some of the evanescent waves with respect to the other evanescent waves. The
novel contributions to the topics of wavemaker theory and wavemakers presented
in this thesis can be broken down as:
1. The phase shift experienced by some of the evanescent waves is presented
for the rst time in Chapter 2, along with a discussion of why this behaviour
occurs, what aect it can have on the wave eld and how it can be used to
help minimise the distortion.
2. Presented for the rst time in Chapter 4, are the hydrodynamics coecients
of piston and
ap wavemakers with two to ten segments, where the segments
3

Chapter 1
are all equal in length. The hydrodynamic coecients for single segment
piston and
ap wavemakers are also presented for comparison.
3. The optimisation of the theoretical segment lengths in both the piston and

ap wavemakers with three to six segments in order to best approximate the
kinematics of a progressive wave is discussed, and the results are presented
for the rst time in Chapter 5. The optimisation results of the geometries of
both the piston and
ap wavemakers with two segments are also presented.
4. The optimisation of the theoretical segment lengths in piston and
ap wave-
makers in order to best reduce the distance between the wavemaker and the
test area in the tank is presented for the rst time in Chapter 5.
5. The optimisation of the theoretical segment strokes in piston and
ap wave-
makers in order to best reduce the distance between the wavemaker and the
test area in the tank is presented in Chapter 6 and the results of the fully
optimised segmented wavemakers are also presented for the rst time.
Contributions (1) and (2) have already been discussed in the publication:
[7] I. Keaney, R. Costello, and J. V. Ringwood, Evanescent Wave Reduc-
tion Using a Segmented Wavemaker in a Two Dimensional Wave Tank,
ASME 2014 33rd International Conference on Ocean, Oshore and Arctic
Engineering , 2014.
1.2 Thesis layout
In Chapter 2, the well-known linear wavemaker theory is developed by solving
the wavemaker boundary value problem. The wavemaker problem is formulated
using complex amplitude notation, the dispersion relation is also derived and the
unique solutions to the dispersion relation are discussed. The behaviour of the
evanescent waves is discussed and the novel concept that some of the evanescent
waves undergo a phase shift is introduced and investigated.
A survey of the wavemaker designs and techniques that have been developed
and used to-date by researchers is presented in Chapter 3, along with their advan-
tages and limitations. Chapter 3 also discusses other areas related to wavemakers,
such as numerical wave tanks, second-order theory and control systems.
Chapter 4 describes the concept of both the piston and
ap segmented wave-
makers, where all the segments are equal in length. The hydrodynamics for the
4

Chapter 1
multi-body problem of the segmented wavemaker are then developed. The de-
grees of freedom of the multi-body system are reduced by applying a programmed
constraint on the motion of the individual segments, specically describing the
motion of all the segments in relation to the segment closest to the free sur-
face. The hydrodynamics of the constrained multi-body system are derived and
presented as functions of the normalised wavenumber k.
In Chapter 5 the lengths of the segments are optimised using two approaches:
to best approximate the kinematics of a progressive wave and to minimise the
distance between the wavemaker and the testable area in the wave tank. The
optimisation is carried out using the Dierential Evolution (DE) optimisation al-
gorithm which is described in Chapter 5. The DE algorithm's control parameters
are tuned in Chapter 5.
In Chapter 6, the strokes of the segments are also optimised by minimising
the distance between the wavemaker and the testable area in the wave tank.
Again the optimisation in Chapter 6 is carried out using the DE algorithm, for
which the control parameters are again tuned. The interference pattern between
the evanescent waves which helps to minimise the distortion is investigated. A
sensitivity analysis is also presented in Chapter 6 to investigate how sensitive the
performance of the segmented wavemaker in reducing the distortion is to errors
in the strokes of the segments. Finally, the ndings of the thesis are concluded
in Chapter 7 and some future work is suggested.
5

Chapter 2
Chapter 2
Wave eld in a tank
The ocean is one of the most chaotic and inaccessible environments on earth,
thus, carrying out any kind of construction in the ocean can be extremely ex-
pensive. As a result when designing oshore structures, such as coastal defences,
telecommunication cables, oil rigs and oshore renewable energy devices, engi-
neers and scientists try to model the interaction between the structures and their
environments as extensively as possible before installation. Numerous theoretical
and physical modelling techniques have advanced greatly over the past 100 years
for studying such interactions and loads experienced by these structures. These
modelling techniques have made testing the performance and survivability, as
well as designing the installation process of oshore structures, more accessible.
Ultimately, these modelling techniques enabled the growth of industries around
utilities that the world is heavily dependent on today, such as energy and global
telecommunication networks.
Many theoretical models, which will be discussed in Chapter 3, have been
developed that have proven to be very useful when predicting the behaviour of
devices in various sea states. Theoretical models have the advantage over physical
modelling of not requiring a large space for experimenting or the construction
of various test rigs, removing a large nancial and time expense. The most
commonly used model in predicting such behaviour of devices in ocean waves
is linear potential
ow theory; this has proven to be remarkably successful in
accurately predicting the behaviour of devices for a practical range of conditions.
Potential
ow theory can be used to describe the
uid's
ow within the domain
by the conditions that exist on the boundaries of the
uid domain. This is a
signicantly more ecient approach that attempting to solve the entire
uid
domain. The problem of describing wave generation using potential
ow theory
is referred to as wavemaker theory and was rst presented by Havelock [8].
Havelock [8] presented a solution to the wavemaker problem by solving the
Laplace equation for the potential
ow problem of wave generation. To satisfy
6

Chapter 2
the boundary condition of the wavemaker problem, Havelock's boundary value
solution is the summation of an innite number of solutions to the Laplace equa-
tion. Of this innite series, only one solution represents a wave which radiates
to innity while the rest represent local standing waves. Biesel and Suquet [9]
extended Havelock's wavemaker theory further by explicitly deriving analytical
models for the traditional piston and
ap wavemakers, illustrated in Parts (a)
and (b) of Figure 2.1 respectively. In Figure 2.1 the coordinate system is dened
z
xz
x
h hS0S0
l(a) (b)
Figure 2.1: (a) Traditional full draft piston wavemaker. (b) Traditional variable
draft
ap wavemaker.
withxandzdenoting the horizontal and vertical axes respectively, S0is the
stroke of the wavemaker at the still free surface level, his the depth of the tank
andlis the height of the
ap's hinge above the tank
oor. Further advances
in Havelock's wavemaker theory were presented by Suh and Dalrymple [10] who
predicted the multi-directional wave eld generated by a number of wavemakers
side-by-side in a wave basin. The second-order irregular wavemaker theory was
developed by Schaer [11]. More recently, Spinneken [12] derived the force trans-
fer functions for Schaer's [11] second order irregular wavemaker theory. In this
chapter the potential wavemaker theory of Havelock [8] is developed following the
convenient notation of Falnes [13]. This chapter takes a deeper look at Havelock's
[8] solutions than has been presented in other sources and an investigation into
the behaviour of the innite summation is presented, particularly the possible
phases for the summation terms representing the local standing waves, which is
a novel contribution.
The rst objective of this chapter is to present a concise and clear development
of wavemaker theory in order to describe the frequency domain wave eld in terms
of the wave amplitude. This is achieved in three parts:
Section 2.1 denes the gravity wave problem by introducing the concept of
linear potential
ow theory, and discussing the assumptions it makes.
Section 2.2 establishes the conditions at the boundaries of the
uid domain,
7

Chapter 2
in order to develop the wavemaker model essential for the work carried out
and presented in this thesis.
Section 2.3 studies the dispersion relation, the nature of its unique solu-
tions1, how these solutions behave and their consequence to the wave eld.
The second and third objectives are dealt with in Section 2.4, which are to de-
velop a convention for quantifying the distortion caused to the wave eld from
the wavemaking process, and to identify a novel approach to minimising this dis-
tortion. Finally, Section 2.5 presents a validation of the in-house code developed
to generate the results presented in this thesis.
2.1 Potential
ow theory
When considering waves on water, the
uid is usually assumed to be incom-
pressible and irrotational. Applying the law of conservation of mass and the
Laplace equation,
r~ v=@~ v
@~ x+@~ v
@z= 0; (2.1)
on to the
uid enforces the assumption of incompressibility while stating that the
volume of
uid
owing into the domain must equal that
owing out, where vis
the velocity of the
uid in a water wave and a vector is indicated by ~. Laplace's
equation is commonly used for the water wave problem as it has been shown to
hold for a wide range of conditions and allows for the problem to be solved from
the conditions imposed on the
uid at its boundaries, making it computationally
ecient. Potential theory has also shown extensively to be reliable for nonlinear
waves [14] including up to 5thorder Stoke waves [15]. The assumption of irrota-
tionality permits the simplication of expressing the
ow velocity as the gradient
of a scalar potential, , referred to as the velocity potential [13],
~ v=r(x;z;t ): (2.2)
Thus, the Laplace equation, Equation (2.1), requires that,
r2= 0 (2.3)
1The term unique solution refers to any solution of the Laplace equation for a unique value
ofk.
8

Chapter 2
As this thesis is only concerned with the waves radiating from a wavemaker,
henceforth, only the radiated wave eld will be considered. Following the method-
ology presented by [13], a further simplication is made by assuming that the
radiated wave eld can be described as the superposition of waves created by
each of the six Degrees Of Freedom (DOF) of an oscillating body,
=6X
j=1uj'j: (2.4)
In Equation (2.4), the index, j, indicates the DOF the body is moving in and u
is dened as
u(z;t) =u0c(z)ei!t; (2.5)
whereu0is the amplitude of the body's velocity oscillating in the
uid and '
is a complex coecient of proportionality. The DOFs of
oating bodies will
be discussed in greater detail in the next section. The complex coecient of
proportionality, 'j, in Equation (2.4) can be thought of as the velocity potential
generated by a device oscillating in the jthDOF and normalised by the device's
velocity in the jthDOF.
To nd a solution that satises Equation (2.1), the principle of separation of
variables is employed [13],
'(x;z) =X(x)Z(z): (2.6)
Applying the Laplace Equation, Equation (2.1), to Equation (2.6) gives the par-
tial dierential problem:
1
X(x)@2X(x)
@x2=1
Z(z)@2Z(z)
@z2: (2.7)
For this to hold true, both sides of Equation (2.7) must be equal to the same
separation constant with opposite signs [13],
@2X(x)
@x2=k2X(x);
@2Z(z)
@z2=k2Z(z):(2.8)
The solutions to Equations (2.8) is given by [13] as,
X(x) =cxeikx+cxeikx; (2.9)
9

Chapter 2
Z(z) =c+ekz+cekz: (2.10)
It should be noted at this point that if kwere to be imaginary, then Z(z) would
become harmonic and X(x) would be hyperbolic. This is a property that will be
of great importance later in this chapter.
2.2 Fluid-structure interaction: Wavemaker
theory
Wavemaker theory deals with some of the fundamental aspects of marine
hydrodynamics and has lent itself to the development of wave energy converters,
\a good wave absorber, must be a good wavemaker" , (Falnes and Bu-
dal, 1978).
In marine hydrodynamics a
oating body can undergo motion in six DOFs (also
referred to as modes of motion), three translational: surge, heave and sway, and
three rotational: roll, pitch and yaw, all of which are illustrated in Figure 2.2.
For the purpose of simplicity in this study we restrict our consideration to a two
SurgeYawHeave
RollSway
Pitch
Propagating Wavexzy
Figure 2.2: Modes of motion of a free body experiencing incident waves.
dimensional wave tank, as described in Figure 2.3. We dene a coordinate system
where the vertical z-axis points positively upwards, with z= 0 set to the mean free
10

Chapter 2
surface elevation, and the horizontal x-axis is dened as pointing positively along
the
uid domain, with x= 0 set at the wavemaker's mean position, (Figure 2.3).
The 2-D wavemaker operates in surge, i.e., a piston wavemaker in Figure 2.3
Part(b), or pitch, i.e., a
ap wavemaker in Figure 2.3 Part(a). The horizontal
Pitching Motion
Surging Motionxz
xz
(a) (b)S0
S0
Figure 2.3: Illustration of the modes of motion of (a) a bottom hinged
ap
wavemaker and (b) a piston wavemaker.
displacement of the wavemaker along the vertical axis is:
S(z;t) =S0c(z)ei!t; (2.11)
whereS0is the wavemaker's stroke at z= 0. The wavemaker's depth prole,
c(z), is the displacement of the wavemaker over znormalised by the wavemaker's
stroke,S0. For a piston, the horizontal displacement is constant over depth, so
the prole function is given as:
c(z) =(
1 for(hl)<z < 0
0 forz <(hl):(2.12)
For a piston wavemaker lis the vertical distance between the bottom of the
wavemaker and the tank
oor. For a
ap wavemaker the horizontal displacement
decreases linearly towards the pivot point, at which the displacement is zero,
c(z) =(
1 +z
(hl)for(hl)<z < 0
0 forz <(hl);(2.13)
11

Chapter 2
In the case of the
ap wavemaker lis the vertical distance between the tank

oor and the pivot point. When the hinge is above the tank
oor lis positive
and in the case where the wavemaker has a \virtual" hinge below the tank's

oorlis negative. A wavemaker with a virtual hinge requires a second point of
articulation, this wavemaker design will be discussed further in Section 3.3.
2.2.1 Boundary value problem
Laplace equation theory is a boundary value problem for which the solutions
depend on the properties of the boundary conditions. The boundaries at the

uid's interfaces in the presented wave tank problem enforce the condition that
the
uid cannot
ow through those boundaries. The changes in the pressure
distribution on the free surface, caused by the
uctuating elevation of the free
surface, impose a condition that must be applied to the
uid at the boundary so
as to maintain a balance between the
uid pressure on the free surface and the
atmospheric pressure [16].
Dynamic free surface boundary condition
The pressure
uctuation on the free surface, (x;t), due to its oscillating
elevation can be described by Bernoulli's equation [16],

gz@
@t+~ v2
2
z=(x;t)=C(t); (2.14)
wheregis the acceleration due to gravity, trepresents the time variable and, on
the free surface, C(t) equals the atmospheric pressure which we can set to zero.
At this point however, we do not know the value for (x;t), so we approximate
Equation (2.14) with the Taylor expansion about the point z= 0. The truncated
form of Equation (2.14) expanded about z= 0 is,

gz@
@t+~ v2
2
z=0+
g@2
@z@t+1
2@~ v2
@z
z=0= 0: (2.15)
Assuming small amplitude waves allows us to linearise Equation (2.15) by ignor-
ing all the non-linear terms, thus the rst order dynamic free surface boundary
condition [16] on z= 0 can then be expressed as:
@
@t
z=0+g= 0: (2.16)
12

Chapter 2
Kinematic boundary condition
A kinematic boundary condition applies at the surface of the wavemaker,
wherex=S(z;t), the
oor of the wave tank, where z=h, and the free
surface, where z=(x;t). We can dene an arbitrary function, G(x;z;t ) = 0, to
describe the surface of the boundary [16]. By setting G(x;z;t ) to zero, [16], we
can conveniently describe the boundaries as,
G(x) =z+h(x) = 0 (2.17)
on the tank
oor,
G(x;t) =z(x;t) = 0 (2.18)
on the free surface and
G(z;t) =xS(z;t) = 0 (2.19)
on the wavemaker's surface. Since the
uid on the boundary moves with the
boundary, the material derivative of the boundary's surfaces with respect to time
is zero, i.e., if we move with the surface, the surface does not change [16],
DG(x;z;t )
Dt=@G
@t+@
@x@G
@x+@
@z@G
@z= 0
G(x;z;t)=0: (2.20)
Dening a normal vector, ~ n, as a unit vector on the boundary's surface and
pointing normally into the
uid domain,
~ n=rG
jrGj; (2.21)
Equation (2.20) can be simplied to:
@G
@t=~ v:rG=~ v:~ njrGj
G(x;z;t)=0; (2.22)
and rewriten as,
~ v:~ n=@G
@t
jrGj
G(x;z;t)=0; (2.23)
Following [16], by applying Equation (2.23) separately to Equations (2.17), (2.18)
and (2.19) the kinematic boundary conditions on each surface can be derived.
13

Chapter 2
Bottom boundary condition
To derive the bottom boundary condition we start by substituting Equa-
tion (2.17) into Equation (2.23). Given that Equation (2.17) is independent
of time, the right hand side of Equation (2.23) goes to zero. Inserting Equa-
tion (2.17) into Equation (2.21), the normal vector is given as,
~ n=@h(x)
@x~i+~kr
@h(x)
@x2
+ 1: (2.24)
Recalling Equation(2.2) and inserting Equation (2.24) into Equation (2.23) gives,
~ v:2
664@h(x)
@x~i+~kr
@h(x)
@x2
+ 13
775=@
@x~i+@
@z~k
:@h(x)
@x~i+~k
= 0; (2.25)
which can be rewriten as a general bottom boundary condition [16],
@
@z=@
@x@h(x)
@x
z=h: (2.26)
Ifh(x) is constant the bottom boundary condition, Equation (2.26), becomes,
@
@z= 0
z=h: (2.27)
Kinematic Free Surface Boundary Condition
In order to derive the kinematic free surface boundary condition the nor-
mal vector to the free surface is given by inserting Equation (2.18) into Equa-
tion (2.21),
~ n=@
@x~i+~kq@
@x
2+ 1: (2.28)
14

Chapter 2
Inserting Equations (2.18) and (2.28) into Equation (2.23) and using Equa-
tion (2.2) we can write,
~ v:2
664@
@x~i+ 1~kr
@h(x)
@x2
+ 13
775=@
@x@
@x+@
@zr
@h(x)
@x2
+ 1=@
@tr
@h(x)
@x2
+ 1
z=(x;t):(2.29)
Then multiplying through provides the kinematic free surface boundary condition
[16],
@
@z=@
@t+@
@x@
@x
z=(x;t): (2.30)
As we are only interested in the rst order solution, Equation (2.30) is lin-
earised in the same manner as the dynamic free surface boundary condition,
Equation (2.16). By expanding Equation (2.30), using Taylor expansion, about
the position z= 0 and ignoring all non-linear terms, the linear kinematic free
surface boundary condition is,
@
@t=@
@z
z=0: (2.31)
For convenience, the dynamic and the kinematic free surface boundary condi-
tions, Equations (2.16) and (2.31) respectively, can be combined by dierentiating
Equation (2.16) with respects of time and substituting into Equation (2.31), giv-
ing a combined free surface boundary condition independent of (x;t) [13],
@2
@t2+g@
@z
z=0=
!2+g@
@z
z=0= 0: (2.32)
Wavemaker boundary condition
To derive the boundary condition on the surface of the wavemaker we rst
nd the normal vector to the wavemaker's surface by inserting Equation (2.19)
into Equation (2.21) giving,
~ n=1~i@S
@z~kq
1 +@S
@z: (2.33)
Substituting Equations (2.19) and (2.33) into Equation (2.23) and multiplying
through, as we did for the kinematic free surface boundary condition in Equa-
15

Chapter 2
tion (2.30), gives the boundary condition [16],
@
@x=@S
@t@
@z@S
@z
x=S(z;t): (2.34)
Again, expanding Equation (2.34) about the point x= 0 and ignoring all the
nonlinear terms provides the linear wavemaker boundary condition,
@
@x=@S
@t
x=0: (2.35)
By substituting Equation (2.11) and (2.4) Equation (2.35) can be writen more
simply as [13],
c(z) =@'
@x(2.36)
2.2.2 The general solution to the wavemaker problem
The complete solution to the Laplace equation, Equation (2.1), is a superpo-
sition of all the unique solutions 'n,
'=NX
n=0Xn(x)Zn(z); (2.37)
whereNis the number of unique solutions. To nd the coecients of Z(z) and
X(x), we follow the methodology used by [13] and apply the bottom boundary
condition, Equation (2.27), to the depth function Equation (2.10), at z=h,
@Z(h)
@z=kc+ekhkcekh= 0: (2.38)
This allows us to eliminate one of the coecients of Z(z),
c+=ce2kh: (2.39)
Substituting Equation (2.39) into Equation (2.10) gives,
Z(z) =c[e2khekz+ekz] =cekh[ek(h+z)+ek(h+z)]: (2.40)
Using the identity, cosh x=ex+ex
2, we have the hyperbolic function:
)Z(z) = 2cekhcosh[k(h+z)]: (2.41)
16

Chapter 2
With a certain amount of foresight and following [13] convention, the coecient
will be denoted as N1=2,
)Z(z) =N1
2cosh[k(h+z)]: (2.42)
The coecient Ncan be found by normalising Equation (2.42)
1
hZ0
hjZ(z)j2dz= 1; (2.43)
)N=2kh+ sinh 2kh
4kh
: (2.44)
It should be noted that since Equation (2.44) is independent of the wavemaker's
depth prole, c(z), it applies to all wave geometries. Since we are only concerned
with waves radiating in the positive direction away from the wavemaker, waves in-
cident on, or travelling behind, the wavemaker are ignored satisfying the radiation
condition discussed by [13, 16]. Then by dropping the subscript Equation (2.9)
becomes,
X(x) =ceikx: (2.45)
The coecient c, sometimes referred to as the Biesel coecient, after F. Biesel
who rst derived it for a piston and
ap wavemaker [13], can now be found by
applying the wavemaker boundary condition, Equation (2.36), to Equation (2.6)
c(z) =@'
@x
x=0=1X
n=0@Xn(0)
@xZn(z): (2.46)
Multiplying across by Z
m(z), where the index mindicates the relevant solution
'm, anddenotes the complex conjugate, then integrating over the depth of the
wave tank gives:
Z0
hc(z)Z
m(z)dz=1X
n=0@Xn(0)
@xZ0
hZ
m(z)Zn(z)dz=@Xm(0)
@xh; (2.47)
where the integral on the right hand side is the orthogonal condition, [13], and
remembering that@Xn(0)
@x=ikc, the Biesel coecient, c, is found as a function
17

Chapter 2
of the wavemaker's depth prole,
c=1
ikhZ0
(hl)c(z)Z(z)dz: (2.48)
2.3 Unique solutions to the dispersion relation
For the purpose of deriving the dispersion relation, the depth function, Equa-
tion (2.10), is normalised by setting Z(0) = 1. This leads to the depth function,
Equation (2.10), being expressed as [13],
Z(z) =cosh[k(h+z)]
cosh(kh); (2.49)
then applying the free surface boundary condition, Equation (2.32) gives,
!2Z(0) =g@Z(0)
@z)!2=gksinh(kh)
cosh(kh)=gktanh(kh); (2.50)
)!2
gk= tanh(kh): (2.51)
Equation (2.51) is referred to as the dispersion relation as it links the wave's
angular frequency, !, to the wavenumber which, as it turns out, is the separa-
tion constant k. Previously, in Section 2.1, we discussed the possibility of the
wavenumber being imaginary, in which case the dispersion relation becomes [13]:
!2
gmn=tan(mnh); (2.52)
wherek=imis the imaginary wavenumber. Plotting both Equations (2.51)
and (2.52) against the normalised wavenumber khin Figure 2.4, it is evident
from the intersection of the blue and green lines that only one solution for Equa-
tion (2.51) and hence only one real value for kcan exist. However, the red lines
intersect the blue line in Figure 2.4 at three dierent points; in fact, if we were
to extendkhto innity we would see that there is an innite number of solutions
to Equation (2.52), and therefore an innite number of imaginary wavenumbers
[16].
Referring back to Equations (2.9) and (2.10) it can be seen that a real value for
18

Chapter 2
Figure 2.4: Solutions to the dispersion relation.
krepresents a wave which is a sine function of x, this is called a progressive wave as
it propagates along the x-axis. However, when the wavenumber is imaginary, the
sine function is dependent on zinstead, shown by Equation (2.53). At this point
it is convenient to adapt the notation mnfor thenthwavenumber, where m0=ik0
represents the progressive wave. With this in mind the functions describing the
solution'are written in a more general form. The depth function becomes [13],
Zn(z) =N1
2ncos[mn(h+z)]; (2.53)
where
Nn=2mnh+ sin 2mnh
4mnh1
2
: (2.54)
The Biesel coecients proportional to the amplitudes of the waves, Equation (2.48),
can be written as [13],
cn=1
mnhZ0
(hl)c(z)Zn(z)dz: (2.55)
Finally, inserting Equations (2.45) into Equation (2.37) and superimposing
19

Chapter 2
all possible solutions gives the core equation that describes the behaviour of the
waves, [13]:
'=1X
n=0cnZn(z)emnx: (2.56)
The rst term of the innite sum in Equation (2.56) is the progressive wave.
All subsequent terms, n1, correspond to the imaginary wavenumbers, which
appear on the surface as standing waves and decay in amplitude exponentially
with distance away from the wavemaker, and are hence called evanescent waves.
2.4 Evanescent waves
Evanescent waves can contaminate the test area in a wave tank distorting
the wave eld from the intended sea state; however, this is generally avoided
by leaving a distance of two to three times the still water depth between the
wavemaker and the test area since we know that the evanescent waves decay
over distance away from the wavemaker (Equation (2.56)). In theory, evanescent
waves could be eliminated by designing a wavemaker so that the depth prole
equates the depth function of the progressive wave [17],
c(z) =cosh[k(h+z)]
cosh(kh): (2.57)
This would cause the Biesel coecients in Equation (2.55) of all the evanescent
waves to vanish. This technique has been successful in removing the evanescent
waves in a numerical wave tank [18], however, building such a physical wavemaker
to eliminate evanescent waves for a range of frequencies has not been possible to
date. This brings us to the rst hypothesis being investigated in this thesis which
was proposed by [5]:
Hypothesis 1:
The closer the depth prole of the wavemaker, c(z), matches that of the
progressive wave, the smaller the amplitude of the evanescent waves,
and hence the lesser the distortion caused by evanescent waves.
It should be noted that the wavemaker depth prole in Equation (2.57) is
dependent on the wavenumber of the progressive wave, hence, it would only be
20

Chapter 2
capable of eliminating evanescent waves when generating progressive waves at the
wavenumber the wavemaker is tuned for.
2.4.1 The eect of evanescent waves on the wave eld
The wavemaker's surface
When the evanescent waves are superimposed to the progressive wave, Equa-
tion (2.56), the resultant
uid motion approximates the motion of the wave-
maker's surface. This can be seen in Figure 2.5, where the horizontal velocity
component of the
uid produced by a piston wavemaker at x= 0 is plotted over
the depth of the tank. What is evident from Figure 2.5 is that, as more evanes-
0 0.2 0.4
v/(S0ω)-1-0.9-0.8-0.7-0.6-0.5-0.4-0.3-0.2-0.10z/h(a)
0.24 0.25 0.26
v/(S0ω)-1-0.9-0.8-0.7-0.6-0.5-0.4-0.3-0.2-0.10z/h(b)
0.248 0.25 0.252
v/(S0ω)-1-0.9-0.8-0.7-0.6-0.5-0.4-0.3-0.2-0.10z/h(c)
Figure 2.5: Horizontal
uid velocity at x= 0 including; (a) only the progressive
wave term, (b) the rst ten terms and (c) the rst fty terms of the truncated
velocity potential generated by a single piston system with a unit stroke, i.e.,
S0= 1 m. [!= 4 rad/sec, h= 0:6 m andkh= 1:1818]
cent waves are included in the truncated series in Equation (2.56), the closer the
motion of the
uid matches that of the wavemaker. Maguire [17] demonstrated
numerically that if the wavemaker's motion matched that of the
uid in a pro-
gressive wave, shown in Figure 2.5 Part (a), the evanescent wave eld vanishes.
21

Chapter 2
Free surface elevation
The main focus of this thesis is how much the evanescent waves distort the
free surface. Rearranging Equation (2.16) and carrying out the derivative, the
free surface can be expressed as [13],
(x;t) =1
g@
@t
z=0=i!
g[]z=0: (2.58)
For convenience the results in this thesis have been evaluated in terms of the
complex amplitude of the free surface elevation, ^ , which is dened so that,
(x;t) = ^ei!t: (2.59)
For a wave generated by a piston wavemaker, the free surface elevation normalised
by the wavemaker's stroke S0, at phases of 0 and , is plotted in Figure 2.6
against the distance xaway from the wavemaker; the velocity potential has been
truncated to include only the rst fty terms. What is highlighted in Figure 2.6
0 1 2 3
x/h-0.2-0.15-0.1-0.0500.050.10.150.2η/S0(a)
Phase = 0
0 1 2 3
x/h-0.2-0.15-0.1-0.0500.050.10.150.2η/S0(b)
Phase = π
Figure 2.6: Free surface elevation created by a piston wavemaker with a phase of
(a) 0 and (b) , calculated including the rst 50 terms of the velocity potential.
[!= 12 rad/sec, h= 1:2 m andkh= 17:6147]
is the nature of the evanescent waves as standing waves rather than transverse
22

Chapter 2
waves, which they can be misconceived as. The evanescent waves result in the
mean position, over a wavelength, of the free surface to be o-set from z= 0.
An extreme wave was chosen for Figure 2.6 as it best demonstrated the nature
of how evanescent waves aect the free surface.
Distortion to the free surface elevation
We can assess the aect of the evanescent waves on the wave eld by the
distortion caused to the free surface elevation. This is dened as the amplitude
of the free surface elevation due to the evanescent waves, ^ evan, as a percentage
of the free surface elevation amplitude due to the progressive wave, ^ 0,
Distortion =j^evanj
j^0j100
1: (2.60)
The free surface elevation due to the evanescent waves, ^ evan, is found by eval-
uating Equation (2.58) for only the summation terms in Equation (2.56) that
represent evanescent waves, i.e., where
'evan=1X
n=1cnZn(z)emnx: (2.61)
Again, the complex amplitude notation is dened so that,
evan= ^evanei!t: (2.62)
The level of distortion caused to the wave eld presented in Figure 2.6 by the
evanescent wave eld, is plotted against the normalised distance, x=h, away from
the wavemaker in Figure 2.7. The distortion at x= 0 caused by a piston wave-
maker is shown as a function of khin Figure 2.8.
2.4.2 Phase of evanescent waves
Each evanescent wave has a wavelength n, which corresponds to the imagi-
nary wavenumber, mn, of the evanescent waves,
n=2
mn: (2.63)
The dispersion relation, Equation (2.52), imposes the condition
(+ 0:5)<h< (+ 1); (2.64)
23

Chapter 2
0 0.5 1 1.5 2 2.5 3
x/h05101520253035Distortion [%]
Figure 2.7: Distortion of the free surface from a piston wavemaker, including the
rst 50 terms of the velocity potential [ != 6rad=sec ,h= 0:6 m andkh= 2:2512].
on the imaginary wavelength n, whereis some positive integer. It can be
seen from Figure 2.4 that if this was not the case, the value!2
mngwould be zero or
negative, which is not possible. Subsequently, integrating the function Zn(z) over
the depth of the tank will always give a positive value for any value of n. The
Biesel coecients, cn, for a wave eld generated by a single piston wavemaker,
i.e.,c(z) = 1, is an example of this,
cn=1
mnhZa
bZn(z)dz; (2.65)
wherea= 0 andb=h. The limits z=aandz=bare the top and bottom of
the wavemaker. However, this does not hold for wavemakers with dierent depth
proles. The reason for this is, as the amplitude of the evanescent wave changes
over depth, it can cause the Biesel coecient to be either positive or negative.
This results in the phase of the evanescent wave being shifted by radians. An
example of this is the
ap wavemaker, where the horizontal velocity components
due to a selected few evanescent waves are plotted in Figure 2.9 over the depth
of the wave tank. Comparing the results shown in Figure 2.9 to the correspond-
ing results for a piston wavemaker in Figure 2.10, the evanescent wave's phase
24

Chapter 2
0 2 4 6 8 10 12
kh020406080100120Distortion [%]
Figure 2.8: Distortion at x= 0 of the free surface caused by a piston wavemaker,
including the rst 50 terms of the velocity potential, where h= 0:6 m.
shift can be identied in Parts (a), (c), (e) and (g). It is the superposition of
the waves in Figure 2.10 that provide the approximation of the piston prole
in Figure 2.5. Not all evanescent waves experience this phase shift, as evident
from Parts (b), (d), (f) and (h) in Figure 2.9. Superimposing the rst fty terms
of the velocity potential, Equation (2.56), generated by a
ap wavemaker and
evaluating the distortion to the wave eld, Equation (2.60), with h= 0:6 m and
kh= 2:2512, in Figure 2.11 it is clear that the distortion is not a monotonic func-
tion ofx. The interference pattern causes the distortion in Figure 2.11 to vanish
at approximately x=h= 0:1 away from the wavemaker and then immediately in-
creases again with x. Similarly, Figure 2.12 shows that the distortion is also not
a monotonic function of frequency or relative depth, kh. This behaviour is due
to an interference pattern occurring from superimposing the natural exponential
functions in Equation (2.56) which have dierent exponents and both positive
and negative coecients. These ndings are in agreement with those of [19] and
[11], whose results show the summation of the evanescent waves' Biesel coe-
cients, Equation (2.55), change from negative to positive over frequency for
ap
wavemakers. Though both [19] and [11]'s results show this behaviour, neither of-
fer an explanation as to why this behaviour occurs or the aect it can have on the
25

Chapter 2
-0.5 0 0.5
v/(S0ω)-1-0.50z/h(a) 2nd term
-4 -2 0 2 4
v/(S0ω) ×10-3-1-0.50z/h(b) 3rd term
-0.05 0 0.05
v/(S0ω)-1-0.50z/h(c) 4th term
-1 -0.5 0 0.5 1
v/(S0ω) ×10-3-1-0.50z/h(d) 5th term
-0.02 -0.01 0 0.01 0.02
v/(S0ω)-1-0.50z/h(e) 6th term
-4 -2 0 2 4
v/(S0ω) ×10-4-1-0.50z/h(f) 7th term
-0.01 -0.005 0 0.005 0.01
v/(S0ω)-1-0.50z/h(g) 8th term
-2 -1 0 1 2
v/(S0ω) ×10-4-1-0.50z/h(h) 9th term
Figure 2.9: The horizontal velocity component of the
uid due to the (a) 2nd,
(b) 3rd, (c) 4th, (d) 5th, (e) 6th, (f) 7th, (g) 8thand (h) 9thterms of the velocity
potential generated by a
ap wavemaker hinged at the wave tank's
oor with a
unit stroke, i.e.,S0= 1 m [h= 0:6 m,!= 1 rad/sec and kh= 0:2499].
26

Chapter 2
-0.02 -0.01 0 0.01 0.02
v/(S0ω)-1-0.50z/h(a) 2nd term
-4 -2 0 2 4
v/(S0ω) ×10-3-1-0.50z/h(b) 3rd term
-2 -1 0 1 2
v/(S0ω) ×10-3-1-0.50z/h(c) 4th term
-1 -0.5 0 0.5 1
v/(S0ω) ×10-3-1-0.50z/h(d) 5th term
-5 0 5
v/(S0ω) ×10-4-1-0.50z/h(e) 6th term
-4 -2 0 2 4
v/(S0ω) ×10-4-1-0.50z/h(f) 7th term
-4 -2 0 2 4
v/(S0ω) ×10-4-1-0.50z/h(g) 8th term
-2 -1 0 1 2
v/(S0ω) ×10-4-1-0.50z/h(h) 9th term
Figure 2.10: The horizontal velocity component of the
uid due to the (a) 2nd,
(b) 3rd, (c) 4th, (d) 5th, (e) 6th, (f) 7th, (g) 8thand (h) 9thterms of the velocity
potential generated by a piston wavemaker with a unit stroke, i.e.,S0= 1 m
[h= 0:6 m,!= 1 rad/sec and kh= 0:2499].
27

Chapter 2
wave eld, probably because it was not the focus of either study. Consequently,
0 0.5 1 1.5 2 2.5 3
x/h00.20.40.60.811.21.41.61.8Distortion [%]
Figure 2.11: Distortion to the free surface against distance, x, away from the
wavemaker, generated by a
ap wavemaker hinged at the wave tank's
oor, with
h= 0:6 m,!= 6 rad/sec and kh= 2:2512.
reducing the distortion in the tank due to the evanescent waves is not necessarily
a matter of trying to create a wavemaker that imitates a progressive wave, which
is not practically possible. Instead we can propose the novel hypothesis:
Hypothesis 2:
The distortion of the wave eld, and more specically the distance of
1% distortion from the wavemaker, can be minimised by developing a
multi-body wavemaker, which is designed to maximise the destructive
interference between the evanescent waves.
The distance of 1% distortion is dened as the distance from the wavemaker
to the point at which the distortion has decreased to 1% and does not increase
above 1% for further increases in xand is denoted in the rest of the thesis as X1%.
The concept of a multi-body wavemaker will be described in detail in Chapter 4.
We aim to achieve this in two ways:
28

Chapter 2
0 2 4 6 8 10 12
kh010203040506070Distortion [%]
Figure 2.12: Distortion at x= 0 of the free surface caused by a
ap wavemaker
hinged at the wave tank's
oor, including the rst 50 terms of the velocity po-
tential, where h= 0:6 m.
1. Optimising the upper and lower edges of each body in the wavemaker which
are given as limits of the integral in Equation (2.65). This study is presented
in Chapter 5.
2. Optimising the stroke amplitude of each individual body in the wavemaker
system to achieve an depth prole, c(z), for the wavemaker that minimises
the distance of 1% distortion from the wavemaker, X1%. This study is
presented in Chapter 6.
Hypothesis 1 and 2 do not strictly agree with each other. Hypothesis 1 sug-
gests that the distortion caused by the evanescent waves can be minimised by
designing a wavemaker's geometry so that its kinematics simulates the kinemat-
ics of the
uid in a progressive wave. On the other hand, Hypothesis 2 proposes
to design the geometry of the wavemaker so as to optimise the phase shifts in
the evanescent waves to cause the greatest level of destructive interference be-
tween evanescent waves, hence, minimising the distortion. Both hypotheses will
be investigated in Chapters 5.
Though it is outside the scope of this thesis a simple experiment could validate
the existence of the interference pattern between the evanescent waves, and thus
29

Chapter 2
validate the phase shift experienced by some of the evanescent waves. By calcu-
lating the free surface elevation along the xaxis at a given frequency generated by
a
ap wavemaker, similar to the free surface elevation shown in Figure 2.11, and
generating an identical wave in a wave tank, the predicted free surface elevation
can be compared to measurements of the free surface elevation taken in the tank,
at specied locations using a wire gauge.
2.5 Validation of code
To validate that the in-house code used to generate the results in this thesis
evaluated the wave eld correctly, the Biesel coecients, c0andcn, are calculated
by the in-house code and are compared with the expressions for the Biesel co-
ecients derived by [17] (also discussed in [13]). For a single piston wavemaker
the Biesel coecients predicted by both the in-house code and Maguire's expres-
sions [17] are plotted as functions of khin Figures 2.13. The Biesel coecients
for a single
ap wavemaker, hinged on the wave tank
oor, also predicted by
both the in-house code and Maguire's expressions [17] are plotted as functions
ofkhin Figures 2.14. In both Figures 2.13 and 2.14, c0is presented in Part (a)
whileP49
n=1cnis presented in Part (b). Figures 2.13 and 2.14 demonstrate that
the wave eld is being evaluated correctly by the in-house code as the results
it generates are identical to those predicted by the expressions derived by [17].
Both sets of results are identical as they are both derived using linear potential
wavemaker theory. In Part (b) of both Figures 2.13 and 2.14 the summation term
was truncated at the 50thterm of the innite summation in Equation (2.56).
30

Chapter 2
Figure 2.13: Comparison of the Biesel coecients, (a) c0, (b)P49
n=1cn, for a
single piston wavemaker calculated by the in-house code and those derived by
[17], where h= 0:6 m.
Figure 2.14: Comparison of the Biesel coecients, (a) c0, (b)P49
n=1cn, for a single

ap wavemaker, hinged on the wave tank's
oor, calculated by the in-house code
and those derived by [17], where h= 0:6 m.
31

Chapter 3
Chapter 3
Designs of wavemakers: A
literature survey
3.1 Introduction
Over the past hundred years wave tank testing facilities have been iteratively
improved as new technologies emerged. Where once engineers were limited to
wavemakers which oered little control over the sea-states generated, today we are
spoiled by the use of rolling seals and eective active absorbers to create precise
reproducible sea-states in wave tanks. The development of Numerical Wave Tanks
(NWTs) has also helped to signicantly improve the iterative prototyping process
of devices. This chapter aims to present a survey of the wavemaker designs that
have been developed to-date, what their purpose is and what their limitations
are.
Section 3.2 provides a background of the development of dierent aspects in
wavemaker theory. Sections 3.3 to 3.12 look at various designs of wavemakers,
what their purpose is and how they perform. The chapter then goes on to discuss
other topics that are relevant to wavemakers. Section 3.13 discusses dierent
techniques that have been used for developing NWTs. Section 3.14 considers
second-order wavemaker theory and Section 3.15 provides a brief discussion on
the use of control systems in wavemakers.
32

Chapter 3
3.2 History of wavemaker theory
Wavemaker theory was rst presented by Havelock [8], who proposed a solu-
tion to the boundary value problem of the wavemaker for an incompressible and
irrotational
uid. Havelock's solution required the superposition of an innite
number of waves in order to satisfy the boundary condition on the wavemaker,
shown in Equation (2.35). This superposition is modelled by the innite sum-
mation series in Equation (2.56), where each term in the series corresponds to
an unique solution to the dispersion relation, discussed early on by [20], Equa-
tion (2.52).
The wave eld generated during the transient starting and stopping stages of
the wavemaker's motion was predicted by Madsen [21], who adopted the velocity
potential presented by Kennard [20] which assumes the motion of the wavemaker
starts from rest. Madsen [21] found that during the transient stages of the wave-
maker's motion, waves are created with larger wave heights than intended. The
eect of the larger wave at the end of a wave train is easily dismissed by simply
ignoring the results from the end of the time series. However, a larger wave at
the start of the wave train can be problematic, as the amplitudes of the waves
following it taper towards the target amplitude. This eect has been shown to be
more pronounced when the wavemaker starts from an extreme point [21]. Mad-
sen [21] was able to minimise this problem by programming the wavemaker with
a soft start, i.e., slowly increasing the amplitude of the paddle's motion towards
the targeted stroke amplitude. Madsen [21] found very good agreement between
their predictions and experimental results when they extended the linear transient
wavemaker theory of [20] to the second-order.
Ursell et. al. [22] examined the limitations for which the assumptions of linear
potential theory remain valid. Ursell et. al. [22] found that for a wave steepness
range of 0:002H=0:03 the experimental results for the wave height were,
on average, 3 :4% lower than the predictions of linear theory, with a scatter of 3%
about the average. For greater wave steepness values of 0 :045H=0:048,
the average error in the experimentally observed wave heights was 10% below the
predicted values [22]. In a wave tank which had a constant depth of h= 10 feet in
front of the wavemaker and then a constant depth of hf= 5 feet after a gradual
transition area, illustrated in Figure 3.1, [14] examined the validity of various wave
theories for wave heights between 0 :5 ft (0:05 m) and 1 :6 ft (0:49 m). Chakrabarti
[14] found that, after averaging the experimental results of the wave lengths that
transitioned to shallower waters, airy wave theory (linear wave theory) holds for
0:56 ft/sec2h=T23:2 ft/sec2.
Although surface tension eects are negligible in linear wave theory, [23] pre-
33

Chapter 3
lhhfxz
Figure 3.1: A wave tank with a gradual transition between two regions of constant
still water depths.
sented a theory to account for surface tension. When predicting the wave eld
generated by a porous wavemaker paddle, [24] found that, using the model de-
veloped by [23], an evanescent wave eld was predicted when the wavemaker's
angular frequency was zero. Chwang [24] provided the following explanation for
this behaviour: \In the absence of surface waves, …, the
uid simply piles up
in front of the wavemaker plate" . Perhaps a more appropriate explanation for
the eect [24] observed is simply capillary action, which is a well known eect of
surface tension on a
uid's free surface near a solid.
Many sources of literature use a parameter referred to as the wavemaker's
gain to assess the performance of the wavemaker. In line with most authors, in
this thesis the gain is dened as the ratio of the progressive wave height to the
amplitude of the wavemaker's stroke,H
S0. The stroke of a piston or
ap is dened
as the displacement of the paddle at the still water level from its mean position
to its extreme position. Some other author's denitions dier slightly, such as,
a
S0orS0
a, whereais the amplitude of the progressive wave, i.e., H= 2a.
3.3 Vertical paddle wavemakers
Flap and piston wavemakers are certainly the most common types of wave-
makers. Their popularity can be attributed to their simple design, the repeatabil-
ity of the sea states they create, that they can be analytically modelled and that
they allow feedback absorption to be very eectively implemented. Many sources
in the literature treat the
ap and piston wavemakers as two separate designs,
though they are closely related. For instance, consider the depth prole of the

ap illustrated in Figure 2.1 Part (b) and described by Equation (2.13); if the
34

Chapter 3
pivoting point of the
ap was set so that l=1, then the paddle's depth prole
would be equivalent to that of a piston, Equation (2.12). The present section
is dedicated to discussing the generalised design of vertical paddle wavemakers,
where the traditional bottom hinged
ap and the piston wavemakers are consid-
ered to be specic paddle designs with points of pivot at l= 0 andl=1,
respectively.
3.3.1 Design parameters
The two dimensional paddle wavemaker problem has two main design param-
eters which can be set: (a) the position of the point of pivot [25] and (b) the draft
of the paddle [26], indicated in Figure 3.2. The selection of both parameters de-
pends on the intended bandwidth of the waves the wavemaker will primarily be
used to generate, as well as space restrictions that may exist in the wave tank.
Two other design parameters can be considered: the submergence of the paddle
(when the wavemaker does not pierce the free surface) and the angle between the
paddle and the horizontal axis, indicated later in Figure 3.11. The submergence
is mostly only relevant when considering the hydrodynamics of a submerged oscil-
lating body. Paddles, for which the mean position is not normal to the horizontal
axis, have been used for studying the generation of waves created by impulses in
near-shore regions due to earthquakes [27] and are considered later in Section 3.8.
The wavemaker's point of pivot is generally chosen so that the motion of the
wavemaker approximates that of the
uid in a progressive wave, thus reducing
the contamination of the wave eld by evanescent waves. It is generally accepted
that piston wavemakers are ideal for generating low frequency, or shallow water
waves, as the amplitude of the
uid's motion in such waves is almost uniform
over depth. Similarly, for modelling high frequency, or deep water waves,
aps
where 0lhare advantageously used. When modelling waves within an
intermediate bandwidth, a
ap can be designed to have a virtual hinge below the

oor of the tank, in this case lbecomes negative. The concept of the virtual hinge
(mentioned in Section 2.2) is achieved by having a second point of articulation
at the bottom of the paddle.
35

Chapter 3
xz
d
l
Pivot point
Figure 3.2: A wavemaker with a virtual hinge and a draft of d.
3.3.2 Consideration of evanescent waves
Biesel [9] extended the theory developed by Havelock [8] dening the depth
prole of both a
ap and a piston wavemaker of full draft, presented in Equa-
tions (2.12) and (2.13), respectively. This allowed for the velocity potential coef-
cients to be explicitly derived, Equation (2.55). Hyun [25] extended the wave-
maker theory further by deriving an expression for the velocity potential created
by a
ap where the pivot point is positioned somewhere between h<z < 0.
Hyun [25] considered the aect that the hinge height has on the amplitude
of the evanescent waves by plotting the normalised horizontal displacement of
the
uid particles on the surface of the wavemaker and the inertial pressure on
the wavemaker, over the depth of the wave tank for a number of dierent hinge
heights. The hydrodynamic inertial pressure and forces acting on the wavemakers
are proportional to the amplitude of the evanescent waves, a relationship that will
become more clear from Equation (4.4) and Section 4.4. Hyun's [25] analysis did
not quantify the distortion to the intended wave eld, caused by the evanescent
waves, but it did show that
aps with shorter drafts reduce the inertial pressure,
over depth, on the paddle at high frequencies. In general, [25] found that as the
36

Chapter 3
frequency decreased, the lowest inertial pressure was achieved with increasing
drafts. Hyun [25] remarked that
aps hinged closer to the free surface require
much greater strokes in order to create waves of the same height; this can lead
to the small angle assumptions and the condition of linearity being violated and
suggests that a non-linear model should be employed. Both [25] and [28] found
that in deep water waves, the hydrodynamic force on the wavemaker is largely
due to the evanescent wave eld. For intermediate and shallow water waves, the
progressive wave is the main contributor to the hydrodynamic force, although the
evanescent waves still have a notable in
uence [25, 28]. The predictions of [28]
were supported by the experimental results from the companion paper [29], which
had an average experimental error of 12 :5% about Hudspeth's [28] prediction for
the wavemaker gain.
3.3.3 Design curves
Probably the rst practical guide to designing a
ap or piston wavemaker
was given by [26], who presented the design curves of the gain and hydrody-
namic forces on a
ap and piston wavemaker of full draft and a piston wave-
maker with variable draft when generating regular (or monochromatic) waves.
Gilbert [26] also provided similar design curves for generating irregular (or non-
monochromatic) sea states, evaluating the maximum values for the wavemaker's
stroke and driving force using the Cartwright method [30]. This work was ex-
tended by [28], who provided the same design curves for a
ap wavemaker with a
variable draft. The inertial force curves presented by [26] and [28] do not follow
a monotonic behaviour over frequency. Neither study provides an explanation
for this behaviour, however, from the discussion in Section 2.4.2, it would appear
that this is due to the interference pattern between the evanescent waves. It
has been well documented that full draft piston wavemakers have greater gains,
producing larger wave heights for a given stroke, as they displace twice as much

uid as a full draft
ap wavemaker [16, 25, 26, 31, 32].
37

Chapter 3
3.3.4 Rolling seal design
To solve the problem of energy leaking around the sides of the paddle, [33]
designed a rolling seal gusset to stop the
ow of water down the sides of the paddle.
The new gusset design had advantages for
ap wavemakers as it meant that
water could be removed from behind the paddle, eectively creating a dry-back

ap wavemaker. The dry-back
ap made it easier to implement force feedback
control, as it removed the unpredictable hydrodynamic forces acting on the back
of the
ap caused by splashing water. Preventing the
ow of
uid through the
sides of the wavemaker also meant that accurate and repeatable directional waves
could be generated by a bank of
aps positioned side-by-side. The use of seals
on
aps has been quite successful at improving the wavemaker's accuracy and
repeatability. Patel and Ionnaou [32] found that for waves with small steepness
values, i.e., 0 :0056<2a
<0:02, whereis the wavelength, the experimental
results of the wavemaker's gain were within 2% of the predictions of potential
theory. For steeper waves, i.e., 0 :02<2a
<0:046, [32] found the experimental
results of the gain were within 3% of those predicted by potential theory. Patel
and Ionnaou [32] suggested that these errors were due to the linear treatment of
the re
ected waves' in
uence on the free surface elevation.
Designing seals for piston wavemakers is however much more dicult. For a
piston wavemaker, [32] reported that the energy leakage around the bottom of
the piston was a signicant issue with the potential theory over-predicting the
experimental results by an average of 9 :6%. Despite using seals, [22] found that
linear potential theory over-predicted the experimental results by 10% for a piston
wavemaker. The wedge-style piston, Figure 3.3, is often used as an alternative to
seals [32], as it eliminates any splashing that would occur behind the paddle and
thus eliminating the hydrodynamic forces from behind the wavemaker, however,
it does increase the total inertia of the piston. A drawback to piston wavemakers
is that they often require linear guides which introduce more friction into the
mechanical system.
38

Chapter 3
xz
d
Figure 3.3: Design of a wedge-style piston wavemaker.
3.3.5 Double articulated paddles
As mentioned earlier in Section 3.3.1, for intermediate water waves a
ap
with a virtual hinge can be used, Figure 3.4. For monochromatic waves both
articulation points will have the same frequency and phase. This can help the
xz
Top Actuator
Bottom Actuator
Figure 3.4: Design of a double actuated paddle.
paddle to better match the kinematics of the
uid in the progressive wave [34].
An extension of the theory derived by [9] for a
ap wavemaker with a virtual
hinge was presented by [34]. By providing a driving signal to the articulation
39

Chapter 3
point at the bottom of the paddle, as well as the top, a wavemaker with a double
actuated paddle can be well suited for generating both high and low frequency
waves creating a buer-tweeter type wavemaker. Such a wavemaker could be
benecial in generating two peaked Ochi-Hubble spectra [35]. When generating
monochromatic intermediate waves, [35] found that their experimental results
for the wavemaker's gain was in good agreement with the predictions of linear
potential theory. However, when generating low frequency waves, the theory over-
predicted the experimental results; [35] attributed this to the waves breaking in
front of the paddle.
3.4 Double-
ap wavemaker
The concept of a double-
ap wavemaker, such as that use at the MARINTEK
Ocean Laboratory and illustrated in Figure 3.5, was investigated by [4], who
used linear wavemaker theory to predict the wave eld generated by the two-
body problem of the double-
ap wavemaker. The angle between the top
ap and
αTα1
α2
l1
l2hxz
Figure 3.5: Design of a double-
ap wavemaker.
the vertical axis is denoted as Twhile the angle between the bottom
ap and
vertical axis is denoted by 2. The acute angle between the
aps, indicated in
Figure 3.5, is 1, giving:T=1+2. The height of the hinges, from the
oor of
the wave tank, are denoted as l1andl2for the top and bottom
aps, respectively.
40

Chapter 3
Hyun [4] found that setting l2=h=2 and increasing the dimensionless parameter
l1=hfrom 0:5 (i.e., a single
ap) to 0 :9 resulted in the height of the generated
wave decreasing as less
uid is being displaced by the wavemaker.
3.4.1 Eect on the evanescent wave eld
Hyun [4] found that, for short and intermediate waves, increasing the angle
ratio,1
2, reduces the inertial pressure, which is caused by a reduction the ampli-
tude of the evanescent waves. Hyun [4] also reported that for
h= 0:25;0:5;1, the
distortion of the wave eld, due to the evanescent waves, at x= 0 decreases as
1
2increases. However, for longer waves, namely
h= 2, [4] found that the single

ap conguration, i.e.,1
2= 0, provided the lowest level of distortion. This result
is hardly surprising given the discussion in Section 3.3.1. Clark and Menken [5]
employed a linear regression method to nd the most suitable angles for 1and
2, in order to minimise the error between the depth proles of the progressive
wave and the double-
ap wavemaker. The double-
ap wavemaker was designed
with the parameters: h= 16 ft,l1= 12:5 ft andl2= 7 ft. Using the linear
regression method, [5] found that for
h<2:2 the top
ap was in phase with
the bottom
ap. However, for
h>2:2 the top
ap was radians out of phase
with the bottom, i.e., they moved in opposite directions. At
h= 2:2, [5] found
that the linear regression method determined the most suitable value for 1to be
zero. When analysing the performance of the double-
ap wavemaker on generat-
ing long waves, [5] observed reductions in the amplitude of the distortion to be
in the range of 33% to 65% of that caused by a single
ap wavemaker. Murdey
[6] also reported that the linear regression method used by [5] for optimising
the strokes of the
aps, provides better reduction of the distortion than setting
2= 0,1= 0 and2=1.
3.5 Plunger wavemaker
A plunger wavemaker refers to the heaving body type wavemakers illustrated
in Figure 3.6. If the plunger is rectangular in shape, Figure 3.6 Part (a), then
the principle of separation of variables, Equation (2.6), can be used [36] and the
potential
ow problem can be solved analytically [37], in the same manner as
the
ap and piston wavemakers. However, this cannot be done for the wedge
or cylinder shaped plungers illustrated in Figure 3.6 Parts (b) and (c) [38, 39],
respectively. Both of these designs must be modelled numerically [36]. It is
41

Chapter 3
(a) (b) (c)
Figure 3.6: Plunger wavemaker designs.
important to dierentiate between wedge-plunger wavemakers and the wedge-
type pistons described in Section 3.3.4. The geometry of a plunger wavemaker
is described by its draft, d, and its length, b(z), which is a function of depth,
Figure 3.7. Plunger wavemakers have the advantage of being very simple to
xz
db(z)
Figure 3.7: Design of a wedge plunger wavemaker.
construct. They remove the need for awkward rolling seals that have tendencies
to leak, and since all of the wavemaker's mechanical components are contained
above the water, they are easily adjusted and maintained.
42

Chapter 3
Plunger wavemaker in wave basins
Many successful wave tanks have been built using plunging-wedge wavemak-
ers, such as the AMOEBA (Advanced Multiple Organised Experimental BAsin)
[2]; a small circular tank with a diameter of 1 :6 m and fty plunging-wedge wave-
makers around its circumference. Each of the wavemakers in the AMOEBA acts
as an absorbing wavemaker and have proven to be very eective at cancelling
out re
ected waves [2]. Comparing the diraction forces experienced by a model
tested in the AMOEBA to similar tests in a larger wave basin, [2] found that both
sets of measurements were in good agreement. This result indicates that small
wave tanks may prove to be a useful tool for early stage testing of o-shore de-
vices. Another advantage of the plunger wavemaker over both the
ap and piston
wavemakers is that, in a circular or curved tank with multiple wavemakers, the
gaps between the plungers can be very small. This cannot be done when using

ap or piston wavemakers in a curved tank, as slightly larger gaps are required
between the wavemakers to ensure the paddles do not hit each other. As a result
of larger gaps between the wavemakers, even with the use of rolling seals, the
problem of energy leakage increases as does the occurrence of cross waves. The
studies discussed in this section have documented the performance of plunger
wavemakers using a variety of numerical modelling techniques. The merits and
limitations of these numerical methods will be discussed later in Section 3.13.
3.5.1 Performance
Studying the radiated wave eld created by a heaving rectangular wavemaker,
[37] found that by increasing the length, b, of the plunger or decreasing the draft,
d, leads to an increase in the wavemaker's gain. These results are echoed by [40]
who also found that increasing the length-to-draft ratio,b
d, of a wedge-plunger
wavemaker provides greater gain. Lee [37] also found that with increasing length-
to-draft ratio the added mass and thus, the amplitude of the evanescent wave eld,
increased. Given that the wavemaking surface of a wedge-plunger wavemaker is
not vertical, like the piston and
ap wavemakers discussed in Section 3.3 in their
mean position, the wave motion is created in the
uid at dierent positions on
thex-axis over the depth. This leads to the
uid motion not having a constant
phase over a vertical slice through the
uid. As a result, a destructive interference
pattern arises, reducing the height of the progressive waves. This eect is not
noticeable at low frequencies, but becomes very pronounced at high frequencies
when the change in b(z) becomes comparable to the wavelength of the radiated
wave. Evidence of this has been reported by [41] and [42], who showed that
43

Chapter 3
the gain initially increases with frequency until reaching a maximum, it then
decreases with further increases in frequency.
Wang [41] estimated that the evanescent wave eld created by wedge-plunger
wavemakers can have an amplitude of 10% of the progressive wave amplitude at
x= 1:4h, for high frequency waves. However, having only one measurement point
means that the distortion cannot be interpolated for other distances away from
the wavemaker; as Section 2.4.2 suggests, the distortion pattern along the x-axis
cannot be easily predicted. Although not discussed, the distortion measurements
presented by [41] are not monotonic over frequency, indicating the existence of an
interference pattern between evanescent waves in anti-phases with one another,
proposed in Section 2.4.2. Wu [38] found that the water depth has a signicant
aect on the wavemaker gain, with the gain decreasing as the still water depth
increased. Similar observations were also made by [22] for the case of a piston
wavemaker. Mikkola [42] found that increasing the length-to-draft ratio, and
thus the wedge angle, increased the non-linearities in the wave form as the wave
became asymmetrical about the horizontal and vertical axes.
The potential theory model developed by [37] proved to be very reliable for the
special case of the heaving rectangular body, when [37] compared the predictions
of the wavemaker's gain with those of a Boundary Element Method (BEM) and
the experimental results of [43]. Comparing the predictions of the added mass
and radiation damping from the analytical model and the BEM also showed
good agreement. Comparing the conformal transformation method developed
by [41] to Wu's [38] Boundary Collocation Method (BCM), both models are
in good agreement with Wang's [41] experimental results of the wavemaker's
gain forb=d= 0:725 (i.e.,d
b= 1:38). However, for b=d= 0:459 (i.e.,d
b=
2:18) the predictions of [38] diverge and become unreliable when kb > 1:5. The
experimental results presented by [41] are scattered about the predictions of the
conformal transformation method by approximately 6 :5%, which is deemed to be
an acceptable experimental error. Both Wang's [41] conformal transformation
and Wu's [38] BCM over-estimate the experimental results for a wedge-plunger
atb
d= 0:625 (i.e.,d
b= 1:6) presented by [44]. Ellix and Arumgam [44] attribute
these discrepancies to energy leaking around the sides of and behind the plungers.
To address this problem, [45] developed a Boundary Element Method (BEM) with
a wedge-plunger wavemaker that accounted for the eects of energy leakage. The
results showed that Wu's[45] BEM is a more superior model than those presented
by [41] and [38] based on a comparison with the experimental results of [44].
44

Chapter 3
3.5.2 Exponentially shaped plunger wavemaker
An exponentially shaped plunging wedge wavemaker was suggested by [46] to
match the kinematics of a progressive deep water wave in an attempt to elimi-
nate the evanescent wave eld. The prole of such a wavemaker, illustrated in
Figure 3.8, would be dened as:
b(z) =sinh[kt(z+h)]
cosh(kth); (3.1)
whereis a small constant of the same order as the free surface elevation [46].
It should be noted that the kinematics of a progressive wave only approximate
an exponential function in deep water waves and that for shallow water waves
Equation (3.1) resembles the uniform depth prole of a piston. The exponential-
plunger can only be eective at eliminating the evanescent wave eld at its ge-
ometrically tuned frequency, with k=kt. Since the exponential-plunger's wave-
making surface is not vertical, it suers from the same inherent problem as the
triangular wedge-plunger, that is, the
uid motion in a vertical slice will not be
in phase over the depth, leading to signicant loss in the wave height for high
frequency waves.
xz
db(z)
Figure 3.8: Design of an exponentially shaped plunger wavemaker.
45

Chapter 3
3.6 Flexible wavemakers
A number of sources in the literature have discussed the concept of a
exible
wavemaker that can bend to have the same depth prole as a progressive wave
[3, 9, 13, 34, 47, 48], Equation (2.57). This allows the wavemaker to generate
progressive waves without an evanescent wave eld. The depth prole of such a

exible wavemaker, illustrated in Figure 3.9 Part (a), would be set as:
c(z) =cosh[kt(z+h)]
cosh(kth); (3.2)
wherektis the geometrically tuned wavenumber [13]. Inserting Equation (3.2)
xz
xz
(a) (b)xz
(c)
Figure 3.9: Proles of
exible wavemakers, (a): Cosh, (b): Cos and (c): panchro-
matic spectrum.
into the expression for the Biesel coecients, Equation (2.55), and then calculat-
ing Equation (2.56), the terms which represent evanescent waves vanish. How-
ever, the geometry of the
exible wavemaker can only be tuned to one frequency.
In order to create progressive waves at other frequencies without any evanescent
waves, the geometry of the wavemaker would need to be changed. Another design
for a
exible wavemaker was proposed by [13] to generate only one evanescent
wave, with an imaginary wavenumber mn, and no progressive wave. This could
be achieved with a wavemaker that has a depth prole, illustrated in Figure 3.9
Part (b) and dened as:
c(z) =cos[mn(z+h)]
cos(mnh): (3.3)
When generating a spectrum containing more than one frequency, the geome-
try of the ideal wavemaker becomes more complicated. As the dierent frequen-
cies in the spectrum are mixed together with random phases, the depth prole
of the wavemaker will be required to change in real time, and can form a more
46

Chapter 3
complicated curve like that shown in Figure 3.9 Part (c). Obviously, it is not
possible to build such a device, and to the authors knowledge there is no infor-
mation available for such a numerical
exible wavemaker generating panchromatic
spectra.
3.6.1 Evaluation of the evanescent wave eld
Maguire and Ingram [48] investigated wavemakers with depth proles de-
scribed by both Equations (3.2) and (3.3), using analytical wavemaker theory;
for the rest of this thesis these wavemakers will be referred to as the cosh and
cos wavemakers, respectively. Maguire and Ingram [48] found that, at the wave-
maker's geometrically tuned frequency, the added mass of the cosh wavemaker
went to zero, indicating that no evanescent waves were generated; an explanation
of this reasoning is provided in Section 4.4. Similarly, [48] also found that at the
tuned frequency of the cos wavemaker the radiation damping of the wavemaker
went to zero, indicating that no progressive wave was generated; again this rea-
soning is explained in Section 4.4. At either side of the tuned frequencies, [48]
found that the added mass and radiation damping of the respective wavemakers
increased quite rapidly. These results demonstrated that there is little value in
constructing a cosh wavemaker that is geometrically tuned to only one frequency
for the purpose of reducing the evanescent waves. However, the cosh wavemaker
design has a lower added mass than a
ap wavemaker which may prove useful for
the purpose of absorbing waves, particularly at high frequencies. Another eect
of the mismatch between the wavemaker's motion and the motion of the
uid in
a progressive wave are spurious free waves which arise in second-order waves and
panchromatic sea-states, spurious waves will be discussed further in Section 3.14.
As far as the author is aware, no study has been done to date to see how the cosh
wavemaker reduces the eect of spurious waves when generating panchromatic
sea-states.
47

Chapter 3
3.6.2 Construction of a
exible design
A cosh wavemaker, constructed as a cantilever beam and geometrically tuned
to a single frequency, was installed in a wave tank in the Water Research Lab-
oratory at the University of New South Wales [47]. However, no information
is available on the performance of the cantilever wavemaker's ability to reduce
the evanescent wave eld. A design similar to the cantilever wavemaker was dis-
cussed in [9], which involved a
exible membrane actuated at several points over
the depth by dierent sized gears. The depth prole of the wavemaker was con-
trolled by choosing correctly sized gears. Biesel and Suquet [9] reported that this

exible membrane generated waves of a rather pure form. However, the inconve-
nience of changing the gear sizes for every frequency the user wanted to generated
and problems with maintaining the seals proved to be too cumbersome [9, 34].
3.7 Pneumatic wavemakers
The design for a pneumatic wavemaker is shown in Figure 3.10. By inject-
ing pressurised air into the chamber and releasing it through valves, the water
surface in the chamber is forced to oscillate, thus, creating a radiating wave. Es-
sentially the concept of a pneumatic wavemaker is an oscillating water column
wave energy converter operating in reverse. An approximate theory for the wave
eld generated by the pneumatic wavemaker was presented in [49] for generating
oscillating and surging waves.
Although pneumatic wavemakers are easy to construct, they have some in-
hibiting characteristics. Keulegan [49] found that it can be dicult to control the
air pressure when the air is rst injected into the chamber. This can cause some
unwanted
uctuations in the free surface, though it only seems to last for a short
period of time. O'Dea and Newman [50] found that non-uniform distribution of
pressure in the chamber when the air is injected can lead to cross waves. Addi-
tionally, the transfer functions for the pneumatic wavemakers can be non-linear
and dicult to calculate [50].
48

Chapter 3
Outlet
Inlet
Figure 3.10: Design of a pneumatic wavemaker.
3.8 Inclined
ap wavemaker
As mentioned in Section 3.3.1, inclined
ap wavemakers, illustrated in Fig-
ure 3.11, have been used to model waves generated by the impulsive movement
of sloping near shore regions, due to earthquakes [27, 51]. The problem of
the inclined
ap wavemaker has some similarities to the wedge-plunger prob-
lem. Firstly, the principles of separation of variables cannot be employed, so the
problem must be solved numerically. Secondly, since the wavemaking surface is
not vertical at its mean position, then in the same manner as discussed in Sec-
tion 3.5.1, there will be a loss in wave height at high frequencies. In Region 1
of the
uid domain, indicated in Figure 3.11, this behaviour becomes more pro-
nounced as
decreases, and hence, the angle between the
ap and vertical axis
increases. In Region 2 of the
uid domain, indicated in Figure 3.11, the loss in
wave height at high frequencies is quite gentle. The greatest value for the wave-
maker's gain is achieved in Region 2 with
= 70[51]. To solve the problem of
the inclined
ap wavemaker, [27] developed a BEM, while [51] used a BCM.
For a
ap wavemaker, with
= 90, the predictions of both Raichlen's [27]
49

Chapter 3
xz
ΩRegion 1 Region 2
Figure 3.11: Design of a inclined
ap wavemaker.
BEM and Wu's [51] BCM for the wavemaker's gain showed good agreement with
the linear wavemaker theory, although the BCM performed slightly better than
the BEM. Comparing the BEM and BCM to experimental results of a
ap wave-
maker's gain with
= 33 :85, [51] found that Raichlen's [27] BEM predicted
the gain slightly more accurately for lower frequencies, while the BCM under-
predicted the gain. For a
ap angle of
= 45, [51] found that both models
drastically over-predicted the experimental results. Given the importance of ac-
counting for energy leakage [45], the errors of the predictions may be considered
to be within an acceptable bracket.
3.9 Duck wavemaker
Salter [52] proposed a wavemaker based on the Edinburgh duck wave energy
converter operating in reverse, illustrated in Figure 3.12. The device combined the
duck's geometry with a force feedback control system, allowing the wavemaker to
act as an active absorber, cancelling out waves incident on the wavemaker. Since
the duck's wavemaking surface is not vertical, the issue of loss in wave height at
high frequencies may arise. However, bearing in mind the ndings of [51] for the
inclined
ap wavemaker's gain in Region 2 of Figure 3.11, it is likely that the
eect of the destructive interference will be negligible for a duck wavemaker.
50

Chapter 3
Pivot point
Figure 3.12: Design of a duck wavemaker.
3.10 Porous wavemakers
The porous wavemaker can be useful for understanding the eectiveness of
damping zones. Sahoo [53] modelled the wave eld generated by an oscillating
cylindrical wavemaker surrounded by an outer porous cylinder. The aim was
to understand the hydrodynamic forces experienced by a pile protected by a
porous damping zone. Chwang [24] developed an analytical model for a porous
piston wavemaker based on Taylor's [54] model of
ow through a porous medium.
The aim was to understand how the generated wave eld was in
uenced by the
dynamic viscosity, which is described as being similar to the Reynold's number
for
ow passing through the pores. Naturally the more porous the wavemaker is,
the smaller its gain [24].
3.11 Sloped Piston
Two models of a piston wavemaker with a sloped face have been developed by
[55] and [56] in order to estimate the hydrodynamic forces experienced by a sloped
dam during an earthquake. The rst approach was based on the Von Karman's
momentum-balance method [55] and the second was a conformal transforma-
tion method used to solve the linear potential theory problem [56]. Comparing
both models, there are signicant discrepancies in the hydrodynamic pressures
as functions of depth. However, after integrating the pressures over depth, the
51

Chapter 3
hydrodynamic forces predicted by both approaches showed excellent agreement
[56].
3.12 Spiral wavemaker
Dalrymple and Dean [57] developed the linear potential theory for a vertical
cylinder moving in small circular patterns uniformly, Figure 3.13 Part (a), or with
the bottom held stationary like a
ap, Figure 3.13 Part (b). This spiral wavemaker
was designed to model sediment transport by waves approaching a beach at an
angle. Dalrymple and Dean [57] found that the wave crest radiated from the
wavemaker could be described by an Archimedean spiral. The predictions for the
wavemaker's gain were in good agreement with experimental results for kh< 2.
Above this limit
ow separation occurs and the assumption of the
uid being
inviscid becomes invalid [57].
(a) (b)
Figure 3.13: Spiral wavemakers.
52

Chapter 3
3.13 Numerical wave tanks
The results presented in this thesis were generated using the analytical wave-
maker theory developed in Chapter 2 and the hydrodynamic theory developed
later in Chapter 4 which was implemented using an in-house code. The benet
of this approach over using a Numerical Wave Tank (NWT) is that it requires
less computation time; this is signicant when optimising the geometry of the
wavemaker where the wave eld is calculated several thousands of times. For
this reason, NWTs are not directly relevant to the work presented in this thesis.
However, given the extensive use of NWTs in o-shore engineering, the author
felt it appropriate to include a section discussing the dierent approaches used
to develop NWTs, with the aim to help inform any engineers who may be new
to the area.
The use of NWTs can provide a lot of
exibility to researchers when prototyp-
ing and modelling o-shore and coastal structures; designs of devices can be easily
and frequently adjusted to test dierent parameters without signicant additional
nancial costs. A benchmark presented by [58] compared the predictions of four
dierent fully nonlinear BEMs, a second order BEM, a fully nonlinear Finite El-
ement Method (FEM) and a fully nonlinear Finite Volume Method (FVM), for a
wave eld generated by a heaving wedge-plunger wavemaker in a NWT to the ex-
perimental results presented by [59]. The results show that all the fully nonlinear
NWTs agree with the experimental results quite well, however, the second order
BEM shows notable discrepancies in predicting the experimental results. Since
NWTs are not restricted to the same limitations as physical tanks, some authors
have found innovative ways of generating waves to eliminate unwanted eects
like evanescent waves, such as the spinning dipole wavemaker [60], discussed in
Section 3.13.1. The following sections present dierent techniques for modelling
NWTs and wavemakers, and discusses the experiences some researchers have had
using them.
3.13.1 Spinning dipole wavemaker
Clement [60] developed a concept for a numerical wavemaker which resem-
bles a spinning dipole in electromagnetism. The spinning dipole wavemaker can
generate wave elds described by the kinematics of a progressive wave, hence
preventing the generation of evanescent waves. Clement [60]'s spinning dipole
wavemaker also does not interact with incident waves, allowing them to pass
straight through without any re
ections. A damping zone behind the wavemaker
53

Chapter 3
can then be used to prevent the waves being re
ected back towards the testing
area [60].
3.13.2 Boundary element method
An advantage of potential theory is that the
ow at any point in the
uid
domain can be evaluated from the information supplied at the boundaries of the
domain [61]. The BEM creates a mesh along the boundary of the domain and
solves the boundary conditions using a boundary integral equation that can be
formulated by Green's identity [45, 61]. An alternative BEM was developed by
[62], for solving transient and non-linear waves using Cauchy's integral theorem.
Dold and Peregrine [62] claimed that this was less computationally expensive.
BEMs can be very useful for modelling more complex features such as com-
plex bathymetry as well as wave-structure interactions for geometries where the
principle of separation of variables, Equation (2.6), cannot be used [61, 63]. Grilli
and Watts [63] used the BEM developed by [61] to model a wave eld generated
by a landslide at a sloped coastal region. O'Dea and Newman [50] simulated a
directional wavemaker in a three dimensional NWT using WAMIT, a commercial
BEM code. A nonlinear BEM was developed by [64] to model the wave eld
generated from a wedge-plunger wavemaker.
Absorption of unwanted waves in a BEM
Absorbing unwanted re
ected waves is as much of a problem in NWTs as it
is in physical wave tanks. A common technique for absorbing short waves is a
numerical beach. In a BEM, this is a damping zone at the end of the NWT,
achieved by adding a dissipative term to the free surface boundary condition
in the damping zone [65]. Long waves can be removed by using an absorbing
piston wavemaker at the far end of the NWT, without much additional cost in
computation time. Clement [65] developed a coupled absorber, consisting of both
a numerical beach and an absorbing piston. The poorest absorption rate that the
coupled absorber achieved for short and long waves was 96% of the wave energy.
In comparison, the absorption rates for the numerical beach and the absorbing
piston, individually, were as low as 10% of the wave energy when absorbing long
or short waves, respectively [65].
54

Chapter 3
3.13.3 Boundary collocation method
A BCM typically nds an exact solution to a problem's governing equation
and then attempts to nd an approximate solution which satises the boundary
value problem. A BCM developed by [51] found exact solutions for the Laplace
equation as well as the free surface, bottom and radiation boundary conditions.
The boundary condition on the surface of the wavemaker was then solved using
the least-squares method. Wu [45] compared the results from the BEM they
developed to the BCM used by [38] and found good agreement between the results;
although, for higher frequencies the predictions of the BCM were slightly less than
those of the BEM.
Advantages of a BCM
The BCM developed by [51] has two main advantages over the BEM approach.
The rst is that the only region of the model that must be solved numerically is
the wavemaker's surface. This signicantly reduces the computation time in com-
parison to a BEM since there are less nodes. The second advantage of Wu's [51]
BCM is that it can handle the corner point problem better, since the only bound-
ary that the corner node needs to be solved numerically for is the wavemaker's
boundary condition.
3.13.4 Conformal mapping
A conformal mapping approach was used by [41] for modelling the wave eld
generated by a heaving wedge-plunger wavemaker and by [56] for modelling the
hydrodynamic forces on a sloping dam in the event of an earthquake. Comparing
the predictions of Wang's [41] model and those of Wu's [38] BCM and Wu's [45]
BEM to the experimental results of [44] for heaving wedge-plunger wavemakers,
the conformal transformation approach seems to be the least reliable for low
frequencies, but improves for high frequencies.
55

Chapter 3
3.13.5 Computational
uid dynamics
Computational Fluid Dynamics (CFD) evaluates the
uid
ow by solving the
Navier-Stokes equation [66, 67]. In contrast to the boundary value solver's dis-
cussed so far, CFD computes the
ow for the entire
uid domain. This of course
makes it much more computationally expensive. However, with the ever increas-
ing processing power of computers, CFD is becoming more practical, particularly
when the
ow characteristics of interest do not fall under the assumptions of
nonlinear potential theory. All of the authors whose results are discussed in this
section, [66, 67, 68, 69], used the same commercial code, ANSYS CFX, to model
a NWT with a
ap wavemaker.
Guidelines for NWTs
A set of guidelines for modelling a reliable NWT in CFD was presented by
[66]. Silva et. al. [66] found that the schemes used for the temporal and spatial
discretization have an impact on the results as well as the size of the time step
and the mesh renement. Silva et. al. [66] reported that for a reliable simulation,
a resolution along the x-axis of 111 nodes per wavelength and along the z-axis of
10 nodes over the wave height were the minimum spatial resolution required. The
temporal resolution should also be less than 100thof the wavemaker's period [66].
It has been reported that downstream from the wavemaker in CFD NWTs, the
wave height can become attenuated [66, 68]. Finnegan and Goggins [69] found
that for shallow water waves, CFD predicted the wave theory quite well but this
agreement did not hold for deep water waves. The deep water predictions did
improve when the height of the hinge was increased from z=htoz=h=2,
though the comparison still remained poor [69].
3.14 Second order theories
Second order wavemaker theories have been developed by a number of authors.
Perhaps the most signicant contribution was provided by [11], who predicted
the second-order Stokes wave and the spurious free waves in the generation of
regular waves, along with the superharmonics and subharmonics that occur when
generating irregular waves. The second-order Stokes waves are regarded to as
\bound" waves, as they travel with the same speed as the rst-order wave. In
contrast, the spurious free waves, mentioned in Section 3.6.1, travel with a speed
slower than the rst-order wave [31]. Superimposing the rst-order wave and the
56

Chapter 3
Stokes wave produces a wave form where the crests are higher and the troughs
are
atter, Figure 3.14. The spurious free waves arise from a mismatch between
Figure 3.14: Stokes waves.
the boundary condition on the wavemaker and the kinematics of a progressive
wave [31, 11]. The speed dierence between the free wave and the rst order wave
causes the wave form to change in both time and space.
The unwanted free wave can be removed from the wave eld by adding a term
to the wavemaker's rst order driving signal to generate a wave component with
the same frequency and wave height as the free wave but in anti-phase, this will
eectively cancel the free wave. Position transfer functions for suppressing free
waves were derived by many researchers. One of the earlier models was presented
by [70], which was limited to relative depths of h=<O (0:1) and waves where:
H2
h3<8
32: (3.4)
Hughes [31] discusses an extension to [70] for a
ap wavemaker with a pivot point
either above or below the bottom of the tank.
Hudspeth and Sulisz [71] developed a theory to describe the Stokes drift and
return
ow of the
uid particles in a tank for monochromatic waves using a gen-
eralised paddle wavemaker design. Both [71] and [11] highlighted the importance
of accounting for the evanescent wave eld when considering second order wave
theory. Hudspeth and Sulisz's [71] theory was extended in [72] to include bi-
chromatic waves, and found that the strength of the evanescent wave eld was
strongly in
uenced by the combination of frequencies as well as the wavemaker
geometry.
A second-order directional wavemaker theory was presented in [10] which dis-
cussed the importance of suppressing free waves. An extension to Schaer's [11]
complete second-order theory for irregular waves was presented in [73] for a multi-
directional wavemaker. A comparison of the free wave amplitude predicted by the
57

Chapter 3
second-order theories of [74], [70] and [75], to experimental results for monochro-
matic waves, was presented by [76]. Buhr Hansen and Svendsen [76] showed that
Madsen's [70] model provided reasonably accurate results for h= < 0:15, while
Fontanet's [74] results were reasonable for h= > 0:2. Daugaard (1972)'s model
proved to be the most suitable over the entire range 0 :1<h=< 0:65.
3.15 Control systems and wave basins
Unwanted waves in the tank can be problematic as they contaminate the
test area and can be dicult to remove. When waves are radiated, re
ected
or scattered from a device back towards the wavemaker, it is essential that the
wavemaker is able to absorb those incident waves. This can be done by generating
additional wave components which will cancel out the waves that re
ect o the
surface of the wavemaker. Both position and force feedback control systems have
received a considerable amount of attention for this purpose.
A position feedback controller presented in [77] and [78] used the signal from
wave gauges, some distance in front of the wavemaker, in order to predict the
waves incident on the wavemaker. The driving signal to the wavemaker was then
adjusted accordingly. The phase of the feedback signal was corrected to account
for the distance between the wavemaker and the gauges, however, there was still a
signicant level of error between the intended and the actual wave eld travelling
away from the wavemaker.
Force feedback allows for the wave eld at the wavemaker's surface to be mea-
sured directly. Salter [79] took the approach of using a force feedback controller
that measured the hydrodynamic forces acting on the wavemaker. This allowed
the incident wave eld to be determined and thus cancel out any re
ected waves.
3.15.1 First and second-order control
Based on Schaer's [11] second-order wavemaker theory, [12] derived the force
transfer functions for a
ap wavemaker and implemented force-feedback control.
In terms of reducing the amplitude of the spurious free waves, [80] demonstrated,
experimentally, that the second-order control theory developed in [12] performs
remarkably well. When absorbing incident waves, [80] found that the rst-order
wavemaker theory proved to be quite eective, with some discrepancies that in-
crease with frequency.
58

Chapter 3
3.15.2 Comparison of force vs. position control
Newman [81] compared the quality of the wave eld, in terms of the free
surface elevation, created by position and force feedback controllers in directional
wave basins using the commercial BEM code, WAMIT. In order for Newman's [81]
position controller to produce a wave eld of high quality, the waves incident on
the active absorber must be a priori. However, when a structure is placed in the
tank that re
ects, scatters or radiates waves, forecasted knowledge of the incident
wave eld is very dicult. Furthermore, [12] also found that force feedback control
was more eective than the position controller at reducing the amplitude of the
spurious waves.
3.15.3 Active absorbers in NWTs
Spinneken et. al. [82] investigated the use of active absorbers in NWTs to
help reduce the computational cost of a simulation, as active absorbers would
remove the need for a large damping zone. This may be dependent on the type
of numerical code used; for instance, with a BEM where only the boundaries are
meshed, the time required to regenerate the mesh would be quite insignicant.
However, this may not be the case for a NWT developed using CFD. Since CFD
is a time series analysis and the wavemaker boundary condition is not linearised
about the position x= 0, the
uid domain changes shape during the simula-
tion. A considerable amount of additional computation is required to regenerate
the mesh every few time steps in the region near the absorber, this could make
[82]'s approach more computationally expensive than the use of a damping zone.
Spinneken et. al. [82] reported that the active absorbers performed rather well
in a NWT, although the performance did drop when simultaneously generating
and absorbing large amplitude or non-linear waves. Finally, [82] found that when
comparing the free surface elevation during consecutive repeat periods of the
spectrum, there was very little deviation in the measured spectrum, indicating
the eectiveness of the active absorbers and the repeatability of the model.
59

Chapter 3
3.15.4 Directional wavemaking in wave basins
Directional wave generation is achieved by using a bank of wavemakers side-
by-side in a wave basin. By controlling the phase dierence between the paddles,
the angle of propagation of the wave-front can be controlled [79]. Dalrymple [83]
predicted the wave eld at the test area in the wave basin based on the waves
re
ected o the side walls of the tank, as well as the waves travelling downstream
from the wavemaker. Mansard et. al. [84] compared the directional theories
developed in [83] and [85] to experimental results; [84] found that the experimental
measurements were within 10% of the target wave height. More recent work
on directional wavemakers in [86] considered the wavemaker's ability to absorb
uni-directional, multi-directional and focused wave events. Spinneken and Swan
[86] noted that the active absorbing wavemaker performed very well with the
measured spectrum being quite close to the target spectrum. The measured
spectrum did begin to dier from the target for short waves, particularly when
the angle of propagation became large. Spinneken and Swan [86] attributed this
to the nite width of the wavemaker paddles and found that a slight ad-hoc
adjustment of the control signal's transfer functions improved the quality of the
result. A second-order directional wavemaker theory for predicting the spurious
free waves was developed in [87].
3.16 Conclusion
A survey of all previous wavemaker designs has been presented. The advan-
tages and limitations of each wavemaker presented have also been discussed. Flap
and piston wavemakers are generally preferred, as they are simple to construct,
they can be modelled analytically and it is easy to implement force feedback
control with
ap and piston wavemakers.
A number of other topics have also been discussed, such as NWTs, control
systems and higher order wave theories. The use of force feedback controllers
has become very popular due to their success at creating repeatable sea states.
However, the problem of evanescent waves and the tank space required in order to
avoid contamination of the test results is still an issue. The proposed segmented
wavemaker investigated in this thesis is designed to deal with this while still
having the desirable characteristics of the
ap and piston wavemakers.
60

Chapter 4
Chapter 4
Segmented wavemaker design: A
multi-body problem
4.1 Introduction
Considering the wavemaker theory developed in Chapter 2 and the discussion
on previous designs of wavemakers' geometries in Chapter 3, it is clear that an
optimal solution is a wavemaker for which the geometry can be easily adjusted
to the optimal conguration for each frequency. Intuitively, a more accurate
approximation of a curve than the stepwise and ane approximation provided
by the piston and
ap respectively, can be achieved by a number of line segments.
Naito [3] proposed this concept in the form of a segmented wavemaker, illustrated
in Figure 4.1 Part (a), which consisted of a number of pistons stacked on top of
each other. The next logical step in the design of a wavemaker is to have the
segments operate as
aps, shown in Figure 4.1 Part (b). Note that the segmented

aps in Figure 4.1 Part (b) are shown to be connected to each other by a hinge.
The use of
aps provides two advantages over the segmented piston design; the
x xz z
(a) (b)
Figure 4.1: (a) Piston and (b)
ap segmented wavemakers.
61

Chapter 4
rst is that it can approximate smooth curves more accurately and the second is
that it would be easier to construct seals to prevent leaking between the segments.
It is expected that the more segments in the wavemaker the better it will be at
approximating any smooth curve.
Hypothesis 2 in Section 2.4.2, suggests that reducing the distortion of the
wave eld may be achieved by controlling the phases of the evanescent waves, via
the wavemaker's geometry, rather than trying to match the motion of the
uid
in a progressive wave with the motion of a wavemaker. This takes advantage of
the
exibility in the segmented wavemaker's geometry, allowing us to study the
potential of the interference pattern between the evanescent waves. Although
[3] proposed the solution of the segmented wavemaker, it does not provide any
analysis of how the system performs. An example of a two segment wavemaker
is that of the dual-
ap [4, 5] used in the MARINTEK laboratory. The dual-

ap wavemaker has been quite useful in creating a wide bandwidth of waves;
however, its aect on the distortion levels are not well understood. The case of the
two segment
ap wavemaker studied in this thesis diers from the MARINTEK
dual-wavemaker, [4, 5], by allowing the bottom segment to have a virtual hinge
below the tank
oor, in a similar manner to the ve segment
ap wavemaker in
Figure 4.1 Part (b).
For irregular sea states, the depth prole of the
uid's horizontal motion is a
superposition of each wave component in the polychromatic spectrum, resulting
in the
uid's depth prole changing continuously over time. For example, the
form of the depth prole can change over one repeat period of the spectrum
from a simple monotonic function of depth to a complicated form that is more
dicult to approximate as previously discussed in Section 3.6 and illustrated in
Figure 3.9, Part (c). The segmented wavemaker gives an obvious advantage for
generating irregular sea states since its geometry can be changed in real time.
However, this falls outside of the scope of this thesis, along with any non-linear
eects that may be induced by the design.
The notation used to describe the segmented wavemaker is explained in Sec-
tion 4.2. The forces that act on the multi-body segmented wavemaker are outlined
and discussed in Section 4.3. The hydrodynamics, including the radiation damp-
ing and the added mass, of a multi-body system such as the segmented wavemaker
are developed in Section 4.4. The hydrodynamic theory presented in Section 4.4
is validated in Section 4.5. In Section 4.6, the DOFs of the system are reduced
to allow the hydrodynamics of the dierent wavemaker systems to be compared.
Section 4.7 explains how each segment's stroke is programmed to approximate
the horizontal velocity prole of a progressive wave. Finally, the chapters ndings
are concluded in Section 4.8.
62

Chapter 4
4.2 Dening a segmented wavemaker
The multi-body system described by a segmented wavemaker is discussed in
this chapter. The segments are labelled from top to bottom as 1 to N. A variable
Adescribing the qthsegment is denoted by the convention, Aq. The variable A
describing some coupling term between the qthandpthsegments is denoted Aqp.
In a wavemaker with Nsegments, where each segment has Mdegrees of freedom
(DOF), the total number of DOFs in the system is @=NM . This thesis will
consider both of the segmented wavemaker congurations in Figure 4.1. In the

ap conguration, Figure 4.1 Part (b), a constraint is imposed on the segments so
that the ends of neighbouring segments must meet. The in
uence of the number of
segments in the system will also be considered. In both congurations, the wave-
maker spans the entire depth of the tank. The potential
ow theory developed in
Chapter 2, specically Equation (2.56), can describe the wave eld generated by
a system of Noscillating bodies by simply superimposing the wave components
generated by each DOF. In this chapter, the hydrodynamics of the segmented
wavemaker systems are considered in order to provide an understanding of the
force requirements of each system. The hydrodynamic coecients are given as
matrices of dimensions @@ , so it can be dicult to compare the hydrodynamic
coecients of wavemakers with dierent numbers of DOF. To make the wave-
makers' hydrodynamics comparable, their hydrodynamic terms are expressed as
single scalar functions of kh. This is achieved by employing the Newton-Euler
equations of motion with Eliminated Constraints (NE-EC) to reduce the DOFs
of the wavemaker, in a similar manner as the DOF's of multi-body wave energy
converters were reduced by [88].
The results presented in this chapter consider segmented wavemakers where
all the segments are equal in length and that the strokes of the segments are
prescribed by the kinematics of the progressive wave, Equation (2.57), i.e., the
qthsegment has a stroke of
Sq=cosh[k(h+(q1)h
N)]
cosh(kh): (4.1)
63

Chapter 4
4.3 Hydrodynamics of the segmented
wavemaker
The hydrodynamics of a wavemaker are similar to those of any rigid body in
marine hydrodynamics. The only dierence is that the wavemaker is provided
with a driving force from some actuator, whereas, in marine hydrodynamics, we
predict how a free body will behave when a wave is incident upon it. Since the late
1970's, force control has become the preferred control strategy for wavemakers as
it provides two advantages [79]:
1. Combining the force and velocity signals from the wavemaker's sensors al-
lows us to control the rate at which energy is passed to the wave, which is
useful for the eective absorption of incident waves.
2. It provides a reliable statistical average of the wave eld across the face of
the wavemaker, which is also key for the eective absorption of incident
waves.
Fr;1
Fr;2
Fr;3
Fr;4M1a1
M2a2
M3a3
M4a4K1x1
K2x2
K3x3
K4x4FD;1
FD;2
FD;3
FD;4xz
Figure 4.2: Forces acting on the wavemaker.
The forces acting on a segmented wavemaker are expressed in a force vector,
denoted F, where the qthcomponent of the vector represents the force experienced
64

Chapter 4
by theqthsegment in the wavemaker. In the force vector,
Fr=0
BBBBBBB@~Fr;1
…
~Fr;q
…
~Fr;N1
CCCCCCCA; (4.2)
the subscript rindicates the radiation force, i.e., the force experienced by the
wavemaker due to the wave eld being generated, and the qthcomponent is the
radiation force vector experienced by the qthbody in a system of Nbodies in the
same direction as the wavemaker's velocity. The vectors and matrices for which
the components are values representing a multi-body system are writen in bold.
Other force vectors describing a multi-body system will take the same form as
Equation (4.2) and are indicated in Figure 4.2. In the case of the segmented
wavemaker, the equation of motion describing the wavemaker is [17]:
FD=Fr+m x+Kx: (4.3)
The driving force, FD, inertial force, m x, and spring force, Kx, are vectors of
the same form as Equation (4.2). The hydrodynamic pressure acting on the
wavemaker system [13],
p=i!; (4.4)
can be integrated over the surface of the body to get the radiation force, Fr;i,
which the wavemaker experiences as it generates a wave eld [13],
Fr;i=ZZ
SpnidS;
=i!ZZ
SrnidS:(4.5)
In Equation (4.5), Sis the wavemaker's wet surface area, of which dSis an
innitesimal area and r='juj.
65

Chapter 4
4.4 Radiation impedance
We now introduce the impedance of the wavemaker due to the radiating wave
[13],
Zij=Fr;i
uj; (4.6)
whereujis the velocity of the wavemaker in the jthmode. The term Zijis the
radiation impedance of the body oscillating in the jthmode, while being acted
on by a wave being generated by a body oscillating in the ithmode. Hence, the
multi-body system's radiation impedance is a square matrix of size @@ ,
Z=0
BBBBBBB@Z11


Z1q


Z1@
…………
Zp1


Zpq


Zp@
…………
Z@1


Z@q


Z@@1
CCCCCCCA: (4.7)
The radiation impedance can be expressed in terms of the velocity potential by
substituting Equation (4.5) into Equation (4.6),
Zij=i!
uiZZ
SrnidS: (4.8)
Using the wavemaker boundary condition, Equation (2.36), gives [13],
Zij=i!
uiZZ
Srci(z)dS: (4.9)
Inserting Equation (2.56) into Equation (4.9) and working though the algebra,
the integral in Equation (4.9) becomes a product of the Biesel coecients c0iand
c0j, thus showing that the radiation impedance matrix, Z, is symmetrical [13].
When we insert the complete solution for the velocity potential, Equation (2.56),
into Equation (4.8), it becomes clear that the radiation impedance is complex
and can be expressed as,
Zr=R(!) +iX(!); (4.10)
whereR(!) is the radiation damping term due to the energy radiating from the
wavemaker in a progressive wave. The radiation reactance, X(!), is related to
66

Chapter 4
the inertia the body experiences as it moves through the
uid,
X=!m(!): (4.11)
This inertia, m(!), referred to as added mass, is due to the inertia force the
wavemaker experiences as it \pushes" the
uid. From our understanding of Equa-
tion (4.8), it is clear that the radiation damping is a result of the progressive wave,
while the added mass is due to the evanescent waves,
R(!) = RefZg;
m(!) =1
!ImfZg:(4.12)
After some extensive algebra the radiation damping and added mass for the
general case of the piston and
ap wavemakers illustrated in Figure 4.3 can be
found. This long process is omitted here, the reader is directed to [13] and [17]
for a more detailed discussion on the derivations, instead we will just state the
expressions here. The radiation damping and added mass for a piston are given
respectively as,
R(!) =!
kh4kh
2kh+ sinh(2kh)sinh[k(h+a)]sinh[k(h+b)]
k2
;(4.13)
m(!) =1X
n=14(sin[mn(h+a)]sin[mn(h+b)])2
m2
n(2mnh+ sin(2mnh)); (4.14)
and for a
ap wavemaker as,
R(!) =!
kh4kh
2kh+ sinh(2kh)sinh[k(h+a)]sinh[k(h+b)]
k
+asinh[k(h+a)]bsinh[k(h+b)]
k
+cosh[k(h+b)]cosh[k(h+a)]
k22
;(4.15)
67

Chapter 4
m(!) =1X
n=14
(2mnh+ sin(2mnh))sin[mn(h+a)]sin[mn(h+b)]
mn
+asin[mn(h+a)]bsin[mn(h+b)]
mn(hl)
+cos[mn(h+a)]cos[mn(h+b)]
m2
n(hl)2
:
(4.16)
For both the piston and
ap wavemaker, aandbare the positions on the zaxis
of the wavemakers top and bottom, respectively, i.e., for a traditional piston or
bottom hinged
ap, a= 0 andb=h. It should be noted that 0 a > b and
bh. From Equation (4.12) we see that the radiation damping matrix, R, and
a
bxz
a
bxz
(a) (b)
Figure 4.3: Illustration of (a)
ap and (b) piston wavemakers which are neither
surface piercing nor full-draft.
the added mass matrix Mare of the same form as the radiation impedance matrix,
Equation (4.7). Substituting Equations (4.10) and (4.11) into Equation (4.6) gives
the radiation force in terms of the hydrodynamic coecients, R(!) andm(!), as
Fr(t) =i!m(!) _x(t) +R(!) _x(t): (4.17)
For the purpose of this study, any parasitic mechanical impedance in the wave-
maker system, i.e., friction, will be ignored.
To understand the behaviour of the segmented wavemaker's hydrodynamic
coecients and what this behaviour can tell us about the wave eld created by a
segmented wavemaker, each term of both the radiation damping and added mass
matrices for segmented piston and
ap wavemakers, Equations (4.13) to (4.16),
are plotted as functions of khin Figures 4.4 to 4.11, where the wavemakers contain
68

Chapter 4
four and ve segments.
4.4.1 Radiation damping
The diagonal terms of the radiation damping matrix of a segmented piston
system and
ap system, Equations (4.13) and (4.15), with four and ve segments
in the system are given as functions of khin Figures 4.4 and 4.5, respectively.
The o-diagonal terms of the radiation damping matrix, Equation (4.12), for
a segmented piston system and
ap system with four and ve segments in the
system are plotted as functions of khin Figures 4.6 and 4.7, respectively. As would
be expected from the nature of the hyperbolic depth function, Equation (2.42),
being greater at z= 0 and decaying over depth, the diagonal components of
the segmented piston wavemaker in Figure 4.4 shows that the closer a segment
is to the free surface, the more eective it is at generating larger wave heights.
The same behaviour can be seen in the diagonal terms of the segmented
ap
wavemaker in Figure 4.5, with the exception of the diagonal terms representing
the bottom segments ( R44in part (a) and R55in part (b)) for which, at low
frequencies, the values of the radiation damping terms are greater than that of the
diagonal term representing the segment above it. This could be expected since,
at low frequencies, the displacement of the bottom segment is almost uniform
over depth, thus, it behaves more like a piston than a
ap.
We see again, with the o-diagonal terms of the radiation damping matrix for
a segmented piston wavemaker with four and ve segments in Figure 4.6, that the
closer a segment is to the free surface, the greater its radiation damping. This can
be interpreted by considering the distance from the mean position of the segments
involved in the coupling term to the free surface. The exception to this behaviour
is the term R23in part (b) of Figure 4.6, which deviates from this behaviour for
high frequencies. In Figure 4.7, the behaviour of the o-diagonal terms in the
radiation damping matrix, for a segmented
ap wavemaker consisting of four and
ve
aps, deviates further from the pattern that is shown in Figure 4.6. The
behaviour of the matrix components in Figure 4.7 for the four and ve segment

ap wavemakers do not present a strong pattern, hence, a reliable description of
this behaviour can not be proposed, as it was with Figure 4.6.
It can be seen that the radiation damping of the segment at the free surface
peaks at a higher frequency than the rest of the segments and that either side of
this peak frequency the radiation damping decreases. Intuitively, it would seem
that the free surface piercing segment has a \tuned" frequency which is dependent
on its geometry. This has also been reported by [26] for piston wavemakers that
69

Chapter 4
are not full draft and by [17] for the Cosh wavemaker. It seems that the radiation
damping peak is a result of the geometry of the wavemaker being better suited
to generating progressive waves at a particular frequency, which is dependent on
the draft of the wavemaker [26].
0 2 4 6 8 10 12
kh020406080100120140160R(ω) [kg/sec](a)
R11
R22
R33
R44
0 2 4 6 8 10 12
kh020406080100120R(ω) [kg/sec](b)
R11
R22
R33
R44
R55
Figure 4.4: Diagonal components of the radiation damping matrix for a: (a) four
piston wavemaker, (b) ve piston wavemaker.
70

Chapter 4
0 2 4 6 8 10 12
kh020406080100R(ω) [kg/sec](a)
R11
R22
R33
R44
0 2 4 6 8 10 12
kh01020304050607080R(ω) [kg/sec](b)
R11
R22
R33
R44
R55
Figure 4.5: Diagonal components of the radiation damping matrix for a: (a) four

ap wavemaker, (b) ve
ap wavemaker.
71

Chapter 4
0 2 4 6 8 10 12
kh020406080100R(ω) [kg/sec](a)
R12
R13
R14
R23
R24
R34
0 2 4 6 8 10 12
kh01020304050607080R(ω) [kg/sec](b)
R12
R13
R14
R15
R23
R24
R25
R34
R35
R45
Figure 4.6: O-diagonal components of the radiation damping matrix for a: (a)
four piston wavemaker, (b) ve piston wavemaker.
72

Chapter 4
0 2 4 6 8 10 12
kh020406080100R(ω) [kg/sec](a)
R12
R13
R14
R23
R24
R34
0 2 4 6 8 10 12
kh0102030405060R(ω) [kg/sec](b)
R12
R13
R14
R15
R23
R24
R25
R34
R35
R45
Figure 4.7: O-diagonal components of the radiation damping matrix for a: (a)
four
ap wavemaker, (b) ve
ap wavemaker.
73

Chapter 4
4.4.2 Added mass
The diagonal terms of the added mass matrix are plotted against khfor the
segmented piston and
ap wavemakers, Equation (4.14) and (4.16) in Figures 4.8
and 4.9, respectively, for wavemakers with four and ve segments. The o-
diagonal terms of the same added mass matrices, Equation (4.12), are plotted
againstkhin Figures 4.10 and 4.11.
Comparing the diagonal added mass matrix terms of the segmented piston
wavemaker in Figure 4.8 to the same for the segmented
ap wavemaker in Fig-
ure 4.9, we see that the segmented
ap design achieves lower values for the added
mass components. It is also evident that the segmented piston wavemaker is
more appropriate for low frequency waves than for high frequency waves, as the
value of the added mass terms increases with frequency for all segments except
the top segment. In contrast to this, for segmented
ap wavemakers, Figures 4.9
and 4.11, with increasing frequency, the added mass decreases asymptotically to
some value close to zero for all the segments, indicating a decline in the strength
of the evanescent wave eld.
74

Chapter 4
0 2 4 6 8 10 12
kh05101520253035m(ω) [kg](a)
M11
M22
M33
M44
0 2 4 6 8 10 12
kh0510152025m(ω) [kg](b)
M11
M22
M33
M44
M55
Figure 4.8: Diagonal components of the added mass matrix for a: (a) four piston
wavemaker, (b) ve piston wavemaker.
75

Chapter 4
0 2 4 6 8 10 12
kh05101520m(ω) [kg](a)
M11
M22
M33
M44
0 2 4 6 8 10 12
kh02468101214m(ω) [kg](b)
M11
M22
M33
M44
M55
Figure 4.9: Diagonal components of the added mass matrix for a: (a) four
ap
wavemaker, (b) ve
ap wavemaker.
76

Chapter 4
0 2 4 6 8 10 12
kh-15-10-505101520m(ω) [kg](a)
M12
M13
M14
M23
M24
M34
0 2 4 6 8 10 12
kh-10-5051015m(ω) [kg](b)M12
M13
M14
M15
M23
M24
M25
M34
M35
M45
Figure 4.10: O-diagonal components of the added mass matrix for a: (a) four
piston wavemaker, (b) ve piston wavemaker.
77

Chapter 4
0 2 4 6 8 10 12
kh-10-505m(ω) [kg](a)
M12
M13
M14
M23
M24
M34
0 2 4 6 8 10 12
kh-8-6-4-20246m(ω) [kg](b)
M12
M13
M14
M15
M23
M24
M25
M34
M35
M45
Figure 4.11: O-diagonal components of the added mass matrix for a: (a) four

ap wavemaker, (b) ve
ap wavemaker.
78

Chapter 4
4.5 Validation of code
In order to verify the results presented in Figures 4.4 to 4.11, the in-house
code used to generate these results was used to generate the radiation damping
and added mass of single piston and
ap wavemakers, where a= 0 andb=h
in Equations (4.13) and (4.16), and compared to the corresponding radiation
damping and added mass coecients derived by [17]. For single piston and
ap
wavemakers Figures 4.12 and 4.13, respectively, plot the radiation damping cal-
culated by the in-house code and that derived by [17] against khin Parts (a) and
the added mass calculated by the in-house code and that derived by [17] against
khin Parts (b) of Figures 4.12 and 4.13. These results can also be compared to
those presented by [17]. Figures 4.12 and 4.13 show that the hydrodynamic coef-
cients calculated by the in-house code and those calculated by [17]'s expressions
are identical, this indicates that the in-house code calculates the hydrodynamic
coecients of the wavemakers correctly. Both sets of results are identical as they
are both derived using linear potential wavemaker theory.
Figure 4.12: Comparison of the radiation damping, (a), and added mass, (b), for
a single piston wavemaker calculated by the in-house code and those derived by
[17], where h= 0:6.
79

Chapter 4
Figure 4.13: Comparison of the radiation damping, (a), and added mass, (b),
for a single
ap wavemaker calculated by the in-house code and those derived by
[17], where h= 0:6.
4.6 Constrained dynamics of a segmented
wavemaker
By imposing programmed constraints via a controller, the strokes of each
segment can be set relative to the stroke of the top segment. The segmented
wavemaker can then be programmed to approximate the ideal
uid motion, Equa-
tion (2.57), as accurately as possible. These programmed constraints allow for
the hydrodynamics of the segmented wavemaker to be simplied so that the hy-
drodynamic functions can be expressed as single scalar quantities rather than
matrices, allowing for the hydrodynamics of wavemakers with dierent DOFs
to be compared. By doing this, the hydrodynamics of systems with dierent
numbers of DOFs can be compared to each other. Of course it would be best
to compare the wave elds generated by the dierent wavemakers, as done in
Chapter 5 and 6, however, the comparison presented in this chapter helps us to
understand how the hydrodynamics of the segmented wavemaker behave. This
is achieved in Section 4.6.2 using the NE-EC to describe the multi-body system
of a segmented wavemaker with less DOFs. First, we look at how the modes of
motion of a wavemaker are dened in Section 4.6.1.
80

Chapter 4
4.6.1 Modes of motion of a wavemaker
Single body wavemakers have one mode of motion, a piston operates in surge
and a
ap operates in pitch, as it pivots about a given point. The motion of a
piston is described by the wavemaker's horizontal displacement while the motion
of a
ap wavemaker is described by the angle between the paddle and the vertical
axis, indicated as in Figure 4.14. However, the horizontal displacement of a
xz
Rotational DOFS0
Translational DOF
θ
Figure 4.14: Motion of a
ap wavemaker decomposed into a translational and a
rotational DOF.
piston and the angle of a
ap are not directly comparable. Instead, we use the
depth proles stated in Equations (2.12) and (2.13). For a piston the depth
prole is simply a stroke of unit length. To derive the depth prole of a
ap
wavemaker we describe the
ap's motion by two dependent DOFs, a translational
and a rotational DOF, dened by the
ap's motion relative to the origin of the
coordinate system. By combining these two dependent DOFs, in the manner
presented below, the
ap wavemaker's motion is described. Starting with the
position of the wavemaker,
~S(z;t) =S0c(z)~iei!t; (4.18)
whereS0is the wavemaker's stroke at z= 0, it then follows that the wavemaker's
velocity is,
~ u(z;t) =u0c(z)~iei!t=i!S 0c(z)~iei!t: (4.19)
81

Chapter 4
The unit vectors along the x,yandzaxis are denoted as ~i,~jand~k, respec-
tively. The wavemakers velocity can be decomposed into the combination of a
translational velocity, ~U, and a rotational (or angular) velocity, ~
, as follows [13],
~ u=~U+ (~
~ s); (4.20)
where~ sis the position vector of the point moving at velocity ~ u. For the
ap
wavemaker, the translational and rotational velocities are dened, respectively,
as:
~U=_~ x=i!S 0~iei!t;
~
=_~=i!~jei!t:(4.21)
Given that ~ sis the position of a point on the surface of the wavemaker and since
the position of the wavemaker is approximated about x= 0, we can say,
~ sz~k: (4.22)
Substituting Equations (4.21) and (4.22) into Equation (4.20) the resultant ve-
locity becomes:
~ u=i![S0~i+~jz~k]ei!t: (4.23)
Utilising the small angle approximation we have,
sin() =S0
h; (4.24)
and substituting Equation (4.24) into Equation (4.23) gives,
~ u=i![S0~i+S0z
h~j~k]ei!t: (4.25)
Carrying out the cross product gives,
~ u=i!S 0[1 +z
h]~iei!t: (4.26)
Comparing Equation (4.26) to Equation (4.19) gives the normalised horizontal
displacement of the paddle as,
c(z) = 1 +z
hl: (4.27)
82

Chapter 4
4.6.2 Newton-Euler equations of motion with eliminated
constraints
When dealing with a multi-body system, such as the segmented wavemaker,
the constrained hydrodynamic functions can be found from the NE-EC [88],
Fr;c=NX
q=1@~ uq
@sFr=PFr; (4.28)
where the subscript cindicates a constrained value, i.e., a value describing a
multi-body system where the DOFs have been reduced, and Nis the number
of bodies in the system. The vector P is a transformation velocity vector. The
independent velocity, s, is a characteristic velocity of which the motion of each
body in a multi-body system can be expressed in relation to. The independent
velocity vector has a size of the number of reduced DOFs in the system. In the
case of the segmented wavemaker, as we are reducing the number of DOFs to
one,shas a single component which we set as the velocity of the top segment,
s=~ u1: (4.29)
The constrained force Fcis also a vector of the same size as s. The general
velocity vector, u, is dened so that the qthcomponent is the velocity vector of
theqthsegment,
u=0
BBBBBBB@~ u1
…
~ uq
…
~ uN1
CCCCCCCA; (4.30)
and is related to the independent velocity [88], s, by
u=PTs: (4.31)
The transformation vector [88], P, dened as
P=
@u1
@s


@uq
@s


@uN
@s
; (4.32)
is the vector that allows us to reduce the DOF, as demonstrated in the NE-EC,
Equation (4.28). It should be noted that the notations in bold indicate vectors
or matrices of which the components represent values associated with individual
83

Chapter 4
segments in a segmented wavemaker, while ~denotes a vector with a direction.
Bear in mind that a vector Amay contain the direction vector ~ aq.
We can use the constrained force, Fcin Equation (4.28), to compare the radia-
tion force experienced by both segmented piston and segmented
ap wavemakers
with dierent numbers of segments, shown in Figures 4.15 and 4.16, respectively.
The force plotted as a function of khin Figures 4.15 and 4.16 is the constrained
force, Fc, on the segmented wavemaker calculated from Equation (4.28), which is
evaluated using the force vector, Equation (4.2), of which the individual compo-
nents are given by Equation (4.5). The forces presented in Figures 4.15 and 4.16
are of the dierent segmented wavemakers generating progressive waves with the
same amplitude as a single piston wavemaker with a unit stroke. It must be en-
sured that the piston and
ap wavemakers are creating the same wave eld when
comparing the radiation force. To achieve this, the strokes of the wavemakers
have been adjusted while generating the results in this thesis so that each wave-
maker is creating a wave with the same wave height as that generated by a single
piston wavemaker with a unit stroke. The need for this stroke correction will be
discussed further in Section 4.6.3, Equation (4.38). In order to approximate the
kinematics of a progressive wave, recall that the strokes of the segmented wave-
maker, in this chapter, are dened in Equation (4.1) by the progressive wave's
depth prole, Equation (2.57). By comparing Figures 4.15 and 4.16, it is clear
that to create the same high frequency waves, a
ap wavemaker requires much
less force than a piston. In Figures 4.15 and 4.16, there is a drastic reduction
in the radiation force of the two segment wavemakers at high frequencies, com-
pared to the single segment wavemakers. However, as the number of segments in
the system is increased further, the reduction in the radiation force is much less
signicant. In fact, there seems to be virtually no improvement after the three
segmented
ap wavemaker. The reduction in the radiation force in the segmented

ap systems compared to the segmented piston systems, and also as the number
of segments in the wavemaker is increased, is understood to be a result of a reduc-
tion in the amplitude of the evanescent waves. Hence, this supports Hypothesis
1, Section 2.4, as the more segments the wavemaker contains the better it ap-
proximates the kinematics of progressive waves and the better it is at reducing
the evanescent wave eld. For the sake of providing a clear idea of the eect of
evanescent waves on the radiation force, Figure 4.17 shows the radiation force
experienced by the wavemakers due to the progressive wave alone as a function
ofkh. Comparing the results presented in Figure 4.17 to Figures 4.15 and 4.16
highlights the evanescent waves' contribution to the radiation force experienced
by a wavemaker. For shallow water waves, it is apparent from Figures 4.15, 4.16
and 4.17 that the contribution to the radiation force from the evanescent waves
84

Chapter 4
is negligible, but can be signicant for wavemakers with three or less segments at
high frequencies.
0 2 4 6 8 10 12
kh00.511.522.533.5Fr [N]×104
1 Pistons
2 Pistons
3 Pistons
4 Pistons
5 Pistons
6 Pistons
7 Pistons
8 Pistons
9 Pistons
10 PistonsDeep water
wavesShallow
water
waves
Figure 4.15: The total radiation force experienced by segmented piston wavemak-
ers generating waves of the same amplitude as a single piston wavemaker with
unit stroke, as a function of khfor ten dierent systems.
4.6.3 Constrained hydrodynamics
Expressing Equation (4.6) in matrix form,
F=Zu; (4.33)
where Zis the radiation damping matrix, it stands to reason that we may express
the constrained radiation force in terms of a constrained radiation impedance [88]:
Fc=Zcs: (4.34)
Substituting Equation (4.33) and (4.31) into Equation (4.28) and equating with
Equation (4.34), the constrained radiation impedance, Zc, can be obtained as:
Zc=PZPT: (4.35)
85

Chapter 4
0 2 4 6 8 10 12
kh0100020003000400050006000Fr [N]1 Flaps
2 Flaps
3 Flaps
4 Flaps
5 Flaps
6 Flaps
7 Flaps
8 Flaps
9 Flaps
10 FlapsDeep water
wavesShallow
water
waves
Figure 4.16: The total radiation force experienced by
ap wavemakers generating
waves of the same amplitude as a single piston wavemaker with unit stroke, as a
function of khfor ten dierent systems.
0 2 4 6 8 10 12
kh0500100015002000250030003500400045005000Fr [N]
Deep water
wavesShallow
water
waves
Figure 4.17: The radiation force experienced by a single piston wavemaker, with
a unit stroke, due to the progressive wave, as a function of kh.
86

Chapter 4
Subsequently, the constrained hydrodynamic coecients are found to be [88]:
Rc(!) =PRPT; (4.36)
and
mc(!) =PMPT: (4.37)
The constrained radiation damping of the segmented piston and
ap wave-
makers are plotted as a function of khin Figures 4.18 and 4.19, respectively, for
wavemakers containing one to ten segments. Comparing Figures 4.18 and 4.19
reveals that the constrained radiation damping is notably less for the segmented

ap wavemakers than the segmented piston wavemakers. This is because the
wavemaker is displacing a smaller volume of water per unit stroke and conse-
quently, is generating waves with smaller wave heights. Recalling Figure 4.1, it
is intuitive that, as the number of segments in the system is increased, the water
displaced by the wavemaker decreases and, thus, so does the radiation damping
and the progressive wave's height. In order for the wave elds created by the
dierent wavemakers to be comparable, the strokes of the wavemaker's must be
corrected. For this purpose, a correction factor is introduced,
='0P1
'0W; (4.38)
where the subscript 0 indicates the rst term of the innite summation in Equa-
tion (2.56), the subscript P1 indicates a single piston wavemaker and the sub-
scriptWrepresents the wavemaker system for which the hydrodynamics are being
evaluated. The constrained velocity potential with the correction factor is:
c=PT: (4.39)
The correction factor, Equation (4.38), does not apply to the radiation damp-
ing and added mass as they are both independent of the wavemaker's motion
amplitude, but is considered in the calculation of Figures 4.15 and 4.16 and the
rest of the results presented this thesis. Figures 4.18 and 4.19 show that at low
frequencies the radiation damping for all the wavemakers tends towards the same
value. This is because each wavemaker begins to operate more like a single piston
wavemaker as the horizontal motion of the
uid in a progressive wave becomes
more uniform over depth.
87

Chapter 4
0 2 4 6 8 10 12
kh050010001500Rc(ω) [kg/sec]1 Pistons
2 Pistons
3 Pistons
4 Pistons
5 Pistons
6 Pistons
7 Pistons
8 Pistons
9 Pistons
10 Pistons
Figure 4.18: The constrained radiation damping of ten segmented piston wave-
makers over kh.
88

Chapter 4
0 2 4 6 8 10 12
kh050010001500Rc(ω) [kg/sec]1 Flaps
2 Flaps
3 Flaps
4 Flaps
5 Flaps
6 Flaps
7 Flaps
8 Flaps
9 Flaps
10 Flaps
Figure 4.19: The constrained radiation damping of ten segmented
ap wavemak-
ers overkh.
89

Chapter 4
The constrained added mass of the segmented piston and
ap wavemakers,
shown in Figures 4.20 and 4.21, respectively, as a function of kh, further supports
Hypothesis 1, Section 2.4, as it is clear that the more segments in the wavemaker
and hence, the better it approximates the kinematics of a progressive wave, the
lower the value of the added mass. The insets in both Figures 4.20 and 4.21
show the behaviour of the constrained added mass for the wavemakers, with two
to ten segments, with greater detail on lower values of the constrained added
mass. The insets reveal that, with the exception of the single piston wavemaker,
the constrained added mass of the segmented wavemakers is not a monotonic
function of kh; these results are similar to the ndings presented in [26]. The
non-monotonic behaviour of these curves is a result of the interference pattern
between the evanescent waves with dierent phases and provides some validation
for Hypothesis 2, discussed in Section 2.4.2.
0 2 4 6 8 10 12
kh020406080100120140160Mc(ω) [kg]1 Pistons
2 Pistons
3 Pistons
4 Pistons
5 Pistons
6 Pistons
7 Pistons
8 Pistons
9 Pistons
10 Pistons
0 5 10
kh00.050.10.150.2Mc(ω) [kg]
Figure 4.20: The constrained added mass of ten segmented piston wavemakers
over frequency as functions of kh. The inset shows the behaviour of the con-
strained added mass with a greater magnication.
90

Chapter 4
0 2 4 6 8 10 12
kh-50510152025Mc(ω) [kg]1 Flaps
2 Flaps
3 Flaps
4 Flaps
5 Flaps
6 Flaps
7 Flaps
8 Flaps
9 Flaps
10 Flaps0 2 4 6 8 10 12
kh-0.0100.010.02Mc(ω) [kg]
Figure 4.21: The constrained added mass of ten segmented
ap wavemakers over
frequency as functions of kh. The inset shows the behaviour of the constrained
added mass with a greater magnication.
91

Chapter 4
4.7 Control parameter
The transformation vector can be programmed to control the amplitude of
motion for each segment in the wavemaker. For instance, the results presented in
this chapter have been calculated with the segments programmed to approximate
the horizontal
uid motion of the progressive waves over the depth of the tank
by setting the independent velocity to:
s= 1; (4.40)
and setting the transformation vector as:
P=0
BBBBBBB@1
:::
cosh[k(h+(q1)h
N)]
cosh(kh)
:::
cosh[k(h+(N1)h
N)]
cosh(kh)1
CCCCCCCA: (4.41)
Later, in Chapter 6, the transformation vector will be programmed to control the
phase shifts of the evanescent waves with the aim of minimising the destructive
interference between the evanescent waves.
4.8 Conclusion
This chapter has presented two designs for a segmented wavemaker, one where
each segment acted as a piston operating in surge mode, and one where each
segment acted as a
ap operating in pitch mode. The hydrodynamics for the
multi-body problem were developed and the DOFs of the segmented system were
reduced so that systems with dierent quantities of segments could be compared.
The results presented of the constrained added mass and radiation forces, Fig-
ures 4.15, 4.16, 4.20 and 4.21, support Hypothesis 1, Section 2.4, i.e., that the
more accurately a wavemaker approximates the horizontal
uid motion in a pro-
gressive wave, the lesser the eect of the evanescent waves. Other than for the
case of the single piston wavemaker, the constrained added mass was shown not
to be a monotonic function of khwhich can be explained by an interference
pattern between evanescent waves that are in anti-phase with each other. This
interference pattern leads to a reduction in the distortion of the wave eld due to
the evanescent waves, Section 2.4.2. It has been found that a drawback with the
92

Chapter 4
segmented wavemakers is that they require larger strokes than a single segment
piston or
ap in order to generate waves with the same wave height.
93

Chapter 5
Chapter 5
Optimisation of segment length
5.1 Introduction
A segmented wavemaker's depth prole, c(z), discussed in Section 2.2 which
describes the wavemaker's geometry, is dependent on two parameters; the length
of each segment in the wavemaker and the segment strokes. The stroke parame-
ters are controlled by the wavemaker's driving signal, and are further considered
in Chapter 6. The segment lengths, however, are decided during the design phase
and remain constant after that. This chapter focuses on optimising the segment
lengths in order to reduce the distortion of the wave eld caused by evanescent
waves. The congurations of the segmented wavemakers which are considered
here are those with two to six segments operating as both pistons and
aps.
The designs are optimised by dening an objective function, a parameter which
relates the segment lengths to the level of distortion caused by the evanescent
waves. An optimisation algorithm then searches for the combination of segment
lengths that gives the minimal value for the objective function. Since the opti-
mised wavemaker design will change over frequency, as discussed in Section 2.4,
the objective functions are averaged uniformly over the range: 0 !14 radi-
ans/sec (0kh12), similar to the operational range of the Omey wave tank
and the wave
ume at the HMRC.
Section 2.4 discusses how a wavemaker which simulates the
uid motion in a
progressive wave eectively eliminates the evanescent wave eld, a concept that
was examined theoretically by [18]. Bearing this in mind, it would be reasonable
to make the assumption that the closer a wavemaker approximates the kinemat-
ics of a progressive wave, the smaller the evanescent waves will be. This was
the logic behind the design of the
exible membrane wavemaker in [9] and the
dual-
ap wavemakers of [5] and [4]. The results for the constrained added mass
of the segmented wavemakers, Figures 4.20 and 4.21 in Section 4.6.3, support this
assumption as they clearly show that, as a segmented wavemaker approximates
94

Chapter 5
the motion of a progressive wave more accurately, the added mass of the system
decreases. The designs of the segmented wavemakers considered in Chapter 4
consist of segments of equal lengths. However, highlighted by Figure 5.1, is a
concept where the kinematics of a progressive wave are better approximated by
segments that decrease in size the closer they are to the free surface. The merit of
this concept lies in the fact that the rate of change of the progressive wave's depth
prole, specied in Equation (2.57), increases exponentially as z!0. Thus, in
order to approximate Equation (2.57) with a better degree of accuracy, shorter
segments are required close to the free surface, as illustrated in Figure 5.1. This
reasoning agrees with the ndings of [4], which reports that the total hydrody-
namic load on the dual-
ap wavemaker is minimised for a particular conguration
where the top
ap was shorter than the bottom
ap. Hyun [4] did not extend
the study to consider the eect on the distortion of the wave eld directly.
Approximating a progressive wave reduces the distortion by minimising the
Biesel coecients [9] of evanescent terms in Equation (2.56). This requires opti-
misation of the wavemaker's depth prole so that the integral in Equation (2.55)
tends to zero. However, achieving a depth prole that will eliminate the evanes-
cent waves would require an innite number of segments, something which is
clearly not feasible for a physical device. Section 2.4.2 suggests the more prac-
tical approach of designing the depth prole to maximise the destructive inter-
ference between the evanescent waves. This is a novel concept that attempts to
nd combinations of values for the Biesel coecients, both positive and negative,
representing the evanescent terms in Equation (2.56), which causes a destructive
interference pattern that minimises the evanescent wave eld amplitude. Fig-
ure 5.2 provides an illustration of how a three segment
ap wavemaker may be
constructed. In Figure 5.2, springs push the
aps to the right, while stepper
motors are pulling them to the left using wires which are coloured purple.
This chapter looks, for the rst time, at the optimisation of segment lengths
in wavemakers with the aim of reducing the area between the wavemaker and the
test area in a tank that is contaminated by evanescent waves. Two approaches
are used to optimise the segment lengths. Approach 1 is to design the geometry
of the segments in a wavemaker so that it provides the best approximation to the
kinematics of a progressive wave. This is similar to the approach taken by [5].
Approach 2 is to nd the optimal segment lengths that minimises the distance
between the wavemaker and the position of 1% distortion directly. A comparison
of both approaches will provide the rst piece of evidence that the destructive
interference pattern between the evanescent waves can help reduce the eect of
distortion in the wave tank, since Approach 2 can utilise this behaviour but
Approach 1 can not.
95

Chapter 5
z
xa1
a2
a3
a4
a5
a6
a7z
xa1a2
a3
a4
a5
a6
a7(a) (b)
Depth profile of progressive wave
Depth profile of wavemaker
Figure 5.1: Illustration of (a) piston and (b)
ap segmented wavemakers approx-
imating the horizontal
uid displacement of a progressive wave.
Section 5.2 looks at the problem of optimising the lengths of the segments
in a wavemaker, including the type of optimisation required, the constraints on
the wavemaker design, the optimisation algorithm and the problem of identify-
ing the objective function's global minima. Sections 5.3 and 5.4 describe two
dierent strategies for optimising the segment lengths and tunes the parameters
of the optimisation algorithm to allow for fast and reliable convergence. Sec-
tions 5.5 and 5.6 present the results of the optimisation of the segment lengths in
a wavemaker using Approach 1 and 2, respectively, while a comparison of both
approaches is provided in Section 5.7. Finally, Section 5.8 presents the concluding
remarks on the optimisation of the wavemaker's segment lengths.
96

Chapter 5
Motor 1
Motor 2Motor 3
Segment 1
Segment 2
Segment 3
Spring 3Spring 2Spring 1Pulley 1
Pulley 2
Pulley 3xz
Figure 5.2: Illustration of how a segmented
ap wavemaker could be constructed.
5.2 Setting up the optimisation problem
Optimisation algorithms have been used widely throughout engineering to
nd the minimum or maximum values of an objective function. The objective
function is specied by the user to describe the eect that one wishes to minimise.
This section describes the problem of nding the optimal segment lengths for a
wavemaker which will minimise the distance between the wavemaker and the
testable area. In this thesis, we consider the testable area in the wave tank to
be where the distortion is no greater than 1%. A level of distortion of 1% is
regarded as an acceptable error, as it is signicantly less than the acceptable
accumulated error from other sources, such as re
ected waves and cross waves.
The performance of the full-draft single segment piston and bottom hinged
ap
wavemakers, commonly used in wave tanks today, will also be presented for the
purpose of comparison with the optimised segmented wavemakers.
Two strategies are proposed for optimising the design of the segmented wave-
maker. The rst one attempts to optimise the lengths of the segments in order
to minimise the dierence between the wavemaker's and the progressive wave's
97

Chapter 5
depth proles, following the idea of matching the progressive waves' kinematics
with that of the wavemaker. This strategy is henceforth referred to in the rest of
this thesis as the kinematic matching approach. The second strategy attempts to
nd a minimum distance away from the wavemaker where the distortion is 1% and
does not increase above 1% thereafter. This allows the optimiser to nd the best
combination of segment lengths that cause an interference pattern that minimises
the untestable area in front of the wavemaker. This strategy is referred to hence-
forth as the minimisation of distortion approach. It obviously makes more sense
to use the second strategy; however, it is far more demanding on computation
time as it requires the computation of the wave eld, which is not necessary for
the kinematic matching approach. Additionally, a comparison of both strategies
allows us to test the hypothesis that optimising the interference pattern between
the evanescent waves may perform better than simply approximating a natural
progressive wave.
5.2.1 Optimisation algorithms
Appropriate choice of an optimisation algorithm is essential for ensuring that
the objective function's global minimum is found in as short a time frame as pos-
sible. If the objective function can be specied as a linear or quadratic function
of the variables of the optimisation problems, the minimum can be found using a
simple optimisation algorithm, such as the Newton-Raphson method [89]. These
types of algorithms search the objective function iteratively, comparing the value
of the current candidate solution to the previous and can nd a minimum in a
relatively short period of time. However, nonlinear objective functions, such as
the ones considered in this thesis, can have other local minima, in addition to the
global minimum, i.e., the lowest possible value for the objective function. Since
quadratic optimisation algorithms cannot dierentiate between local and global
minima, they are not appropriate for nonlinear optimisation problems. Evolution-
ary Algorithms (EAs) [90] and Genetic algorithms (GAs) [91] have been proven
to be quite eective at nding global minima, though at the cost of signicant
computational time. EAs and GAs work by creating an initial random set of
candidate solutions, referred to as a population. The algorithm then iteratively
modies the properties of the population's members until all the candidate so-
lutions in the population appear to converge towards a single optimal solution.
Following typical convention each iteration is called a generation [90]. The dif-
ference between EAs and GAs is how they create the new population. Both EAs
and GAs can have many control parameters, such as population size, mutation
98

Chapter 5
rate, etc., all of which in
uence the algorithm's ability to nd a global minimum
and the rate of convergence. For each optimisation problem, the optimisation
algorithm's control parameters need to be tuned to ensure eective, reliable and
fast convergence towards an optimised solution. The optimisation algorithm used
in this thesis, Dierential Evolution (DE), is an elegant EA algorithm, which has
been shown to have relatively fast convergence, with few parameters that require
tuning [90]; this will be discussed in more detail in Section 5.2.3.
5.2.2 Wavemaker constraints
The variables of the optimisation problem are dened as the vertical position
of the top and bottom edges of each segment, in both piston and
ap segmented
wavemakers, indicated in Figure 5.1 as ai. The edges of the segments are labelled
so thatairepresents the top edge of the ithsegment and the bottom edge of the
(ith1) segment. In a wavemaker with Nsegments, the a1andaN+1edges are held
constant at 0 and h, respectively, enforcing the condition that the wavemaker
covers the entire depth of the tank. Thus, the optimisation problem has N1
variables. The optimisation algorithm is programmed with the constraint:
ai+1aiai1; (5.1)
so that the candidate solutions are feasible and do not contain overlapping seg-
ments. The purpose of the equality condition in Equation (5.1) is to allow the
optimisation algorithm to eliminate segments if it nds that a lower objective
function can be achieved with fewer segments, although it is not expected that
this will occur. For the purpose of optimising the lengths of the segments, the
stroke of each segment is dened so that the horizontal displacement of each
segment joint is given by:
xi=cosh[k(h+ai)]
cosh[kh]: (5.2)
For the piston wavemakers, Equation (5.2) gives the horizontal displacement at
the top of the segment, while for the
ap wavemakers Equation (5.2) gives the
horizontal displacement xiat the joint aias denoted in Figure 5.1. The optimi-
sation of the segment strokes will be considered in Chapter 6.
99

Chapter 5
5.2.3 Dierential evolution algorithm
The DE algorithm [90] has been selected for the optimisation of the segmented
wavemakers as it provides fast convergence towards a solution and is relatively
simple to implement [92]. The DE algorithm evaluates the objective function for
each member of the current population and then creates a mutant population in
which each member is a mutation of the corresponding member in the current
population. A trial population is then created as a crossover between the current
and mutant populations by randomly selecting members from both to become
members of the trial population. If a member of the trial population has a
objective function value which is less than or equal to that of the corresponding
member in the current population, then that trial member is selected for the
new generation, otherwise the member from the current population is kept on for
another generation. This selection process makes the algorithm more focused on
converging and reduces the run time, but it can also mean that the algorithm
does not search as wide of a search space, thus limiting its application.
The simple implementation of the DE algorithm is due to it having only
four parameters which need to be tuned: the population size; the maximum
number of generations the algorithm will evaluate, or the threshold value for the
objective function; a scale factor, which controls the scale of the mutation; and a
crossover probability, which is the probability of a mutant being selected for the
trial population. Increasing either the population size or the maximum number
of generations will increase the algorithm's ability to converge to a minimum,
however, they will also increase the computation time. In this thesis, a trial
and error approach was used to nd values for both of these parameters, for
which little-to-no improvement to the objective function value can be found by
increasing them further; this was carried out while the author was learning how to
implement the DE algorithm. Generally the scale factor, which must be positive,
should have a value between 0 and 1; however, [93] shows that the scale factor
should be no less than 0 :3. For problems where the variables cannot be optimised
independently from each other a good crossover probability is often found between
0:9 and 1 [92].
The DE algorithm was implemented using the Matlab toolbox based on [90]
(http://www1.icsi.berkeley.edu/ storn/code.html) and was modied by the au-
thor to impose the variable constraints in Equation (5.1). The author also mod-
ied the DE Matlab toolbox to enable the algorithm to compute in parallel,
dividing the calculations up over eight cores in the computer, hence, making the
optimisation process run signicantly faster.
100

Chapter 5
5.2.4 Finding a global minimum
Often, objective functions can have several local minima in addition to the
global minimum, so when a minimum of the objective function is found it can
be dicult to be sure that it is a global minimum, that is to say, no lower values
of the objective function are possible. However, it can be sucient to simply
show that a solution found by the optimiser appears to be the minimum in a
signicantly large area of the search space. Some foresight is used to determine
that the optimisation experiments carried out in this thesis should be repeated
six times. If the standard deviations of the solutions found by the optimiser over
the six optimisation experiments is suciently low, then it can be concluded with
a reasonable amount of condence that the optimiser has converged towards the
minimum of the objective function. Each optimisation experiment will be referred
to as an optimisation run.
5.3 Approach 1: Matching the progressive
wave kinematics
The kinematic matching approach attempts to minimise the distortion in the
wave tank by approximating the velocity depth prole of the progressive wave
with the segmented wavemaker's velocity depth prole. The DE algorithm is used
to nd the optimal lengths of each segment that provides the most accurate ap-
proximation of the progressive wave's kinematics. This is similar to the approach
taken by [5] who used a linear regression method based on least squares estima-
tion to nd the segment lengths in a dual-
ap wavemaker that would provide
the best approximation of the progressive wave's depth function. The objective
function for the kinematic matching approach is dened as:
r=NzX
i=1j(ct(zi)cwm(zi))j; (5.3)
where the dierence term is the dierence between the depth prole of the pro-
gressive wave, ct(z), and depth prole of the wavemaker, cwm(z). The bar above
the dierence term indicates that it has been averaged uniformly over the range
of frequencies, 0!14 radians/sec, and Nzis the number of elements, zi,
that the depth of the tank is divided uniformly into over depth.
101

Chapter 5
5.3.1 Tuning the scale factor and crossover probability
The performance of the optimiser is considered by its ability to converge to
the objective function's global minimum and the time required to do so; this is
largely determined by the parameters of the optimiser discussed in Section 5.2.3.
This section looks at tuning the parameters of the DE algorithm to achieve the
best performance when optimising the segment lengths in the wavemaker using
the kinematic matching approach.
It was found that a population size of 100 provides good convergence and since
the kinematic matching approach does not require a lot of computation time,
the algorithm was allowed to run for 1000 generations. With these settings, the
algorithm took 5.5 mins to terminate. The DE algorithm was tuned by optimising
the six-segmented piston wavemaker using a range of values for the scale factor
and crossover probability. As discussed in Section 5.2.4, each optimisation run
is repeated six times to test the reliability of the optimiser's setup. The results
of tuning the DE algorithm for the kinematic matching approach are presented
in Figure 5.3 for each combination of the scale factor, Fw, and the crossover
probability, Cp.
Figure 5.3 shows how the DE algorithm performs at converging to a minimum
value for the objective function over a range of values for the scale factor and
cross probability. Figure 5.3 shows that for a scale factor greater than 0 :5, the
DE algorithm is quite reliable over all the values of the crossover probability, with
very little variation between the objective function values in each run. Due to the
approximate nature of numerical solvers, some variation can be expected. The
most consistently low values for the objective functions, presented in Figure 5.3,
appear to occur with a scale factor of 1 and a crossover probability between 0 :3
and 0:7. Based on this analysis, the segment lengths of the wavemakers will be
optimised using the kinematic matching approach in Section 5.5 with the scale
factor and crossover probability of the DE algorithm set to 1 and 0 :5, respectively.
102

Chapter 5
00.2 0.4 0.6 0.8 1200250300350400Fw = 0.3
CpObjective Function: r

Run 1
Run 2
Run 3
Run 4
Run 5
Run 6
00.2 0.4 0.6 0.8 1240260280300320340Fw = 0.4
CpObjective Function: r

Run 1
Run 2
Run 3
Run 4
Run 5
Run 6
00.2 0.4 0.6 0.8 1245250255260265270275Fw = 0.5
CpObjective Function: r

Run 1
Run 2
Run 3
Run 4
Run 5
Run 6
00.2 0.4 0.6 0.8 1249.86249.88249.9249.92249.94249.96249.98250Fw = 0.6
CpObjective Function: r

Run 1
Run 2
Run 3
Run 4
Run 5
Run 6
00.2 0.4 0.6 0.8 1249.86249.88249.9249.92249.94249.96Fw = 0.7
CpObjective Function: r

Run 1
Run 2
Run 3
Run 4
Run 5
Run 6
00.2 0.4 0.6 0.8 1249.86249.87249.88249.89249.9249.91249.92249.93Fw = 0.8
CpObjective Function: r

Run 1
Run 2
Run 3
Run 4
Run 5
Run 6
00.2 0.4 0.6 0.8 1249.86249.88249.9249.92249.94249.96249.98250Fw = 0.9
CpObjective Function: r

Run 1
Run 2
Run 3
Run 4
Run 5
Run 6
00.2 0.4 0.6 0.8 1249.86249.87249.88249.89249.9249.91249.92249.93Fw = 1
CpObjective Function: r

Run 1
Run 2
Run 3
Run 4
Run 5
Run 6
Figure 5.3: Results of tuning the scale factor, Fw, and crossover probability, Cp,
for the kinematic matching approach using a six segment piston wavemaker, for
six optimisation runs.
103

Chapter 5
5.4 Approach 2: Minimising the wave eld
distortion
The advantages of using the minimisation of distortion approach is that it
allows the optimiser to nd the best wavemaker design for reducing the distance
between the wavemaker and the testable area in the tank. Unlike the kinematic
matching approach, it does not assume that the best way to reduce the distor-
tion is to simulate a progressive wave. Instead, it can utilise the interference
pattern between the evanescent waves, discussed in Section 2.4.2, to minimise
the distortion. It is expected that this approach will produce better results than
the kinematic matching approach, described in the previous section. So far it is
unknown as to whether or not the interference pattern between the evanescent
waves can help to reduce the distortion caused by the evanescent wave eld. A
comparison of the results from both approaches will provide validation of this.
5.4.1 Objective function: position of 1%distortion away
from the wavemaker
The aim of the minimisation of distortion approach is to minimise the region
in front of the wavemaker which is too contaminated with evanescent waves for
meaningful tests to be performed on devices. The objective function, r, used is
dened as the distance from the wavemaker to the position where the distortion
is 1% and does not increase above 1% for greater values of x. This will be referred
to as the position of 1% distortion and denoted as X1%. The interference between
the evanescent waves makes it cumbersome to predict the distortion pattern along
thex-axis. Hence, neither the total nor the maximum distortion values would
make good objective functions, since they provide no information as to how far
away from the wavemaker the testable area is. The position of 1% distortion is
a pragmatic function as it provides the nearest location to the wavemaker where
testing can be carried out while avoiding serious contamination from evanescent
waves. As we are interested in the wavemaker's performance within the range
0!14 radians/sec, the objective function is the position of 1% distortion
averaged uniformly over this range.
The position of 1% distortion was found by use of a numerical solver. The nu-
merical solver was initialised at the position x= 5h, and then iteratively worked
towards the wavemaker, calculating the distortion of the candidate solution at
each iteration, to nd the 1stoccurrence of the distortion having a value of 1%.
This approach ensures that the distortion could not be greater than 1% for any
104

Chapter 5
distance from the wavemaker greater than that reported as the position of 1%
distortion. From the author's experience, it is not likely that the position of 1%
distortion would be greater than 5 h.
5.4.2 Tuning the scale factor and the crossover
probability
The DE algorithm behaves dierently with dierent objective functions. Thus,
the parameters of DE must be tuned again to allow it to converge eciently to-
wards the minima of the objective function, the position of 1% distortion. Unlike
the kinematic matching approach, the minimisation of distortion approach re-
quires Equation (2.56) to be calculated, up to the rst 50 terms of the summation,
for each candidate solution, at 10 uniformly distributed frequencies as it is being
averaged. This can leave the optimisation very time consuming. The objective
function is averaged over 10 frequencies as this provides reasonable distribution
over the frequency range without requiring too much additional computation
time. For the purpose of tuning the scale factor, the population size was set to
100 and the algorithm was limited to evaluate only 100 generations. When tuning
the crossover probability, the population size was set to 10 and the algorithm was
limited to evaluate only 10 generations. Although these settings did not provide
convergence to a particular solution, they did allow us to determine suitable val-
ues for scale factor and the crossover probability. These values were determined
by a trial and error basis, where it was found that increasing either parameter
further did not result in dierent values of the scale factor and crossover probabil-
ity being selected. To alleviate some of the time expense the tuning process was
limited to save computation time. Rather than evaluating numerous combina-
tions of both parameters like in Section 5.3.1, the scale factor was rst tuned for a
xed crossover probability of 0 :98. This crossover probability value was selected
by an educated guess, backed up by the discussion in Section 5.2.3. When the
preferred scale factor was found, the crossover probability was then tuned for the
chosen scale factor.
The large variance in the results in Figure 5.5 is due to the optimisation
algorithm being terminated before it was allowed to nd a minimum. Since, at
this stage, we are only interested in nding a value for the crossover probability
that provides the fastest convergence towards a minimum, we can terminate the
optimisation algorithm after just 10 generations, and observe which values are
beginning to converge faster. This allowed us to reduce the computation time
of tuning the parameters of the DE algorithm. By repeating this process six
105

Chapter 5
times, we can identify which values for the scale factor and crossover probability
consistently provide the fast convergences.
The results in Figure 5.4 show that the DE algorithm performs best for scale
factors above 0 :6. Considering these results along with the recommendations
of [90], discussed in Section 5.2.3, a scale factor value of 1 was chosen for the
minimisation of distortion approach.
The crossover probability was then tuned using a scale factor of 1 and the
results are presented in Figure 5.5. The crossover probability was tuned more
nely between the values 0 :9 and 1, as [92] found this to be the best range
of values for an optimisation problem where the variables cannot be optimised
independently, discussed in Section 5.2.3. Based on the results in Figure 5.5, a
crossover probability value of 0 :98 is chosen for optimising the segment lengths
using the minimisation of distortion approach. This value is selected as it gives
the lowest value for the objective function and has quite a low variance between
each run.
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Scale Factor0.170.180.190.20.210.220.230.240.250.26X1%/hRun1
Run2
Run3
Run4
Run5
Run6
Figure 5.4: Results of tuning the scale factor, Fw, for the minimisation of dis-
tortion approach using a six segment piston wavemaker.
106

Chapter 5
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Crossover Probability0.150.20.250.30.350.40.450.50.550.6X1%/hRun1
Run2
Run3
Run4
Run5
Run6
Figure 5.5: Results of tuning the crossover probability, Cp, for the minimisation
of distortion approach using a six segment piston wavemaker, with increased
resolution between 0 :9 and 1.
5.5 Optimisation results: kinematics matching
This section presents the results of the segment lengths for wavemakers opti-
mised using the kinematic matching approach discussed in Section 5.3. The scale
factor and crossover probability were selected in Section 5.3.1 to be 1 and 0 :5,
respectively, while the population size was set to 100. The algorithm was allowed
to evaluate 1000 generations which proved to be more than enough time for it
to converge to a solution and only took 5.5 mins. The optimisation algorithm
iteratively searches for the objective function dened in Equation (5.3), which
measures how well the prole of the wavemaker matches the depth prole of the

uid in a progressive wave averaged over 10 frequencies uniformly distributed
between 0!14 radians/sec. The performance of the optimised wavemaker
is assessed by the position of 1% distortion, normalised by h, away from the
wavemaker which is plotted against the number of segments in the wavemaker
as shown in Figure 5.6. The segment lengths of the optimised wavemakers are
illustrated in Figure 5.7. The standard deviation of the objective function was
taken over the six optimisation runs for each wavemaker system and are listed in
Table 5.1. The standard deviations, which are 0 for wavemakers with two to four
segments and of the order of 1014for wavemakers with ve or six segments, pro-
107

Chapter 5
Standard deviation over six optimisation runs
Number of segments Piston Flap
2 0 0
3 0 0
4 0 0
5 6 :226910146:22691014
6 3 :113410141:12571014
Table 5.1: Standard deviation of the six optimisation runs for the piston and
ap
wavemakers with two to six segments optimised using the kinematic matching
approach, corresponding to the results presented in Figure 5.6.
vide condence that the DE algorithm found the minimum within a large basin
of attraction of the objective function.
The results presented in Figure 5.6 compare how the optimised segmented
wavemakers perform against each other and the single segment wavemakers in
terms of reducing the position of 1% distortion away from the wavemaker. The
position of 1% distortion is normalised by the still water depth hand is denoted
asX1%=h. The objective function was found by taking the dierence between
the wavemaker prole, c(z), and the progressive wave's depth prole, denoted
in Equation (5.3) as cwm(z) and dened in Equation (2.57), and averaging over
frequency, as shown in Equation (5.3). Traditionally, for a single segment wave-
maker, the rule of thumb has been to allow a distance of 2 hto 3hbetween the
wavemaker and the test area [16], which is justied in Figure 5.6. It is clear from
Figure 5.6 that a wavemaker with multiple segments can provide a signicant im-
provement on reducing the tank space in front of the wavemaker contaminated by
evanescent waves. The results in Figure 5.6 supports the hypothesis that adding
more segments to the wavemaker will always allow the wavemaker to further re-
duce the eect of the distortion caused by the evanescent waves. Given that the
reduction becomes less signicant as more segments are added to the wavemaker,
this analysis does not include wavemakers with more than six segments. Un-
surprisingly, since
ap segments can approximate the kinematics of progressive
waves better than piston segments, the
ap type wavemakers performed better
at reducing the distortion caused by the evanescent waves.
The designs of the segmented wavemakers optimised using the kinematic
matching approach are illustrated in Figure 5.7. Parts (a) and (b) show the
piston and
ap wavemakers, respectively, for wavemakers with two to six seg-
ments as labelled. The lengths of the optimised segments, normalised by h, are
indicated and although the drawings are not strictly to scale, it provides a good
representation of the actual lengths of the segments. The normalised segment
108

Chapter 5
1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6
Number of Segments00.20.40.60.811.21.41.61.82X1%/hSegmented Pistons
Segmented Flaps
Figure 5.6: Normalised position of 1% distortion, X1%=h, averaged over frequency,
against the number of segments in the wavemaker for both segmented piston and

ap wavemakers, with segment lengths optimised using the kinematic matching
approach.
lengths of the wavemakers, shown in Figure 5.7, agree with the hypothesis in
Section 5.1 which proposes that the segments' lengths would decrease in size the
closer they are to the free surface. This is because, as the value of zincreases
fromhtowards zero, the rate of change of the progressive wave's depth prole,
Equation (2.57), increases and therefore should be approximated with shorter
straight lines, as illustrated in Figure 5.1.
109

Chapter 5
0:2605
0:73950:1430
0:2595
0:59750:09760:0738
0:1478
0:4395
2 Segments 3 Segments 4 Segments 5 Segments 6 Segments
0:2731
0:72690:1570
0:2747
0:56830:11040:760
0:1470
0:2507
0:50470:1006
0:23820:0592
0:1024
0:1464
0:2250
0:3909
0:0852
0:1634
0:38490:0878
0:1620
0:2665
0:46110:1140
0:25250:0694
0:1160
0:1618
0:2360
0:3289
2 Segments 3 Segments 4 Segments 5 Segments 6 Segments(a)
(b)
Figure 5.7: Optimised lengths of the segments, normalised by h, in (a) piston
and (b)
ap wavemakers optimised using the kinematic matching approach.
110

Chapter 5
5.6 Optimisation results: minimisation of
distortion
This section presents the results for the optimisation of the segment lengths
in a wavemaker by the minimisation of distortion approach, discussed in Sec-
tion 5.4, using the DE algorithm. As with the kinematic matching approach, the
segment lengths are optimised for both piston and
ap segmented wavemakers.
The strokes of the segments were prescribed to match the displacement of a
uid
in a progressive wave, given by Equation (5.2). The computational demand of
the minimisation of distortion approach was discussed in Section 5.2. To reduce
the time required to converge to a solution, the algorithm, which was set to have
a population size of 100, was terminated after 150 generations. These parameters
were determined with an trial and error approach and were found to be suciently
large enough to allow the algorithm to converge. The scale factor and crossover
probability are set to 1 and 0 :98, respectively, as determined in Section 5.4.2.
The position of 1% distortion for the wavemakers optimised using the minimi-
sation of distortion approach is presented in Figure 5.8 along with that of the
single segment piston and
ap wavemakers for comparison. The reliability test,
discussed in Section 5.2.4, was also performed and Table 5.2 gives the standard
deviation of the objective function over the six optimisation runs to determine if
the algorithm has converged towards the best available minima. The lengths of
the optimised segments are presented in Figure 5.9.
The values of the standard deviations presented in Table 5.2 show a supe-
rior convergence towards a minimum value of the objective function than the
results presented in Figure 5.5. This is because, unlike the process of tuning the
crossover probability, for which the results are presented in Figure 5.5, while op-
timising the segment lengths, the DE algorithm was allowed to converge towards
a minimum value of the objective function. In contrast to this, when tuning the
crossover probability, the DE algorithm was terminated prematurely as discussed
in Section 5.4.2.
As seen in Figure 5.6, the results presented in Figure 5.8 show that
ap seg-
mented wavemakers are better at reducing the position of 1% distortion than
piston wavemakers and that further improvement is achieved by adding more
segments to the wavemaker. The position of 1% distortion presented in Fig-
ure 5.8 is normalised by the still water depth h. What is most interesting about
the results from Figure 5.8 is that, for the
ap wavemakers with ve and six seg-
ments, the level of distortion caused by the evanescent waves falls to 1% almost
immediately in front of the wavemaker. As it is hoped that further improvement
111

Chapter 5
may be achieved by optimising the strokes of the individual segments, this is a
very promising result. It is worth noting that, for the relatively simple design
of the
ap wavemaker with three segments the position of 1% distortion is al-
most one fth of the water depth in Figure 5.8. For some relatively shallow wave
tanks it may not be possible, for practical reasons, to install a test device within
a distance of 0 :2hfrom the wavemaker. To provide context, when generating
a wave, with kh= 2:73 and a wave height of 0 :2 m in a tank with a depth of
0:6 m, a single bottom hinged
ap wavemaker would require a normalised stroke
ofS0=h= 0:2590 at the still water level. In such a case, further improvement on
the performance of the
ap wavemaker with three segments would not be of any
signicance.
1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6
Number of segments00.20.40.60.811.21.41.61.82X1%/hSegmented Pistons
Segmented Flaps
Figure 5.8: Normalised position of 1% distortion, X1%=h, averaged over frequency,
against the number of segments in the wavemaker for both segmented piston and

ap wavemakers, with the segment lengths optimised using the minimisation of
distortion approach.
The standard deviation over the six optimisation runs was calculated for each
wavemaker, and are presented in Table 5.2. The values reported in Table 5.2 being
of the order 107m or less, allow us to conclude condently that the algorithm
has found the minima of the objective function, at least within a large basin of
attraction.
The designs of the segmented wavemakers optimised using the minimisation
of distortion approach discussed above are illustrated in Parts (a) and (b) of Fig-
112

Chapter 5
Number of segments Piston Flap
2 1 :172910141:46191014
3 1 :928310142:63981015
4 8 :767610159:10111016
5 1 :597110122:63001010
6 6 :73721091:4622107
Table 5.2: Standard deviation of the six optimisation runs for the piston and
ap
segment wavemakers with two to six segments optimised using the minimisation
of distortion approach, corresponding to the results presented in Figure 5.8.
ure 5.9 for the piston and
ap, respectively, and normalised by h. Again, the
drawings in Figure 5.9 are not to scale, but do provide a good representation of
the correct segment lengths, which are indicated. It is clear from Figure 5.9 that
the segment lengths do not strictly decrease the closer they are to the free sur-
face. Unlike the case of the
ap type wavemaker optimised using the kinematic
approach, it is believed that the segment lengths, optimised using the minimi-
sation of distortion approach, provides a destructive interference pattern which
achieves the shortest attainable distance between the wavemaker and the position
of 1% distortion.
113

Chapter 5
0:3489
0:65110:2501
0:2833
0:46660:20170:1720
0:1511
0:34080:0972
0:1792
0:2347
0:38440:1286
0:20740:1474
0:1072
0:1360
0:1923
0:3198
2 Segments 3 Segments 4 Segments 5 Segments 6 Segments
0:4010
0:59900:2578
0:2714
0:47090:1800:1269
0:1491
0:38030:0685
0:1606
0:2483
0:41110:1042
0:23950:0870
0:0939
0:1482
0:2415
0:3610
2 Segments 3 Segments 4 Segments 5 Segments 6 Segments(a)
(b)
Figure 5.9: Optimised lengths of the segments, normalised by h, in (a) piston and
(b)
ap wavemakers optimised using the minimisation of distortion approach.
114

Chapter 5
5.7 Comparison of results
The following section is a comparison of the results from both the kinematic
matching and the minimisation of distortion approaches in Sections 5.5 and 5.6,
respectively. A comparison is made to discuss the performance of both methods,
the role of the interference pattern between the evanescent waves and the prac-
tical implementation of both approaches in terms of computational time. Both
optimisation strategies were performed on the same PC with an i7, 2.9 GHz core
with eight logical processors and 16 GBs of RAM.
It is clear from Figures 5.6 and 5.8 that the minimisation of distortion ap-
proach can yield better results. Intuitively, as discussed in Section 5.1, it could
be expected that the segment lengths in a wavemaker would decrease the closer
they are to the free surface. However, both types of segmented wavemakers op-
timised by the minimisation of distortion approach, shown in Figure 5.9, deviate
from this behaviour. This is unsurprising since this approach utilises the phase
shifts in the evanescent waves to cause as much destructive interference as pos-
sible in order to minimise the distortion, rather than simply approximating the

uid motion in a progressive wave. There still remains a clear trend in Figure 5.9
where the segments closer to the free surface tend to be shorter than those fur-
ther away. This provides a compelling argument that the interference pattern
between the evanescent waves, which was introduced in Section 2.4.2, can have
a noticeable in
uence on the optimal design of a wavemaker when reducing the
distance between the wavemaker and the testable area in the tank.
In terms of computation time, the kinematic matching approach is a great deal
more ecient, terminating after an average of 5 :5 mins per run compared to the
minimisation of distortion approach which takes, on average, 2 :5 hours per run
to terminate. For the reader who wishes to optimise the design of their own seg-
mented wavemaker with a given number of segments, by following the guidelines
presented in this chapter the total time required to optimise the system should
be approximately 33 mins using the kinematic matching approach and 15 hours
using the minimisation of distortion approach. Even though it takes 27 :3 times
longer to compute, the time required to optimise a segmented wavemaker using
the minimisation of distortion approach is reasonable in context with the time
required to design the mechanics of the wavemaker and to develop the code to
predict the hydrodynamic model.
115

Chapter 5
5.8 Conclusion
The optimisation of the lengths of the segments in a wavemaker with the
aim of reducing the eect of distortion in a wave tank caused by the presence of
evanescent waves, has been discussed. Two separate strategies were employed to
optimise the geometry of the wavemaker and the results from both approaches
were presented. Approach 1 was simply to approximate the motion of the
uid in
a progressive wave as accurately as possible with the motion of the wavemaker.
Approach 2 was to minimise the distance between the wavemaker and the testable
area in the tank directly. This approach aims to nd the wavemaker design which
creates a destructive interference pattern between the evanescent waves that min-
imises the amplitude of the evanescent wave eld. This is possible by causing a
phase shift of radians to some of the evanescent waves. For both strategies, the
DE algorithm was used to search for the segments' lengths which gave the lowest
value of the objective functions. An extensive procedure was followed to ensure
that the minima found by the DE algorithm were either the global or the best
available minima of the objective function. The results show that minimising
the distance between the wavemaker and the testable area yields notably better
results, though this was expected. The comparison between both methods in-
dicates that the phases of the individual evanescent waves can be optimised to
create a destructive interference pattern which minimises the distortion in a wave
tank.
116

Chapter 6
Chapter 6
Optimisation of segment stroke
6.1 Introduction
The aim of this chapter is to nd the optimal strokes of each segment in the
wavemaker which minimises the distance between the wavemaker and the testable
area in the tank. Hypothesis 2, Section 2.4.2, proposed that the optimal geometri-
cal design for a rigid-body wavemaker for minimising the distortion would lead to
some evanescent waves experiencing a phase shift of radians in relation to other
evanescent waves. The premise of Hypothesis 2 is that these phase shifts would
then lead to a destructive interference pattern, hence, reducing the magnitude of
the collective evanescent wave eld.
The results presented in Chapter 5 conrmed Hypothesis 2 where it was
demonstrated that, when optimising the segment lengths in a wavemaker, the
interference pattern could be optimised to minimise the wave eld distortion.
Other evidence supporting Hypothesis 2 are the results of the constrained added
mass for the segmented wavemakers in Figures 4.20 and 4.21, which were dis-
cussed in Section 4.6.3.
Although the results in Section 5.6 of the optimised segment lengths showed
signicant improvement over the traditional piston and
ap wavemakers, the
strokes of each segment were prescribed by the kinematics of the progressive
wave, Equation (2.57), Section 2.4. According to Hypothesis 2, however, there is
no reason that suggests the optimal strokes will be provided by Equation (2.57).
Thus, it is conceivable that further improvement may be achieved by optimising
the segment strokes to utilise the evanescent waves' interference pattern.
As discussed in Section 5.4.1, minimising the total distortion directly is di-
cult and cumbersome. Instead, the same objective function is employed that was
proposed in Section 5.4.1 for the minimisation of distortion approach; that is the
distance from the wavemaker to where the distortion level is 1% and does not
increase to be greater than 1% for further increases in x, this is referred to as
117

Chapter 6
the position of 1% distortion and denoted by X1%. Certainly, we may presume
that a wavemaker in which both the segments' lengths and strokes are optimised
will provide the best performance in terms of reducing the position of 1% distor-
tion. Yet, a question that emerges is, which has a greater in
uence on minimising
the distortion: optimising the segment lengths or the segment strokes? The mo-
tivation behind this question is to understand which aspect is more important
when designing a wavemaker for reducing the distortion, the physical geometry
of each segment or the wavemaker's control system, which controls the strokes
of each segment. To answer this question, the segment strokes are optimised for
two cases: (1) where the segment lengths are equal and (2) where the segment
lengths were optimised by the minimisation of distortion approach and are given
in Figure 5.9, Section 5.6. The results from Case (1) can then be compared to
those presented in Figure 5.8, where the segmented wavemakers have optimised
segment lengths, but the strokes were determined by the kinematic matching
approach, Equation (5.2). This comparison will allow us to determine whether
the optimal segment lengths or strokes are more eective at reducing the wave
eld distortion. The results from Case (2) will provide the best achievable re-
duction of the distance between the wavemaker and the testable tank area using
the segmented wavemakers. Unlike the segment lengths, it is feasible to tune the
segment strokes for each individual frequency. It is to be expected that the poor-
est performance, in terms of minimising the distortion, will arise at the highest
frequency considered; however, it is also useful to understand how the wavemaker
performs over an operating frequency range.
The DE algorithm used in Chapter 5 is selected to optimise the segment
strokes, as it proves to be eective at nding the minima of the objective function
in a relatively short period of time and is rather simple to implement, having
few parameters to tune [92]. The DE algorithm is demonstrated, by the results
presented in this chapter, to be quite reliable at nding the lowest obtainable
values of the objective function, the position of 1% distortion. From the results
presented in Figure 5.8, it is clear that no further signicant improvement in
the reduction of the wave eld distortion can be achieved by adding more than
six segments into the
ap wavemaker. Hence, our investigation in this chapter
is restricted to both the piston and
ap wavemakers with two to six segments.
The performance of a single segment piston and
ap, in terms of the position of
1% distortion, is also presented to allow for a comparison between the optimised
segmented wavemakers and the traditional wavemakers.
In order to improve the performance of the DE algorithm during the optimi-
sation process, the variable search space is limited by constraining the maximum
amplitude of the segments' strokes to not exceed 2 unit strokes (a unit stroke is a
118

Chapter 6
stroke of 1 m). This does not aect the results in any way, as the distortion of the
wave eld is not strictly dependent on the amplitude of the segments' strokes,
but rather the ratio between the amplitudes of the progressive and evanescent
wave elds. Reducing the search space of each variable allows the DE algorithm
to converge to a solution much faster. It could certainly be argued that, in order
to achieve the true optimal wavemaker conguration, the segment lengths and
strokes should be optimised simultaneously. However, this optimisation problem
becomes much more dicult, having roughly twice the number of variables. Bear-
ing in mind the quality of results obtained from optimising the segment lengths
alone, it seems reasonable to assume that no signicant improvement would be
achieved by doing so.
Presented in Section 6.2 is the analysis for tuning the parameters of the DE
algorithm for the problem of optimising the strokes of the segmented wavemakers.
In Section 6.3 the strokes are optimised for segmented wavemakers where all the
segments are equal in length. A comparison is then made with the results of the
segmented wavemakers with optimised segment lengths and strokes prescribed
by Equation (5.2) in Figure 5.8, Section 5.6. The optimisation of the strokes
for the segmented wavemakers designs with optimised segment lengths, shown
in Figure 5.8, is presented in Section 6.4. The depth proles of the segmented
wavemakers where both the lengths and strokes are optimised is also presented
in Section 6.4 to conrm the validity of Hypothesis 2, Section 2.4.2. Section 6.5
looks at the aect of the evanescent waves on the wave eld when the strokes and
lengths of the segments in the wavemaker are optimised. A sensitivity analysis as
to how errors in the wavemaker's strokes aect the performance of the wavemaker
in terms of the distortion to the wave eld and the delity of the progressive wave
height is presented in Sections 6.6.1 and 6.6.2, respectively. Finally, the ndings
of the optimisation of segment strokes in segmented wavemakers are concluded
in Section 6.7.
6.2 Tuning the optimisation parameters
To ensure ecient and reliable convergence to a solution for the problem of
optimising the segment strokes, the parameters of the DE algorithm must be
tuned, as they were in Chapter 5. It was found that a population size of 100 solu-
tion candidates provided good convergence speeds with little benet arising from
larger population sizes which require more computation time. When optimising
the
ap segmented wavemakers at kh= 12 and the piston segmented wavemaker
119

Chapter 6
over the entire range of 0 kh12, there was no further improvements to the
solutions found by the optimiser after 300 generations. When optimising the
ap
segmented wavemakers for kh10:8 it was found that the DE algorithm did not
nd any better solutions after 120 generations. As explained in Section 5.2.4, it
is important to ensure that the minimal value for the objective function found
by the optimiser is in fact the lowest value that can be achieved. Here a little
foresight is used by choosing to repeat the optimisation runs six times. This will
be justied by the results presented in this chapter, which indicate that six op-
timisation runs is sucient in determining that the DE algorithm is converging
towards the same best available minimum.
While tuning the DE algorithm's scale factor, Fw, the population size was
set to 100 and the crossover probability was set to a value of 1. For a range of
values between 0 :3 and 1, the scale factor was tuned for a piston type wavemaker
with six segments. The DE algorithm was terminated after 100 generations.
The six independent optimisation runs for the tuning process are presented in
Figure 6.1. The results in Figure 6.1 clearly indicate that the fastest convergence
0.20.30.40.50.60.70.80.9 100.0050.010.015
Scale FactorObjective function

Run 1
Run 2
Run 3
Run 4
Run 5
Run 6
Figure 6.1: Results of tuning the scale factor, Fw, for optimising the strokes of
each segment.
to a minimum is achieved with a scale factor value of 0 :3. For reasons explained
in Section 5.2.3 scale factor values less than 0 :3 were not considered.
120

Chapter 6
Following this, the crossover probability, Cp, was tuned between values of 0 :3
and 1, using a scale factor of 0 :3. Again the population size was set to 100 while
the algorithm was terminated after 50 generations. Between values of 0 :9 and 1,
the resolution for tuning the crossover probability was increased, since the most
appropriate value is often found within this range, as discussed in Section 5.2.3.
The results of tuning the crossover probability, for the purpose of optimising
the segments' strokes, are presented in Figure 6.2. Figure 6.2 indicates that
the crossover probability value which nds the most reliably low value for the
objective function is 1.
0.20.30.40.50.60.70.80.9 10.010.0120.0140.0160.0180.020.0220.0240.0260.0280.03
Crossover ProbabilityObjective function

Run 1
Run 2
Run 3
Run 4
Run 5
Run 6
Figure 6.2: Results of tuning the crossover probability, Cp, for optimising the
strokes of each segment.
Following from Figures 6.1 and 6.2 the optimisation of the segment strokes
carried out in this chapter is done with the scale factor and crossover probability
of the DE algorithm set to 0 :3 and 1, respectively.
As with the results presented in Figure 5.5, there is a large variance in the
results presented in Figures 6.1 and 6.2. The reason for this was described in
Section 5.4.2, which is that for the purpose of tuning both the scale factor and the
crossover probability, the optimisation algorithm was terminated before it fully
converged to a minimum. By terminating the optimisation algorithm early, we
were able to reduce the computational cost, while still being able to determine the
121

Chapter 6
appropriate values for the scale factor and the crossover probability that provided
the fastest convergence for the optimisation problem.
6.3 Case (1): Optimisation of strokes for
segments of equal lengths
This section will address the problem of nding the optimal strokes for each
segment in a wavemaker where all the segments are of equal lengths, as proposed
in Section 6.1, Case (1). The segment stokes are optimised for a number of
frequencies, for both piston and
ap wavemakers with two to six segments. To
provide condence that the minima found are indeed the lowest available values of
the objective function, the optimisation runs are repeated six times, as discussed
in Section 6.2.
The lowest values for the objective function, the position of 1% distortion,
obtained over the six optimisation runs are plotted against khin Figures 6.3
and 6.4 for the piston and
ap wavemakers, respectively. In both Figures 6.3
and 6.4, the yaxis is the normalised position of 1% distortion, while khis rep-
resented by the xaxis. Immediately, we see in both Figures 6.3 and 6.4 that
optimising the strokes of the segmented wavemakers provides a drastic reduction
in the level of distortion in the tank compared to the traditional single segment
wavemakers. Comparing Figures 6.3 and 6.4 we see that, just like the results
presented in Figures 5.6 and 5.8, Chapter 5, the
ap wavemakers are still more
successful at reducing the eect of the wave eld distortion caused by the evanes-
cent waves. To answer the question as to which provides a greater reduction in
the wave eld distortion: optimising the segment lengths or strokes, Figures 6.3
and 6.4 are compared to the respective results for the piston and
ap wavemakers
in Figure 5.8, where the segment lengths have been optimised but the strokes are
prescribed by Equation (5.2). It is evident from Figures 6.3, 6.4 and 5.8 that
optimising the segment strokes yields better results for reducing the wave eld
distortion than optimising the segment lengths.
To provide context, we look at the two segment piston and
ap wavemakers
in Figure 5.8, where the segment lengths are optimised and the segment strokes
are prescribed by the kinematics of a progressive wave. The normalised positions
of 1% distortion, X1%=h, atkh= 12 are 1.1412 and 0.6951, for the piston and

ap wavemakers, respectively. The corresponding values presented in Figures 6.3
and 6.4 for the normalised positions of 1% distortion are 0 :4464 and 0:08594 at
kh= 12 for the piston and
ap segmented wavemakers, respectively, where the
122

Chapter 6
2 3 4 5 6 7 8 9 10 11 12
kh00.511.52X1%/h1 Segments
2 Segments
3 Segments
4 Segments
5 Segments
6 Segments
Figure 6.3: Lowest value of X1%=hobtained by optimising the segments' strokes
for piston wavemakers where the segments are of equal length, where h= 0:6.
segments are of equal length. An argument can be made that this is partly due
to the segment lengths not being optimised for each frequency like the strokes
were; however, having the segment lengths change for each frequency is clearly
impractical, even when monochromatic waves are being considered. Perhaps what
is most encouraging about the results in Figure 6.4 is that the normalised position
of 1% distortion for a two segment
ap wavemaker goes to zero for kh4:9. For

ap wavemakers with more than three segments in Figure 6.4, the normalised
position of 1% distortion is zero for the entire range of kh. This suggests that the
wave eld distortion can eectively be eliminated using a rather simple wavemaker
design.
Regarding the generation of random waves, by simply applying the principle
of superposition we could work out the resulting distortion pattern. However,
by optimising the segment strokes of the wavemaker specically for the spec-
trum being generated, it may be possible that the evanescent waves generated at
one frequency would help cancel out the evanescent waves generated at another
frequency, reducing the eect of the distortion further.
The minimum values for the objective function found by the optimiser during
each optimisation run are presented in Tables A.1 to A.5 for the segmented piston
123

Chapter 6
2 3 4 5 6 7 8 9 10 11 12
kh00.20.40.60.811.21.41.61.8X1%/h1 Segments
2 Segments
3 Segments
4 Segments
5 Segments
6 Segments
Figure 6.4: Lowest value of X1%=hobtained by optimising the segments' strokes
for
ap wavemakers where the segments are of equal length, where h= 0:6.
congurations, and Tables A.6 to A.10 for the segmented
ap congurations in
Appendix A. The standard deviations of the six independent runs for each wave-
maker are presented in Tables A.11 and A.12 in Appendix A for the piston and

ap wavemakers, respectively. Tables A.1 to A.12, in Appendix A, are provided
for completeness and demonstrate the consistency of the minimum objective func-
tion value found by the optimisation algorithm. The optimised segment strokes
that resulted in the lowest positions of 1% distortion achieved by the piston con-
guration wavemakers, at each frequency in Tables A.1 to A.5, are presented in
Tables A.13 to A.17 in Appendix A. Similarly, the optimised segment strokes for
the
ap wavemakers are given in Tables A.18 to A.22 in Appendix A. Bear in
mind that the segmented
ap wavemaker is actuated at the joints between the

aps as well as at the positions z= 0 andz=h. Hence, a
ap wavemaker
withNsegments has ( N+ 1) strokes. The values for segment strokes presented
in Tables A.13 to A.22, in Appendix A, can be used as inputs for a controller
designed to operate the segmented wavemaker. In situations where the optimiser
reports an objective function value of zero, we can be certain that the true min-
imum has been found, since a negative value would indicate a location behind
the wavemaker. For the cases when the lowest value for the objective function
124

Chapter 6
is not zero, the standard deviation of the results from the independent optimisa-
tion runs presented in Tables A.11 and A.12 are suciently low to conclude with
condence that the optimiser has found the best available minimum.
6.4 Case (2): Optimisation of strokes for
optimised segment lengths
The second problem considered in this chapter is the optimisation of the
strokes for the segmented wavemakers illustrated in Figure 5.9, where the seg-
ment lengths have been optimised by the minimisation of distortion approach
in Section 5.6. The ambition of this section is to nd the conguration of each
segmented wavemaker which provides the lowest value of the position of 1% dis-
tortion that can be achieved by the wavemakers. For the rest of this thesis the
wavemakers optimised in this section will be referred to as \completely opti-
mised" segmented wavemakers since both the segment lengths and strokes have
been optimised.
As before, the strokes are optimised for piston and
ap wavemakers with two
to six segments. The best obtained values of the position of 1% distortion over
the six optimisation runs are plotted for each optimised segment piston and
ap
wavemaker in Figures 6.5 and 6.6, respectively. It is apparent, from comparing
Figures 6.5 and 6.6 to the results presented in Figures 6.3 and 6.4, that the com-
pletely optimised segmented wavemakers provide only a small improvement from
wavemakers where the strokes alone were optimised. It is reasonable to determine
that the most important factor, in terms of reducing the eect of distortion in
the wave tank, to be considered when designing a segmented wavemaker, is the
strokes of the segments. Although, as constructing the wavemakers illustrated
in Figure 5.9 presents no additional challenges in comparison with wavemakers
where the all segments are equal in length, there is no obvious reason as to why
the optimal segment lengths should not be used. It is worth noting from Fig-
ure 6.6 that the three segment
ap wavemaker achieves a distortion level less than
1% at the surface of the wavemaker for the entire range of kh.
The results of the six independent optimisation runs for each segmented wave-
maker are presented in Tables A.23 to A.27 for the piston wavemakers and Ta-
bles A.28 to A.32 for the
ap wavemakers, in Appendix A. The standard de-
viations over the six optimisation runs are presented in Tables A.33 and A.34
for the piston and
ap wavemakers, respectively, in Appendix A. The optimised
125

Chapter 6
2 3 4 5 6 7 8 9 10 11 12
kh00.511.52X1%/h1 Segments
2 Segments
3 Segments
4 Segments
5 Segments
6 Segments
Figure 6.5: Lowest value of X1%=hobtained by optimising the segments' strokes
for piston wavemakers with optimised segment lengths, where h= 0:6.
strokes which provided the lowest position of 1% distortion of the six optimisa-
tion runs for each wavemaker are presented in Tables A.35 to A.39 for the piston
wavemakers and Tables A.40 to A.44 for the
ap wavemakers. Again, recalling
from Section 6.3 that a
ap wavemaker with Nsegments has ( N+ 1) strokes.
For the instances where an objective function value of zero was not obtained, the
maximum standard deviation presented in Tables A.33 and A.34 are so low that
it seems safe to conclude that the minimum values of the objective function was
found in each case.
To provide graphical context of the completely optimised segmented wave-
makers, the depth proles, normalised by the stroke of the top segment, for each
wavemaker at k= 2, 10 and 20 m1are illustrated in Figures 6.7 and 6.8 for
the piston and
ap wavemakers, respectively. The depth proles in Figures 6.7
and 6.8 highlight how dierent the optimised wavemakers' depth proles are to
that of the progressive waves, Equation (2.57). The signicance of the results
presented in this chapter is not solely the success in reducing the level of distor-
tion in the wave eld, but also in the validation of Hypothesis 2, Section 2.4.2.
The depth proles of the segmented wavemakers in Figures 6.7 and 6.8 suggest
that the success in reducing the distortion was achieved by utilising the inter-
126

Chapter 6
2 3 4 5 6 7 8 9 10 11 12
kh00.20.40.60.811.21.41.61.8X1%/h1 Segments
2 Segments
3 Segments
4 Segments
5 Segments
6 Segments
Figure 6.6: Lowest value of X1%=hobtained by optimising the segments' strokes
for
ap wavemakers with optimised segment lengths, where h= 0:6.
ference pattern between the evanescent waves. This contradicts the traditional
assumptions that, to reduce the distortion, a wavemaker should approximate the
kinematics of a progressive wave, as stated in Hypothesis 1, Section 2.4. It should
be noted that the segments in the wavemakers in Figures 6.7 and 6.8 are either
in phase or in anti-phase with each other. It is worth mentioning that the wave-
makers illustrated in Figures 6.7 and 6.8 don't displace as much
uid for a given
stroke,S0(i.e., the stroke at the free surface elevation), resulting in smaller wave
heights being generated. To account for this, the amplitude of the all the segments
strokes can be increased using a correction factor, Equation (4.38) Section 4.6.3.
127

Chapter 6
-5 0 5
Normalised horizontal displacement-1-0.50z/h(c) 6 Segment Piston
-1 -0.5 0 0.5 1
Normalised horizontal displacement-1-0.50z/h(b) 6 Segment Piston
-1 0 1
Normalised horizontal displacement-1-0.50z/h(a) 6 Segment Piston-1 0 1
Normalised horizontal displacement-1-0.50z/h(a) 5 Segment Piston
-1 -0.5 0 0.5 1
Normalised horizontal displacement-1-0.50z/h(b) 5 Segment Piston
-5 0 5
Normalised horizontal displacement-1-0.50z/h(c) 5 Segment Piston-2 -1 0 1 2
Normalised horizontal displacement-1-0.50z/h(c) 4 Segment Piston
-1 0 1
Normalised horizontal displacement-1-0.50z/h(b) 4 Segment Piston
-1 0 1
Normalised horizontal displacement-1-0.50z/h(a) 4 Segment Piston-1 0 1
Normalised horizontal displacement-1-0.50z/h(a) 3 Segment Piston
-1 -0.5 0 0.5 1
Normalised horizontal displacement-1-0.50z/h(b) 3 Segment Piston
-1 -0.5 0 0.5 1
Normalised horizontal displacement-1-0.50z/h(c) 3 Segment Piston-1 -0.5 0 0.5 1
Normalised horizontal displacement-1-0.50z/h(c) 2 Segment Piston
-1 -0.5 0 0.5 1
Normalised horizontal displacement-1-0.50z/h(b) 2 Segment Piston
-1 -0.5 0 0.5 1
Normalised horizontal displacement-1-0.50z/h(a) 2 Segment Piston
Figure 6.7: Depth proles of completely optimised segmented piston wavemakers,
normalised by the stroke of the top segment, for (a) != 4 rad/sec, (b) !=
9:9 rad/sec and (c) != 14 rad/sec.
128

Chapter 6
-1 -0.5 0 0.5 1
Normalised horizontal displacement-1-0.50z/h(c) 6 Segment Flap
-1 -0.5 0 0.5 1
Normalised horizontal displacement-1-0.50z/h(b) 6 Segment Flap
-1 0 1
Normalised horizontal displacement-1-0.50z/h(a) 6 Segment Flap-1 0 1
Normalised horizontal displacement-1-0.50z/h(a) 5 Segment Flap
-1 -0.5 0 0.5 1
Normalised horizontal displacement-1-0.50z/h(b) 5 Segment Flap
-1 -0.5 0 0.5 1
Normalised horizontal displacement-1-0.50z/h(c) 5 Segment Flap-1 -0.5 0 0.5 1
Normalised horizontal displacement-1-0.50z/h(c) 4 Segment Flap
-1 -0.5 0 0.5 1
Normalised horizontal displacement-1-0.50z/h(b) 4 Segment Flap
-1 0 1
Normalised horizontal displacement-1-0.50z/h(a) 4 Segment Flap-2 0 2
Normalised horizontal displacement-1-0.50z/h(a) 3 Segment Flap
-1 -0.5 0 0.5 1
Normalised horizontal displacement-1-0.50z/h(b) 3 Segment Flap
-1 -0.5 0 0.5 1
Normalised horizontal displacement-1-0.50z/h(c) 3 Segment Flap-1 -0.5 0 0.5 1
Normalised horizontal displacement-1-0.50z/h(c) 2 Segment Flap
-1 -0.5 0 0.5 1
Normalised horizontal displacement-1-0.50z/h(b) 2 Segment Flap
-1 0 1
Normalised horizontal displacement-1-0.50z/h(a) 2 Segment Flap
Figure 6.8: Depth proles of completely optimised segmented
ap wavemakers,
normalised by the stroke of the top segment, for (a) != 4 rad/sec, (b) !=
9:9 rad/sec and (c) != 14 rad/sec.
129

Chapter 6
6.5 Analysis of the evanescent wave eld
(sample results)
The aect of the interference pattern on the wave eld distortion is illustrated
in Figures 6.9 and 6.10, where the distortion is plotted against the normalised
distance away from the wavemaker, x=h. The distortion patterns shown in Fig-
ures 6.9 and 6.10 are those caused by the wavemakers' depth proles presented
in Figures 6.7 and 6.8, respectively. The horizontal line in Figures 6.9 and 6.10
highlights the level of 1% distortion, the threshold distortion level considered in
this thesis.
If none of the evanescent waves underwent a phase shift, the distortion func-
tions presented in Figures 6.9 and 6.10 would decrease in a exponential manner
with increasing x, as shown in Section 2.4.1, Figure 2.7 and by [19]; however, this
is obviously not the case. Instead, the distortion functions shown in Figures 6.9
and 6.10 are the results of the superposition of 49 dierent exponential functions,
Equation (2.56), containing various exponents and with positive and negative
coecients. The peaks in the distortion function are indicative of areas of con-
structive interference between the evanescent waves, while the occurrences of the
distortion level going to zero arise from destructive interference. In each plot in
Figures 6.9 and 6.10 after the last distortion peak (i.e., the peak furthest away
from the wavemaker's surface at x= 0), the distortion does appear to exhibit
asymptotic behaviour by approaching zero as x! 1 .
130

Chapter 6
0 1 2 30.20.40.60.81
x/hDistortion(a) 2 Segment Piston
0 1 2 351015
x/hDistortion(b) 2 Segment Piston
0 1 2 310203040
x/hDistortion(c) 2 Segment Piston
0 1 2 30.20.40.60.81
x/hDistortion(a) 3 Segment Piston
0 1 2 312345
x/hDistortion(b) 3 Segment Piston
0 1 2 35101520
x/hDistortion(c) 3 Segment Piston
0 1 2 30.20.40.60.81
x/hDistortion(a) 4 Segment Piston
0 1 2 30.20.40.60.81
x/hDistortion(b) 4 Segment Piston
0 1 2 3246810
x/hDistortion(c) 4 Segment Piston
0 1 2 30.20.40.60.81
x/hDistortion(a) 5 Segment Piston
0 1 2 30.20.40.60.81
x/hDistortion(b) 5 Segment Piston
0 1 2 31234
x/hDistortion(c) 5 Segment Piston
0 1 2 30.20.40.60.81
x/hDistortion(a) 6 Segment Piston
0 1 2 30.20.40.60.81
x/hDistortion(b) 6 Segment Piston
0 1 2 30.20.40.60.81
x/hDistortion(c) 6 Segment Piston
Figure 6.9: Wave eld distortion [%] against the distance away from the wave-
maker normalised by the tank depth, x=h, created by the depth proles of
completely optimised segmented piston wavemakers for (a) != 4 rad/sec, (b)
!= 9:9 rad/sec and (c) != 14 rad/sec. The dashed line marks the 1% distortion
level.
131

Chapter 6
0 1 2 30.20.40.60.81
x/hDistortion(a) 2 Segment Flap
0 1 2 30.20.40.60.81
x/hDistortion(b) 2 Segment Flap
0 1 2 3246810
x/hDistortion(c) 2 Segment Flap
0 1 2 30.20.40.60.81
x/hDistortion(a) 3 Segment Flap
0 1 2 30.20.40.60.81
x/hDistortion(b) 3 Segment Flap
0 1 2 30.20.40.60.81
x/hDistortion(c) 3 Segment Flap
0 1 2 30.20.40.60.81
x/hDistortion(a) 4 Segment Flap
0 1 2 30.20.40.60.81
x/hDistortion(b) 4 Segment Flap
0 1 2 30.20.40.60.81
x/hDistortion(c) 4 Segment Flap
0 1 2 30.20.40.60.81
x/hDistortion(a) 5 Segment Flap
0 1 2 30.20.40.60.81
x/hDistortion(b) 5 Segment Flap
0 1 2 30.20.40.60.81
x/hDistortion(c) 5 Segment Flap
0 1 2 30.20.40.60.81
x/hDistortion(a) 6 Segment Flap
0 1 2 30.20.40.60.81
x/hDistortion(b) 6 Segment Flap
0 1 2 30.20.40.60.81
x/hDistortion(c) 6 Segment Flap
Figure 6.10: Wave eld distortion [%] against the distance away from the wave-
maker normalised by the tank depth, x=h, created by the depth proles of
completely optimised segmented
ap wavemakers for (a) != 4 rad/sec, (b)
!= 9:9 rad/sec and (c) != 14 rad/sec. The dashed line marks the 1% distor-
tion level.
132

Chapter 6
To illustrate the evanescent wave phase shift, the contributions to the
uid's
horizontal velocity component due to each evanescent wave is presented in Fig-
ure 6.11 for the six segment piston generating a wave with kh= 1:2. Parts (a)
through to (j) in Figure 6.11 illustrate the rst ten terms of the innite summa-
tion series in Equation (2.56), respectively, where Part (a) represents the pro-
gressive wave. In a similar manner to Figures 2.9 and 2.10, in Figure 6.11 the
depth is represented by the y-axis of the graphs while the horizontal velocity
component is represented by the x-axis. As anticipated, the 1st, 4th, 5thand 9th
evanescent waves, Parts (b), (e), (f) and (j) of Figure 6.11, respectively, are 
radians out of phase at z= 0 with the 2nd, 3rd, 6th, 7thand 8thevanescent waves,
Parts (c), (d), (g), (h) and (i) of Figure 6.11, respectively, thus, creating an in-
terference pattern. Another observation is that the amplitude of the evanescent
waves vary in a manner which appears random. For instance, the amplitude of the
eighth summation term in Part (h) Figure 6.11, is greater than the amplitudes of
the second, third, fourth, fth, sixth and seventh terms. This is a unique result
as most sources of literature only consider the case of the single piston wave-
maker when analysing the evanescent waves, in which case the amplitude of the
evanescent waves decrease with larger imaginary wavenumbers (i.e., the higher
its index in the innite summation series in Equation (2.56)). The importance of
this result is that, when evaluating a truncated version of Equation (2.56), care
must be taken to not ignore some of the larger amplitude evanescent waves. For
this thesis it has been determined, via trial and error, that considering the rst
fty terms of the innite series seems to suce in providing accurate evaluations
of Equation (2.56).
133

Chapter 6
-0.1 -0.05 0 0.05 0.1
vx/(S0 ω)-1-0.50z/h(j)
-0.02 -0.01 0 0.01 0.02
vx/(S0 ω)-1-0.50z/h(i)-0.06 -0.04 -0.02 0 0.02 0.04 0.06
vx/(S0 ω)-1-0.50z/h(g)
-0.15 -0.1 -0.05 0 0.05 0.1 0.15
vx/(S0 ω)-1-0.50z/h(h)-0.15 -0.1 -0.05 0 0.05 0.1 0.15
vx/(S0 ω)-1-0.50z/h(f)
-0.04 -0.02 0 0.02 0.04
vx/(S0 ω)-1-0.50z/h(e)-0.01 0 0.01
vx/(S0 ω)-1-0.50z/h(c)
-0.1 -0.05 0 0.05 0.1
vx/(S0 ω)-1-0.50z/h(d)-0.03 -0.02 -0.01 0 0.01 0.02 0.03
vx/(S0 ω)-1-0.50z/h(b)
0.2 0.25 0.3
vx/(S0 ω)-1-0.50z/h(a)
Figure 6.11: The contribution of the rst ten terms in the velocity potential's
summation series, (a) to (j), to the horizontal velocity component of the
uid
on the wavemaker's surface, for a piston wavemaker with six segments at !=
4 rad/sec.
134

Chapter 6
6.6 Sensitivity Analysis
So far in this thesis it has been assumed that the wavemakers are constructed
and operate with perfect accuracy. Yet, it can be certain that when building a
physical segmented wavemaker, the actual device will be
awed and errors will
creep in. As a result, the performance of the wavemaker will dier and the dis-
tortion levels will increase from the minimum. For example, the lengths of the
segments may not be manufactured accurately or the actuators driving the seg-
ments may not be precise. It has already been determined in Section 6.3 that
small variations in the segment lengths will have little consequence on the distor-
tion level compared to the segment stroke. Instead, the attention of this sensitiv-
ity analysis is directed to errors in the segment strokes of the completely optimised
segmented wavemakers. Aside from the distortion level, another concern is the
delity of the progressive wave's wave height, H. From the relationship between
the velocity potential and the wavemaker's motion amplitude, Equation (2.4), it
is easy to see how changes in the segment strokes may impact the height of the
progressive wave. It may be reasonable to think that if more actuators are used,
the wavemaker will have a greater sensitivity to errors in the segments' strokes.
However, as shown by the analysis of the segmented wavemaker's constrained
radiation damping, Figures 4.18 and 4.19, the deeper the segment is, the less
in
uence it has on the progressive wave. Presented in this section is a sensitivity
analysis which assesses the degree to which the wave eld will be aected by
relative errors in the strokes of each segment in the wavemaker.
Relative errors between the segment strokes of 1 103m, 5104m and
5105m were used in carrying out this sensitivity analysis, meaning that the
position of the actuator may have a error of 0:5103m,2:5104m
and2:5105m. During the sensitivity analysis, constraints of the McCowan
breaking wave height limit, 0 :78h[16], and the wave steepness limit, H= = 0:142
(orHk= 0:8922) [31], were imposed on the wave heights. The sensitivity analysis
was performed over a range of khand normalised wave height, H=h, values in
a wave tank of depth, h= 0:6 m. The sensitivity analysis was carried out by
evaluating all possible combinations of relative errors between the segment strokes
for each value of H=h andkh. The combination of relative error in the segments'
strokes that result in the greatest errors in the wave eld are then presented
below.
135

Chapter 6
6.6.1 Sensitivity of the distortion level
The maximum positions of 1% distortion occurring due to relative errors of
1103m, 5104m and 5105m between the segment strokes, has been eval-
uated for both piston and
ap wavemakers with two to six segments. The results
are presented as contour plots where the contour levels indicates the normalised
position of 1% distortion, X1%=h, withkhrepresented on the x-axis and the nor-
malised wave height, H=h, represented on the y-axis. Figures 6.12 and 6.16 show
the position of 1% distortion for segmented piston wavemakers with two to six
segments, respectively, allowing for a relative error between the segments' strokes,
while Figures 6.17 and 6.21 show the same for the segmented
ap wavemakers.
The position of 1% distortion allowing for a relative error of 1 103m is presented
in Part (a) of Figures 6.12 to 6.21, while Parts (b) and (c) show the position of 1%
distortion allowing for relative errors of 5 104m and 5105m, respectively.
For the segmented piston wavemakers, Figures 6.12 to 6.16, it seems that, in gen-
eral, the position of 1% distortion, accounting for relative errors, increases with
khand decreases with increasing numbers of segments in the wavemaker. For the
segmented
ap wavemakers, Figures 6.17 to 6.21, the position of 1% distortion,
accounting for relative errors, does decrease as more segments are added to the
wavemaker, however, it does not follow the pattern of increasing with kh. All
the segmented
ap wavemakers perform better than the corresponding segmented
piston wavemakers at reducing the position of 1% distortion, even when a rela-
tive error between the segment strokes is considered. Part (c) of Figure 6.18 has
been omitted, when allowing for a relative error of 5 105m, as the distortion
caused by a three segment
ap is less than 1% for the entire ranges of khand
H=h considered.
The results for the three segment
ap wavemaker are very encouraging, as
even with the presence of relative errors of 1 103m and 5104m, the posi-
tion of 1% distortion remains close to zero for almost all of the khandH=h ranges
and reaches a maximum value of approximately 0 :4h. For the segmented piston
wavemaker, the results presented in Figures 6.12 to 6.16 indicate that wavemak-
ers with more segments still perform better overall at reducing the position of
1% distortion. It should be noted that the two segment piston wavemaker still
performs quite well with a maximum position of 1% distortion of approximately
0:5hwith a relative error of 1 103m. It seems from this analysis that the
best choice in wavemaker conguration is the three segment
ap wavemaker, as
it reduces the level of distortion to below 1% for much of the H=h andkhrange
and at most, the position of 1% distortion increases to 0 :4hfrom the wavemaker,
while allowing for errors of up to 1 103m in the segments' strokes. The three
136

Chapter 6
segment
ap wavemaker also remains relatively simple to implement.
137

Chapter 6
Figure 6.12: Contour plot of the position of 1% distortion against khandH=h
for a two segment piston wavemaker with relative errors between the segment
strokes of (a) 1103m, (b) 5104m and (c) 5105m.
138

Chapter 6
Figure 6.13: Contour plot of the position of 1% distortion against khandH=h
for a three segment piston wavemaker with relative errors between the segment
strokes of (a) 1103m, (b) 5104m and (c) 5105m.
139

Chapter 6
Figure 6.14: Contour plot of the position of 1% distortion against khandH=h
for a four segment piston wavemaker with relative errors between the segment
strokes of (a) 1103m, (b) 5104m and (c) 5105m.
140

Chapter 6
Figure 6.15: Contour plot of the position of 1% distortion against khandH=h
for a ve segment piston wavemaker with relative errors between the segment
strokes of (a) 1103m, (b) 5104m and (c) 5105m.
141

Chapter 6
Figure 6.16: Contour plot of the position of 1% distortion against khandH=h for
a six segment piston wavemaker with relative errors between the segment strokes
of (a) 1103m, (b) 5104m and (c) 5105m.
142

Chapter 6
Figure 6.17: Contour plot of the position of 1% distortion against khandH=h for
a two segment
ap wavemaker with relative errors between the segment strokes
of (a) 1103m, (b) 5104m and (c) 5105m.
143

Chapter 6
Figure 6.18: Contour plot of the position of 1% distortion against khandH=h for
a three segment
ap wavemaker with relative errors between the segment strokes
of (a) 1103m, (b) 5104m and (c) 5105m.
144

Chapter 6
Figure 6.19: Contour plot of the position of 1% distortion against khandH=h for
a four segment
ap wavemaker with relative errors between the segment strokes
of (a) 1103m, (b) 5104m and (c) 5105m.
145

Chapter 6
Figure 6.20: Contour plot of the position of 1% distortion against khandH=h for
a ve segment
ap wavemaker with relative errors between the segment strokes
of (a) 1103m, (b) 5104m and (c) 5105m.
146

Chapter 6
Figure 6.21: Contour plot of the position of 1% distortion against khandH=h for
a six segment
ap wavemaker with relative errors between the segment strokes of
(a) 1103m, (b) 5104m and (c) 5105m.
147

Chapter 6
6.6.2 Sensitivity analysis: progressive wave height
During wave tank testing, the repeatability and delity of waves produced
by the wavemaker is critical for developing an understanding as to how a device
responds to various wave conditions. Hence, the sensitivity of the progressive
wave height to errors in the segments' strokes is of great interest. The results of
the progressive wave heights' sensitivity to relative errors of 1 103m, 5104m
and 5105m in the segments' strokes are presented for segmented piston
wavemakers with two to six segments in Figures 6.22 to 6.26 and similarly in
Figure 6.27 and 6.31 for segmented
ap wavemakers with two to six segments,
respectively.
For the contour plots in Figures 6.22 and 6.31, the x-axis represents khand the
y-axis represents the intended normalised wave height, H=h. The contour levels
in Figures 6.22 to 6.31 represent the percentage error between the generated wave
heights, accounting for a relative error between the segmented wavemakers and
the intended wave height. Parts (a), (b) and (c) present the percentage error in
the wave height given a relative error between the segment strokes of 1 103m,
5104m and 5105m, respectively.
Figures 6.32 and 6.33 show the average percentage error occurring with two to
six segments in piston and
ap wavemakers, respectively. The error bars in both
Figures 6.32 and 6.33 indicate the maximum and minimum percentage error that
arises, over the range of khand normalised wave heights in Figures 6.22 to 6.31,
when errors exist in the segment strokes. Parts (a), (b) and (c) present the
percentage error of the wave height with relative errors of 1 103m, 5104m
and 5105m, respectively, in the segment strokes. Figures 6.32 and 6.33 indicate
that the delity of the progressive wave height is acceptable when relative errors
of 1103m, 5104m and 5105m are present in the segment strokes for
all the wavemakers. The maximum percentage error of 2 :2% occurs for the
ap
wavemaker with four segments, Figure 6.33 Part (a). It is interesting to note that
the delity of the progressive wave heights is slightly poorer for the segmented

ap wavemakers than the segmented piston wavemakers. Figures 6.32 and 6.33
alleviate the concern that more segments in the wavemaker would result in too
much unreliability in the progressive wave heights, Section 6.1.
148

Chapter 6
Figure 6.22: Contour plot of the percentage error between the generated wave
height to the intended wave height, H, againstkhandH=h for a two segment
piston wavemaker with relative errors between the segment strokes of (a) 1 
103m, (b) 5104m and (c) 5105m.
149

Chapter 6
Figure 6.23: Contour plot of the percentage error between the generated wave
height to the intended wave height, H, againstkhandH=h for a three segment
piston wavemaker with relative errors between the segment strokes of (a) 1 
103m, (b) 5104m and (c) 5105m.
150

Chapter 6
Figure 6.24: Contour plot of the percentage error between the generated wave
height to the intended wave height, H, againstkhandH=h for a four segment
piston wavemaker with relative errors between the segment strokes of (a) 1 
103m, (b) 5104m and (c) 5105m.
151

Chapter 6
Figure 6.25: Contour plot of the percentage error between the generated wave
height to the intended wave height, H, againstkhandH=h for a ve segment
piston wavemaker with relative errors between the segment strokes of (a) 1 
103m, (b) 5104m and (c) 5105m.
152

Chapter 6
Figure 6.26: Contour plot of the percentage error between the generated wave
height to the intended wave height, H, againstkhandH=h for a six segment
piston wavemaker with relative errors between the segment strokes of (a) 1 
103m, (b) 5104m and (c) 5105m.
153

Chapter 6
Figure 6.27: Contour plot of the percentage error between the generated wave
height to the intended wave height, H, againstkhandH=h for a two segment
ap
wavemaker with relative errors between the segment strokes of (a) 1 103m,
(b) 5104m and (c) 5105m.
154

Chapter 6
Figure 6.28: Contour plot of the percentage error between the generated wave
height to the intended wave height, H, againstkhandH=h for a three segment

ap wavemaker with relative errors between the segment strokes of (a) 1 103m,
(b) 5104m and (c) 5105m.
155

Chapter 6
Figure 6.29: Contour plot of the percentage error between the generated wave
height to the intended wave height, H, againstkhandH=h for a four segment

ap wavemaker with relative errors between the segment strokes of (a) 1 103m,
(b) 5104m and (c) 5105m.
156

Chapter 6
Figure 6.30: Contour plot of the percentage error between the generated wave
height to the intended wave height, H, againstkhandH=h for a ve segment
ap
wavemaker with relative errors between the segment strokes of (a) 1 103m,
(b) 5104m and (c) 5105m.
157

Chapter 6
Figure 6.31: Contour plot of the percentage error between the generated wave
height to the intended wave height, H, againstkhandH=h for a six segment
ap
wavemaker with relative errors between the segment strokes of (a) 1 103m,
(b) 5104m and (c) 5105m.
158

Chapter 6
Figure 6.32: The maximum, minimum and average percentage error of the gener-
ated wave heights to the intended wave heights, H, overkhandH=h in segmented
piston wavemakers with two to six segments and with relative errors between the
segment strokes of (a) 1 103m, (b) 5104m and (c) 5105m.
159

Chapter 6
Figure 6.33: The maximum, minimum and average percentage error of the gener-
ated wave heights to the intended wave heights, H, overkhandH=h in segmented

ap wavemakers with two to six segments and with relative errors between the
segment strokes of (a) 1 103m, (b) 5104m and (c) 5105m.
160

Chapter 6
6.7 Conclusion
With the aim of minimising the distortion to the wave eld caused by evanes-
cent waves when generating monochromatic waves, the segment strokes of both
piston and
ap segmented wavemakers with two to six segments have been opti-
mised for two dierent cases.
In Case (1) the segments in the wavemaker are all equal in length. This
allowed for a comparison between the performance of a wavemaker where just the
segment strokes were optimised, Figures 6.3 and 6.4, and a wavemaker where just
the segment lengths were optimised, Figure 5.8. It was clear from this comparison
that optimising the strokes of the segments yields a notably better reduction in
the wave eld distortion than optimising the segment lengths. However, it was
suggested that this could be due to the fact that the strokes are optimised for
each individual frequency.
In Case (2) the optimisation of the strokes for wavemakers with optimised seg-
ment lengths was carried out. In this case the completely optimised two segment

ap wavemaker seemed to be promising as the maximum position of 1% distortion
was only 0:06827hatkh= 12, and for kh5:9, the distortion never increases
above 1%, Figure 6.6. The sensitivity analysis performed in Section 6.6.1 indi-
cates that, with the presence of errors in the segment strokes of up to 1 103m,
the performance of the two segment
ap wavemaker decreased signicantly. For
instance, with a relative error of 5 105m between the segments, the maximum
position of 1% distortion increased to over 0 :4h; although, this is still better than
a single segment
ap for which the maximum position of 1% distortion is 1 :8h.
The preferred wavemaker design is the completely optimised three segment

ap, as it has relatively few segments and it reduces the distortion level to be
less than 1% over the range 0 kh12. Even with the presence of a relative
error of 1103m between the segments' strokes, the position of 1% distortion
remains quite low.
161

Chapter 7
Chapter 7
Conclusion
The objective of this thesis was to minimise the distortion in a wave tank caused
by the presence of evanescent waves using a segmented wavemaker. Chapters 2
and 4 look at the relevant wavemaker theory and the hydrodynamics of the seg-
mented wavemaker, respectively. Chapter 3 gives an overview of the work carried
out by other researchers who have contributed towards the development of wave-
makers, along with the technologies surrounding wavemakers, including feedback
control systems, second order wavemaker theory and rolling seals. The focus of
this thesis was to investigate Hypotheses 1 and 2, stated in Chapter 2 respectively
as:
1. The closer the depth prole of the wavemaker matches that of the progres-
sive wave, the smaller the amplitude of the evanescent waves, and hence
the lesser the distortion caused by evanescent waves.
2. The distortion of the wave eld, and more specically the distance of 1%
distortion from the wavemaker, can be minimised by developing a multi-
body wavemaker which is designed to maximise the destructive interference
between the evanescent waves.
Chapter 4 presents the hydrodynamics of piston and
ap segmented wavemakers
consisting of one to ten segments, where all the segments in the wavemaker are
equal in length. The geometries of the wavemakers were optimised in accordance
with both Hypothesis 1 and 2 separately. Chapter 5 deals with the optimisation
of the segment lengths while Chapter 6 discusses the optimisation of the segment
strokes. The results presented in Chapters 4, 5 and 6 show that all congurations
of the segment wavemaker provide substantial reduction in the wave eld distor-
tion in comparison to the single segment wavemakers. While Hypothesis 1 and 2
both hold true, the results that prove Hypothesis 2, in Chapters 5 and 6, demon-
strate that a greater reduction in the distortion can be achieved by utilising the
162

Chapter 7
interference pattern between the evanescent waves, rather than approximating
the kinematics of the progressive wave.
Previous to this work, it was always assumed that the innite number of
evanescent waves created during the wavemaking process were in phase with
each other at z= 0. A consequence of this assumption is that the evanescent
wave eld's amplitude would decay exponentially with distance away from the
wavemaker. Another assumption made by previous researchers was that the
amplitude of the evanescent waves decrease as their imaginary wavenumber in-
creases in value. In Section 2.4.2, aspects of wavemaker theory were presented,
for the rst time, which show that neither of these assumptions hold for wave-
makers whose depth proles vary over z. The theory predicted that some of the
evanescent waves experienced a phase shift of radians, which leads to destruc-
tive interference between the evanescent waves. As a result of this phase shift,
the distortion pattern can be very dierent from the exponential function of x
that was expected by other researchers. It was demonstrated in Figure 2.9, Sec-
tion 2.4.2, and in Figure 6.11, Section 6.5, that for a wavemaker prole which is
not constant over depth, the amplitude of the evanescent waves does not decrease
with increasing imaginary wavenumbers as expected. This is an entirely novel
concept.
It was observed that optimising the segment lengths and strokes using the
minimisation of distortion approach achieved superior results than the kinematic
matching approach. This was demonstrated by comparing the results presented
in Figures 6.5 and 6.6 to those presented in Figure 5.6, which illustrated the
performance of the segmented wavemakers in reducing the distortion when opti-
mised using the minimisation of distortion approach and the kinematic matching
approach, respectively.
To understand whether optimising the segment lengths or strokes of the pis-
ton and
ap wavemakers is more benecial, a comparison was made between
Figure 5.8, where the segment lengths were optimised using the minimisation
of distortion approach and where the strokes were prescribed using kinematic
matching, to the results presented in Figures 6.3 and 6.4, where the segments
are of equal lengths and where the minimisation of distortion approach was used
to optimise the segment strokes. The comparison shows that optimising the
strokes is more benecial than optimising the segment lengths. This result can
be largely attributed to the fact that the segment strokes can be optimised for
each frequency, while the segment length cannot. This is an important result
as it is much easier for testing facilities to control the strokes of their segments
rather than the segment lengths.
The results presented in Figures 4.20 and 4.21, Section 4.6.3, for the con-
163

Chapter 7
strained added mass of the segmented wavemaker, conrmed the fundamental
hypothesis of this thesis, that the amplitude of the evanescent waves is reduced
when more segments are added to the wavemaker. The signicance of the re-
duction in the added mass lessens as more segments are added. For most of the
segmented wavemakers considered in Chapter 4, the constrained added mass of
the wavemakers is not monotonic over kh; this is a due to the interference between
the evanescent waves. Without the occurrence of the phase shift, the constrained
added mass would be monotonic over khfor each of these wavemakers, similar to
the single piston wavemaker.
It was proposed in Hypothesis 2, that the interference between the evanes-
cent waves could be optimised in order to increase the destructive interference
and hence, minimise the distortion. Hypothesis 2 was investigated in Chapters 5
and 6, where the lengths and strokes of the segmented wavemaker were opti-
mised in order to minimise the distance between the wavemaker and the testable
area in a wave tank. This approach allows the optimiser to nd the segmented
wavemaker conguration which achieves the minimal level of distortion by op-
timising the destructive interference between the evanescent waves. In order to
draw a comparison between Hypothesis 2 and the traditional ideas, the segments'
lengths and strokes were also optimised to approximate the kinematics of a pro-
gressive wave, Hypothesis 1. Comparing the two approaches for optimising the
segmented wavemaker, in both Chapters 5 and 6, clearly demonstrates that utilis-
ing the interference between the evanescent waves is more successful than simply
approximating the kinematics of a progressive wave. The signicance of this re-
sult is that a short wave tank can be equipped with a wavemaker that drastically
reduces the distortion caused by the evanescent wave eld.
A sensitivity analysis was presented in Section 6.6.1 to see how errors in
the actuator's position can aect the ability of the segmented wavemaker to
reduce the distortion in the wave tank. This analysis was carried out by allowing
for errors between the relative segment strokes of 1 103m, 5104m and
5105m. The results presented in Figures 6.12 to 6.21 demonstrated that
the segmented wavemaker designs perform quite well even with relative errors of
1103m, 5104m and 5105m between the segment positions. Another
sensitivity analysis is presented in Section 6.6.2 which considers how errors in the
segment position aected the wave height of the progressive wave. The results
in Figures 6.22 to 6.31 show that relative errors of 1 103m, 5104m
and 5105m between the segment positions have very little aect on the
wave height of the progressive wave. Furthermore, the variance in the errors
in the progressive wave height between the dierent piston and
ap segmented
wavemakers, presented in Figures 6.32 and 6.33, respectively, was very small.
164

Chapter 7
The nal conclusion of this thesis is that a
ap wavemaker with three seg-
ments is the most appropriate design of segmented wavemaker for reducing the
evanescent wave eld in a wave
ume, as it eectively reduces the position of
1% distortion signicantly over the ranges of khand wave height values consid-
ered. Additionally, the three segment
ap wavemaker is not very susceptible to
actuator errors and remains relatively simple to construct.
7.1 Suggestions for future work
Although we have demonstrated the success of the segmented wavemakers
in reducing the distortion caused by evanescent waves in regular waves, there
remains some areas for which the suitability of the segmented wavemaker is still
unknown. First is the aect the segmented wavemaker has on the spurious free
waves created by the mismatch between the motion of the wavemaker and that of
the
uid in a progressive wave. Second is the practicality of generating irregular
waves and to what extent the segmented wavemaker can reduce the distortion in
an irregular wave eld. It would also be interesting to see how the segmented
wavemaker will perform, when its ability to reduce the distortion is averaged over
frequency with a non-uniform weighting. The third suggestion of future work is
to perform the experimental validation of the existence of the interference pattern
between the evanescent waves, discussed in Section 2.4.2.
It would also be interesting to see how useful an understanding of the inter-
ference pattern between the evanescent waves and thus, the minimisation of a
devices' added mass, will be to other topics, such as the absorption of unwanted
waves and wave energy conversion.
7.2 Closing remarks
For the past hundred years, wavemakers have been iteratively improved as
new technologies have emerged. The topic of generating waves has helped to
deepen our understanding of how structures interact with waves, namely wave
energy converters. Over the last thirty years, as the use of feedback controllers
has become more widespread, some truly impressive wave tanks have been built.
The author is eager to see how wavemakers progress in the future and would
recommend any young engineers to become involved in this enlightening topic.
165

Chapter A
Appendix A
Results of complete optimised
wavemakers
166

Chapter ARun number Wavenumber
2m14m16m18m110m112m114m116m118m120m1
1 0 0.0676 0.1302 0.1718 0.2012 0.2225 0.2385 0.2507 0.2603 0.2678
2 0 0.0676 0.1301 0.1718 0.2012 0.2225 0.2385 0.2507 0.2603 0.2678
3 0 0.0676 0.1302 0.1718 0.2012 0.2225 0.2385 0.2507 0.2603 0.2678
4 0 0.0676 0.1302 0.1718 0.2012 0.2225 0.2385 0.2507 0.2603 0.2678
5 0 0.0676 0.1302 0.1718 0.2012 0.2225 0.2385 0.2507 0.2603 0.2678
6 0 0.0676 0.1302 0.1718 0.2012 0.2225 0.2385 0.2507 0.2603 0.2678
Table A.1: Lowest obtained values of the position of 1% distortion, in meters, for a two segment piston wavemaker with segments of equal
lengths and optimised segment strokes, where h= 0:6 m.
Run number Wavenumber
2m14m16m18m110m112m114m116m118m120m1
1 0 0 0.007501 0.02899 0.04600 0.05955 0.07050 0.07945 0.08685 0.09301
2 0 0 0.007501 0.02899 0.04600 0.05955 0.07050 0.07945 0.08685 0.09301
3 0 0 0.007501 0.02899 0.04600 0.05955 0.07050 0.07945 0.08685 0.09301
4 0 0 0.007501 0.02899 0.04600 0.05955 0.07050 0.07945 0.08685 0.09301
5 0 0 0.007501 0.02899 0.04600 0.05955 0.07050 0.07945 0.08685 0.09301
6 0 0 0.007501 0.02899 0.04600 0.05955 0.07050 0.07945 0.08685 0.09301
Table A.2: Lowest obtained values of the position of 1% distortion, in meters, for a three segment piston wavemaker with segments of
equal lengths and optimised segment strokes, where h= 0:6 m.
167

Chapter ARun number Wavenumber
2m14m16m18m110m112m114m116m118m120m1
1 0 0 0 0 0.003769 0.01224 0.01974 0.02628 0.03196 0.03693
2 0 0 0 0 0.003769 0.01224 0.01974 0.02628 0.03196 0.03693
3 0 0 0 0 0.003769 0.01224 0.01974 0.02628 0.03196 0.03693
4 0 0 0 0 0.003769 0.01224 0.01974 0.02628 0.03196 0.03693
5 0 0 0 0 0.003769 0.01224 0.01974 0.02628 0.03196 0.03693
6 0 0 0 0 0.003769 0.01224 0.01974 0.02628 0.03196 0.03693
Table A.3: Lowest obtained values of the position of 1% distortion, in meters, for a four segment piston wavemaker with segments of equal
lengths and optimised segment strokes, where h= 0:6 m.
Run number Wavenumber
2m14m16m18m110m112m114m116m118m120m1
1 0 0 0 0 0 0 0.001124 0.005298 0.009307 0.01292
2 0 0 0 0 0 0 0.001119 0.005279 0.009257 0.01291
3 0 0 0 0 0 0 0.001174 0.005425 0.009221 0.01291
4 0 0 0 0 0 0 0.001121 0.005271 0.009255 0.01293
5 0 0 0 0 0 0 0.001120 0.005312 0.009328 0.01291
6 0 0 0 0 0 0 0.001147 0.005334 0.009245 0.01292
Table A.4: Lowest obtained values of the position of 1% distortion, in meters, for a ve segment piston wavemaker with segments of equal
lengths and optimised segment strokes, where h= 0:6 m.
Run number Wavenumber
2m14m16m18m110m112m114m116m118m120m1
1 0 0 0 0 0 0 0 0 0 0.002984
2 0 0 0 0 0 0 0 0 0 0.003525
3 0 0 0 0 0 0 0 0 0 0.002858
4 0 0 0 0 0 0 0 0 0 0.003616
5 0 0 0 0 0 0 0 0 0.0004538 0.003870
6 0 0 0 0 0 0 0 0 0.0001429 0.002500
Table A.5: Lowest obtained values of the position of 1% distortion, in meters, for a six segment piston wavemaker with segments of equal
lengths and optimised segment strokes, where h= 0:6 m.
168

Chapter ARun number Wavenumber
2m14m16m18m110m112m114m116m118m120m1
1 0 0 0 0 0.003904 0.01746 0.02877 0.03796 0.04544 0.05156
2 0 0 0 0 0.003905 0.01746 0.02877 0.03796 0.04544 0.05156
3 0 0 0 0 0.003908 0.01746 0.02877 0.03796 0.04544 0.05156
4 0 0 0 0 0.003909 0.01746 0.02877 0.03796 0.04544 0.05156
5 0 0 0 0 0.003912 0.01746 0.02877 0.03796 0.04544 0.05156
6 0 0 0 0 0.003905 0.01746 0.02877 0.03797 0.04544 0.05156
Table A.6: Lowest obtained values of the position of 1% distortion, in meters, for a two segment
ap wavemaker with segments of equal
lengths and optimised segment strokes, where h= 0:6 m.
Run number Wavenumber
2m14m16m18m110m112m114m116m118m120m1
1 0 0 0 0 0 0 0 0 0.0006257 0.004500
2 0 0 0 0 0 0 0 0 0.0008378 0.004500
3 0 0 0 0 0 0 0 0 0.0006869 0.004500
4 0 0 0 0 0 0 0 0 0.0006786 0.004501
5 0 0 0 0 0 0 0 0 0.0006372 0.004500
6 0 0 0 0 0 0 0 0 0.0005590 0.004500
Table A.7: Lowest obtained values of the position of 1% distortion, in meters, for a three segment
ap wavemaker with segments of equal
lengths and optimised segment strokes, where h= 0:6 m.
169

Chapter ARun number Wavenumber
2m14m16m18m110m112m114m116m118m120m1
1 0 0 0 0 0 0 0 0 0 0
2 0 0 0 0 0 0 0 0 0 0
3 0 0 0 0 0 0 0 0 0 0
4 0 0 0 0 0 0 0 0 0 0
5 0 0 0 0 0 0 0 0 0 0
6 0 0 0 0 0 0 0 0 0 0
Table A.8: Lowest obtained values of the position of 1% distortion, in meters, for a four segment
ap wavemaker with segments of equal
lengths and optimised segment strokes, where h= 0:6 m.
Run number Wavenumber
2m14m16m18m110m112m114m116m118m120m1
1 0 0 0 0 0 0 0 0 0 0
2 0 0 0 0 0 0 0 0 0 0
3 0 0 0 0 0 0 0 0 0 0
4 0 0 0 0 0 0 0 0 0 0
5 0 0 0 0 0 0 0 0 0 0
6 0 0 0 0 0 0 0 0 0 0
Table A.9: Lowest obtained values of the position of 1% distortion, in meters, for a ve segment
ap wavemaker with segments of equal
lengths and optimised segment strokes, where h= 0:6 m.
Run number Wavenumber
2m14m16m18m110m112m114m116m118m120m1
1 0 0 0 0 0 0 0 0 0 0
2 0 0 0 0 0 0 0 0 0 0
3 0 0 0 0 0 0 0 0 0 0
4 0 0 0 0 0 0 0 0 0 0
5 0 0 0 0 0 0 0 0 0 0
6 0 0 0 0 0 0 0 0 0 0
Table A.10: Lowest obtained values of the position of 1% distortion, in meters, for a six segment
ap wavemaker with segments of equal
lengths and optimised segment strokes, where h= 0:6 m.
170

Chapter ANumber of
segments in
WavemakerWavenumber
2m14m16m18m110m112m114m116m118m120m1
2 0 2 :18910155:78210166:64510165:40910152:01810158:61910161:46710161:24610153:1091015
3 0 0 2 :07410127:15610131:41710122:26410121:81610121:26810121:19310129:9621013
4 0 0 0 0 3 :97610083:01910071:13110071:48610071:86210072:9981007
5 0 0 0 0 0 0 2 :20810055:62010054:01810051:0391005
6 0 0 0 0 0 0 0 0 1 :82810045:2471005
Table A.11: Standard deviation of the lowest obtained values of the position of 1% distortion for a segmented piston wavemaker with
segments of equal lengths and optimised segment strokes, corresponding to the results presented in Tables A.1 to A.5.
Number of
segments in
WavemakerWavenumber
2m14m16m18m110m112m114m116m118m120m1
2 0 0 0 0 3 :02710061:93210062:37010062:33910061:42310062:6001012
3 0 0 0 0 0 0 0 0 9 :36510051:2991007
4 0 0 0 0 0 0 0 0 0 0
5 0 0 0 0 0 0 0 0 0 0
6 0 0 0 0 0 0 0 0 0 0
Table A.12: Standard deviation of the lowest obtained values of the position of 1% distortion for a segmented
ap wavemaker with
segments of equal lengths and optimised segment strokes, corresponding to the results presented in Tables A.6 to A.10.
171

Chapter AStroke num-
berWavenumber
2m14m16m18m110m112m114m116m118m120m1
1 1.4197 -1.6848 -1.5558 1.7969 1.3603 -1.4166 -0.3493 1.1809 -0.4511 -0.5022
2 0.8047 -0.2572 0.1268 -0.3801 -0.3951 0.4848 0.1322 -0.4780 0.1916 0.2211
Table A.13: Optimised strokes, in meters, for a two segment piston wavemaker with segments of equal lengths, where h= 0:6 m.
Stroke num-
berWavenumber
2m14m16m18m110m112m114m116m118m120m1
1 -1.8474 1.4063 1.8260 1.1410 1.7140 1.6300 -1.2037 1.1716 -0.8223 -1.1600
2 -1.1747 -0.0714 -0.7002 -0.6415 -1.2346 -1.4030 1.1844 -1.278 0.9735 1.4659
3 -1.2538 0.7241 0.9111 0.5822 0.9691 1.03525 -0.8504 0.9065 -0.686 -1.0305
Table A.14: Optimised strokes, in meters, for a three segment piston wavemaker with segments of equal lengths, where h= 0:6 m.
Stroke num-
berWavenumber
2m14m16m18m110m112m114m116m118m120m1
1 -0.7976 1.2957 -0.9704 -0.9373 -0.4119 0.4848 0.4689 -0.4133 -0.3917 -0.3256
2 -0.8144 1.1883 0.4671 0.7547 0.5593 -0.7213 -0.7631 0.7325 0.7515 0.6715
3 0.5420 -1.8177 -1.5998 -1.7982 -1.4308 1.7951 1.8766 -1.7948 -1.8423 -1.6500
4 -1.2939 1.9577 0.8455 0.9948 0.9431 -1.2025 -1.2695 1.2214 1.2581 1.1289
Table A.15: Optimised strokes, in meters, for a four segment piston wavemaker with segments of equal lengths, where h= 0:6 m.
172

Chapter AStroke num-
berWavenumber
2m14m16m18m110m112m114m116m118m120m1
1 -1.8166 1.9570 0.9090 -1.5368 -1.0521 0.0757 0.07127 -0.0412 -0.0315 -0.0419
2 -1.5877 0.3465 -0.0931 0.4308 0.7336 -0.1748 -0.1915 0.1139 0.0901 0.1251
3 -1.7315 1.9767 0.1426 -1.1191 -0.9851 0.7685 0.8813 -0.5253 -0.4184 -0.5869
4 0.02041 -0.5283 1.8953 -0.0256 -1.5185 -1.4624 -1.7823 1.0791 0.8706 1.2342
5 -1.8952 0.6166 -1.4353 0.2737 1.7820 0.8591 1.0797 -0.6581 -0.5338 -0.7597
Table A.16: Optimised strokes, in meters, for a ve segment piston wavemaker with segments of equal lengths, where h= 0:6 m.
Stroke num-
berWavenumber
2m14m16m18m110m112m114m116m118m120m1
1 1.8503 -1.9292 -1.8770 -1.4271 -1.8423 0.2717 0.3057 -0.0173 0.0063 0.0098
2 1.0699 0.4681 -0.1733 -0.0312 0.6754 -0.3244 -0.5063 0.0582 -0.0279 -0.0415
3 1.5728 -1.9920 -0.3543 -0.8170 -1.5255 1.0350 1.6477 -0.3670 0.2002 0.2916
4 1.3753 -1.3896 -1.7348 0.9114 -0.7530 -0.9876 -1.8695 1.1886 -0.7248 -1.0443
5 0.4279 -1.6943 -0.0606 -1.3577 1.9143 -0.6013 -0.1424 -1.8873 1.2423 1.7714
6 1.5721 1.9802 0.7383 0.5025 -0.6281 0.9048 0.8378 1.0108 -0.6922 -0.9809
Table A.17: Optimised strokes, in meters, for a six segment piston wavemaker with segments of equal lengths, where h= 0:6 m.
173

Chapter AStroke num-
berWavenumber
2m14m16m18m110m112m114m116m118m120m1
1 0.9277 -1.3844 1.9679 -1.8770 1.9257 1.4144 -0.7414 1.8946 1.1227 -0.6345
2 1.3200 -0.3984 -0.1682 0.3742 -0.6477 -0.5939 0.3631 -1.040 -0.6733 0.4081
3 -0.0898 -0.2126 0.6954 -0.6927 0.9589 0.8180 -0.4864 1.3791 0.8901 -0.5397
Table A.18: Optimised strokes, in meters, for a two segment
ap wavemaker with segments of equal lengths, where h= 0:6 m.
Stroke num-
berWavenumber
2m14m16m18m110m112m114m116m118m120m1
1 1.3464 -1.7733 1.7041 1.9715 1.4396 -1.9539 -0.8170 -0.5661 -0.4572 0.5280
2 0.4218 -1.4056 0.5710 0.0840 -0.0379 0.4231 0.2851 0.2743 0.2713 -0.3451
3 1.4964 -1.8923 -0.3544 0.4722 0.1039 -0.8076 -0.7796 -0.7285 -0.7678 0.9697
4 -0.3819 0.4995 0.9873 -0.4715 0.0819 0.8264 1.2081 1.0935 1.1815 – 1.4929
Table A.19: Optimised strokes, in meters, for a three segment
ap wavemaker with segments of equal lengths, where h= 0:6 m.
Stroke num-
berWavenumber
2m14m16m18m110m112m114m116m118m120m1
1 1.0950 1.3048 1.6239 1.7817 1.6849 -0.7473 -1.8450 1.1053 -1.9971 -1.2989
2 0.9581 1.0700 0.8921 0.1945 0.2945 0.0187 0.1474 -0.2544 0.5891 0.4828
3 0.3390 -0.3493 -0.4607 0.8426 0.0006 -0.2624 -0.3804 0.6652 -1.0258 -0.7685
4 1.9282 1.4763 1.4418 -0.9765 0.0407 0.2269 0.5636 -0.8093 0.6923 0.0343
5 -1.3844 -0.9878 -1.3926 1.0933 0.1095 -0.0624 -0.8532 0.6712 -0.2351 0.8224
Table A.20: Optimised strokes, in meters, for a four segment
ap wavemaker with segments of equal lengths, where h= 0:6 m.
174

Chapter AStroke num-
berWavenumber
2m14m16m18m110m112m114m116m118m120m1
1 1.5119 -1.9729 -1.4298 1.9402 1.8227 -1.8419 -1.9851 -1.6498 -1.3395 -1.9756
2 1.8816 -1.8665 -0.4692 1.0237 0.4686 -0.3854 -0.1081 0.1944 0.1355 0.3165
3 0.5339 0.2943 -0.4439 -0.7866 0.3611 0.3663 -0.4277 -1.1085 0.0619 -0.0598
4 1.9180 -1.4718 -0.4323 1.9927 -0.3928 -1.5981 0.7781 1.8258 -1.4545 -1.4165
5 0.9034 0.5347 -0.4296 -1.9281 0.4770 1.8166 -1.3298 -0.9004 1.6872 1.5760
6 -0.5793 -1.9331 1.1929 1.9351 -0.4849 -1.2955 1.6396 -0.1155 -0.7835 -0.7322
Table A.21: Optimised strokes, in meters, for a ve segment
ap wavemaker with segments of equal lengths, where h= 0:6 m.
Stroke num-
berWavenumber
2m14m16m18m110m112m114m116m118m120m1
1 -1.2225 1.9832 1.5025 1.8256 1.9531 -1.7402 1.8915 -1.5686 0.9076 0.9607
2 -1.0959 1.5858 0.7960 1.2342 0.9323 -0.6236 0.0619 -0.1705 -0.1073 -0.1375
3 -1.4697 .7846 0.9871 1.6510 1.3343 0.8472 1.1281 -0.02576 0.7293 -0.2481
4 -1.5002 1.7512 1.4853 0.0628 0.3565 -1.8152 -1.8078 0.2709 -1.2508 0.6536
5 -0.5485 -1.3091 -1.6470 -0.0446 -0.4561 -0.0500 1.7115 -1.5802 0.9146 -1.7001
6 -1.5809 1.8889 0.0390 -0.3338 0.7069 1.9440 -0.5535 1.1746 -0.3565 1.2225
7 1.9204 -0.7771 1.2851 0.5194 -1.2032 -1.9481 -0.3134 0.2031 0.2692 -0.1464
Table A.22: Optimised strokes, in meters, for a six segment
ap wavemaker with segments of equal lengths, where h= 0:6 m.
175

Chapter ARun number Wavenumber
2m14m16m18m110m112m114m116m118m120m1
1 0 0.01399 0.06153 0.09986 0.1298 0.1527 0.1701 0.1837 0.1943 0.2030
2 0 0.01399 0.06153 0.09986 0.1298 0.1527 0.1701 0.1837 0.1943 0.2030
3 0 0.01399 0.06153 0.09986 0.1298 0.1527 0.1701 0.1837 0.1943 0.2030
4 0 0.01399 0.06153 0.09986 0.1298 0.1527 0.1701 0.1837 0.1943 0.2030
5 0 0.01399 0.06153 0.09986 0.1298 0.1527 0.1701 0.1837 0.1943 0.2030
6 0 0.01399 0.06153 0.09986 0.1298 0.1527 0.1701 0.1837 0.1943 0.2030
Table A.23: Lowest obtained values of the position of 1% distortion, in meters, for a two segment piston wavemaker, with both the
segments' strokes and lengths optimised, where h= 0:6 m.
Run number Wavenumber
2m14m16m18m110m112m114m116m118m120m1
1 0 0 0 0.01158 0.02517 0.03642 0.04581 0.05370 0.06038 0.06609
2 0 0 0 0.01158 0.02517 0.03642 0.04581 0.05370 0.06038 0.06609
3 0 0 0 0.01158 0.02517 0.03642 0.04581 0.05370 0.06038 0.06609
4 0 0 0 0.01158 0.02517 0.03642 0.04581 0.05370 0.06038 0.06609
5 0 0 0 0.01158 0.02517 0.03642 0.04581 0.05370 0.06038 0.06609
6 0 0 0 0.01158 0.02517 0.03642 0.04581 0.05370 0.06038 0.06609
Table A.24: Lowest obtained values of the position of 1% distortion, in meters, for a three segment piston wavemaker, with both the
segments' strokes and lengths optimised, where h= 0:6 m.
176

Chapter ARun number Wavenumber
2m14m16m18m110m112m114m116m118m120m1
1 0 0 0 0 0 0.004068 0.009728 0.01498 0.01971 0.02393
2 0 0 0 0 0 0.004068 0.009728 0.01498 0.01971 0.02393
3 0 0 0 0 0 0.004068 0.009728 0.01498 0.01971 0.02393
4 0 0 0 0 0 0.004068 0.009728 0.01498 0.01971 0.02393
5 0 0 0 0 0 0.004068 0.009728 0.01498 0.01971 0.02393
6 0 0 0 0 0 0.004068 0.009728 0.01498 0.01971 0.02393
Table A.25: Lowest obtained values of the position of 1% distortion, in meters, for a four segment piston wavemaker, with both the
segments' strokes and lengths optimised, where h= 0:6 m.
Run number Wavenumber
2m14m16m18m110m112m114m116m118m120m1
1 0 0 0 0 0 0 0.0002104 0.002252 0.004533 0.006933
2 0 0 0 0 0 0 0.0002096 0.002250 0.004526 0.006903
3 0 0 0 0 0 0 0.0002119 0.002261 0.004519 0.006903
4 0 0 0 0 0 0 0.0002084 0.002256 0.004510 0.006925
5 0 0 0 0 0 0 0.0002088 0.002259 0.004534 0.006924
6 0 0 0 0 0 0 0.0002078 0.002252 0.004510 0.006907
Table A.26: Lowest obtained values of the position of 1% distortion, in meters, for a ve segment piston wavemaker, with both the
segments' strokes and lengths optimised, where h= 0:6 m.
Run number Wavenumber
2m14m16m18m110m112m114m116m118m120m1
1 0 0 0 0 0 0 0 0 0 0
2 0 0 0 0 0 0 0 0 0 0
3 0 0 0 0 0 0 0 0 0 0
4 0 0 0 0 0 0 0 0 0 0
5 0 0 0 0 0 0 0 0 0 0
6 0 0 0 0 0 0 0 0 0 0
Table A.27: Lowest obtained values of the position of 1% distortion, in meters, for a six segment piston wavemaker, with both the
segments' strokes and lengths optimised, where h= 0:6 m.
177

Chapter ARun number Wavenumber
2m14m16m18m110m112m114m116m118m120m1
1 0 0 0 0 0 0.008092 0.01860 0.02747 0.03483 0.04096
2 0 0 0 0 0 0.008092 0.01860 0.02746 0.03483 0.04096
3 0 0 0 0 0 0.008093 0.01860 0.02746 0.03483 0.04096
4 0 0 0 0 0 0.008094 0.01860 0.02746 0.03483 0.04096
5 0 0 0 0 0 0.008092 0.01860 0.02746 0.03483 0.04096
6 0 0 0 0 0 0.008091 0.01860 0.02746 0.03483 0.04096
Table A.28: Lowest obtained values of the position of 1% distortion, in meters, for a two segment
ap wavemaker, with both the segments'
strokes and lengths optimised, where h= 0:6 m.
Run number Wavenumber
2m14m16m18m110m112m114m116m118m120m1
1 0 0 0 0 0 0 0 0 0 0
2 0 0 0 0 0 0 0 0 0 0
3 0 0 0 0 0 0 0 0 0 0
4 0 0 0 0 0 0 0 0 0 0
5 0 0 0 0 0 0 0 0 0 0
6 0 0 0 0 0 0 0 0 0 0
Table A.29: Lowest obtained values of the position of 1% distortion, in meters, for a three segment
ap wavemaker, with both the
segments' strokes and lengths optimised, where h= 0:6 m.
178

Chapter ARun number Wavenumber
2m14m16m18m110m112m114m116m118m120m1
1 0 0 0 0 0 0 0 0 0 0
2 0 0 0 0 0 0 0 0 0 0
3 0 0 0 0 0 0 0 0 0 0
4 0 0 0 0 0 0 0 0 0 0
5 0 0 0 0 0 0 0 0 0 0
6 0 0 0 0 0 0 0 0 0 0
Table A.30: Lowest obtained values of the position of 1% distortion, in meters, for a four segment
ap wavemaker, with both the segments'
strokes and lengths optimised, where h= 0:6 m.
Run number Wavenumber
2m14m16m18m110m112m114m116m118m120m1
1 0 0 0 0 0 0 0 0 0 0
2 0 0 0 0 0 0 0 0 0 0
3 0 0 0 0 0 0 0 0 0 0
4 0 0 0 0 0 0 0 0 0 0
5 0 0 0 0 0 0 0 0 0 0
6 0 0 0 0 0 0 0 0 0 0
Table A.31: Lowest obtained values of the position of 1% distortion, in meters, for a ve segment
ap wavemaker, with both the segments'
strokes and lengths optimised, where h= 0:6 m.
Run number Wavenumber
2m14m16m18m110m112m114m116m118m120m1
1 0 0 0 0 0 0 0 0 0 0
2 0 0 0 0 0 0 0 0 0 0
3 0 0 0 0 0 0 0 0 0 0
4 0 0 0 0 0 0 0 0 0 0
5 0 0 0 0 0 0 0 0 0 0
6 0 0 0 0 0 0 0 0 0 0
Table A.32: Lowest obtained values of the position of 1% distortion, in meters, for a six segment
ap wavemaker, with both the segments'
strokes and lengths optimised, where h= 0:6 m.
179

Chapter ANumber of
segments in
WavemakerWavenumber
2m14m16m18m110m112m114m116m118m120m1
2 0 8 :811310188:683310171:367710154:116810170 1 :454410152:482510170 4 :08691015
3 0 0 0 1 :857910121:774810121:008110121:846410121:564110122:373610124:66121013
4 0 0 0 0 0 8 :941210087:726610083:916910088:796410083:13541008
5 0 0 0 0 0 0 1 :493310064:318610061:055010051:32221005
6 0 0 0 0 0 0 0 0 0 0
Table A.33: Standard deviation of the lowest obtained position of 1% distortion for completely optimised segmented piston wavemaker,
corresponding to the results presented in Tables A.23-27.
Number of
segments in
WavemakerWavenumber
2m14m16m18m110m112m114m116m118m120m1
2 0 0 0 0 0 8 :238410071:750410063:537410069:980910074:95561013
3 0 0 0 0 0 0 0 0 0 0
4 0 0 0 0 0 0 0 0 0 0
5 0 0 0 0 0 0 0 0 0 0
6 0 0 0 0 0 0 0 0 0 0
Table A.34: Standard deviation of the lowest obtained position of 1% distortion for completely optimised segmented
ap wavemaker,
corresponding to the results presented in Tables A.28-32.
180

Chapter AStroke Num-
berWavenumber
2m14m16m18m110m112m114m116m118m120m1
1 1.7221 -1.1230 -0.8468 -1.6663 0.6234 -1.5191 1.6455 -1.1513 -0.7753 0.3158
2 1.1146 -0.2918 -0.0676 0.0275 -0.0464 0.1709 -0.2290 0.1827 0.1345 -0.0584
Table A.35: Optimised strokes, in meters, for a two segment piston wavemaker with optimised segment lengths, where h= 0:6 m.
Stroke Num-
berWavenumber
2m14m16m18m110m112m114m116m118m120m1
1 -1.6050 -1.6112 1.8132 -1.5026 -1.2598 -0.8125 1.7194 1.7398 -1.6568 1.6329
2 -1.8281 -0.7413 -0.0557 0.2798 0.3912 0.3411 -0.8869 -1.0437 1.1158 -1.2049
3 -0.8940 -0.3785 0.4161 -0.3013 -0.2550 -0.1781 0.4153 0.4615 -0.4777 0.5060
Table A.36: Optimised strokes, in meters, for a three segment piston wavemaker with optimised segment lengths, where h= 0:6 m.
Stroke Num-
berWavenumber
2m14m16m18m110m112m114m116m118m120m1
1 1.8368 1.8963 1.8982 1.6004 -0.8404 0.9428 -1.0589 -0.9504 0.7489 -0.7230
2 1.8427 1.1600 -0.4395 -0.7648 0.7928 -1.0792 1.3314 1.3060 -1.1175 1.1635
3 1.8313 0.3070 1.6739 1.5780 -1.2575 1.6344 -1.9418 -1.8634 1.5753 -1.6297
4 0.9232 0.6363 -0.2492 -0.4279 0.4039 -0.5705 0.6961 0.6784 -0.5787 0.6017
Table A.37: Optimised strokes, in meters, for a four segment piston wavemaker with optimised segment lengths, where h= 0:6 m.
181

Chapter AStroke Num-
berWavenumber
2m14m16m18m110m112m114m116m118m120m1
1 -1.6657 1.8146 -1.9331 1.9929 1.9837 1.8110 -0.1448 0.1932 0.2258 0.2133
2 -1.5121 1.3656 -0.2511 -0.4398 -0.8708 -1.8039 0.4340 -0.6155 -0.7378 -0.7188
3 -1.6987 0.6930 -1.9999 0.8786 0.4220 1.9748 -1.0652 1.5377 1.8420 1.7998
4 0.2709 -0.3793 1.1348 0.6067 1.6059 0.5882 0.9694 -1.4692 -1.7919 -1.7766
5 -1.5960 0.9424 -0.6169 -0.1491 -0.6560 -0.5942 -0.2723 0.4282 0.5277 0.5274
Table A.38: Optimised strokes, in meters, for a ve segment piston wavemaker with optimised segment lengths, where h= 0:6 m.
Stroke Num-
berWavenumber
2m14m16m18m110m112m114m116m118m120m1
1 1.3717 -1.5523 -1.8911 1.1578 -1.9871 1.3114 -1.7035 0.7296 -0.2992 -0.3233
2 0.5857 -1.6487 -1.9730 -0.1011 0.3544 -0.609 1.5553 -1.2154 0.7305 0.9340
3 1.9345 1.6777 1.4703 1.6608 -0.5897 0.5123 -1.2298 1.5410 -1.4953 -1.7337
4 -0.9455 -1.9451 -0.5007 -1.8613 -0.8104 0.3374 -1.2532 0.4644 1.0290 0.5927
5 1.8782 -1.7645 -1.7910 1.3905 -0.1336 0.1470 1.0546 -1.4495 -0.1498 0.7722
6 0.6354 0.3955 0.3210 -0.1841 0.1998 -0.0932 -0.1961 0.5102 -0.0412 -0.3913
Table A.39: Optimised strokes, in meters, for a six segment piston wavemaker with optimised segment lengths, where h= 0:6 m.
182

Chapter AStroke Num-
berWavenumber
2m14m16m18m110m112m114m116m118m120m1
1 -1.9018 -1.3591 1.5193 1.7862 1.9936 1.8555 -1.2433 -1.8614 -0.8485 -1.8142
2 -1.6789 -0.4849 0.1808 -0.0445 -0.3136 -0.4522 0.3812 0.6697 0.3438 0.8063
3 -0.8528 -0.2388 0.1685 0.2570 0.5194 0.6115 -0.4763 -0.8068 -0.4068 -0.9453
Table A.40: Optimised strokes, in meters, for a two segment
ap wavemaker with optimised segment lengths, where h= 0:6 m.
Stroke Num-
berWavenumber
2m14m16m18m110m112m114m116m118m120m1
1 -0.8258 -1.2149 1.8610 1.0666 -0.5555 -1.8800 -1.7326 1.5886 -1.0573 1.3779
2 -1.5710 -0.2006 0.5122 0.2815 -0.0881 0.0446 0.1770 -0.3450 0.3270 -0.5778
3 -0.0045 -1.0570 0.5584 -0.0183 0.0598 -0.3438 -0.5116 0.5390 -0.5880 1.1145
4 -0.7198 0.7106 -0.1383 0.1583 -0.1419 0.3245 0.6523 -0.5092 0.6867 -1.3500
Table A.41: Optimised strokes, in meters, for a three segment
ap wavemaker with optimised segment lengths, where h= 0:6 m.
Stroke Num-
berWavenumber
2m14m16m18m110m112m114m116m118m120m1
1 1.9427 1.1209 1.6378 -1.9481 1.4718 -1.9623 -1.7286 -1.9353 -1.9973 1.4669
2 1.9878 1.4372 1.3670 -1.0002 0.8830 -0.2307 -0.3246 0.1352 0.1811 -0.1390
3 1.9343 -0.7576 -0.4001 0.09335 -0.8080 -1.0217 0.3919 -0.7555 -0.4106 0.3441
4 0.7979 -1.2829 1.5345 -0.6892 1.1384 1.5235 -1.1450 0.4801 -0.1894 -0.3294
5 1.7215 1.3634 -1.4268 0.6069 -0.9723 -1.7931 1.4661 -0.2240 0.5185 0.3172
Table A.42: Optimised strokes, in meters, for a four segment
ap wavemaker with optimised segment lengths, where h= 0:6 m.
183

Chapter AStroke Num-
berWavenumber
2m14m16m18m110m112m114m116m118m120m1
1 -1.4680 -1.6837 -1.1000 -1.8467 1.9877 1.9844 -1.9380 1.8278 1.9200 1.9642
2 -0.9843 -1.6294 -0.8217 -1.5938 0.7952 1.8107 -0.7568 0.9396 0.3977 0.2580
3 -0.2118 -1.3499 -1.9921 0.6109 0.8144 -1.5015 0.5801 1.5038 0.0276 0.1142
4 -1.5778 0.2000 0.6093 -1.2118 -0.0088 1.1486 -1.5476 1.0723 0.3651 -0.0040
5 -0.4480 -1.2078 -0.4671 -0.3120 -0.0427 0.7126 1.5823 -1.8366 -0.4540 0.1925
6 -0.8135 0.1676 -0.1870 0.6960 0.2120 -1.3240 -1.6390 1.8253 0.4948 -0.1824
Table A.43: Optimised strokes, in meters, for a ve segment
ap wavemaker with optimised segment lengths, where h= 0:6 m.
Stroke Num-
berWavenumber
2m14m16m18m110m112m114m116m118m120m1
1 1.8829 1.9500 -1.8757 1.4163 -1.9535 -1.9051 1.9107 -1.9819 -1.8990 1.9015
2 1.8104 1.9078 -1.2629 1.0821 -1.3791 -1.5241 1.2052 -1.8605 -0.7002 1.0009
3 -0.6771 1.8418 -0.0666 1.4695 -1.7880 0.3578 0.0081 1.3061 -0.8698 -0.6729
4 1.9679 1.3743 -1.5065 1.0408 -1.6507 -0.6506 0.6525 -1.5755 -0.8467 0.9856
5 1.9951 1.6562 -0.4906 1.3702 1.6199 -0.4353 -0.3726 -1.8811 1.2296 -0.0304
6 1.9753 1.0225 0.0791 -0.2811 -1.1114 0.2703 0.6033 1.8928 -0.8620 -0.6653
7 -1.1290 -0.5385 -0.1465 0.0203 0.6641 -0.1960 -0.6025 -1.9238 0.5635 0.9311
Table A.44: Optimised strokes, in meters, for a six segment
ap wavemaker with optimised segment lengths, where h= 0:6 m.
184

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