STABLE, THREEDIMENSIONAL, BIPERIODIC WAVES
IN SHALLOW WATER
BY
NORMAN WAHL SCHEFFNER
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN
PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1987
ACKNOWLEDGEMENTS
The author wishes to gratefully acknowledge his committee chairman
and advisor, Professor Joseph L. Hammack, for his guidance, assistance,
and enthusiasm during the entirety of this project. Without this untir
ing dedication, the successful completion of this investigation would
not have been possible. The following members of the author's super
visory committee are gratefully acknowledged for their advice and
support: Professors Ashish J. Mehta, Robert G. Dean, ChenChi Hsu,
Tom IP. Shih, and Thomas T. Bowman. Professor James T. Kirby is also
thanked for his comments on the final manuscript.
The writer would also like to thank Dr. Harvey Segur for his pro
found knowledge of the KP equation and his genuine interest in the suc
cess of this project. He not only provided the software used to compute
and generate exact KP solutions graphically, but also provided invalu
able assistance and guidance during the data analysis phase of the proj
ect. This assistance is greatly appreciated.
The author would especially like to thank those who assisted in the
experimental phase of this investigation. This assistance and advice
extended from the initial stages of attempting to generate waves through
the collection and storage of data. Hardware malfunctions, software
bugs, logistical difficulties, and other seemingly insurmountable prob
lems were almost routinely overcome with the help of the following dedi
cated personnel of the US Army Engineer Waterways Experiment Station at
Vicksburg, Mississippi: Larry A. Barnes, Michael J. Briggs, Mary L.
(Dean) Hampton, and Kent A. Turner of the Wave Processes Branch, Wave
Dynamics Division, Coastal Engineering Research Center; Lonnie L. Frier,
Homer C. Greer III, and Barry W. McCleave of the Operations Branch,
Instrumentation Services Division; and Charles E. Ray of the Photography
Branch, Information Products Division.
This research investigation was funded through a Department of the
Army InHouse Laboratory Independent Research (ILIR) program. The
author would like to thank the Department of the Army and the members of
the ILIR selection committee for funding this project.
Last, but certainly not least, I would like to thank Gail A. Bird
for her continuous support of this educational endeavor.
TABLE OF CONTENTS
Page
ACKNOWLEDGEMENTS ............................ .................... .. ii
LIST OF TABLES ................................................... vi
LIST OF FIGURES ..................................... .............. vii
ABSTRACT ......................................................... x
CHAPTERS
1. INTRODUCTION ............................................. 1
2. LITERATURE REVIEW ........................................ 6
3. THE KADOMTSEVPETVIASHVILI (KP) EQUATION ................. 17
3.1 Derivation of the KP Equation ....... ................ 17
3.2 Solutions of the KP Equation in terms of Riemann
Theta Functions of Genus 2 .......;.................. 34
3.3 The Construction and Properties of Genus 2
Solutions ........................................... 43
4. LABORATORY FACILITIES AND EXPERIMENTAL PROCEDURES ........ 49
4.1 The Wave Basin .............................. ......... 49
4.2 The Directional Spectral Wave Generator ............. 52
4.3 A Methodology for Generating Waves .................. 58
4.3.1 The Generation of Cnoidal Waves .............. 58
4.3.2 The Generation of Genus 2 Waves .............. 72
4.4 The Measurement of Waves ............................ 76
4.4.1 The Photographic System ...................... 77
4.4.2 The Wave Gages ............................... 80
5. A COMPARISON OF GENUS 2 THEORY WITH EXPERIMENTAL
WAVES ........................................... ....... 87
5.1 The Free Parameters of a Genus 2 Solution ........... 87
5.1.1 Sensitivity analysis for the parameter b ..... 89
5.1.2 Sensitivity analysis for the parameter p ..... 91
5.1.3 Sensitivity analysis for the parameter X ..... 92
5.2 The Dimensional Genus 2 KP Solution ................. 94
5.3 A Methodology for Relating Genus 2 Solutions to
Observed Waves ..................................... 97
5.4 Presentation and Discussion of Results .............. 109
6. CONCLUSIONS .............................................. 125
APPENDICES
A. ELLIPTIC FUNCTION SOLUTIONS TO THE KdV EQUATION .......... 128
B. EXPERIMENTAL DATA AND EXACT GENUS 2 KP SOLUTIONS ......... 134
REFERENCES ....................................................... 183
BIOGRAPHICAL SKETCH .............................................. 186
LIST OF TABLES
Number Description Page
4.1 The Experimental Waves ................................. 73
5.1 Free parameters of the genus 2 KP solution for the
experimental program .................................. 115
5.2 Comparison of measured and computed wave parameters ..... 117
5.3 Comparison of the average rms error for the typical
wave and the composite wave ............................ 119
5.4 Small parameters defining nonlinearity,
dispersiveness, and threedimensionality for the
experimental program .................................... 121
LIST OF FIGURES
Number Description Page
3.1 Schematic diagram of flow domain ........................ 18
3.2 Example genus 2 solution (b = 1.5, p = 0.5, X = 0.1) ... 45
3.3 Example genus 2 solution (b = 3.5, P = 0.5, X = 0.1) ... 45
3.4 A basic period parallelogram ............................ 47
4.1 Schematic drawing of the wave basin ..................... 50
4.2 Bathymetry of the wave basin ............................ 51
4.3 The directional spectral wave generator ................. 53
4.4 Schematic diagram of a wave generator module ............ 54
4.5 Schematic diagram of a wave board ....................... 55
4.6 System console block diagram ............................ 56
4.7 Servocontroller block diagram .......................... 56
4.8 The computer system .................................... 59
4.9 Wave generation phase plane ............................. 61
4.10 A comparison between a generated wave and cnoidal wave
theory ............................................... 70
4.11 Wave profiles from the nine wave gages for a uniformly
generated cnoidal wave ................................. 71
4.12 Measured wave profile in the saddle region of
experiment CN2015 ...................................... 75
4.13 Measured wave profile in the saddle region corresponding
to an exact solution generation of experiment CN2015 .... 75
4.14 The photographic system ................................ 78
4.15 Horizontal measurement distortion ....................... 79
4.16 Schematic diagram for wave gage placement ............... 81
4.17 Schematic diagram of parallelrod resistance
transducer ........................................... 83
4.18 Parallelrod wave sensor ................................ 85
4.19 Waverod calibration .................................... 86
5.1 Sensitivity of the parameters w, fmax, and v
to the parameter b ...................................... 89
5.2 Example wavefields demonstrating the effect of the
parameter b with X = 0.100 and p = 0.500.
a) b = 2.000, v = 0.629, fmax = 2.522, w = 3.197
b) b = 6.000, v = 0.277, fmax = 0.116, w = 0.350 .... 90
5.3 Sensitivity of the parameters w fmax and v to the
parameter V ......... ... .......... ..................... 91
5.4 Example wavefields demonstrating the effect of the parameter
v with b = 3.000 and X = 0.100.
a) v = 0.400, v = 0.291, fmax = 0.572, w = 0.713
b) p = 0.800, v = 1.163, fmax = 2.286, w = 5.705 ... 92
5.5 Sensitivity of the parameters w fmax and v to the
parameter X ......................................... 93
5.6 Example wavefields demonstrating the effect of the parameter
X with b = 3.000 and P =0.500.
a) X = 0.300, v = 0.218, fmax = 0.908, w = 0.541
b) X = 0.800, v = 0.032, fmax = 0.492, w = 0.121 ... 94
5.7 Mosaic photograph of the experimental wave field in
experiment CN3007 ...................................... 98
5.8 Wave profiles for the nine wave gages in experiment
CN3007 ............................................... 101
5.9 Sixteen KP wave profiles for the halfparallelogram
solution corresponding to experiment CN3007 ............. 104
5.10 Sixteen KP wave profiles for the halfparallelogram
solution corresponding to experiment CN2015 ............. 105
5.11 Theoretical and experimental wave profiles for
experiment CN3007 ..................................... 110
5.12 Theoretical and experimental wave profiles for
experiment CN2015 ....................................... 111
5.13 Normalized contour map of the theoretical solution for
experiment CN3007 ..................................... 112
viii
5.14 Threedimensional view of the theoretical solution for
experiment CN3007 ..................................... 113
5.15 Normalized contour map of the theoretical solution for
experiment CN2015 ....................................... 113
5.16 Threedimensional view of the theoretical solution for
experiment CN2015 ....................................... 114
A.1 Schematic diagram of the fluid domain ................... 128
B.1 Mosaic photographs of the experimental waves ............ 135
B.2 Experimental wave profiles .............................. 147
B.3 Theoretical and experimental wave profiles .............. 159
B.4 Normalized contour map and threedimensional view
of the KP solutions for the experimental waves .......... 171
Abstract of Dissertation Presented to the
Graduate School of the University of Florida
in Partial Fulfillment of the Requirements
for the Degree of Doctor of Philosophy
STABLE, THREEDIMENSIONAL, BIPERIODIC WAVES
IN SHALLOW WATER
By
Norman Wahl Scheffner
May 1987
Chairman: Joseph L. Hammack, Jr.
Cochairman: Ashish J. Mehta
Major Department: Engineering Sciences
Waves in shallow water are inherently threedimensional and non
linear. Experiments are presented herein which demonstrate the exist
ence of a new class of long water waves which are genuinely three
dimensional, nonlinear, and of (quasi) permanent form. These waves are
referred to as biperiodic in that they have two real periods, both tem
porally and spatially. The waves are produced in the laboratory by the
simultaneous generation of two cnoidal wave trains which intersect at
angles to one another. The resulting surface pattern is represented by
a tiling of hexagonal patterns, each of which is bounded by wave crests
of spatially variable amplitude. Experiments are conducted over a wide
range of generation parameters in order to fully document the waves in
the vertical and two horizontal directions. The hexagonalshaped waves
are remarkably robust, retaining their integrity for maximum wave
heights up to and including breaking and for widely varying horizontal
length scales.
The KadomtsevPetviashvili (KP) equation is tested as a model for
these biperiodic waves. This equation is the direct threedimensional
generalization of the famous KortewegdeVries (KdV) equation for weakly
nonlinear waves in two dimensions. It is known that the KP equation
admits an infinite dimensional family of periodic solutions which
are defined in terms of Riemann theta functions of genus N. Genus 2
solutions have two real periods and are similar in structure to the
hexagonallyshaped waves observed in the experiments. A methodology is
developed which relates the free parameters of the genus 2 solution to
the temporal and spatial data of the experimentally generated waves.
Comparisons of exact genus 2 solutions with measured data show excellent
agreement over the entire range of experiments. Even though near
breaking waves and highly threedimensional wave forms are encountered,
the total rms error between experiment and KP theory never exceeds 20%
although known sources of error are introduced.. Hence, the KP equation
appears to be a very robust model of nonlinear, threedimensional waves
propagating in shallow water, reminiscent of the KdV equation in two
dimensions.
CHAPTER 1
INTRODUCTION
The propagation of waves in shallow water is a phenomenon of sig
nificant practical importance. Shallow water waves are especially im
portant to the field of coastal engineering where their effects on
beaches, harbors, inlets, coastal structures, etc. are both economical
and aesthetic concerns. The ability to model realistic wave character
istics such as their vertical height distribution, surface pattern,
fluid velocities, and wave speed is essential for developing engineering
solutions to problems in the coastal zone. Difficulties in making such
predictions arise from the fact that the equations governing the physics
of flow, i.e. the conservation laws of Newtonian physics and the appro
priate boundary conditions, cannot be solved exactly. The inability to
solve these equations in closed form is due to the nonlinear terms con
tained in the governing equations. In order to circumvent these diffi
culties, a variety of simplifying approximations is made. For example,
the nonlinear terms are often neglected, giving rise to a linear wave
theory. Both the omission of nonlinear terms and threedimensionality
are especially severe restrictions for nearshore problems and result in
solutions which do not realistically model many situations.
Natural waves experience dramatic changes in their appearance as
they propagate from deep water into shallow water regions. In the
shallow areas, the waves become steep with high crests and long shallow
troughs. This transformation in shape can be attributed primarily to
the decrease in water depth. Additional boundary conditions, such as
irregular shoreline features, nonuniform variations in bathymetry, and
the presence of coastal structures result in the refraction, diffraction
and reflection of the incident wave; hence, the resulting wave field is
not only nonlinear in shape but also spatially threedimensional. For
wave fields which can be reasonably approximated in two dimensions,
cnoidal wave theory, first published by Korteweg and deVries (KdV) in
1895, has been found to be descriptive of the nonlinear features ob
served in shallow water. The linear wave approximation, most commonly
used for threedimensional coastal engineering applications, assumes
that the nonlinear terms in the governing equations are negligible.
Unfortunately, this theory does not predict the nonlinear three
dimensional features which are often of importance in shallow water
regions. Therefore, a realistic analytical model which describes both
nonlinear and threedimensional waves in shallow water is not available
currently.
A recent advance in the theoretical description of three
dimensional, nonlinear waves in shallow water is presented by Segur
and Finkel (SF, 1985). They present an explicit, analytical solution
for threedimensional, weakly nonlinear wave forms. These solutions
are biperiodic in that the waves have two independent spatial and
temporal periods. Biperiodic waves are an exact solution of the
KadomtsevPetviashvili equation (KP, 1970) and represent a natural
threedimensional generalization of the twodimensional cnoidal waves
of the KdV equation.
The analytical solution of the KP equation described by Segur and
Finkel represents a somewhat abstract mathematical formulation which has
never been applied to actual wavefields. If, in fact, these solutions
model nonlinear waves accurately, they will represent a significant ad
vancement in the field of nonlinear wave mechanics and a powerful new
tool for the coastal engineer. Herein are presented laboratory experi
ments which document the existence of a new class of long water waves
which are truly threedimensional, biperiodic and nonlinear. The exper
imentally generated waves are used to test the validity of the KP solu
tions presented by SF. In order to verify these solutions as a model
for the experimental wave fields, the mathematical parameters of the
exact solution first must be related to the physical characteristics of
the waves measured in the laboratory. Secondly, an experimental program
must be developed that provides a reasonably comprehensive test of KP
theory. Additionally, parameter limits are sought in order to establish
the stability and range of applicability of the biperiodic solutions.
An experimental test of the KP equation as a viable model for
threedimensional, periodic, and nonlinear waves requires the success
ful completion of several tasks. For example, even though the qualita
tive features of the surface pattern for biperiodic waves are documented
by Segur and Finkel, procedures are not available which would provide a
formal basis for applying KP theory to practical situations. Instead,
SF present a series of conjectures which suggest a methodology for in
ferring the free mathematical parameters of the exact solution from
certain physical measurements of an observed threedimensional wave
field. An initial task of this study is to utilize the conjectures
of SF and develop a technique for calculating exact KP solutions from
measured wave characteristics. Secondly, an experimental laboratory
program is developed for generating threedimensional waves (with two
dimensional surface patterns) which are qualitatively similar to those
presented by Segur and Finkel. Following the generation of the proper
wave patterns, a methodology is developed for measuring the spatial and
temporal characteristics of the wave field necessary for determining the
solution parameters. Finally, a comparison of measured data and best
fit theoretical solutions is made in order to establish the stability
and range of validity of KP theory over a wide parameter range.
A brief review of twodimensional nonlinear wave theory in shallow
water is presented in Chapter 2 in order to provide a proper perspective
for the extension of the theory into three dimensions. This chapter
begins with a discussion of the first experimental documentation of per
manent form shallow water waves by John Scott Russell in 1844. The for
mal derivation of the KP equation is presented in Chapter 3. The exact
biperiodic solutions presented by Segur and Finkel (1985) are also de
scribed in this chapter. Chapter 4 describes the laboratory facilities
and the experimental procedures developed in order to accomplish the
goals of this study. The experimental procedures include the method
used to generate threedimensional wave patterns and the data acquisi
tion techniques employed to quantify the resulting wave fields. A
methodology for relating KP theory to wave measurements is presented in
Chapter 5. This chapter includes an investigation of the parameters
in the KP solution and their relationship to experimental wave charac
teristics. Conclusions of this study are presented in Chapter 6. A
5
presentation of the elliptic functions used for the generation of waves
in the laboratory is shown in Appendix A. All of the spatial and
temporal data used in this study are presented in Appendix B.
CHAPTER 2
LITERATURE REVIEW
In the middle 1800s, a controversy arose as to whether or not a
single, localized wave of elevation could propagate at constant velocity
with permanent form, neither steepening nor dispersing. The argument
was prompted by the observation in 1834 and subsequent laboratory veri
fication in 1844 of a permanentform wave by John Scott Russell. This
wave has since been termed the "solitary wave" and, more recently, a
"soliton." At that time, no known mathematical solutions for the equa
tions of fluid motion existed which adequately described the solitary
wave. Linear (inviscid) theory described a wave form which dispersed
into sinusoidal spectral components because of the dependence of the
computed phase speed on the wave length. Although these waves were of
permanent form, they were not of the shape observed by Russell. The
existing theory advocated by Airy did account for nonlinearity but did
not account for dispersion of the wave. This theory described waves of
elevation which steepened in time but did not disperse; i.e., they were
not of permanent form and contradicted Russell's observations. Even
though Russell meticulously documented the existence of the solitary
wave, his findings were essentially ignored by Airy. In fact, a certain
amount of contemptuousness and jealousy appears to have existed between
the two scientists because in 1845, just one year after Russell's labo
ratory verification, Airy published a theory of long waves in which he
specifically addressed the propagation of smallbutfinite amplitude
waves. Airy's interest in the subject was somewhat biased in that his
wave theory did not admit permanent form solutions. His attitude was
reflected in the published theory in which he concluded that solitary
waves of permanent form, such as those reported by Russell, do not
exist!
Fortunately, mathematicians and fluid mechanicians other than Airy
were interested in the solitary wave which seemed to contradict all pre
viously existing theories of fluid motion. Subsequently, intense ef
forts were directed at deriving an approximate governing equation which
would successfully model the waves observed by Russell. During this
time, several theories were advanced which explained the existence of
solitary waves. Boussinesq in 1871 and, independently, Rayleigh in 1876
first derived theories which admitted solitary waves as solutions. The
most concise mathematical treatment for the solitary wave was presented
in 1895 by Korteweg and deVries. They derived an approximate evolution
equation for a wave field which admits both solitary and periodic solu
tions. This remarkable equation is now known as the KortewegdeVries
(KdV) equation and has the form
ft + 6 ffk + fxji = 0 2.1
The KdV equation was derived as a model for the propagation of a wave
which is both weakly nonlinear and weakly dispersive. In the nondi
mensionalized equation 2.1, f represents a suitably scaled wave
amplitude, t is time and x is the direction of wave propagation.
The periodic solutions of the KdV equation were termed "cnoidal waves"
(in analogy with sinusoidal waves) by Korteweg and deVries. These
periodic solutions can be written in the following form:
f(U,) = 22m2 cn2 (y,m) 2a 1 + m 2.2
where cn is the Jacobian elliptic cosine function and y is a phase
argument (to be described at a later point). The functions K(m) and
E(m) represent the complete elliptical integrals of the first and
second kind. The argument m is the Jacobian elliptical parameter with
a modulus of the form 0 < m < 1 The amplitude parameter a is the
following function of the nondimensionalized wavelength X :
2K(m)
(A presentation of the complete cnoidal wave solution in an alternate,
but equivalent, form of Equation 2.2 is made in Appendix A.) The above
solution recovers sinusoidal waves as m approaches zero. As the wave
length becomes infinitely large, m approaches unity and the solitary
wave solution is recovered with the form
f(0,) = sech2(y) 2.3
The specific point of interest here is that the exact periodic solution
is written completely in terms of wellknown analytic functions and can
therefore be used for analyzing the characteristics of naturally occur
ring twodimensional waves. The practical application of cnoidal wave
theory was recognized by Wiegel (1960) who developed a set of figures
which made the calculation of cnoidal wave solutions in terms of mea
surable wave quantities an easy task. This development was a signifi
cant contribution to the field of coastal and oceanographic engineer
ing since it provided design engineers with the first usable two
dimensional, nonlinear, shallow water wave model. Until this time,
linear wave theory was used primarily for the majority of coastal
applications, regardless of its applicability to the problem. Even
though cnoidal wave theory is only twodimensional, descriptive of one
dimensional or longcrested waves, a marked improvement over linear
solutions was made possible for the practicing engineer.
The development of an adequate understanding of solitary (aperi
odic) and cnoidal (periodic) waves required about 50 years, extending
from Russell's observations to the publication of KdV theory. The
explanation given by KdV for the existence of the soliton wave was then
apparently overlooked by most subsequent researchers. This lack of
understanding is evidenced in the literature as manifest by the refer
ences to the "long wave paradox" which questions the theoretical basis
for the propagation of a nonlinear wave that neither steepens nor dis
perses. Ursell (1953), apparently unaware of the results of Korteweg
and deVries, provided a clear explanation of this paradox in terms of
the parameter (now referred to as the Ursell parameter),
2
U = a2.4
h3
In equation 2.4, a is a dimensional measure of wave amplitude, L is
the dimensional wavelength, and h is the depth of water. Ursell demon
strated that this parameter represented a ratio of weakly nonlinear
effects (measured by a/h) to weakly dispersive effects (measured by
h2/L2) which can be used to distinguish between flow regimes. Inter
pretive examples of the relative magnitude of this parameter are common.
For example, when the wave in question has a Ursell parameter of order
unity, U = 0(1), then the effects of nonlinearity and dispersion are
comparable and a balance is possible between the two effects. A perma
nent form wave can result when these weak effects are balanced. When
the parameter is small, U << 1 nonlinearity is negligible and the
waves are essentially linear. The wave then disperses into sinusoidal
components, each of which is a permanentform solution of linear theory.
When the parameter becomes large, U >> 1 the governing equation is of
the type advocated by Airy (1845) which does not admit permanent form
solutions. These nonlinear waves experience steepening and stretching
due to the effect of the wave amplitude on the wave speed. (This effect
is known as amplitude dispersion.) Since the Ursell parameter does suc
cessfully predict the flow regime for a wave with given dimensions, it
is commonly used in engineering practice.
It is interesting to note that Ursell was not the first to use the
parameter of Equation 2.4. In fact, the first reference to the Ursell
parameter was much earlier in a paper by Stokes (1847). Stokes demon
strated that a secondorder, permanentform solution could be derived
for the fluid motion if an approximation method was used in which this
parameter is taken to be small. Unfortunately, Stokes apparently did
not recognize the significance of his observation for explaining that
the existence of a permanentform nonlinear wave in shallow water was
due to the balance of opposing steepening and dispersion effects. For
example, in the same paper, he agreed with Airy's conclusion by making
the statement that "a solitary wave can not be propagated." Although
Stokes later recognized that this conclusion was erroneous, he never
again referred to the parameter. The next reference to the Ursell
parameter was made by Korteweg and deVries (1895) who demonstrated that
their cnoidal wave solutions reduced to Stokes' secondorder solution
when the elliptic modulus became small. Furthermore, KdV related the
elliptic modulus of their solution to the Ursell parameter and showed
that a correspondingly small value resulted in a sinusoidal solution.
This differentiation between wave regimes; i.e., cnoidal or sinusoidal,
based on the relative size of the Ursell parameter demonstrated that
Korteweg and deVries were certainly cognizant of the impact of the
parameter on the resulting wave solution.
Following the introduction of the KdV equation with its solitary
and cnoidal solutions, no new applications appear to have been reported
until 1960, at which time the equation reemerged in a study of
collisionfree hydromagnetic waves (Gardner and Morikawa, 1960).
Related studies by Kruskal and Zabusky (1963) again resulted in the
derivation of the equation. It was in this new research context that
physicists and mathematicians began to discover applications of the KdV
equation which would significantly impact the scientific community.
These discoveries led to the formulation and development of the Inverse
Scattering Transform (IST) by Gardner, Green, Kruskal and Miura (1967).
Their landmark paper outlined a revolutionary solution technique which
can be used to predict the exact number of solitary waves, or "soli
tons," which emerge from arbitrary periodic initial conditions. In
fact, solutions that describe any finite number of interaction solitons
can also be expressed in closed form.
The significance of the IST was far more profound than was initi
ally realized. Zakharov and Shabat (1972), using a technique introduced
by Lax (1968), demonstrated that the IST provided an exact solution for
the nonlinear Schrodinger equation, which describes nonlinear waves in
deep water. Their work demonstrated that the solution technique was not
an accident which was only applicable to the KdV equation. Soon, many
physically significant nonlinear partial differential equations (PDEs)
were found to be solvable by the IST, firmly demonstrating the power and
versatility of the solution technique. Ablowitz, Kaup, Newell, and
Segur (1973,1974) extended the applicability of.the transform by
employing Lax's (1968) approach to develop criteria which made it
possible to derive equations which could be solved by the IST. An
enormous amount of theoretical interest had been generated by the
introduction of the transform, so much so, that specialized research
applications were beginning to emerge. One area of particular impor
tance to the study herein relates to the case of periodic boundary
conditions and solutions.
An important contribution to the theory of nonlinear equations with
periodic boundary conditions was made by McKean and van Moerbeke (1975)
and Marchenko (1977). Their work established a connection between the
spectral theory of operators with periodic coefficients and algebraic
geometry, the theory of finitedimensional completely integrable
Hamiltonian systems and the theory of nonlinear equations of the KdV
type (Dubrovin, 1981). They showed that the KdV equation admitted an
infinitely dimensional family of solutions which could be written in
terms of Riemann theta functions of the form
2
f(A,t) = 2 In 9(1, .2' 4N; B) 2.5
where 0 is a theta function of genus N. The theta function contains
N onedimensional (in the horizontal plane) phase variables and a
scalar parameter B. They showed that the genus 1 solution was equi
valent to the cnoidal solution shown in Equation 2.2 and was the only
permanent form solution of the KdV equation.
The generalization and extension of this theory to three
dimensional systems was made by Krichever (1976). He developed a
methodology for solving the threedimensional generalization of the KdV
equation, the KadomtsevPetviashvili (KP) equation. This equation,
which was first proposed by KP (1970) and is formally derived in
Chapter 3, can be written in the scaled form:
(ft + 6ff + fxx) + 3fy = 0 2.6
where (x,y) are orthogonal coordinates in the plane of the quiescent
water surface with x representing the primary direction of wave propa
gation. The equation is based on the assumptions of weak nonlinearity
and weak dispersion, as in the derivation of the KdV equation, and on
weak threedimensionality. Each effect is assumed to be of an equal
order of magnitude. The previous statement that the KP equation is a
direct threedimensional generalization of the KdV equation can be
seen. The equation reverts to the KdV equation when no crestwise or
A
variations in the ydirection occur.
Krichever (1976) showed that the KP equation admits an infinitely
dimensional family of exact periodic (or quasiperiodic) solutions. The
concepts employed by Krichever in his solution methodology were adapted
and further extended by Dubrovin (1981) in order to express these
periodic solutions in the following form:
a2
f(,9, = 2 n 9(i' 2' ..' N; B) 2.7
where 8 is a Riemann theta function of genus N, composed of N two
dimensional phase variables and an N X N symmetric Riemann matrix
B. Genus 1 solutions are exactly equivalent to cnoidal waves; i.e.,
they are singly periodic, twodimensional, nonlinear waves which propa
pate at some angle to the Sdirection. Genus 2 solutions are the sub
ject of the investigation herein. These solutions are biperiodic, truly
threedimensional, nonlinear waves which propagate with permanent form
at a constant velocity. The resulting twodimensional surface pattern
therefore appears stationary to an observer translating with the waves
at the correct velocity. Genus 3 and higher order solutions are multi
periodic solutions which cannot be characterized as permanent form since
no translating coordinate system exists that allows the solutions to
become stationary.
Dubrovin's detailed treatment of the subject culminated, for our
purposes, in an analysis of the genus 1, 2, and 3 solutions to the KP
equation. He presented a series of theorems, lemmas, and corollaries
which proved the existence and uniqueness of solutions to the KP equa
tion. He also developed the basic guidelines which are required for
actually constructing genus 1 and genus 2 solutions although he pre
sented no explicit examples for doing so. Dubrovin's paper laid the
theoretical foundation for extending the theory from a highly abstract
mathematical proof into a computationally effective tool. The formid
able task of utilizing Dubrovin's theory in the development of an analy
tical wave model capable of yielding exact, truly threedimensional,
biperiodic genus 2 solutions of the KP equation was successfully accom
plished by Segur and Finkel (1985). A detailed.description of the math
ematical machinery developed by SF for genus 2 KP solutions is presented
in Chapter 3.
Although exact biperiodic wave solutions for shallow water have
only recently been presented, threedimensional approximations have been
studied and reported in the literature. Solutions for interacting waves
have been reported by Miles (1977), Bryant (1982), Melville (1980), and
Roberts and Schwartz (1983). Each of these investigations show non
linear coupling of two intersecting waves which are in qualitative
agreement with the exact solutions and with the observed behavior of
interacting waves. Since each of these results is produced by approx
imation methods, they are not relatable to the observed characteristics
of intersecting waves. The exact solutions presented by Segur and
16
Finkel described herein represent the first exact biperiodic solution
which can be quantitatively compared to observed waves.
CHAPTER 3
THE KADOMTSEVPETVIASHVILI (KP) EQUATION
This chapter is intended to provide a background for the study of
genus 2 solutions of the KP equation. It begins with a formal
derivation of the KP equation in order to document the procedures used
and the assumptions underlying this approximate model equation. Follow
ing the derivation, a complete presentation of the analytical genus 2
solution, as derived by Segur and Finkel (1985), is presented. The po
tential relevancy of this solution as a wave model is made through the
presentation of several graphical examples demonstrating the three
dimensional nonlinear structure of these exact solutions. The following
sections provide the background for developing the experimental portion
of the study and the determination of the correspondence between exact
solutions and measured waves.
3.1 Derivation of the KP Equation
The KP equation was first proposed, but not formally derived, by
Kadomtsev and Petviashvili (1970). Their interest in the equation was
a consequence of their study on the stability of solitary waves to
transverse (crestwise) perturbations. The formal derivation of the
KP equation, which closely parallels that of the KdV equation, begins
by defining the fluid and its boundaries. Consider for example a three
dimensional, inviscid, incompressible, flow domain as shown in
Figure 3.1.
Schematic diagram of flow domain
The equations governing this flow are Euler's equations for the
conservation of linear momentum
au au au
 + *u y + v
au
+w
8V 8v 8v av
S+u +v +w
aw aw aw aw
S+ u2 +V +w
5T 53ra
pa
1 ap
p Y
1 Ia
p g
and the continuity equation for the conservation of mass
Figure 3.1
au av aw
a3 + + a
3.2
In addition, the assumption of irrotational motion yields the following
qualities:
aw av aw au av au
aY ~ T = T ay = 0
3.3
In the above dimensional equations, E represents time and u, v,
and w represent the Eulerian velocity components in the orthogonal
i, y, and 2 directions. Additional terms include the fluid density
p, the fluid pressure p, and the acceleration of gravity g. It fol
lows from Equation 3.3 that the velocity field is derivable from a po
tential i which can be written in the following form:
~u 4 1=
ay v, 3'i
A kinematic boundary conditions for the free surface of the flow regime
shown in Figure 3.1 can be written as
a + u L+ v w 0 on (a,y,h+c,t) 3.5
TU 8x 37 W
whereas the corresponding boundary condition for a horizontal bottom is
written as
w = 0 on (',7,0,Z) 3.6
where e represents the elevation of the free surface measured from the
quiesent fluid level. A dynamic condition for the free surface boundary
can be written by combining Equations 3.1 through 3.4 to find
+ i lv2 + g = 0 on (I,y,h+r,t) 3.7
where the linear operator v = (a, aQ, a ) is used and the pressure on
x 2
the free surface is assumed constant. (Since this constant value can be
absorbed into the velocity potential, the pressure is conveniently set
to zero in the above derivation.)
The equations can now be consolidated to define a boundary value
problem for the motion of the fluid domain shown in Figure 3.1 subject
to the defined boundary conditions. For example, equations 3.2 and 3.4
are combined to yield Laplace's equation for the velocity potential
which determines the threedimensional velocity distribution of the
fluid domain; i.e.,
V2 = 0 .
3.8
The fluid motion defined by the velocity potential is not only
required to satisfy equation 3.8 at all points in the flow domain but
also to satisfy the boundary conditions defined by Equations 3.5, 3.6,
and 3.7 on the upper and lower boundaries. These conditions are rewrit
ten in terms of the velocity potential and surface elevation to yield
the kinematic free surface boundary condition
C 0 c Y Cy + 0: = 0 on (9,y,h+S,E)
t x yy z
3.9
the kinematic bottom boundary condition
4 : = 0
z
on (iy,0,Z) 3.10
and the dynamic free surface boundary condition
+ 1 12 + g.= 0
4t 2
on (0,y,h+,_E)
3.11
The governing equations and associated boundary conditions repre
sented by Equations 3.8 through 3.11 cannot be solved analytically in
their present form; however, a solution can be obtained if certain sim
plifying assumptions are made. For example, if all of the nonlinear
terms in the governing equations and boundary condition equations are
assumed negligible, the resulting linear system of equations becomes
solvable. Of course, this results in linear wave theory in which
velocities and surface elevations are constructed in terms of the
normalmode solutions; i.e., sine and cosine functions.
The derivation of the nonlinear KdV and KP equations requires a
more systematic approach since the nonlinear subtleties of these solu
tions are lost in the linear approximation. The decision as to which
terms are retained and which are omitted is made through a systematic
study of the relative magnitude of each term in the equation based on
the existence and subsequent ordering of certain small parameters. This
approximation is accomplished through the use of power series expansions
in terms of the small parameters.
The formal derivation of the KP equation first requires the scaling
of all dimensional quantities by introducing the following "scales." A
global length scale for the wave, usually considered to be the wave
length, is defined as L, for which a corresponding wavenumber k = 2w/L
is defined. For threedimensional flow, k represents a vector wave
number with i' and 7 components. The magnitude of this wavenumber
is defined by the relationship IkI = (12 + 2)1/2 where 1 represents
the idirection wavenumber and m represents that in the 7direction.
An amplitude scale, descriptive of the wave crest height, is defined
as a. A vertical scale h is defined as the depth of flow in which
the wave is propagating.
These three representative scales (k, a, and h) are similar to
those used by Stokes (1847), Korteweg and deVries (1895), and Ursell
(1953). One additional scale is introduced in order to define a
reference speed of propagation for the wave. This scale is simply de
fined as the celerity of a shallow water wave, as found in linear wave
theory; i.e.,
C =T
The purpose of defining representative scales for a given flow
regime is to enable one to characterize the wave behavior in a systema
tic manner similar to the approach described by Ursell (1953). This
characterization is made by analyzing the relative magnitude of selected
combinations of the representative scales for that wave. Three of these
combinations are used for defining the characteristics of the KP equa
tion. Each of these resulting "scaled parameters" will be used in the
formal derivation in order to insure that the derived evolution equation
will describe a wave field which will behave in a manner consistent with
the defined relative magnitudes of the scaling parameters. The first of
these parameters, given below,
a
a h
defines a wave amplitude to depth parameter which provides an indication
of the degree of nonlinearity of the wave. Smallness of this param
eter implies weak nonlinearity and, in the limit a 0 linear wave
theory is recovered. The second parameter
8 = (kh)2
provides a measure of the length of the wave with respect to the depth
of flow in which the wave is propagating. Smallness of this parameter
implies shallowwater conditions so that dispersion is weak. The third
parameter provides a measure of the threedimensionality of the wave.
This parameter, shown below,
(M)2
indicates the direction of propagation of the wave field with respect to
a defined orthogonal coordinate system. Smallness of the parameter, for
example, indicates that the primary direction of propagation is in the
'direction and that the wave is weakly threedimensional. When the
parameter vanishes, the flow becomes the twodimensional flow field
governed by the KdV equation.
The formal derivation of the KP equation is based on the assumption
that each of the defined parameters are small (i.e. << 1) which implies
a weakly nonlinear, weakly dispersive, and weakly threedimensional
flow. The relative magnitudes of each of these parameters will be
chosen in a subsequent analysis. The derivation begins with the scaling
of the governing equation and associated boundary conditions. This is
accomplished by introducing the following nondimensional quantities:
x = k
y =ky
h
S=
a
3.12
3.13
3.14
3.15
3.16
t =Ck
* Ck
ga
3.17
Substitution into Laplace's equation (Equation 3.8) results in the
following relationship:
3.18
0 (0 + ) + = 0 .
xx yy zz
In a similar manner, the kinematic free surface boundary condition
of Equation 3.9 is written
1
yn + a n + a 0 = 0
t x x yy 8 z
3.19
and the corresponding kinematic bottom boundary condition of Equa
tion 3.10 takes the form
S=0 .
z
The dynamic free surface boundary condition of Equation 3.11 becomes
1 2 1 2 1 +
I + a O + a 2 + 2 ( z + n = 0
t 2 x 2 y 28 z
3.20
3.21
Equations 3.183.21 now represent the complete nondimensional equations
governing the flow.
Next, each of the dependent variables is represented in a power
series expansion in terms of a small parameter. For the velocity poten
tial, we assume the following form
27
*(x,y,z,t;8) = Sm m (x,y,z,t) 3.22
m=O
which is substituted into Equation 3.18. Collecting all terms with mul
tipliers of like order of powers of 8 yields the form below.
0 O zz) 1Oxx + Oyy + 1zz) 3.23
2
+ 2(I xx + lyy + 2zz) + =
Since each sum of terms in Equation 3.23 is ordered by powers of the
small parameter 8, the overall equation is satisfied if, and only if,
each sum of terms is zero. Hence, the original single equation in terms
of 9 is replaced by an infinite set of equations for m. The equa
tions resulting from Equation 3.23 are shown below.
0(80) effects: Ozz = 0 3.24
0(81) effects: Oxx + Oyy + lzz = 0 3.25
0(82) effects: 1xx + yy + 2zz = 0 3.26
0(8= 0yy.26
Integration of Equation 3.24 with respect to z yields
*0 = G(x,y,t)z + *0(x,y,t)
where G(x,y,t) and *0(x,y,t) are integration constants. Application
of the bottom boundary condition of Equation 3.20 (i.e. z = 0)
requires
G(x,y,t) = 0
so that
o0(x,y,z,t) = o((x,y,t) .
3.27
Similar integration of Equations 3.25 and 3.26 and application of the
bottom boundary condition result in the following two relationships:
S(xy.zt) (0 yy) z2 1
1 (x,y,z,t) : (Oxx + %Oyy) + I
3.28
and
*2(x,y,z,t) (mOxxxx + 2Oxxyy + 0yyyy)
3.29
2! xx + yy) z2 + 2
Substitution of these results into equation 3.22 yields the following
expansion for 9, the velocity potential, correct to the third order.
+ = I0 ( xx + 0yy)z2 + 2 2 2 (1xx
:0 2 [02 C21 (0yxx
+ 1yy)z2
S24 Oxxxx + 2xxyy + Oyyyy + 3.30
The further analysis requires the introduction of a slow time scale.
This new time scale will permit the suppression of secular terms that
arise in the analysis of the dynamic free surface boundary condition.
Define
T = et
3.31
where e represents the small parameter defined previously. In addi
tion, we will make a Galilean transformation to a uniformly translating
coordinate system by letting
X = x t .
3.32
Differentiation between the different length scales in the x, y and z
directions will also be made by explicitly defining the following:
Y 1/2y
Z = z
3.33
3.34
The new scales of Equations 3.31 through 3.34 are substituted into Equa
tion 3.30 to obtain the following slow time representation for the
second order correct velocity potential.
*(X,Y,Z,T; 8) = + OXX + e2 YY) z2+
2 1 2 2
+ 82 2 XX + e 2, ) Z2
2 2 .XX
+ (2 2 + Z 4.4 z] + O(83 ) 3.35
24 OXXXX + 2OXXYY + OYYY 3) 335
We now introduce the following power series expansion representation of
the free surface displacement in terms of the new slow time scale
parameter.
3.36
n(X,Y,Z,T;e) = em n (X,Y,Z,T)
m=0
The kinematic and dynamic free surface boundary conditions of Equa
tions 3.19 and 3.21 respectively can now be written in terms of the slow
time scale. This substitution results in the following two equations
for the velocity potential and surface displacement:
1
nX + enT + axnx + ac n Z = 0
1 2 1 2 1 a 2
X + EO + 2 a0 + 2 aEY + 2 a *Z + n = 0 .
X T 2 X 2 Y 2 0 Z
3.37
3.38
Note that the new governing equations now contain all three small param
eters (a, 8, and e ) which have been introduced to allow for the
specific ordering of the final wave solution. The key to the derivation
of the KP equation is the assumption that each of the parameters are of
an equivalent order of magnitude. This assumption is made by letting
0(a) = O(B) = 0() .
3.39
Substitution of the series expansions for the velocity potential and the
free surface displacement (Equations 3.35 and 3.36) into the boundary
equations 3.37 and 3.38, expansion, and consolidation of ordered terms
in e yields the following two relationships:
and
COe( + ) + e1 ( OXXX 1IX + + 0 OT + 1) + 0(E2) = 0 3.40
and
(nOX + 'OXX)+ 1(nX nOT + (OXOX + OYY
3.41
+ X OXXXX + oXX + 0(2) = 0 .
Analysis of the 0( 0) terms show that
n0 = OX
A similar analysis of the 0(e ) terms yields
1 1 2
1 1X OT 2 OXXX 2 OX
and
nX IXX = nOX OX + "OOXX 6 OXXXX + OYY + nOT
3.42
3.43
3.44
Now equating the partial derivative (with respect to X) of Equation 3.43
with Equation 3.44, again taking the X partial derivative of the entire
result, and consolidating terms yields
OTX OXXXX " OXOXX OX~OX "OOXX nOT)X OYX = 0. 3.45
Substitution of Equations 3.4, and 3.42 into Equation 3.45 results in
the KadomtsevPetviashvili equation,
(uOT + 3UUoX + OXXX + uOYY = 0
3.46
where uo =OX = nO A final transformation of variables is now
required in order to write Equation 3.46 in the form used by Segur and
Finkel (1984). Let
S= X
Y 1
ST
6
f u=
2 0
The substitution of these variables into Equation 3.45 results in the
following form of the KP equation which will be used extensively in the
remainder of this study.
(ft + 6ffA + fq) + 3f% = 0 3.47
3.2 Solutions of the KP Equation in terms of Riemann Theta Functions of
Genus 2
Krichever (1976) showed that the KP equation admitted an infinitely
dimensional family of exact quasiperiodic solutions which could be
written in terms of Riemann theta functions of genus N. The techniques
employed by Krichever were extended by Dubrovin (1981) to specifically
address the genus 1, 2, and 3 solutions. The solutions relevant to this
study are the biperiodic genus 2 solutions which are truly three
dimensional and have two real periods, both spatially and temporally.
Dubrovin provided the necessary existence and uniqueness criteria re
quired for computing these solutions. The task of actually applying
Dubrovin's criteria and solution approach to compute an exact genus 2
solution of the KP equation was first completedby Segur and Finkel in
1985. This, of course, required the development of a considerable
amount of mathematical machinery to implement Dubrovin's outline. The
purpose of this section is to present, and describe, the machinery which
was presented by SF to compute these genus 2 solutions.
Genus 2 solutions of the KP equation can be written as
2
f(x,,=t) = 2 In ( $2, B) 3.48
ax
where 9 is the genus 2 Riemann theta function, composed of a 2
component phase variable V and a (2 X 2) realvalued Riemann matrix B.
The construction of this solution begins with the introduction of the
two phase variables
)1 1= + + wl1 + 10
and 3.49
*2 2 p2 + + t 20
The parameters Vi, V2 and v v2 are wave numbers in the x and 9
directions, respectively, while wl, w2 represents the angular fre
quencies of the wave with respect to the translating coordinate system
in which the KP equation operates. The constants *10' *20 represent
a constant shift in phase and are of no dynamical significance. A much
more thorough description of these coefficients will be presented later.
The second ingredient involves the specification of a symmetric, real
valued, negative definite 2 X 2 Riemann matrix of the form shown below.
b11 b12
B = 3.50
b12 b22
Negative definiteness is assured by requiring
b < b22 < b b22 b2 > 0 3.51
11 22<0 11 22 12
The role of the phase variables and the Riemann matrix in the specifi
cation of the theta function can now be shown. A genus 2 Riemann theta
function can be defined by a double Fourier series (Segur and Finkel,
1985)
0 (4'I2Y B) = exp i m B1i + imif 3.52
m1 = m2=2
where = (ml, m2) and the products are defined by
2 2
B 1= mb11 + 2m1m2b12 2+ 22
and
IT = m i1 + m2 2
The theta function requires two additional refinements in order to
assure a unique genus 2 solution. For example, SF (1985) showed that
two different Riemann matrices could result in identical theta func
tions. These two matrices are therefore equivalent and can be related
to each other by the appropriate transformation. The existence of equi
valent matrices which produce identical solutions introduces a question
as to whether or not the solution is unique. In order to resolve this
ambiguity, SF (1985) introduced the concept of a basic Riemann matrix.
They chose the following parameters to be natural representations for a
basic Riemann matrix:
b = max (b11, b22)
x = b2/b
12
d = det B/b
3.53
where both b and d are negative and X is real. Segur and Finkel
(1984) chose the basic Riemann matrix to be of the form
3.54
b bX
B =b
bX bX + d
where the requirement that the matrix is basic and negative definite is
satisfied by
3.55
b < 0, 2 < d b (1 X2)
4'
Under these conditions, a basic Riemann matrix generates one and only
one theta function. Another difficulty with the general definition of
the theta function as given by Equation 3.52 results when the off diag
onal terms of the matrix become zero. Diagonal matrices are referred to
as decomposable, otherwise, they are indecomposable. Dubrovin (1981)
proved that nontrivial genus 2 solutions of the KP equation only result
from indecomposable matrices. Although Dubrovin (1981) gave an explicit
test for decomposability, Segur and Finkel (1985) provided a simpler
test in terms of their parameters for a basic Riemann matrix. A basic
Riemann matrix is decomposable if, and only if, X = 0
A realvalued, negative definite, indecomposable theta function has
been associated with its corresponding basic Riemann matrix of the form
given by Equation 3.54. The requirements imposed on that matrix, are
that the parameters b, d, and X are real, and that X is non
zero. The basic definition of a genus 2 Riemann theta function can now
be written in terms of these new parameters.
exp 1 dm2 exp b m(1 + m2 )2{
Sm 1=1
3.56
e(lp' 2', B) =
m2=Cc
x cos (m101 + m22)
The above definition assures the existence of a real valued, indecompos
ible theta function, but it does not assure that the resulting theta
function will provide a solution to the KP equation. This assurance re
quires the development of two additional concepts as noted by Dubrovin
(1981). The new ingredients are thetaconstants and two additional
phase variables.
The concept of thetaconstants begins with the definition of a two
component vector which can assume any one of the following four
values:
39
p (0 (1/2) (1/2
P P2) 0 1' \1/2 /2 "
3.57
These values correspond to the four halfperiods of a theta function
(Dubrovin, 1981). Every Riemann matrix generates a fourcomponent
thetaconstant (SF, 1985) which can be written in the following form
A fl
e~ji5J=
exp (i + ) B ( + )
where m = (ml, m2). Equation 3.58 can be written in terms of the
basic parameters as
0[P] = m exp d(m2 + 2
mexp b [m + pl + X(m2 + )2
M,1=L'
3.58
3.59
where each thetaconstant is differentiable with respect to the param
eters b, d, and X.
Secondly, two new phase variables *3 and *4 are defined in
terms of the previous phase variables according to
3.60
44 = 2 X X ,1' 43 X 2
S=" m2=
where
b
, 12 bX
22 bX + d
3.61
Wavenumbers and angular frequencies for these new phase variables can be
written analogous to Equations 3.60 as
4 P 2 X1 = 1 X~ 2
v4 = 2 XV1 V3 = V1 X2
Uq = 2 Xl 9 3 = W1 XKW2
3.62
All of the components needed to state Dubrovin's main theorem have now
been established. The theorem requires that a function in the form of
Equation 3.48 is a solution to the KP equation if, and only if, the
following matrix equation is satisfied:
3.63
MX = 4SV
where the components of this matrix notation are
132
X V= 14 + 4 + 6vI( = 1'Vi' 3.64
V1 4 + 3 4 3
2 1 "4
qmy4 + 3v4 3
4
and
/2 1a 1 ap
(p) TTX) 2 p '
S: 2 a2b 2 P ab ad '
\ab0
d b (2)) 3.65
The parameter D shown in Equations 3.64 represents a constant of inte
gration with no physical significance. The system of four equations
represented by Equation 3.63 can always be solved if the Riemann matrix
is indecomposable. The matrix equations of Equation 3.63 can be in
verted to yield the following four relationships corresponding to the
four possible values of the two component vector p. The resulting
relationships are
Il + 3 1 4)
(14 + U4w1 + 6vI 1
3.66
2/4 : P34
4')4 + 3 14 3= 4)
where the parameters on the right hand side (P1, P2' P3, P4) represent
welldefined fourth order polynomials in the variable N4/Ul. (The
polynomials in Equations 3.66 are obtained by inverting M.) .The con
stant of integration D is arbitrary so that its equation can be ignored.
The two angular frequencies, wl and w4, can be eliminated from Equa
tions 3.66 to yield the following single relationship:
4 1)2 4 (6 P 3.67
(v4 U01) = 3 61(U4 6)
where Pg is a welldefined polynomial of degree 6. The left hand side
of Equation 3.67 is realvalued; therefore, in order for Equation 3.67
to be satisfied, the polynomial must be positive or zero; i.e.,
P6 6) 2 0 3.68
All existence and uniqueness criteria have now been presented for
genus 2 solutions of the KP equation. The results are summarized as
follows: Equation 3.48 represents a realvalued solution of the KP
equation if,and only if, the associated Riemann theta function satisfies
the criteria that 1) the phase variables, defined by Equations 3.49 are
realvalued, 2) the associated Riemann matrix is basic and indecompos
able, and 3) the polynomial relationships represented by Equations 3.63
are satisfied. Provided these criteria are met, the following section
demonstrates the computation of genus 2 solutions.
3.3 The Construction and Properties of Genus 2 Solutions
The construction of a genus 2 solution of the KP equation requires
the specification of the following eleven parameters:
I' 2' V 1' 2'2, 2 1, 2' 10' *20' b, d, X
The first eight of these parameters define the phase variables of Equa
tions 3.49 while the remaining three are contained in the basic Riemann
matrix defined by Equations 3.54. Dubrovin's theorem of Equation 3.63
provides three relationships among the eleven parameters; hence, there
are only eight independent parameters required to specify a genus 2
solution. Of these, I10 and 20 serve only to determine the origin
of the coordinate system and do not impact the dynamics of the solution.
Thus, the most general genus 2 solution of the KP equation contains only
six dynamical parameters which may be chosen freely. In order to
provide insight into the structure of the genus 2 solutions and to be
able to assess the effect of each parameter on the wave form, it is
useful to specify the six dynamical parameters and calculate some typi
cal solutions. In the experiments to follow, spatial and temporal
symmetry will be exploited in order to expedite the measurement program.
The symmetry of the generated waves provides three additional relation
ships among the six free parameters of the genus 2 solution; i.e.,
l = 2 = v2 v, Il = w2 w a
so that only three free parameters are available for specification. In
addition, the experimental measurements make it convenient to choose b,
p, and X for the free parameters. Making use of these additional
constraints on the family of genus 2 solutions, two examples are calcu
lated and presented in Figures 3.2 and 3.3. These figures show perspec
tive views of the water surface at a fixed time when the parameter b is
varied while p and X are held constant. (A more detailed examina
tion of the solution sensitivity to each of the free parameters will be
presented in Chapter 5.)
The exact solutions shown in Figures 3.2 and 3.3 are typical of all
of the symmetric subfamily of genus 2 solutions. The surface wave pat
tern consists of a single, basic structure which repeats in a tiling of
the entire water surface. A typical, basic structure can be isolated as
in Figure 3.4 by the construction of a "period parallelogram." Inside
the period parallelogram the wave crests form two Vshapes, pointing in
f A
y
ic"
*^
Figure 3.2 Example genus 2 solution (b = 1.5, v = 0.5, X = 0.1)
A
x f
Example genus 2 solution (b = 3.5, v = 0.5, X = 0.1)
Figure 3.3
opposite directions, and connected by a single, straight crest. Here
after, the Vshaped region will be referred to as the "saddle region"
while the straight crest between the V's will be termed the "stem."
(The motivation for both names will become apparent shortly.) Note that
crest amplitudes are largest in the stem region. The entire wave pat
tern propagates at a constant speed in a direction normal to the stem
region. The sides of the period parallelogram coincide with lines of
constant phase defined by the phase parameters noted in Figure 3.4. The
periodicity in each of these two directions is increased by 2w across
the period parallelogram. Specific relationships between other mathema
tical parameters and the wave structure inside the period parallelogram
have not been established for the general case. However, SF examine the
limit case of b,d 0 and prove that the actual wave crests of the sad
dle region coincide with lines of constant *3 and 4. The wave pat
tern in the limit b,d 0 is similar to that of.Figure 3.2; mathema
tically, the solution appears as two KdV solitons, propagating at angles
to one another and producing a third wave (the stem region) in a manner
that is well known from other investigations (e.g. see Miles, 1977). In
addition to the exact correspondence of *3 and 4 with individual
wave crests in the saddle region, the interpretation of the genus 2
solutions as two intersecting wave trains is especially important to
the experimental study and to the application of these solutions to
actual ocean waves. (Interestingly, a stimulus for the interest by
Segur in these waves was experiments on intersecting waves by Hammack,
1980.) The examination of the twosoliton limit solution also estab
lished that the two parameters X and K are a measure of the rotation
of the individual wave crests from the directions of periodicity; i.e.,
Period Parallelogram
Direction of Propagation
V.
0 Wave Generator
Stem Region
Saddle Region
Figure 3.4 A basic period parallelogram
*J2 = C2 + 27r
43 = C3
44= C4
*1 = C1 + 27
Sand 2". Alternatively, this rotation is related to the amount of
"phase shift" a wave experiences as a consequence of passing through a
region of interaction with another wave. All of these aspects of the
genus 2 solution will be made more explicit in Chapter 5.
CHAPTER 4
LABORATORY FACILITIES AND EXPERIMENTAL PROCEDURES
This chapter describes the laboratory facilities and experimental
procedures used to generate the threedimensional wave fields for com
parison with exact genus 2 solutions of the KP equation. This chapter
begins with a detailed description of the wave basin and wave generator.
A basic knowledge of the wave making capability is essential to the for
mulation of an approach for generating candidate waves for comparison
with genus 2 solutions. The wavegeneration methodology follows the
description of the physical facility. Due to the threedimensional
nature of the wave forms required for this study, considerable detail is
presented for the datagathering program to quantitatively measure the
temporal and spatial structure of the wave field.
4.1 The Wave Basin
A wave basin measuring 98.0 ft wide, 184.0 ft long, and 2.5 ft deep
is used for the experimental portion of the study. The walls of the
basin are constructed of concrete filled, nonreinforced, cinder blocks
resting on the concrete slab that forms the bottom of the basin. A
schematic diagram of the wave basin is shown in Figure 4.1.
The concrete slab was poured by standard construction procedures to
normally acceptable tolerances. The topography of the tank bottom is
shown in Figure 4.2 and reveals a maximum variation of +/ 0.5 inch.
98.0 Feet 
; 184.0 Feet
Gage, Array
9 8 7 6 4 3 2 1
40.0 'eet
Wave Geerator
90.0 Feet
Figure 4.1 Schematic drawing of the wave basin
High and low areas resulted which can be identified in the figure. As
will be discussed in a later section, the effects of these irregular
zones were evidenced in the measured wave height patterns. The inset
numbers shown in Figures 4.1 and 4.2 refer to the location of wave gages
in the basin which will be described subsequently.
The downstream end of the wave basin, opposite the wave generator,
is lined with rubberized horsehair to a depth of approximately 2.0 ft,
extending out a distance of approximately 6.0 ft from the wall. The
purpose of this absorption material is to both reduce reflections from
the rear wall of the basin during testing and to dissipate the oscil
lation of waves within the basin following testing. Sidewalls are not
lined with the wave absorption material. The 90 ft wide wave generator,
95.0
S0
,.j 5
0.0
0.0 47.5
X (ft)
7 CONTOURS
CONTOUR LEVELS FROM .300 TO .300
CONTOUR INTERVAL OF .100 (inches)
Figure 4.2 Bathymetry of the wave basin
which nearly spans the basin width, is located to the right of the gages
in Figure 4.2.
4.2 The Directional Spectral Wave Generator
A wave generator capable of generating single or multiple wave
forms of variable shape and direction is located at the US Army Engineer
Waterways Experiment Station's Coastal Engineering Research Center
(CERC) located in Vicksburg, Mississippi. This directional spectral
wave generator is shown in Figure 4.3. It was designed and constructed
for CERC by MTS Systems Corporation of Minneapolis, Minnesota, based on
design specifications provided by CERC.
The directional spectral wave generator is composed of 60 indivi
dually programmable wave paddles. The generator was designed in a port
able configuration of 4 separate, self contained modules (Chatham,
1984). Each of these modules is composed of 15 separate wave boards
constructed on a steel frame as shown on the schematic drawing of Fig
ure 4.4. Each module is equipped with six adjustable mounting pads for
leveling purposes and can be moved by using four dollies at each of four
lifting posts, two located in the front and two in the rear.
The wave boards, measuring 1.5 ft wide and 2.5 ft in height each,
are individually driven in a pistonlike motion by a 0.75 horsepower,
directdrive servomotor located at the articulated joint between
adjacent boards. The joint structure consists of a fixed and linked
hinge as shown in Figure 4.5. Extremity points (left edge of paddle 1
and right edge of paddle 60) are driven by single fixed hinges. The
connections between adjacent wave boards are smoothed by means of a
flexibleplate seal which slides in slots located on each wave board.
Figure 4.3 The directional spectral wave generator
ELECTRONIC ASSEMBLY
DRIVE / s y
ASSEMBLY/ Y
DRIVE
PLATE
Figure 4.4 Schematic diagram of a wave
generator module (Outlaw, 1984)
The maximum stroke of a wave board is 1.0 ft, corresponding to a
+/ 0.5 ft displacement from the midpoint position. Each wave board
can be operated up to and including 180 degrees out of phase with the
adjacent board. As already noted, the boards are operated in a piston
like motion and are not sealed at the floor. The displacement of each
paddle is controlled by a beltdriven carriage assembly connecting the
drive assembly to the belt drive as shown in Figure 4.4. A transducer
is located on each wave board to monitor displacement and provide a
feedback signal to the wave generator console. The servocontroller
module for each servomotor monitors this position feedback and gener
ates a strokelimit and displacementerror detection signal which stops
further displacement of the wave board if either limit is exceeded. The
DRIVE
ARTICULATED
SEAL WAVEBOARD
SEAL/
WAVEBOARD
Figure 4.5 Schematic diagram of a wave board (Outlaw, 1984)
system console block diagram is shown in Figure 4.6 and the servo
controller block diagram is presented in Figure 4.7. Enclosures are
mounted on the top of each module for containing the motor and trans
ducer power and signal equipment. The cables required for the trans
mission of wave board displacement signals and the position transducer
feedback signals are located on three cable reel assemblies adjacent to
the equipment enclosures.
Each paddle of the four portable modules is electronically con
trolled and electromechanically operated according to the input com
mand signal received from each associated control channel. This re
quires a total of 61 control channels corresponding to the push points
Figure 4.6
MODULE
TRANSDUCER
CONODIONE R
FUNCTION
GENERATOR
(OPTIONAL)
CONVERTER
RAMPIRE
SYS CONTROL
PANEL
TO MOORl
O/A SERVO
ONVERTER aomluM CONTROLLER ONaCd
(1611 [XffoWAa p(06
r
NNCTION U
I GENERATOt i
(PROPOSED)

SYS .smsor
CONTROL ulnl..ommT.
PANEL .ITr Toms
fnUl lITo cowls
V / oi.t&
CO5S I O ORIOx
6V PAIR
IV PWR
ILY WI
System console block diagram (Outlaw, 1984)
Figure 4.7 Servocontroller block diagram (Outlaw, 1984)
(articulated joints) for each of the 60 paddles (A single control
channel provides the common signal for the joint between adjacent
paddles). Independent control of each paddle in the system is provided
by an Automated Data Acquisition and Control System (ADACS). The ADACS
system was developed for the directional spectral generator through the
modification of an existing controlfeedback system (Whalin et al.,
1974) reported by Durham and Greer (1976). This hardware/software
interface allows the user a 20 update per second per wave board command
control signal to the wave generator. This control capability is per
formed by the wave generator console which provides the digital to
analog (D/A) conversion of the programmed signal such that 61 channels
of control signal are simultaneously output to each of the 61 wave
paddle servos. The sampling and storage of data at a rate of 50 samples
per second per gage for up to 128 gages through multiplexed channels of
analog to digital (A/D) conversion is provided by the system. The re
sponse of each wave board to the individual control signals is monitored
so that when either the stroke or displacement limits have been ex
ceeded, disable signals can be issued to the respective paddle. In
addition, signals are provided to a calibration/test indicator located
on the system console so that adjustments of the servo controllers can
be made when necessary. Details of the system are reported by Turner
and Durham (1984).
The computer system supporting the ADACS is a Digital Equipment
Corporation (DEC) VAX 11/750 central processing unit (CPU). The system
is equipped with an IEEE 448 interface for the D/A conversion of the
usersupplied digital control signal. Peripherals to the basic CPU
include 121 megabytes of fixeddisk mass storage, 10 megabytes of
removabledisk mass storage, two 125 inchpersecond 800/1200 BPI mag
netic tape drives, two line printers, a Versatec printer/plotter, and a
Tektronix 4014 CRT unit equipped with hardcopy capabilities. The com
puter system is shown in Figure 4.8.
4.3 A Methodology for Generating Waves
Genus 2 solutions of the KP equation were shown in Chapter 3 to
describe a threedimensional, nonlinear wave pattern. The development
of these solutions by Segur and Finkel was partially a consequence of
experiments by Hammack (1980) which indicated qualitatively similar sur
face patterns resulting from the interaction of incident and reflected
waves. A similar interpretation of genus 2 waves was presented in Chap
ter 3. The development of an experimental procedure which would result
in the evolution of surface wave patterns qualitatively similar to genus
2 solutions was achieved by experimentally reproducing the conditions
reported by Hammack, i.e. interacting waves. In view of this interpre
tation, the interacting wave trains used for the experiments were chosen
to be cnoidal waves, since the periodic extension of a solitary wave is
a cnoidal wave. This section will first describe the methodology used
for generating cnoidal waves and then discuss the technique of evolving
an appropriate wave form through the generation of simultaneously inter
secting cnoidal wave trains. The indirect procedure of wave form evolu
tion outlined here instead of the exact generation of genus 2 waves will
be addressed at the end of this section.
4.3.1 The Generation of Cnoidal Waves
The generation of a cnoidal wave with the directional spectral
wave generator is accomplished by utilizing the wave generation
4i37
_ 1IIIIIIC
Figure 4.8 The computer system
technique presented by Goring in 1978. Goring's method prescribes the
displacementtime history required of a single piston wave generator to
generate a long, permanent form wave. Because of the similarities in
both the wave form and wave paddle motion, the generation approach is an
ideal one for the present application. Therefore, the identical tech
nique is used here to program the directional spectral wave generator
with the added complexities of 60 paddles (with 61 push points) and pro
visions for phase lagging between adjacent paddles necessary for the
subsequent generation of oblique waves. The basic theory is presented
below.
Goring's wave generation methodology provides a means of relating
the vertical displacement of the water surface profile of a known free
wave to the horizontal wave paddle motion required to generate that
wave. The principal idea is to equate the velocity of the paddle to the
velocity beneath the wave surface at the location of the moving wave
paddle. By knowing the time history of the desired free wave, the time
history of the wave paddle motion necessary for generating that wave can
be computed. Figure 4.9 was presented by Goring to demonstrate the way
in which the generation equation is obtained.
The inset diagram (c) represents the desired water surface profile.
In this example a linear sinusoidal surface displacement has been spe
cified. The wave has an amplitude a and is propagating to the right
with a wave celerity of c. The corresponding horizontal velocity time
history is shown in the inset diagram (a). It can be seen that the
velocity and surface time series are in phase, consistent with linear
wave theory. Desired is the time history of the displacement of the
wave paddle required to generate a sinusoidal wave. This desired
61
(c)
17(x,t) c
a
o
t
I I I
I I
t1
t I Ic
T '
S(ot)
V WAVE PLATE
TRAJECTORY ^(t)
_/(o,t)_ ______________________________
V V x
Wave generation phase plane (Goring, 1978)
Figure 4.9
displacement (t), termed the "trajectory" by Goring, is written in the
following form,
dt =
where u(S,t) represents the depth averaged velocity written as a func
tion of the time varying trajectory of the wave board. Since we are
dealing exclusively with long waves, the assumption is made that the
particle velocity is constant throughout the water column.
The above representation for the velocity produces a distortion of
the trajectory from what would be observed at a fixed location. For
example, if u(O,t) were used in Equation 4.1, the velocity would be
only a function of time and the resulting trajectory would simply be
sinusoidal in shape. The point of maximum trajectory, = S would
occur at the time t = T/2 When the velocity representation of Equa
tion 4.1 is used, the maximum trajectory is achieved at a time of t
=T/2 + S/c In Goring's words (1978) "Thus the time taken for the
plate to travel forward to its full extent is time S/c longer than it
would be if the trajectory were sinusoidal and consequently the time
taken for the plate to travel back to its original is time S/c shorter
than it would be if the trajectory were sinusoidal." Physically, if the
wave paddle position is not considered, thereby ignoring the celerity of
the wave, secondary waves will be produced at the wave generator paddle.
This occurs because the crests and troughs, which are not traveling at
the exact speed of the paddles, reflect off the paddles to produce the
secondary wave effect.
63
For waves of permanent form it was shown (Svendsen, 1974) by con
tinuity that the velocity averaged over the depth is
u(x,t) C(xt) d
h + n(x,t) dt
4.2
where n represents the surface displacement. It is assumed that this
displacement can be written in the following form:
A A
ri(&,t) = We()
where H represents a wave amplitude and f(8) is the appropriate
function sinusoidall, cnoidal, etc.) of the phase variable
0 = k(ct ) .
The total derivative of Equation 4.4 is written as
A
de *
dt k(c) .
dt
4.3
4.5
64
By using the chain rule, the time derivative on the right hand side can
be written as follows:
d = *.d8 dk
dt di T i nk(cE)
dt "d6 dt ade
4.6
d k(ck)
By using the relationships of Equations 4.1, 4.2, and 4.3; Equation 4.6
can be simplified to the following
Hf(e)
do kh
Integration with respect to the phase variable yields
A
H e
((t) = f f f(w)dw
4.7
where w represents a dummy variable and the phase variable
given by Equation 4.4. The resulting equation for the paddle
is implicit in that the phase variable on the right hand side
A
6 is
trajectory
is also a
function of the trajectory; therefore, the equation must be solved
numerically. The solution technique selected by Goring was Newton's
method, also referred to as the NewtonRaphson method. A general ex
pression for this numerical procedure can be written for an arbitrary
function F as a function of a phase variable 9 as
Ai+l i F(_ )
^i
Fe(0 )
The superscript i represents the iteration number. The iterative
^i Ai+1
procedure is to select an initial 0 and compute e This is
Ai Ai+1
repeated until the quantity 1 e is adequately small. The
solution scheme is a rapidly convergent one for most well behaved
A
functions and results in an accurate approximation for e. The
arbitrary function can be defined by writing the phase function of
Equation 4.4 in the following identity.
F 0 k(ct 0) 0
Substitution of this identity into Equation 4.7 results in
A
F = A kct + h f f(w)dw 4.8
0
Now, the partial derivative with respect to the phase variable
yields the form
6 F8 = 1 + h
Equations 4.8 and
A
solution for 8.
phase variable at
4.9 are the precise form necessary for a Newton method
Substitution yields the following solution for the
the i+1 iteration:
A
H 0/
Ai+l Ai ikct + h ( (w)dw
0 0  pA
1 + (e)
h
4.10
The iteration of this relationship to the desired level of convergence
A
will result in an accurate approximation for 8 at time t. Then, the
paddle displacement can be determined by rewriting Equation 4.4 in the
form
A
S= ct .
Equation 4.10 represents an implicit solution method for the phase
variable of an arbitrary wave form. We are now interested in the
specific wave form of a cnoidal wave. The surface displacement function
for a cnoidal wave can be written as
4.9
67
()A yt h
f() = t + cn2Im) 4.11
where h represents the depth of flow, yt represents the distance
from the wave trough to the bottom boundary, cn is the Jacobian
elliptic function, m is the elliptic parameter, and
S= 2k t 4.12
T L/
is the phase variable (the sign has been changed for convenience ac
cording to Goring's paper) written in terms of the first complete ellip
tic integral K(m), the wave period T, and the wavelength L. This
form is exactly equivalent to that shown by Equation 2.2. The integral
of this function, necessary for the evaluation of Equation 4.10, can be
written in closed form (from Abramowitz and Stegun, 1970) as
A E( m) ml
0m m
f f(w)dw = M1 4.13
where E(9m) is the second incomplete integral and mI is the
complimentary elliptic parameter defined as
mI = 1 m .
Substitution of Equations 4.11 and 4.13 into the Newton approximation
results in the following relationship:
2Kht 1 ^Ai+ H i
i+l Ai T + (Y + E(Im)
i il T t m m 4.14
t + H cn2 ( m)
Note that the negative sign in the first term of the numerator (2Kht/T)
was inadvertently omitted by Goring. A thorough description of the
methods used to evaluate the various elliptic functions is provided in
Appendix A. Upon completion of an adequate number of iterations to
achieve the desired level of accuracy, the paddle displacement at time
t can be written from Equation 4.12 as
(t) = L 4.15
The programming of the wave generator to produce these displacements is
accomplished in the following sequence of operations. Reference is now
made to Figure A.1 in Appendix A. A wavelength and maximum water sur
face elevation is specified for each desired cnoidal wave. Based on
this wavelength and wave elevation data, values for n1, n2, 1, ml,
T, and the first K(m) and second E(m) complete and the second
E(~Im) incomplete integrals are computed. The wave period is divided
into 360 time segments corresponding to 361 discrete values (0360).
For each time value, the phase variable of Equation 4.12 is defined and
usedin the Newton iteration method to compute a displacement for the
paddle corresponding to each of the 360 degree representations of the
period. This procedure is repeated for each of the 61 push points of
the 60 wave generator paddles. A magnetic tape is generated which con
tains the control signal for the displacement of each push point for the
time series corresponding to a control signal update of 20 updates per
second per paddle. The wave generator control software program is
executed and the waves are produced on the wave generator corresponding
to the input signal on the magnetic tape.
An example of the generated cnoidal waves can be seen in Fig
ure 4.10 in which a single period of a cnoidal wave with a wavelength of
7.0 feet and a maximum wave height of 1.84 inches is shown. Discre
pancies between theory and measurement are due to the variations in the
basin topography as evidenced in Figure 4.2. This spatial variation in
depth produces an approximate +/ 25% variationfrom the mean of the
measured total wave heights for a cnoidal wave uniformly generated
perpendicularly from the axis of the wave generator. This effect can be
seen in the nine wave gage traces shown in Figure 4.11. The shoaling of
the wave is obvious in the traces of gages 3, 6, and 7 which can be seen
from Figure 4.2 to be located behind shallow areas. If these three
gages are omitted, the variation is on the order of 14%. Fortunately,
this shoaling effect is much less pronounced in the evolved waves which
are used for verification of the KP equation. This is probably due to
the fact that the test waves result from the nonlinear interaction of
two separate waves generated from separate directions. The influence of
the basin floor on the verification will be further addressed. The
waves of interest, the candidate genus 2 waves, will now be discussed.
70
CNOO07
0
0* C PTE
0
l ,
CI
COMPUTED
m nEASURED
0 i I i
0.CO .20 .0 0.50 0.80
Figure 4.10 A comparison between a generated wave and
cnoidal wave theory
CNDIDOI. TEST CN0007
GAGE I
3.0.
0.0 0 130.0 15.0 10.0 25.0 30.0
TIME ISECS)
0." CNOIDAL TEST CN0007
GAGE 4
3.0
2.0.
3 0
3.0
3.0 J 10.0 IS 3 20.0 25.0 30.0
TIME ;SECS)
5.3 j .0 0 s 3 25.0 300.
TIME ISECSI
2. i
TIEuES
4. CNOIOL TEST CN0007
GRGE 5
3.0.
20.
3.0
1.0
2.0
3.0.
t.O
0.0 5.0 13.3 15.0o 0. 0 25 o 10.0
TIME (SECSI
S'0 CNOI OAL TEST CN0007
GrGE 3
.o0
3.0.
2.0
0.0
0.0 5.0 10 0 15.0 20.3 25 0 30.0
I DE b3El)
a
0.0 5.0 10.0 15.0 20.3 25 3 30.0
TIME ISECSI
S1 CNOIDAL TEST CN0007
GAGE 9
3.0
2.0
.... ,if!,,lffi ^
1.0.
t.o.
0.0 5.0 3 15.3 3) 2' 5 ju 3
TIME 1SECSI
Figure 4.11 Wave profiles from the 9 wave gages for a uniformly
generated cnoidal wave
4.3.2 The Generation of Genus 2 Waves
Genus 2 wave forms were produced in the wave basin by evolving the
proper form rather than by directly generating it. The reason for this
approach will be discussed at the end of this section. The evolution
technique is as follows. The procedures described for generating
cnoidal waves were modified such that a single cnoidal wave could be
generated at an angle to the axis of the wave generator. A second wave
was then simultaneously generated at an equal but negative angle such
that the two separate waves are generated at a predetermined angle of
intersection which is symmetric to the wave generator. In order to
fully investigate the validity and limits of applicability of the KP
equation as a model for threedimensional nonlinear waves, an experi
mental program was devised to generate a variety of wave patterns which
span a wide range of nonlinearity and threedimensionality.
A broad range of nonlinearity of the basicwave shape is achieved
by generating three basic cnoidal wave trains. These waves are gen
erated with heights of approximately 1.0 inch and wavelengths of
7.0 ft, 11.0 ft, and 15.0 ft, corresponding to an elliptic parameter
m of 0.44, 0.73, and 0.89 respectively. Water depth was maintained at
1.0 ft. Variations in the threedimensionality of the resulting wave
patterns was achieved by intersecting each of the three cnoidal wave
trains at a variety of angles. These angles of intersection are ob
tained by programming a phase shift between adjacent wave paddles. A
positive shift for one wave and a negative shift for the other wave
results in the generation of the desired symmetrically intersecting
waves. This phase shift is approximately equivalent to the angle of the
wavecrest with respect to the axis of the generator. A wide range of
angles of intersection were used in order to completely cover the range
of weak to strong interaction of the two basic waves.
Twelve wave fields were selected for generation to test the KP
equation. The generation components of each are shown on Table 4.1.
The angle indicated in the table shows the approximate (linear wave
relationship) correspondence between the phase shift and the angle of
propagation.
Verification of the KP equation as a model for threedimensional
nonlinear waves will be successfully accomplished by reproducing the
wave patterns indicated in Table 4.1 with exact solutions. Reproduction
requires the development of a unique correspondence between the free
parameters of the genus 2 solution and the physical characteristics of
the observed wave field. Correspondence is developed in Chapter 5.
Table 4.1 The Experimental Waves
Test
Number Wavelength (ft) Phase Shift (deg) Angle (deg) Period (sec)
CN1007 7.0 10.0 7.45 1.378
CN1507 7.0 15.0 11.21 1.378
CN2007 7.0 20.0 15.03 1.378
CN3007 7.0 30.0 22.89 1.378
CN4007 7.0 40.0 31.23 1.378
CN1011 11.0 10.0 11.75 1.947
CN1511 11.0 15.0 17.79 1.947
CN2011 11.0 20.0 24.04 1.947
CN3011 11.0 30.0 37.67 1.947
CN1015 15.0 10.0 16.12 2.553
CN1515 15.0 15.0 24.62 2.553
CN2015 15.0 20.0 33.75 2.553
Prior to addressing the free dynamical parameters of the exact
solution, a comment on the generation technique utilized for this
investigation is necessary. Waves were generated in the wave tank by
evolving an approximate genus 2 wave as described above. This approach
was first adopted because the relationship between the free parameters
of the exact solution and the physical characteristics of the desired
wave form were unknown at the onset of the investigation. For example,
an appropriately shaped wave is first required in order to develop a
means of relating the free solution parameters to that observed wave.
These parameters could then be used to compute an exact solution which
would emulate the observed wave. Following the successful completion of
this task, the logical extension would be to generate the exact solution
and analyze the resulting wave. This was in fact accomplished, but with
disappointing results.
The finite dimensions of the 1.5 ft wide paddle proved to introduce
strong perturbations in the small features of the resulting wave. An
example result from experiment CN2015, described in Chapter 5, will be
used here to illustrate this problem. The stem region of experimental
wave CN2015 is on the order of 3.5 ft in length. It is physically im
possible to generate this region exactly with 1.5 ft wide paddles.
Examples of the perturbations introduced are shown in Figures 4.12 and
4.13. Figure 4.12 shows a wave trace in the saddle region for the
evolved wave of experiment CN2015. Note the symmetrical peaks and uni
form wave shape. An exact solution corresponding to this wave field was
computed. Figure 4.13 demonstrates a similarly located wave trace for
that generated exact solution. The perturbations are evident from the
nonuniform shape of the resulting wave which actually evolves a third
5.00 10.000 5.00 20.00 25.00
IIM. IN btLONU5
Figure 4.12
Measured wave profile in the saddle
experiment CN2015
V P f
0
1
I
ui
u1
z
j3
A
_*c
''^ '
0.00 5.00 10.00 15.00 20.00 25.00
IIME IN tLONU5
30.00 35.00
Figure 4.13 Measured wave profile in the saddle region corresponding
to an exact solution generation of experiment CN2015
0'.00
30.00 35.00
region of
VVY
''
51
r I
peak. Repeated attempts at generating exact waves always failed to gen
erate a clean wave form. The conclusion of this exercise was that a
relatively clean genus 2 wave could be continuously evolved but could
not be discretely generated by existing facilities.
4.4 The Measurement of Waves
The difficulty of quantifying threedimensional wave phenomena with
twodimensional instrumentation is well recognized. Furthermore, the
presentation of twodimensional data in a concise yet definitive form
for effectively demonstrating threedimensional effects is difficult.
The measurement program developed here can best be motivated by looking
at the basic features of the generated waves. Figure 3.3 shows a typi
cal wave form produced by the technique described above. Symmetry of
the wave pattern was achieved by generating identical cnoidal waves
(equal wavelength and height) at symmetric angles. The period paral
lelogram, discussed in Chapter 3 and shown in Figure 3.4, was described
as a basic surface pattern which repeats to form a global surface wave
field. The complete.specification of this area will define the surface
pattern and be sufficient for verification of the KP solution. The
basis for choosing symmetric waves can now be seen, a symmetric period
parallelogram is generated which propagates in a direction perpendicular
to the axis of the wave generator.
Two separate means of data collection were used to quantitatively
measure the parameters of the basic parallelogram. First, a photogra
phic technique was devised to measure the spatial distribution of the
generated wave field. Photographs provided a visual representation of
both the physical size of the resulting period parallelograms and of the
internal features, such as the stem and saddle regions. These data were
used to determine the placement and spacing of a single fixed linear
array of recording wave gages which would be capable of quantifying the
vertical, horizontal, and temporal distribution of each of the period
parallelograms. These two collection techniques are described below.
4.4.1 The Photographic System
Measurement of the twodimensional geometry of the surface wave
patterns was found to be highly beneficial in that it provided both
quantitative and qualitative information on the spatial structure of the
period parallelogram. This procedures is described. Two Hasselblad
Model 500 EL/M 70mm cameras were each equipped with a 50 mm lens, an
automatic advance 50 exposure film canister, and a remote control expo
sure capability. The two cameras were installed approximately 23.0 feet
above the floor of the wave basin, located on either end of an approxi
mately 20.0 foot long 3 X 5 inch aluminum box beam which was clamped to
an existing catwalk and cantilevered out over the wave basin. This pro
cedure resulted in a final placement of the cameras centered on the wave
generator a distance of 40.0 feet from the axis of the wave boards. Be
cause of the focal length of the lenses, the field of vision of each
cameras was approximately 23 X 23 feet. The resulting two photographs
could then be combined in a mosaic to form a 23 X 40 foot picture. Il
lumination of the basin area beneath the cameras was by means of 2
Ascor, 8000 wattsecond strobe lights with remote control activation
capability.
Both cameras and strobes were connected to a remote control activa
tion panel which, when activated, operated both simultaneously. The
control panel was located adjacent to the wave generator console in the
computer room. A single gage was centrally placed 55.0 feet from the
wave generator, beyond the viewing range of the cameras. A schematic
diagram of the photographic setup is shown in Figure 4.14. Gage output
was monitored with a Tektronix Model 5111A dual trace oscilloscope, also
located adjacent to the generator console, to provide the operator with
a means of determining when to activate the cameras and strobes. It was
assumed that when the wave front first became visible on the oscillo
scope screen, the wave field would be fully developed in the camera
viewing area. A photograph was taken at this time followed by four more
at approximately 5.0 second intervals. This procedure was used for all
experimental wave patterns. A total of 240 photographs, representing
120 mosaics, of surface wave patterns were taken for the study. A rep
resentative photograph of each wave pattern used for analysis is in
cluded in Appendix B.
Wave Gage
t t
15.0 Feet
Camera # 1 Camera # 2 55.0 Feet
40.0 Feet
Strobe Lights
Wave Generator
I Remote Control Panel
Figure 4.14 The photographic system
The photographic technique described above proved to be an invalu
able tool for understanding and interpreting the qualitative features of
the generated wave fields. Without the aid of these photographs, the
successful formulation of a data collection program would have been
extremely difficult.
A problem which exists with photographic data is that of distor
tion. Although the photographs were primarily used in a qualitative
sense, this problem is addressed here. Horizontal measurements from the
photographs are based on grid marks painted on the basin floor for that
purpose. Since the waves are actually photographed on the surface (one
foot above the bottom), a discrepancy between actual and measured dis
tances is experienced which increases with distance from the camera
lens. An example is shown in Figure 4.15 to illustrate this effect.
Camera Lens
23.0 Feet
Figure 4.15 Horizontal measurement distortion
Assume a wave crest is photographed which is actually 23.0 feet below
and 10.0 feet from the camera. Due to the diffraction of light (assum
ing an index of refraction of 1.3330) a distance of 10.313 feet will
be measured from the floor scale. This amounts to an error of 3% in
10 feet (6% for the entire viewing area). Directly under the camera,
the error is zero. Because of this variable horizontal discrepancy,
error limits for horizontal measurements were determined. These limits
will be further addressed in Chapter 5.
4.4.2 The Wave Gages
The second set of required data are quantitative water surface
elevations which will relate the vertical structure of the observed
waves to the exact genus 2 solutions of the KP equation. These data
were used to quantify certain wave characteristics, such as the hori
zontal variation in height and shape within the period parallelogram.
Measurement of the required threedimensional distribution of the wave
field was greatly simplified by the selection of the symmetrically
intersecting waves. As previously mentioned, the resulting permanent
wave form, bounded by the basic period parallelogram, propagates perpen
dicular to the face of the wave generator. The period of the generated
wave is easily measured with wave gages and the width of the period
parallelogram is measured from the photographs. These two data deter
mine the propagating velocity of the permanentform wavefield. By know
ing the period and velocity, a time series measured from a stationary
gage for one period can easily be converted to a spatial water surface
distribution across one horizontal wavelength.
The simplification achieved by symmetry can now be demonstrated.
As can be seen in the schematic of Figure 3.4, the axis of the stem
region of interaction is parallel to the face of the wave generator. An
array of nine recording wave gages was located in the wave basin paral
lel to this same line. The gages were placed a distance of 40.0 feet
from the face of the wave generator, spaced 2.5 feet apart. The entire
array was centered on the generator such that gages 1 and 9 were each
10.0 feet from the generator centerline as shown in Figures 4.1 and
4.2. The placement of these gages with respect to the hexagonal wave
forms and period parallelograms is shown in Figure 4.16.
The sample wave pattern shown graphically now demonstrates the
advantages of generating symmetrical waves. For example, it can be seen
that a common point exists in the center of each hexagonal figure which
represents the common apex of two period parallelograms. It can be seen
that the location of each gage can be uniquely identified within a half
Period
S ".% "/Parallelogram
9 8 7,6 5 4A 3 2 1
Zero
Point
Wave Generator
Figure 4.16 Schematic diagram for wave gage placement
parallelogram by referencing it according to its distance from the
common, or zero point. Because of the symmetry, the left half of the
right parallelogram is exactly equivalent to the right half of the left
parallelogram. The determination of just one half parallelogram is then
sufficient to completely describe the entire period parallelogram and
hence the entire global wave field. The data collection scheme was
specifically aimed at this goal by mapping each of the nine gages into a
common half period parallelogram. In the example shown; gages 6 and
4, 7 and 3, 8 and 2, and 9 and 1 are equivalent since each pair are
equidistant from the zero point. Since the actual location of that
point with respect to the gage line axis varies for each test run, the
first estimated relationships between the zero point and the gage loca
tions were determined from the mosaic photographs. Subsequent adjust
ments were made by shifting the solution origin by varying I10 and
20 of Equations 3.49. An example of the gagezero point correspon
dence will be presented in Chapter 5.
The water level gages used for this study are watersurface
piercing, parallelrod, conductance type gages. They are identical to
those for which the original ADACS was developed. Use of the gages made
it possible to utilize existing calibration, storage, and plotting
software. Each gage is associated with a Wheatstone bridge, shown
schematically in Figure 4.17. Operationally, a transducer measures the
conductance of the water between the two vertically mounted parallel
rods. This measured conductance is directly proportional to the depth
of submergence of the rods. The output from each gage is sent to the
ADACS through shielded cables. The accuracy of the gages was reported
EXCITATION
VOLTAGE
\/ x MODEL BOTTOM \
Figure 4.17
Schematic diagram of parallelrod resistance transducer
(Durham and Greer, 1976)
by Durham and Greer (1976) to be within 0.001 ft. A typical wave gage
is shown in Figure 4.18.
The actual process of taking data was based on the procedures de
veloped and the software written for the ADACS described in Section 4.3.
The operational steps are as follows.
Each wave gage is calibrated prior to the generation of waves. The
calibration process entails the monitoring of the output voltages from
the linearposition potentiometer located on each gage. This is accom
plished by system software/hardware interfacings which move each paral
lel rod sensor into and out of the water a known distance. Each sensor
is systematically moved to 11 quasiequally spaced (within the practical
limits of the geartrain driven electric motor) locations over a user
specified range. During this movement, 21 voltage samples are taken
from which an average value for each of the 11 locations are computed.
A schematic diagram of the calibration process is shown in Figure 4.19.
The averaged 11 values for each gage are fitted to a least squares
linear fit to determine the calibration curve. If the maximum deviation
from this linear fit exceeds a userspecified tolerance, a quadratic fit
is performed. A cubic spline can be applied if the quadratic fit is
outside tolerances. The final resulting calibration curve relating
voltages to water surface displacements for each gage is then stored in
disk memory for later use by system software.
The control signal for a desired wave combination is used to
generate an experimental wave field. The location of the wave front in
the basin is determined by the operator by simply monitoring the output
of any two of the nine gages with the dual channel oscilloscope. When
it has been determined that the wave field is fully developed at the
Figure 4.18 Parallelrod wave sensor
COMPUTER
LINEAR POSITION
POTENTIOMETER
CALIBRATION AND
ROTA ION
TRANSLATION
TRANSFER ELECTRIC MOTOR
2 PARALLEL RODS
+E  AH +E
J 7+A
SWL +A  T I
c 
E 
c
E
I I I I
I 6 II 16 21
VOLTAGE SAMPLES
Figure 4.19 Waverod calibration (Turner and Durham, 1984)
array of wave gages, the operator initiates the sampling of data.
Sampling extends for a userspecified period oftime. The data, along
with the corresponding calibration curves, are stored on disk. The time
series for each gage is automatically plotted on a Versatec printer/
plotter and written into disk storage for subsequent analysis. The
length of data sampling used for this study was 30.0 seconds. With a
sampling rate of 50 samples per second, 13500 data points were collected
and stored for all nine gages for each experimental wave.
The data collected for this project are presented graphically in
Appendix B. The results of the verification of the KP equation to the
12 generated wave fields are presented in Chapter 5.
CHAPTER 5
A COMPARISON OF GENUS 2 THEORY WITH EXPERIMENTAL WAVES
This chapter relates the exact genus 2 solutions of the KP equation
described in Chapter 3 to the wave characteristics measured in the
twelve laboratory experiments described in Chapter 4. The development
of this relationship requires the detailed assessment of the free param
eters in the solution. In particular, insight into the sensitivity of
the solution to each of these free parameters must be established since
the spatial and temporal features of the solution are linked non
linearily to these parameters. Once a basic understanding of the coupl
ing between parameters is established, a methodology is developed for
selecting and optimizing the solution such that a "bestfit" to measured
data is achieved. The quantitative assessment of the comparisons be
tween bestfit genus 2 waves and measured data for each of the twelve
experiments of Table 4.1 will demonstrate the capability of the KP equa
tion to model a wide range of laboratorygenerated wave phenomena.
5.1 The Free Parameters of a Genus 2 Solution
The calculation of a general genus 2 solution of the KP equation
requires the specification of six dynamical parameters and two nondynam
ical parameters. (These parameters were noted in Section 3.4.) The ex
perimental program described in Sections 4.3 and 4.5.2 employs symmetr
ical waves in order to evolve a period parallelogram which is symmetric
about both the x and yaxis as was shown in Figures 3.4 and 4.16. A
symmetrical parallelogram was desirable so that a fixed linear wave gage
array could be used to measure all experimental waves. Symmetry intro
duces the following relations:
UI = 12 V >' =I = _2 V' Wl = W2 W 5.1
Hence, the number of free parameters for the symmetric subset of
solutions is reduced to five, with only three of dynamical signifi
cance. These three free parameters are truly independent and can be
arbitrarily selected from the nine dynamical parameters of the general
genus 2 solution. The remaining six dependent parameters are computed
from Dubrovin's theorem of Equation 3.66 and the relationships shown in
Equations 5.1. The free parameters chosen for this investigation are
b, v, and X. These were selected because their specification resulted
in a rapidly convergent algorithm for computing a bestfit with measured
data. The algorithm consists of an interactive program which was speci
fically developed to compare computed and measured wave characteristics.
In order to gain insight into the effects of changing parameter values,
a sensitivity analysis is made to demonstrate the impact of each of the
independent free parameters on the computed waves.
In each of the following analyses, two of the independent variables
are held fixed while the third is allowed to vary. The relative effect
of the single parameter is then measured by changes in the nondimen
sional maximum computed wave elevation fmax, the angular frequency
89
w, and the ydirection wavenumber v. These parameters were selected
because their values yield the measurable quantities of maximum wave
elevation, period, and ydimension length of the period parallelogram.
5.1.1 Sensitivity analysis for the parameter b
As already noted in Section 3.4, the parameter b provides a mea
sure of the nonlinearity of the wave field. There it was shown that for
b+O the waves appeared as two KdV solitons whose interactions were
highly localized in space. For b more negative, the wave heights
decrease and a wave profile measured through the stem region becomes
more sinusoidal. More detailed insight into the effects of b on the
genus 2 waves is provided in Figure 5.1 which shows the effects of vary
ing b on w, fmax, and v when X and u are fixed. It can be seen
from Figure 5.1 that a 3fold increase in b (6. to 2.) produces a
6.0 
fmax
4.0
2.0
v
0.0
0.0 2.0 4.0 6.0 8.0 10.0
b
Figure 5.1 Sensitivity of the parameters w fmax and v to
the parameter b
