Citation
Stable, three-dimensional, biperiodic waves in shallow water

Material Information

Title:
Stable, three-dimensional, biperiodic waves in shallow water
Creator:
Scheffner, Norman Wahl, 1945- ( Dissertant )
Hammack, Joseph L. ( Thesis advisor )
Mehta, Ashish J. ( Reviewer )
Dean, Robert G. ( Reviewer )
Hsu, Chen-Chi ( Reviewer )
Shi, Tom I-P. ( Reviewer )
Bowman, Thomas T. ( Reviewer )
Place of Publication:
Gainesville, Fla.
Publisher:
University of Florida
Publication Date:
Copyright Date:
1987
Language:
English
Physical Description:
xi, 186 leaves : ill. ; 28 cm.

Subjects

Subjects / Keywords:
Cnoidal waves ( jstor )
Mathematical independent variables ( jstor )
Mathematical variables ( jstor )
Parallelograms ( jstor )
Shallow water ( jstor )
Signals ( jstor )
Velocity ( jstor )
Waveforms ( jstor )
Wavelengths ( jstor )
Waves ( jstor )
Dissertations, Academic -- Engineering Sciences -- UF
Engineering Sciences thesis Ph. D
Water waves -- Mathematical models ( lcsh )
Genre:
bibliography ( marcgt )
non-fiction ( marcgt )
theses ( local )

Notes

Abstract:
Waves in shallow water are inherently three-dimensional and non-linear. Experiments are presented herein which demonstrate the existence 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 temporally 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 hexagonal-shaped waves are remarkable robust, retaining their integrity for maximum wave heights up to and including breaking and for widely varying horizontal length scales. The Kadomtsev-Petviashvili (KP) equation is tested as a model for these biperiodic waves. This equation is the direct three-dimensional generalization of the famous Korteweg-deVries (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 hexagonally-shaped waves observed in the experiments. A methodology is developed which related 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 three-dimensional 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, three-dimensional waves propagating in shallow water, reminiscent of the KdV equation in two dimensions.
Thesis:
Thesis (Ph. D.)--University of Florida, 1987.
Bibliography:
Bibliography: leaves 183-185.
General Note:
Typescript.
General Note:
Vita.
Statement of Responsibility:
by Norman Wahl Scheffner.

Record Information

Source Institution:
University of Florida
Holding Location:
University of Florida
Rights Management:
Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
Resource Identifier:
030367811 ( ALEPH )
16904870 ( OCLC )
AEQ8991 ( NOTIS )

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Full Text












STABLE, THREE-DIMENSIONAL, 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, Chen-Chi Hsu,

Tom I-P. 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 In-House 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 KADOMTSEV-PETVIASHVILI (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 three-dimensionality 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 Servo-controller 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 parallel-rod resistance
transducer ........................................... 83

4.18 Parallel-rod 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 half-parallelogram
solution corresponding to experiment CN3007 ............. 104

5.10 Sixteen KP wave profiles for the half-parallelogram
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 Three-dimensional view of the theoretical solution for
experiment CN3007 ..................................... 113

5.15 Normalized contour map of the theoretical solution for
experiment CN2015 ....................................... 113

5.16 Three-dimensional 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 three-dimensional 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, THREE-DIMENSIONAL, 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 three-dimensional 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 hexagonal-shaped waves

are remarkably robust, retaining their integrity for maximum wave

heights up to and including breaking and for widely varying horizontal

length scales.








The Kadomtsev-Petviashvili (KP) equation is tested as a model for

these biperiodic waves. This equation is the direct three-dimensional

generalization of the famous Korteweg-deVries (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

hexagonally-shaped 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 three-dimensional 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, three-dimensional 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 three-dimensionality

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 three-dimensional. 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 three-dimensional 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 three-dimensional 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 three-dimensional, 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

Kadomtsev-Petviashvili equation (KP, 1970) and represent a natural








three-dimensional generalization of the two-dimensional 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 three-dimensional, 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

three-dimensional, 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 three-dimensional 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 three-dimensional 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 two-dimensional 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 three-dimensional 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 permanent-form 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 small-but-finite 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 Korteweg-deVries

(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 well-known analytic functions and can

therefore be used for analyzing the characteristics of naturally occur-

ring two-dimensional 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 two-dimensional, descriptive of one-

dimensional or long-crested 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 permanent-form 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 second-order, permanent-form 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 permanent-form 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' second-order 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 re-emerged in a study of

collision-free 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 finite-dimensional 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 one-dimensional (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 three-dimensional generalization of the KdV

equation, the Kadomtsev-Petviashvili (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 three-dimensionality. Each effect is assumed to be of an equal

order of magnitude. The previous statement that the KP equation is a

direct three-dimensional generalization of the KdV equation can be

seen. The equation reverts to the KdV equation when no crest-wise or
A
variations in the y-direction occur.

Krichever (1976) showed that the KP equation admits an infinitely

dimensional family of exact periodic (or quasi-periodic) 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, two-dimensional, nonlinear waves which propa-

pate at some angle to the S-direction. Genus 2 solutions are the sub-

ject of the investigation herein. These solutions are biperiodic, truly

three-dimensional, nonlinear waves which propagate with permanent form

at a constant velocity. The resulting two-dimensional 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 three-dimensional,

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, three-dimensional 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 KADOMTSEV-PETVIASHVILI (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 (crest-wise) 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 8-x 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 lv|2 + 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 three-dimensional 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

normal-mode 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 three-dimensional 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 i-direction wavenumber and m represents that in the 7-direction.

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 shallow-water conditions so that dispersion is weak. The third

parameter provides a measure of the three-dimensionality 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 three-dimensional. When the

parameter vanishes, the flow becomes the two-dimensional 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 three-dimensional

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 non-dimensional 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 3.18-3.21 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 Kadomtsev-Petviashvili 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


S-T
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 quasi-periodic 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 completed-by 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) real-valued 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 B-1i + imif- 3.52
m1 =- m2=-2




where = (ml, m2) and the products are defined by




2 2
-B- 1= mb11 + 2m1m2b12 2+ 22

and

I-T = 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 real-valued, 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 theta-constants and two additional

phase variables.

The concept of theta-constants 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 half-periods of a theta function
(Dubrovin, 1981). Every Riemann matrix generates a four-component
theta-constant (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 theta-constant 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
well-defined 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 well-defined polynomial of degree 6. The left hand side
of Equation 3.67 is real-valued; 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 real-valued 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

real-valued, 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 V-shapes, 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 V-shaped 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 two-soliton 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 three-dimensional 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 wave-generation methodology follows the

description of the physical facility. Due to the three-dimensional

nature of the wave forms required for this study, considerable detail is

presented for the data-gathering 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, non-reinforced, 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 horse-hair 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 piston-like motion by a 0.75 horsepower,

direct-drive servo-motor 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

flexible-plate 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 mid-point 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 belt-driven 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 servo-controller

module for each servo-motor monitors this position feedback and gener-

ates a stroke-limit and displacement-error 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
(1-611 [XffoWAa p(0-6--

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 Servo-controller 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 control-feedback 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

user-supplied digital control signal. Peripherals to the basic CPU

include 121 megabytes of fixed-disk mass storage, 10 megabytes of








removable-disk mass storage, two 125 inch-per-second 800/1200 BPI mag-

netic tape drives, two line printers, a Versatec printer/plotter, and a

Tektronix 4014 CRT unit equipped with hard-copy 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 three-dimensional, 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

displacement-time 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(c-E)
dt -"d6 dt ade


4.6


d k(c-k)


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 Newton-Raphson 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 i-kct + 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 = M-1-- 4.13




where E(9|m) 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 (0-360).

For each time value, the phase variable of Equation 4.12 is defined and







used-in 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% variation-from 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.


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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 three-dimensional nonlinear waves, an experi-

mental program was devised to generate a variety of wave patterns which

span a wide range of nonlinearity and three-dimensionality.

A broad range of nonlinearity of the basic-wave 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 three-dimensionality 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 three-dimensional

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 three-dimensional wave phenomena with

two-dimensional instrumentation is well recognized. Furthermore, the

presentation of two-dimensional data in a concise yet definitive form

for effectively demonstrating three-dimensional 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 two-dimensional 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 watt-second 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 three-dimensional 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 permanent-form 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 gage-zero point correspon-

dence will be presented in Chapter 5.

The water level gages used for this study are water-surface-

piercing, parallel-rod, 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 parallel-rod 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 linear-position 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 quasi-equally spaced (within the practical

limits of the gear-train 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 user-specified 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 Parallel-rod 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 user-specified period of-time. 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 "best-fit" to measured

data is achieved. The quantitative assessment of the comparisons be-

tween best-fit 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 laboratory-generated 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 y-axis 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 best-fit 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 y-direction wavenumber v. These parameters were selected

because their values yield the measurable quantities of maximum wave

elevation, period, and y-dimension 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 3-fold 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




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