Instabilities of gravity-capillary water waves

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Instabilities of gravity-capillary water waves
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xi, 129 leaves : ill. ; 28 cm.
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Perlin, Marc, 1951-
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Gravity waves -- Mathematical models   ( lcsh )
Water waves -- Mathematical models   ( lcsh )
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theses   ( marcgt )
non-fiction   ( marcgt )

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Thesis:
Thesis (Ph. D.)--University of Florida, 1989.
Bibliography:
Includes bibliographical references (leaves 125-128).
Statement of Responsibility:
by Marc Perlin.
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Typescript.
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Vita.

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INSTABILITIES OF GRAVITY-CAPILLARY WATER WAVES


By

MARC PERLIN

















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


1989








ACKNOWLEDGMENTS


I must express my gratitude to Dr. J. L. Hammack for providing a truly state-

of-the-art facility, a tremendous working environment and encouragement when

need be. To the person who introduced me to the study of water waves and coastal/

ocean engineering. Dr. R. G. Dean, I am eternally indebted. I thank my friend,

colleague, and committee member, Dr. J. T. Kirby for his support, both analyti-

cally and in general. Dr. H. Segur provided much needed insight to the dynamics

of water waves, both through Dr. Hammack and directly. And to Dr. U. H.

Kurzweg, who served as a member of my committee and is responsible for helping

me bring my mathematics up-to-par, 1 am thankful. I would also like to thank my

long-time associates and former fellow students. Dr. W. R. Dally and Dr. D. L.

Kriebel, for their friendship.

The above persons are responsible for helping me with the technical part of

the dissertation and education. but the people who really suffered are my family.

My wife, Terry, not only had to listen to my trials and tribulations, but had to do

without me so much of the time. (Of course, some people would probably say that

she was better off, but I dare say she doesn't feel that way.) Believe it or not. she

often brought dinner to me to save me the trouble of going home during my quali-

fier studying days. There is no way to properly express my gratitude to her. My

two children, Ian and Sarah, also suffered, but hardly realized it since they knew

little else. And finally, to my parents. I am grateful. I thank them for giving me, in

addition to all of the usual parental guidance and love. something so vital in life's

survival, a near perfect work ethic.















TABLE OF CONTENTS




ACKNOWLEDGMENTS ............ ... ...... ................. ii

LIST O F FIG U RES ................................ ............... v

ABSTRACT ....................................................... x

CHAPTERS

1. GENERAL INTRODUCTION ................................ 1

2. INTRODUCTION TO GRAVITY-CAPILLARY INSTABILITIES:
PA RT 1 .............. ........................ ........... 3

3. THEORETICAL CONSIDERATIONS FOR PART 1 ................ 8

3.1 Review of the NLS Model ................. ............ 10
3.2 The Stationary, NLS Model ............................... 18

4. EXPERIMENTAL FACILITIES, PROCEDURES AND IMAGE
ANALYSIS FOR PART 1 ................................... 24

4.1 The Imaging and Computer Systems ........................ 28
4.2 Data Analysis Procedures ................................ 29
4.3 Lighting and Image Effects ................................ 33

5. RESULTS AND DISCUSSION OF PART 1 ....................... 39

5.1 Evolution of Quasi-steady Wavefields ....................... 39
5.2 Experimental Results and Comparison with sNLS Theory ..... 47
5.2.1 The 25 Hz W avetrain ............................... 47
5.2.2 The 17 Hz Wavetrain ............................... 55
5.2.3 The 13.6 Hz Wavetrain ............................. 61


iii








5.2.4 The 9.8 Hz Wavetrain, Wilton's Ripples ................ 66
5.2.5 The 8 Hz W avetrain ................................ 72

6. SUMMARY AND CONCLUSIONS OF PART 1 ................... 78

7. INTRODUCTION TO THREE-WAVE RESONANCE: PART 2 ....... 80

8. RESONANT INTERACTION THEORY .......................... 83

9. RIT AND RIPPLES ..................... ...... ....... ............. 87

10. EXPERIMENTAL FACILITIES FOR PART 2 ..................... 94

11. REVIEW OF PREVIOUS EXPERIMENTS ON RIPPLE TRIADS ..... 97

11.1 The 25 Hz Wavetrain: Spatial Data ...................... 97
11.2 The 25 Hz Wavetrain: Temporal Data .................... 100
11.3 Other Wavetrains with Frequencies Exceeding 19.6 Hz ........ 103

12. RESONANT TRIAD SELECTION .......................... 107

13. EVOLUTION OF A RIPPLE WAVETRAIN IN THE PRESENCE
OF RANDOM WAVES ...................................... 115

14. SUMMARY AND CONCLUSIONS OF PART 2 ................... 121

15. GENERAL SUMMARY AND CONCLUSIONS ................. 123

REFERENCES ...........................................125

BIOGRAPHICAL SKETCH .......................................... 129















LIST OF FIGURES


Figure Description Page


3.1 The fluid domain and a definition sketch. ..................... 9

3.2 Parameter space of the nonlinear Schroedinger equation showing
regions of stability (S) and instability (U) of a wavetrain to long-
itudinal modulations. Curve Ci corresponds to zeros of X; curves
C4 and C5 correspond to zeros of v; Curves C2 and C3 correspond
to singularities of v. The location of parameters (*) for each exper-
iment is also shown. (Hammack, private communication. 1989). 15

3.3 The dispersion relation of (3.30) calculated for a 25 Hz wavetrain
with a=0.04 cm. The (p.q) coordinate system is shown along with a
degenerate, side-band quartet. The dashed line is a circle, centered
at (0,0), with a radius equal to the parabolic vertex coordinate..... 21

4.1 The experimental facility and coordinate system. An elevation view
is shown in (a) and a plan view in (b). ....................... 25

4.2 A typical quiescent background image (taken prior to each exper-
iment) with a frame superposed to depict the region used by the
two-dimensional transform. ................. ................ 30

4.3 An overhead image of an 8 Hz experiment (GC0806). ........... 30

4.4 Demonstration of the "criss-cross" effect caused by the numerical
two-dimensional FFT. These contour maps of vwavenumber spectra
are for sine waves with: (a) k=6.325 and -=420. (b) k=6.325 and
E=450, (c) k=6.325 and ==480. and (d) k=6.200 and E=450. ...... 34

4.5 Average intensity level, K. versus inclination angle of light from
vertical. A. for a quiescent water surface (shown by asterisks).
Diamonds represent the plane projection light intensity of the light
source versus angle. ....................................... 36









Figure Description Page

5.1 Two seconds of a time series from experiment GC2504 .......... 40

5.2 Demonstration of the quasi-steadiness of the 25 Hz wavefield of
experiment GC2504. These contour maps of wavenumber spectra
represent the wavefield (a) 1/25 sec, (b) 1/5 sec. (c) 1/2 sec. and (d)
1 sec after the contour map shown in figure 5.9. ................ 41

5.3 A time series of a 24 Hz wavetrain from wavemaker startup ....... 43

5.4 A time-sequence of contour maps of wavenumber spectra for a 24 Hz
wavetrain approaching quasi-steadiness: (a) 3 sec, (b) 9 sec, (c) 21
sec, (d) 33 sec. (e) 45 sec, (f) 46 sec. (g) 47 sec, and (h) 57 sec
(theoretical (- -) prediction of the sNLS eq.) into the experiment 44

5.5 Overhead images of experiments (a) GC2502. (b) GC2503, and
(c) GC2504. ........................ ................... 49

5.6 The amplitude-frequency spectra for experiments (a) GC2502,
(b) GC2503, and (c) GC2504. ................................ 50

5.7 Contour map of the amplitude-wavenumber spectrum (see inset for a
perspective view) for the 25 Hz wavetrain of experiment GC2502.
Dispersion relation (- -) of the sNLS equation: the region of
spreading (- -) and the most unstable sidebands (X) as
predicted by the linear stability analysis. ..................... 51

5.8 Contour map of the amplitude-wavenumber spectrum (see inset for a
perspective view) for the 25 Hz wavetrain of experiment GC2503.
Dispersion relation (- --) of the sNLS equation: the region of
spreading ( --) and the most unstable sidebands (X) as
predicted by the linear stability analysis. ..................... 52

5.9 Contour map of the amplitude-wavenumber spectrum (see inset for a
perspective view) for the 25 Hz wavetrain of experiment GC2504.
Dispersion relation (- -) of the sNLS equation: the region of
spreading (- -) and the most unstable sidebands (X) as
predicted by the linear stability analysis. ..................... 53









Figure Description Page

5.10 The amplitude-frequency spectra for experiments (a) GC1702,
(b) GC1704. and (c) GC1706. .............................. 56

5.11 Contour map of the amplitude-wavenumber spectrum (see inset for a
perspective view) for the 17 Hz wavetrain of experiment GC1702.
Dispersion relation (- -) of the sNLS equation: the region of
spreading (- -) and the most unstable sidebands (X) as
predicted by the linear stability analysis. ...................... 57

5.12 Contour map of the amplitude-wavenumber spectrum (see inset for a
perspective view) for the 17 Hz wavetrain of experiment GC1704.
Dispersion relation (- -) of the sNLS equation: the region of
spreading ( -) and the most unstable sidebands (X) as
predicted by the linear stability analysis. .......... ......... 58

5.13 Contour map of the amplitude-wavenumber spectrum (see inset for a
perspective view) for the 17 Hz wavetrain of experiment GC1706.
Dispersion relation (- -) of the sNLS equation: the region of
spreading (- -) and the most unstable sidebands (X) as
predicted by the linear stability analysis. ....................... 59

5.14 The amplitude-frequency spectra for experiments (a) GC13601.
(b) GC13605, and (c) GC13606. .............................. 62

5.15 Contour map of the amplitude-wavenumber spectrum (see inset for a
perspective view) for the 13.6 Hz wavetrain of experiment GC13601.
Dispersion relation (- -) of the sNLS equation: the region of
spreading ( ) and the most unstable sidebands (X) as
predicted by the linear stability analysis. ....................... 63

5.16 Contour map of the amplitude-wavenumber spectrum (see inset for a
perspective view) for the 13.6 Hz wavetrain of experiment GC13605.
Dispersion relation (- -) of the sNLS equation: the region of
spreading ( --) and the most unstable sidebands (X) as
predicted by the linear stability analysis. ...................... 64








Figure Description Page

5.17 Contour map of the amplitude-wavenumber spectrum (see inset for a
perspective view) for the 13.6 Hz wavetrain of experiment GC13606.
Dispersion relation (- -) of the sNLS equation: the region of
spreading (- -) and the most unstable sidebands (X) as
predicted by the linear stability analysis. ....................... 65

5.18 The amplitude-frequency spectra for experiments (a) GC9801.
(b) GC9802, (c) GC9804, and (d) GC9810. ..................... 67

5.19 Contour map of the amplitude-wavenumber spectrum (see inset for a
perspective view) for the 9.8 Hz wavetrain of experiment GC9801. 68

5.20 Contour map of the amplitude-wavenumber spectrum (see inset for a
perspective view) for the 9.8 Hz wavetrain of experiment GC9802. 69

5.21 Contour map of the amplitude-wavenumber spectrum (see inset for a
perspective view) for the 9.8 Hz wavetrain of experiment GC9804. 70

5.22 Contour map of the amplitude-wavenumber spectrum (see inset for a
perspective view) for the 9.8 Hz wavetrain of experiment GC9810. 71

5.23 The amplitude-frequency spectra for experiments (a) GC0802,
(b) GC0804, and (c) GC0806. ................................ 73

5.24 Contour map of the amplitude-wavenumber spectrum (see inset for a
perspective view) for the 8 Hz wavetrain of experiment GC0802. .. 74

5.25 Contour map of the amplitude-wavenumber spectrum (see inset for a
perspective view) for the 8 Hz wavetrain of experiment GC0804. .. 75

5.26 Contour map of the amplitude-wavenumber spectrum (see inset for a
perspective view) for the 8 Hz wavetrain of experiment GC0806.
Dispersion relation (- -) of the sNLS equation for a 24 Hz
wavetrain. ......................... .............. ...... 76

9.1 Graphical solution of the kinematic resonance conditions (after
Simmons 1969) for collinear triads of a 25 Hz test wave.
(Henderson and Hammack. 1987). ............................ 89









Description


9.2 Theoretical interaction coefficients for ripple triads. (a) Y2 'vs. f2 for
a series of test waves. (b) Interaction coefficients and initial growth
rate for a 25 Hz test wave- Y: 2;- ---3;
100. withe = :r/2, al = 0.02 mm (Henderson and
H am m ack, 1987)...................................... 92

11.1 Overhead view showing spatial evolution of a 25 Hz wavetrain with
increasing ks1 -values of (a) 0.04, (b) 0.23, (c) 0.29. (d) 0.46, (e) 0.67
and (f) 1.08. (Henderson and Hammack. 1987). ................. 99

11.2 Temporal wave profiles and corresponding periodograms for the 25
Hz wavetrains of figures 11.1 (b), (c), (d) and (e). respectively.
(Henderson and Hammack. 1987). ............................ 101

11.3 Periodograms for 8 wavetrains. Generation parameters and results
are presented in Table 11.1. (Henderson and Hammack. 1987) ..... 104

12.1 Amplitude spectra of computer generated command signals for a 25
Hz sine wave with an amplitude of 1.2 v. (a) DEC MicroPDP-11
(b) DEC VAXstation II. .................................... 109

12.2 Amplitude spectral evolution for a 25 Hlz wavetrain seeded with a 57
Hz wavetrain with 1/10 of its amplitude at the wavemaker ......... 113

13.1 Average amplitude spectra at .r = 7 cm for (a) natural background
noise in the wave tank and (b) random waves generated by the
wavemaker. ................ .... ................... ........ 118

13.2 Average amplitude spectra showing the transfer of energy by non-
linear interactions between a 25 Hz test wave and random waves
(white noise) generated with a S/N = 10. ............ .... ..... 119


Figure















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

INSTABILITIES OF GRAVITY-CAPILLARY WATER WAVES

By

MARC PERLIN

August 1989


Chairman: Joseph L. Hammack. Jr.
Major Department: Aerospace Engineering. Mechanics, and
Engineering Science

In Part 1 of this study, gravity-capillary waves, generated mechanically on

clean, deep water in a wide channel, are shown to be spatially unstable due to a

degenerate four-wave interaction. This instability occurs for wavetrains with fre-

quencies greater than 9.8 Hz. and manifests itself as quasi-steady in time but spa-

tially disordered. Experimental measurements are made using a high-speed imag-

ing system which allows the wavenumber spectrum of the water surface to be

calculated. Wavetrains with small to moderate amplitudes and frequencies greater

than 9.8 Hz show directional spreading of energy during propagation. This transfer

of energy is found to be modeled accurately by solutions of a stationary, nonlinear

Schroedinger (NLS) equation involving two (surface) space coordinates. Experi-

mental wavetrains with cyclic frequencies of 25. 17. 13.6. 9.8 and 8 Hz are investi-

gated. The 9.8 Hz wavetrain (Wilton's ripples) is unstable due to second harmonic

resonance; however, it exhibits some directional spreading of energy as well. The 8








Hz wavetrain is stable until a detuned, third harmonic resonance occurs, and then

the third harmonic (24 Hz) becomes spatially unstable. Reasonable agreement be-

tween quantities predicted by a linear stability analysis (i.e., the region of spread-

ing and the most unstable spatial sidebands) of the stationary, NLS equation and

the measured data is shown for all the wavetrains with frequencies greater than 9.8

Hz.

The second part of the study concentrates on experimental waves with frequen-

cies greater than 19.6 Hz where a continuum of resonant three-wave interactions

are theoretically admissible. Rather than the entire continuum being excited, tem-

poral data show that a dominant resonant triad(s) evolves due to the presence of

exceedingly small background waves at distinct frequencies. If. in addition to a

primary wavetrain, a broad-banded frequency spectrum of background waves is

generated. the entire continuum of admissible triads is excited in a manner consis-

tent with resonant interaction theory for a single triad.















CHAPTER 1


GENERAL INTRODUCTION



For over a hundred years, water waves have been studied by observation in

nature, experiments, mathematical analyses, and, more recently. by numerical

techniques. Original interest appears to have been motivated by the study of the

tides. More recently, interest has broadened to include the entire spectrum of

ocean surface waves which includes many length scales. Waves for which surface

tension and gravitation are important driving forces are known as gravity-capillary

(G-C) waves or simply ripples. These waves have typical frequencies in the range

of 5 through 50 Hertz (Hz) and wavelengths of about 5 through 0.5 cm. In the last

few years, remote sensing of the ocean surface to obtain information about larger-

scale gravity waves has actually focused attention on these smaller-scale waves.

They seem to play a very important role in the manner in which the high-frequency

radar waves are.scattered, apparently because of the proximity of their respective

wavelengths. Herein, we will investigate two different instabilities of G-C water

waves.

Previous research has demonstrated that plane. G-C waves generated in a labo-

ratory are fraught with instabilities which occur rapidly. even for wavetrains of

sensible (low to moderate wave steepnesses) amplitudes. These initially one-

dimensional surface patterns quickly evolve into two-dimensional surfaces. These

instabilities are investigated both experimentally and analytically by two distinct

approaches; a degenerate, four-wave interaction and a resonant three-wave interac-








tion. In the first analysis, Part 1, it is shown that the stationary, nonlinear

Schroedinger 'equation governs the spatial disordering of the wavetrain. Thus, a

mechanism for the directional spreading of energy is established. This analysis

encompasses Chapters 2 through 6.

In Part 2, Chapters 7 through 14, the resonant three-wave (triad) equations are

presented and reviewed. Experimental evidence of the selection mechanism for

triads is presented. It is shown that one or more resonant triads can be excited by a

high frequency seed wave of exceedingly small amplitude. The evolution of a rip-

ple wavetrain in the presence of random waves is also addressed. It is shown that if

no seed waves are present, the entire continuum of admissible triads is excited in a

manner consistent with resonant interaction theory for a single triad.

Since ripple dynamics are similar in many respects to other nonlinear dynami-

cal systems, this work should be of interest to persons in these related areas. In

fact, several of the governing equations derived in the context of G-C waves, such

as the three-wave resonance equations and the nonlinear Schroedinger equation,

are nearly identical to those in other areas of waves in dispersive media; the differ-

ences arise in the coefficients, only. These areas include optics, rigid-body me-

chanics, hydrodynamic stability, and plasma physics.

The second part of this study is essentially a reprint of a research paper by

Hammack, Perlin, and Henderson (to appear, 1989), which was presented by Pro-

fessor Hammack during a series of lectures at the International School of Physics

"Enrico Fermi," Course CIX. This research was conducted prior to installation of

the imaging system that is used extensively in the study of Part 1.















CHAPTER 2


INTRODUCTION TO GRAVITY-CAPILLARY INSTABILITIES: PART 1


Waves on the oceans' surface have stimulated the curiosity of observers for

many years. Contemporary scientific interests in ocean waves appears to have be-

gun with the study and prediction of tides. Airy (1845) was the first to analyze

tides using the equations of motion derived from Newtonian mechanics. Since the

seminal work of Airy, water waves have been studied intensely using observations,

experiments, mathematical analyses, and. more recently. numerical techniques.

Interests have broadened to include the entire spectrum of the oceans' surface

waves, especially those generated by wind. When wind blows over a water surface,

waves are generated whose primary restoring forces are gravitation for the longer

waves and surface tension for the shorter waves. When both restoring forces are

important, the waves are termed gravity-capillary (G-C) waves, or simply ripples.

These are the waves of interest in this study.

Ripples are omnipresent on the ocean surface. In addition to wind generation,

they result from breaking gravity waves, rain, and other disturbances which have

length-scales of about 1 cm. Kelvin (1871) first examined infinitesimal G-C waves

theoretically, and obtained their "dispersion" relation for deep water (i.e. water

depth much greater than 1 cm). This relation between the radian frequency, w and

the wavenumber magnitude. k for a wavetrain of infinitesimal amplitude was given

by Kelvin as










to = gk + Tk- (2.1)

where g represents the acceleration of gravity and T is the kinematic surface ten-

sion. According to (2.1), the cyclic wave frequency, f = w/2x at which gravity

and surface tension effects are equal on a clean surface. T = 73 cm3/sec2 (T=73

dyn/cm per mass density.p), is 13.6 Hz. The corresponding wavelength, 21r/k is

about 1.7 cm. Rayleigh (1890) investigated ripples on clean and greasy surfaces

both experimentally and analytically, and used (2.1) to determine the surface ten-

sion, T from measured values of o and k. Faraday (1883) studied ripples, or

"crispations" as he termed them, by vibrating a glass plate vertically which was

covered with a layer of water. This vibration generated standing waves on the

water surface whose crests were orthogonal: the excited waves had a period which

was twice that of the vibration. In these studies, an implicit assumption was made

that the wavetrains are stable.


Unfortunately. small-amplitude G-C wavetrains are unstable, i.e. they do not

propagate without significant distortions. The first example of a ripple instability

was found by Harrison (1909) and explained by Wilton (1915). Wilton showed that

a wavetrain of 9.8 Hz (on clean, deep water) is unstable to its second, or 19.6 Hz.

harmonic. In fact, there exists a countable set of these "internal resonances" for

G-C waves, where each wave is unstable to one of its higher harmonics. One ex-

ample of importance herein is an 8.38 Hz wavetrain which is unstable to its third

harmonic (25.14 Hz).


In addition to internal resonances, small-amplitude trains of water waves are

subject to a variety of instabilities that arise from interactions with small, back-

ground perturbations. Perhaps. the best-known instability of this type is that which

results from longitudinal perturbations in the amplitudes of plane gravity waves.






5


The existence of this instability was first recognized by Benjamin and Feir (1967).

Whitham (1967), and Lighthill (1965), and is known as the Benjamin-Feir (B-F)

instability. A more detailed analysis of the Benjamin-Feir instability was presented

by Benney and Newell (1967), Zakharov (1968) and Hasimoto and Ono (1972),

who derived the nonlinear Schroedinger (NLS) equation for the evolution of an

unstable one-dimensional (plane) wavetrain. This equation plays an especially im-

portant role in this study, and has the form (e.g. Ablowitz and Segur, 1979)



iA, + A.,, + | A i A = 0. (2.2)


In (2.2), A represents a suitably normalized complex-amplitude "envelope" of the

wavetrain, t is the time, and +x is the direction of propagation. The NLS equation

can be solved exactly, for arbitrary initial data. using the methods of "Inverse

Scattering Theory" as long as the envelope decays to zero at infinity. This solution

shows that the Benjamin-Feir instability leads to very ordered "soliton" behavior

and recurrence.

Typically, plane G-C wavetrains theoretically manifest the Benjamin-Feir in-

stabilitiy. Also, G-C waves are subject to other instabilities when the perturbations

occur obliquely to the direction of propagation. These instabilities result from

three-wave, four-wave, etc., resonances which are described by the fundamental

ideas of resonant interactions formulated by Phillips (1960). Resonant triads are

the first to occur on clean, deep water when the frequency of a G-C wavetrain is

greater than 19.6 Hz. McGoldrick (1965) showed that standing G-C waves may

display three-wave resonances (resonant triads). Simmons (1969) presented a more

general derivation for propagating waves which included spatial variations. (In

fact, Wilton's ripples are a degenerate three-wave resonance.) Experiments on

three-wave resonance have been presented by McGoldrick (1970). Bannerjee and









Korpel (1982), and Henderson and Hammack (1987). Ma (1982) investigated the

bifurcation from two-dimensional G-C waves into three-dimensional waves. Chen

and Saffman (1985) investigated the three-dimensional stability and bifurcation of

capillary and gravity waves. Zhang and Melville (1986 and 1987) have numerically

studied three-dimensional instabilities of two-dimensional, G-C waves. Aside from

the interactions due to the NLS equation, waves with frequencies of less than 19.6

Hz exhibit resonant four-wave interactions; they do not admit resonant triads.

While resonant quartets have been studied in considerable detail for gravity waves,

there has been very little attention given to four-wave interactions for G-C waves.

(In fact, the Benjamin-Feir instability of gravity waves can be viewed as a degener-

ate four-wave resonance.) Yet. there is clear experimental evidence that these

waves are indeed unstable (Henderson and Lee (1986) and Henderson, 1986).

When a wavetrain with a frequency less than 19.6 Hz is generated mechanically in

a wide channel, a spatially disordered wavefield occurs. Modulations are present in

both surface directions; however, frequency spectra show that the energy present is

concentrated at the input wave and superharmonic frequencies only.

The objective of the first part of this study is to examine this instability, both

experimentally and analytically. Experimental measurements of G-C wavetrains

are made using a high-speed imaging system. These spatial data are used to calcu-

late the wavenumber spectrum. Wavetrains are generated with small to moderate

amplitudes; all wavetrains with frequencies greater than 9.8 Hz show that the input

energy spreads directionally during wave propagation. This transfer of energy is

modeled accurately by solutions of a NLS equation similar to (2.2). in which the

independent variables (t,.r) are replaced by the surface coordinates (.,y). This in-

stability occurs for all wavetrains on clean, deep water with frequencies greater

than 9.8 Hz.









An outline of Part 1 of this dissertation is as follows. Chapter 3 presents theo-

retical considerations including an outline of the derivation of the two space dimen-

sions plus time (2+1) NLS equation, and the most unstable sidebands, spatial

growth rates, and the limiting angles of the directional spreading which can occur

according to a linear stability analysis. Chapter 4 describes the experimental facili-

ties and procedures, with an emphasis on imaging techniques and the interpreta-

tion of wavenumber spectra. Chapter 5 presents the experimental results for

wavetrains with frequencies of 25, 17, 13.6, 9.8 and 8 Hz and a comparison with

the prediction of the stationary, nonlinear Schroedinger equation. Finally, Chapter

6 summarizes the results and conclusions of this study.















CHAPTER 3

THEORETICAL CONSIDERATIONS FOR PART 1


Consider the fluid domain shown in figure 3.1 which consists of a water layer

with uniform, quiescent depth h, that is bounded below by a rigid impermeable

boundary and bounded above by a free surface with a tension, T. A Cartesian

coordinate system is located such that the x and v axes lie in the quiescent water

surface and the z axis is vertically upward, parallel to a downward gravitational

force per unit mass, g. Vertical deformations of the water surface from the quies-

cent position are denoted by (.r.y,t). It is well known (e.g. Lamb, 1932) that the

inviscid. irrotational, incompressible motions of this fluid layer can be formulated

in terms of a velocity potential. <(x,y.z,t) as


V 2 = 0,


-h < z < '(,v.V,t)


with boundary conditions



= 0. = -/h



+ 0., x + 0v = : .


(3.1)


(3.2a)


(3.2b)


1 = T|x.(l + C) + ,v-(1 + C) 2. v C.
g + 0, + IV2(i = T( 2+ .2
2 (1 + x + 12)3/2


, z = (3.2c)


: = ,






















































Figure 3.1


The fluid domain and a definition sketch.








Equations (3.1) and (3.2) are analytically intractable. Therefore, approximations

are used. Herein, the approximation of interest is an evolution equation for the

envelope of a modulated wavetrain on water of finite depth, the nonlinear

Schroedinger equation.

3.1 Review of the NLS Model

The characteristic length scales of the water wave problem of (3.1) and (3.2)

are the reciprocal wavevector magnitude, k~', the wave amplitude, a, and the water

depth, h. The important dimensionless parameters determined from these scales

are the wave steepness, ka, and the relative water depth, kh. Herein, we represent

the two-dimensional wave vector by k = (1,m) and use ko to represent the wavenum-
ber of the plane wave input during experiments.

In order to derive the general NLS equation (following Ablowitz and Segur,

1979), it is necessary to make the following assumptions; (1) small amplitude

waves (E = ka < 1), (2) slowly varying modulations (61/k < 1 ), (3) almost

one-dimensional waves (ImI/k < 1), and (4) a balance of these effects. To avoid

shallow water, we require (kh)2 > E. If we now expand the dependent variables.

4 and of (3.1) and (3.2), in terms of a power series in the small parameter, E,

and employ a Taylor series to represent functions in terms of their behavior at the

known reference position, z=0, we obtain an infinite set of linear equations ordered

in powers of E to replace (3.1) and (3.2). The first order, linear, homogeneous

problem has the solution



J cosh k(z + h) l
cosh k ep (k -t) ) cc] (3.3)
L cosh kh


with the dispersion relation









w2 = (gk + Tk3) tanh kh (3.4)


(Herein, "cc" denotes the complex conjugate.)


In order to allow slow modulations of the wavetrain at second and higher or-

ders (i.e. a non-uniform wavetrain which is narrow-banded), the method of multi-

ple scales is used to expand independent variables (x.y,t). In addition, 4 and t are

expanded such that the amplitude and mean motion (which arises at second order

for a uniform wavetrain) are allowed to vary slowly. Slow scales are introduced as

follows,


x E = Ex, yi = Ey, tt = Et, t2 = E2t (3.5)


and the expansions of 4 and t are


C = c,,yi,, t, ) shk(z+ h) |t(. 1y,,,.2) exp(i(k.v -wt)) + cc] )+ O(2)
Scosh kkh



C (g iw A exp (i(kx-wt)) + cc+O(2) (3.6)



where <( represents the mean motion. At 0(e2), the removal of secular terms pro-

duces an evolution equation which requires that the wave envelope travel with its

group velocity (Cs = dw/dk).


A,, + C, Ax, = 0. (3.7)



An additional equation for the mean motion is obtained (see Ablowitz and Segur,

1979).








To obtain the NLS equation, the expansion is carried to the next order, O(e3).
When secularity is removed, and the results are cast into dimensionless form using
the following scaling

E = Ek(x-Ct). 'i = eky, r = E2(gk)/2t
(3.8)
A = k2(gk)-'2A, 4 = k'(gk)-'/2 ,

the evolution equations are


i Ar + 2 Ag + pi A, = IAX A + xi A (t (3.9a)

a i +


where a = tanh kh, T = k2 T/g (3.10a)

w2 = gk( 1+ t ), wo = gk. A = k2- / 2wo (3.1Ob)


S= k2( / 2w = -kCg (3.1Oc)
()7m2j 2o w


o(o (1- r?) (9 ) + P7(3 ) (7-r)
4 (wo{(-2)( T3- o) +

(00J8 o2 2 (1 -2)2 (1 +T) (3.10Od)
S4wJ T

Z1 = 1 + (1 2w

a = (gh C) /gh (3.1Of)

f 1 ) + (3.10g)
o kJ w k 1 + T


(3.10h)


1 = X X1f /a








Equations (3.9a) and (3.9b) are the NLS equation in two space dimensions and

time, i.e. (2+1), coupled with a long-wave equation for the mean motion. The NLS

equation in two spatial dimensions for gravity waves was derived by Zakharov

(1968), Benney and Roskes (1969), and Davey and Stewartson (1974). Djordjevic
and Redekopp (1977) and Ablowitz and Segur (1979) derived the equations in the

form (3.9), which includes surface tension. The coefficients of (3.9) given in (3.10)

are calculated using the wavevector k = (lo, mo = 0) of the underlying wavetrain.

Misprints in Djordjevic and Redekopp (1977) and Ablowitz and Segur (1979) have

been corrected in (3.9) and (3.10).

Equation (3.9) does not appear to be analytically tractable (Ablowitz and
Segur, 1979); however, a variety of approximations are. For example, neglecting

transverse variations, i.e. a/aq = 0, (3.9b) can be integrated, and thus (3.9) yields




i Ar + AA = rA I A (3.11)

I = -[3/a I A |2. (3.12)



Equation (3.11) is the NLS equation in one space and one time dimension (1+1),

and it describes the evolution of a wavetrain whose envelope is modulated in the

longitudinal direction. This equation was derived by Benney and Newell (1967) and
Hasimoto and Ono (1972). Zakharov and Shabat (1972). found a scattering prob-
lem that allowed (3.11) to be solved exactly using Inverse Scattering Theory. They

showed that initial data, which vanishes sufficiently fast as 0. -s ., evolves into a

finite number of envelope solitons and a continuous spectrum of oscillatory waves.

The solution of (3.11) depends critically on the signs of the coefficients of the

equations. The critical dividing lines are established by the sign of the quantity,








(Xv). When Xv<0, the region is unstable (this represents all three unstable regions

and is known'as the Benjamin-Feir (B-F) instability); conversely, when Xv>0, the

region is stable. A map of parameter space showing where sign changes occur and

regions of stability (S) and instability (U) is presented in figure 3.2. A similar

figure was used by Djordjevic and Redekopp (1977), Ablowitz and Segur (1979),

and Henderson and Lee (1986). For large kh, X changes sign at approximately 6.4

Hz (or 7= 0.1547) or across curve Ci. Along curve C2, v has a singularity which

corresponds to second harmonic resonance known as Wilton Ripples. The curve,

C3, across which a and v change sign and v is singular, signifies that a long wave-
short wave resonance has occurred. (This case has been investigated by Djordjevic

and Redekopp, 1977). The parameter locations for the experiments presented

herein, are also shown in figure 3.2; they occur in three separate regions. The 8 Hz

experiments are in a stable region, while the 9.8 Hz wavetrains. located on curve,

C2, correspond to Wilton ripples. The remaining experimental wavetrains are in a
B-F unstable region.

We present an outline of the stability analysis of the NLS equation which fol-

lows that of Hasimoto and Ono (1972). Transforming (3.11) to a dimensional

equation using = E2(gk)/b' c = Ek, and A = k0(gk)-'/ A. yields


i Ai + AA-, = t IA 2 A (3.13)


where = v k4(gk)-/2 and A = 2 k-(gk)'/. First. we seek a nonlinear plane

wave solution of the NLS equation which has an oscillatory envelope of the form

A(r, A,) = A e ,L, t (3.14)


where A0 is a constant (possibly complex) amplitude, 02 is the modulation fre-


































2-




1.1105


1 -




0.1547


0-
(








Figure 3.2


* 25 HZ

24 HZ


S17 HZ



13.6 HZ
S


9.8 HZ


1.3628


Parameter space of the nonlinear Schroedinger equation showing
regions of stability (S) and instability (U) of a wavetrain to long-
itudinal modulations. Curve C1 corresponds to zeros of A; curves
C4 and C5 correspond to zeros of v: Curves C2 and C3 correspond
to singularities of v. The location of parameters (e) for each exper-
iment is also shown. (Hammack, private communication, 1989).








quency, and K is the modulation wavenumber. Substituting (3.14) into (3.13) yields
the dispersion relation for the wave envelope

Q = -2A K IAo|. (3.15)


If we set K=0 and Ao = i(g + Tk')ao/2w, where ao represents the physical wave

amplitude, then (3.15) gives the dispersion relation for a uniform wavetrain with a

nonlinear correction to its linear frequency. For gravity waves (T=0), this wavetrain

represents the well-known Stokes solution with the third order frequency correction

(,i = w +(1/2) k2a ).

To determine whether a wavetrain is stable, we introduce a perturbation of the
form


A(i, A) = (Ao + e B) e ,ui + 'hi (3.16)


and determine its time evolution. In (3.16), B = B(ri.) is a pure imaginary pertur-

bation of the complex amplitude, b = b(rT, ) is a real perturbation of wave phase,

and e < 1 is a small ordering parameter. Substitution of (3.16) into (3.13), and

retaining terms through first order, yields two equations. The zeroth order equation

is satisfied identically. The first order equation, for which the real and imaginary

parts must be satisfied separately, gives


-i Bi + 2 (Ao( b. = 0 (3.17a)

iA B i 2 V IAJ2 B + IAio br = 0 (3.17b)



Equations (3.17a) and (3.17b) are a coupled set of linear differential equations

with constant coefficients; hence, we assume solutions of the form










B(i, ) = Bo e' (cX wil cc (3.18a)

b(i, ) = bo e' (W wi) + cc (3.18b)



where B0 and bo are taken to be real constants and K and W are the perturbation
wavenumber and frequency. When W is pure imaginary and positive, the perturba-
tion will grow at an exponential rate, IWI. Substitution of (3.18) into (3.17) yields


-W Bo JA AoI K-2 bo = 0 (3.19a)

( 2 v IAo12 + A ) B0 + W IA()bo = 0. (3.19b)



The determinant of (3.19) must vanish for nontrivial solutions. This condition
leads to a dispersion relation for the perturbation wavenumber and frequency,
namely


S= A: A2 ( + ( 2Ai, IA)/I. ) (3.20)


When AV is positive, W is real and the perturbations do not grow. However, when

Av is negative, W can be imaginary and positive so that the pertubation grows, i.e.
the wavetrain is unstable. Instability occurs when


A < (-2i,/A)'/- A,,I. (3.21)


Also, we define the limiting wavenumber as /max- = (- 1;/2)1/ |IAol Therefore.

as shown by Hasimoto and Ono (1972), gravity waves (T = 0) in the regime

where A; is negative (kh > 1.3628) and (3.21) is valid, are unstable. For the modu-









nation wavenumbers which satisfy (3.21) and for which Af2 is negative, the growth

rate is


IW = (- 22 i, IAo2 K A2 k4 2. (3.22)


Finally, the most unstable sideband, Km, and its growth rate, IWI, occur where the

derivative of (3.22) with respect to kA vanishes. This calculation yields



S= ( -'/ )1/2 A, (3.23)

I W m = -li' A,2 (3.24)


Obviously, the maximum unstable wavenumber. the maximum growth rate, and

the most unstable sidebands are functions of the wavetrain amplitude. (Also, note

that kmnx = v72 km ).


3.2 The Stationary. NLS Model


The motivation for the following model is based on experimental measure-

ments in which the envelopes appear to be steady. Also, the transverse modula-

tional wavenumbers are larger than the longitudinal values. If the latter of these

two observations is assumed, the terms in (3.9), which contain derivatives in (, are

negligible at leading order. These equations decouple, and (3.9a) becomes


i A, + p A,, = X I A 2 A (3.25)


Transforming (3.25) to laboratory coordinates and dimensional form. using the

transformation









x = e-'k-' + -2C,(gk)-1/2r y = -lk-l


t = e-2(gk)-'/2 A = k2(gk)-l/2A (3.26)


we obtain


i (A, + C, A, ) + (ik-2(gk)'/2) A = ( k(gk)- /2) A12 A 2 (3.27)


Experimental results show that our wavefields are quasi-steady (herein, quasi-
steady describes a process which is steady (has no temporal dependence) over
many periods of the underlying wave); therefore, the time derivative term in (3.27)
is negligible. These approximations result in a stationary NLS (sNLS) equation
with the form


i A, + /i Ayy = X IA 2 A (3.28)



where ji = u k-2(gk)'1/Cg-1 and X = k (gk)-'2Cg-' Equation (3.28) has the

same form as (3.13); however, the independent variables (it.) have been replaced

by (x,y), and the coefficients, (A, f), have been replaced by (yl, _). Since the equa-
tions are analogous, we may glean our results directly from the stability analysis in
Section 3.1.

If we assume oscillatory modulations in space of the form


A(., y) = A0, e -" -q v)


(3.29)









we obtain the dispersion relation for the wave envelope, analogous to (3.15) as



q2 + p IoI (3.30)


Figure 3.3 shows this dispersion relation, which is a parabola in the wavenumber

plane, for the 25 Hz experiment presented in figure 5.1. The shape of the parabola

is nearly circular in the vicinity of the input wavevector as shown by the dashed,

circular arc in figure 3.3. This approximation is reasonably accurate to angles of

30*. Since the magnitudes of the wavenumbers along this portion of the curve

are nearly constant, the frequencies calculated from (3.4) are nearly equal. There-

fore, although many waves propagating in different directions may be present,

frequency spectra will only show energy at one frequency.


Any coordinate along the parabola in figure 3.3 represents a wavevector,

which forms a degenerate quartet (four-wave resonance) with the input wavevec-

tor; one quartet is shown in figure 3.3. (Note that the input wavevector does not lie

on the parabola.) If the transverse wavevector, q, is zero, a degenerate, collinear

quartet is present. In the derivation of the NLS equation one assumes weak, two

dimensionality, and so equation (3.30) should only be valid for small values of p

and q.


For the sNLS equation, (3.28), we present analogous results to the linear sta-

bility analysis of Hasimoto and Ono (1972) presented in Section 3.1. Rather than

determining the most unstable wavenumber. the maximum wavenumber of the

instability, and the temporal growth rate, the analysis now predicts the most unsta-

ble wavenumber in the y direction, the maximum wavenumber of the instability in

the y direction, and the spatial growth rate in the x direction. We will use perturba-

tion wavenumbers, P and Q, for the x and y directions, respectively. According to
























(cm-)


0. 4. 8.


S(cm-')


Figure 3.3


The dispersion relation of (3.30) calculated for a 25 Hz wavetrain
with a=0.04 cm. The (p,q) coordinate system is shown along with a
degenerate, side-band quartet. The dashed line is a circle, centered
at (0,0), with a radius equal to the parabolic vertex coordinate.








the linear stability analysis, P must be imaginary and positive for perturbations to

grow in space, and therefore, it restricts the admissible wavenumbers in the y

direction. We set q=0 so the wavetrain represents a nonlinear plane, G-C wavetrain

in space (with Ao = [- i(g + Tk')/(2w)J ao for a real, physical wave amplitude, a0,

and a slightly different wavenumber than for the linear wavetrain), and then seek

instabilities which grow in space. The following inequality (and the maximum

sideband value) must be valid in order that the perturbation grow and an instability

result


Q < (- 2 X/t)'1/2 IA,) max (3.31)

The most unstable sideband and maximum spatial growth rate are


m = (- X/li)/2 |Ao and IPI,, = IA, (3.32)


Herein, Q,,,ax will be shown in figures to indicate the allowable spreading of

wave energy according to this linear stability analysis. The angle associated with

this spreading is termed 0 As noted by Yuen and Lake (1982), the fact that the

instability of the waves can only occur in a fairly narrow band (about the input k),

preserves the assumption of a narrow-banded spectra used to derive the evolution

equations.

For gravity waves, the analogous equation to (3.28) has a different sign for the

nonlinear term (ie. X changes sign and has a positive value). This form of the NLS

equation cannot be unstable. Thus, the instability we present is uniquely applicable

to G-C waves.

While three-dimensional instabilities of G-C waves have been studied previ-

ously, e.g. by Zhang and Melville (1986 and 1987) and Chen and Saffman (1985),








the stationary, spatial instability described above appears to have been overlooked.

This is a consequence of examining instabilities by positing perturbations whose

temporal behavior is used to define stable and unstable regimes.

In our experiments, amplitudes vary significantly in space due to viscous
damping (and also due to the instability we are investigating); however, we will use
only one measured amplitude, au as a representative value. Since the unstable

sidebands depend on ao and growth rates depend on u02, some discrepancies be-

tween theoretical and experimental results are anticipated. To predict an amplitude

at a specific location using a value from another location requires a viscous model.

Henderson and Lee (1986) have shown that an immobile-surface model is appro-
priate when the facilities and procedures used herein are employed. According to
this model, the amplitude envelope decays as


A(x) = Ao exp (-6x ) (3.33)

( ko(v w/2)'/2cosh2koh
where 6 = F + k Csinh 2k'1 J
C. sinh 2koh

S= ( )/2 ( 2koh + C + (3.34)
2 2 sinh 2koh b

b is the channel width, and v. is the kinematic viscosity of water. Hence, the

e-folding distance for viscous damping is 1/8. In addition to damping, viscosity

causes wavelengths to decrease slightly from those predicted by the inviscid disper-
sion relation.















CHAPTER 4


EXPERIMENTAL FACILITIES, PROCEDURES AND IMAGE ANALYSIS FOR
PART 1


The experimental facility was specifically designed and constructed to study

gravity-capillary (G-C) waves. It consists of six systems: (1) the wave tank, (2) the

wave generator and its electronics, (3) the water supply and filtering system, (4)

the wave gauge and its attendant electronics, (5) the computer system, and (6) the

high-speed imaging system. Systems (1) through (4) have been discussed in detail

by Henderson and Hammack (1987). and they are described briefly herein. The

computer and high-speed imaging systems have been recently integrated into the

laboratory. These two systems provide unique measurements of spatial wave data

which is of special importance to the results reported herein. Hence, they are

described in detail. Data analysis techniques are also discussed with an emphasis on

the spatial data obtained by the imaging system.

The wave tank. shown schematically in figure 4.1, is 91 cm wide, 183 cm long

and 15 cm deep. It consists of tempered glass sidewalls and bottom in an alumi-

num frame with aluminum end walls. A sheet of white plexiglass is located be-

neath the glass bottom. (This sheet is required for imaging purposes.) Temporary

wave guides made of (wetted) aluminum are set adjacent to the wavemaker paddle

to create a wave channel 30.5 cm wide within the 91 cm width of the basin. The

tank is supported by a steel frame which rests on vibration-isolation pads. The

entire inside of the tank is cleaned before and after experiments with ethyl alcohol.























































Figure 4.1


The experimental facility and coordinate system. An elevation view
is shown in (a) and a plan view in (b).
























































Figure 4.1--continued









The wavemaker paddle is supported by an electrodynamic shaker connected to

another steel frame which straddles the wave tank and rests on vibration-isolation

pads as shown in figure 4.1. The paddle is an aluminum right-angled wedge with a

length of 30.4 cm, a height (H) of 0.90 cm. and a width (w) of 0.44 cm. In figure

4.1 an enlarged view of the paddle is shown, including its position with respect to

the water surface. It should be emphasized that the paddle submergence, s, must

be duplicated among experiments in order to obtain repeatable results. Herein, all

experiments have a depth of paddle submergence of 0.24 cm. An eddy-current

displacement transducer monitors the vertical motion of the paddle and provides a

position-feedback signal to a servo-controller. The controller compares the pro-

grammed displacement signal with the feedback signal, and corrects wavemaker

motion to strictly follow the desired motion. The maximum displacement of the

wavemaker is 6 mm (which corresponds to a 6-volt command signal): however for

the experiments reported herein, displacements never exceed 3 mm. For the ex-

periments reported in Part 1, all wavemaker input signals are sinusoidal.


The water supply system contains 80 gallons of water stored in two closed

polyethylene containers that are connected to the wave basin by PVC piping and

valves. The Sybron filtration system consists of a 0.2 micron particle filter, a car-

bon-absorption filter, and an organic removal filter. The water is gravity-fed into

the wave tank, and pumped very slowly through the filtration system at the conclu-

sion of an experiment.


A capacitance-type wave gauge is used to record the water surface elevation as

a function of time at a fixed location. (xo.Y0). in the basin. The water-penetrating

probe is a sealed glass tube enclosing a conductor and has diameter of 1.17 mm.

Unlike the probe used by H-H, the dynamic response of this gauge is essentially

constant until about 30 Hz. The wave gauge is calibrated dynamically by oscillating









it vertically with a known frequency and amplitude, and then comparing this ampli-

tude to the voltage of the output signal. (See H-H for details of the calibration

procedure.)

4.1 The Imaging and Computer Systems

The computer system used in these experiments is a DEC VAXstation 11. This

32-bit system contains an analog-output system (AAV11-DA), with up to 4 chan-

nels of output, and an analog input system (ADVI 1-DA) with up to 16 channels of

input. Two real-time clocks (KWVI 1-C) are used in conjunction with the digital-to-

analog output and the analog-to-digital input. Data acquisition and analysis, as well

as command signals to the wave generator, are performed by the computer system

via the command language of Signal Technology's Interactive Laboratory System

(ILS) software. A command procedure of ILS instructions, once initiated, auto-

matically acquires time series data and analyzes it in near real-time. Numerical

and graphical programs from the DEC VAXlab software are also used. A 71 mega-

byte (RD53) system disk and a 456 mega-byte (RA81) storage disk are used for

data acquisition and processing. In addition to the wave gauge data. the command

and feedback signals of the wavemaker are also recorded simultaneously and ana-

lyzed. The computer is interfaced to the imaging system through an RS-232 con-

nection which allows transfer of control commands and data between the two sys-

tems.

The high-speed video imaging system is a Kodak Ektapro 1000, computer-

based device, capable of processing data from two imagers at framing rates up to

1000 Hz. Each frame can be divided into as many as 6 pictures to produce up to

6000 pictures per second. Each image is formed from a 240 (horizontal) by 192

(vertical) pixel-array with a resolution of 256 gray levels (intensity) per pixel. Im-

ages are recorded on one-half inch instrument tape and, during playback, are then








digitized by the processor at a rate of 30 frames per second. Both imagers are used

in the experimental program (though not necessarily concurrently); one imager is

located directly above the water surface and the other is located at the end of the

tank to give an oblique view of the wavefield. A 25 mm TV lens, with an f/stop as

low as 0.95, is used with the overhead imagery. while an 11-66 mm zoom lens is

used with the other imager.

Using Kodak's command language, digitized images can be downloaded

through the RS-232 connector from the Ektapro 1000's computer memory to the

VAXstation II system. Since it is desirable to maintain the array size as a multiple

of "2" in order to be compatible for numerical analysis, a 128 x 128 pixel array is

transferred. Figure 4.2 shows a typical overhead image of the quiescent water

surface when viewed on the system monitor. A rectangular border has been

marked to delineate the area of the image which is transferred and processed in

the analyses. Note that this sampling area is oriented at an angle of about 450 to

the channel axis (also see figure 4.1). The reason for this orientation will be dis-

cussed in the next section. The channel sidewalls are also visible in figure 4.2.

Figure 4.3 shows a typical overhead image from an experiment in which an 8 Hz

wavetrain has been generated. The dominant light and dark bands transverse to the

channel axis represent the crests and troughs, respectively, of the 8 Hlz wavetrain.

(Even the 24 Hz superharmonic of the 8 Hz wavetrain is visible in figure 4.3!)

More details concerning the interpretation of these overhead images will be pre-

sented shortly.

4.2 Data Analysis Procedures

Wave gauge data are digitized and analyzed by the computer in near real-time.

These data are analyzed to produce amplitude-frequency spectra (amplitude

periodograms). The time series of water surface elevation. (o0,j0), are low-pass



























Figure 4.2


A typical quiescent background image (taken prior to each exper-
iment) with a frame superposed to depict the region used by the
two-dimensional transform.


An overhead image of an 8 Hz experiment (GC0806).


Figure 4.3








analog (Butterworth) filtered with a cutoff frequency of 100 Hz, digitized at a rate

of 250 Hz, amplified by 20 decibels, high-pass analog filtered at a cutoff frequency

of 1 Hz, and then amplified by another 20 decibels. (A 20 decibel amplification

corresponds to a factor of 10.) The duration of the time series is 65.532 seconds,

or equivalently, 16,384 samples. The temporal resolution is 0.004 sec, and the

resolution in the amplitude-frequency spectra is 0.0153 Hz. Thus, the Nyquist

(folding) frequency is 125 Hz. Therefore, analog filtering at 100 Hz removes any

corruption due to aliasing of the amplitude-frequency spectra. The command signal

to the wavemaker is provided by analog output from a 2500 Hz digital signal. The

command and feedback signals of the wavemaker system are also digitized and

analyzed to provide a check on the experimental input. Note that all frequency

spectra herein are presented in terms of decibel wave amplitudes, instead of cm-

wave amplitudes. The decibel amplitude is 20 logi0 (amplitude signal). The fre-

quency spectra are computed by using fast Fourier transforms (FFT) of the time

series.

The overhead images, such as the one shown in figure 4.3. are analyzed using

a two-dimensional FFT to obtain the amplitude-wavenumber spectrum of the

wavefield. An array of 27 x 27 (16,384) samples is used by the transform. Prior to

the analysis of the wavefield image, an image of the quiescent water surface, ob-

tained prior to each experiment (e.g. see figure 4.2), is subtracted from the

wavefield data. This removes any variation in background lighting and/or in-

homogeneities in the response of the imager sensors. Second. a mean-value of the

gray-levels for the wavefield image is subtracted from each pixel value. Of a possi-

ble 256 gray-levels available per pixel, experiments, herein, used as many as 225

gray-levels. The resolution of the images is 0.174 cm which gives a wavenumber

resolution of 0.284 rad/cm. The folding (Nyquist) wavenumber is 18.055 rad/cm,









or equivalently, a free wave with a frequency of 106.5 Hz. Although no method for

analog filtering of the image is available, it is believed that no significant wave

activity is present at, or beyond, this frequency in our experiments. The image

resolution (0.174 cm for the experiments, herein) was determined by viewing a

linear scale located in the plane of the quiescent water surface. Then, the imager's

reticle was used to determine the number of pixels along a known distance on the

scale. This procedure was then repeated in a direction perpendicular to the first

orientation to insure that both axes of the imager gave identical resolutions. The

framing rate of the imager for the experiments presented herein is 125 Hz.


As already shown in figures 4.2 and 4.3, the imaged area of the water surface

was oriented at an angle of about 450 to the axis of the channel. This orientation is

advantageous due to the manner in which the FFT algorithm computes the ampli-

tudes corresponding to a given wavenumber. The algorithm only associates ampli-

tudes with the (+l,+m) quadrant of the wavenumber plane. Hence, it "folds" ampli-

tudes for wavevectors oriented in the -m direction (i.e. waves propagating such

that they have a component wavenumber in the -y direction, across the channel)

into the +m-direction. Similarly, it folds amplitudes in the -/ direction. Hence,

orienting the image at an angle of about 450 allows us to distinguish wave direc-

tions up to 450 from the +x-channel axis. Since the waves in these experiments

are concentrated within an angle of 450 to the channel axis, the folding does not

contaminate the computed spectra in a significant manner.


One problem with numerical calculations, such as the use of FFT's, is the

"smearing" of information which results from representing continuous functions

with discrete functions. In the calculation of the wavenumber spectra, amplitudes

are smeared to adjacent wavevectors. In order to determine how this discretization

affects wavenumber spectra, an analytical sinusoidal wavetrain was digitized and









used as direct input to the two-dimensional FFT. This effect is shown in figure 4.4

where computed spectral contour maps in the wavenumber plane are presented.

The first three maps have been calculated with an input wavenumber of 6.325 (25

Hz) and input angles with respect to the abscissa. E. of 420. 450 and 480. Note that

the smearing for the 420 wavevector is mostly in the abscissa direction while the

smearing for the wavevector oriented at 480 is mostly in the ordinate direction. The

"criss-cross" effect is present in the case of a 450 wavevector, although smaller,

and, in fact, this contour map very much resembles the 17 Hz, small-amplitude

case (see figure 5.7). The fourth map in the figure is also a 450 wavevector, how-

ever its input wavenumber magnitude is now 6.20. By simply changing the

wavenumber so that it falls near the midpoint of adjacent wavenumber "bins", the

"criss-cross" effect has been significantly increased, both in extent and breadth. It

is noted that this effect would be reduced by decreasing the discretization interval

between image gray-levels (i.e. a larger number of known gray-levels for the same

physical area). This "criss-cross" effect will appear in many of the contour maps.

It is important to remember the numerical smearing (caused by the transform)

when interpreting the contour maps of wavenumber space, since the spreading of

wave amplitude caused by the physics of water waves is the main focus of Part 1.


4.3 Lighting and Image Effects


The major factor affecting images of the ripple surface is lighting. The images

are very sensitive to the manner in which the surface is illuminated. Various light-

ing techniques were tried: however, the rear lighting is believed to be superior and,

consequently, is used (note the position of the lights in figure 4.1). Two mechani-

cally coupled, 600 watt halogen lights, in reflector housings. provide the lighting.

The lights are centered on the wave channel. If the lighting is too direct, the imagery













(cml)


4.


Lx


8. 0.


-1
' (cm )


-1
1' (cm )


Figure 4.4


Demonstration of the "criss-cross" effect caused by the numerical
two-dimensional FFT. These contour maps of wavenumber spectra
are for sine waves with: (a) k=6.325 and E=420, (b) k=6.325 and
==450, (c) k=6.325 and ==480, and (d) k=6.200 and ==450.


-1
(cm)









is "overpowered" (i.e. the gray-level values reach 255). Too little light results in

the opposite effect. Also, duplicate experiments, with varied lighting gave slightly

different results. It is believed that the dominant wavenumber properties of the

surface are being measured.


An explanation of the genesis of the image is now presented. Again, reference

is made to figure 4.1; the bottom of the tank is shown to be tempered glass under-

laid with a white, plastic sheet. The plastic sheet is used to help diffuse light. It is

conjectured that light which penetrates the water surface and the glass bottom, is

diffused by the plastic sheet; then, as it passes back through the water surface, it is

refracted (i.e. water wave crests focus light while troughs reduce light intensity).

Thus, an image has bright bands along wave crests and dark bands along troughs

(see figure 4.3). Occasionally, images have extremely small, bright spots present,

believed to be specular reflection. If the white plastic sheet is removed, the video

image is essentially destroyed.


In order to demonstrate that the image is a consequence of diffuse rather than

specular light, a simple, rather crude experiment was conducted. Using the over-

head imager, and a quiescent water surface, the angle of the halogen lights, with

respect to the vei-tical, was measured. An image of the water surface was recorded,

transferred to the computer, and an average gray-level was computed. The angle of

incident light was then changed and the process was repeated. The data from this

experiment are represented by the asterisks in figure 4.5. The angle of the light

source, A, as measured from the vertical is shown versus the average image inten-

sity, K. A significant reduction of light intensity from the maximum value meas-

ured is noted. (If the water surface was specular, the water surface would always

appear black except when the light was originating from the imager direction.)

Next, a light-intensity meter was used to measure the intensity of light versus de-














250.





200.





150.


100.





50.





0.


A (deg)


Figure 4.5


Average intensity level. K. versus inclination angle of light from
vertical, A, for a quiescent water surface (shown by asterisks).
Diamonds represent the plane projection light intensity of the
light source versus angle.


0 0
*








o o
0 0




0 0
S* *
*









aviation angle along the vertical centerline of a plane projection. The maximum

intensity levels measured using the light meter were then normalized by the maxi-

mum average intensity obtained from the video images. These values are repre-

sented by the diamonds in figure 4.5. It is clear that the "surface" is diffuse reflec-

tive because the two curves are similar. The white plastic sheet is apparently re-

sponsible for the surface appearing to be diffuse. The intensity level differences

between the two measurements is due to the difference between projecting light

onto a perpendicular surface and onto an inclined surface and is consistent.


The following tests were performed in order to gain additional insight into the

nature of the data obtained from the imaging system. as well as the interpretation

of their wavenumber spectra. First, simple square-wave, planform patterns were

drawn with black ink on white paper, placed in the wave tank, imaged and trans-

formed. These patterns have known Fourier transforms which could then be com-

pared with the results from the imaging. Reasonable agreement was found. Sec-

ond, various known wavefields in terms of sea-surface elevation were generated

numerically by superposing sinusoidal waves with different (known) wavevectors.

For each wavefield, the maximum sea level was given a gray-level intensity of 255

and the minimum sea level a value of 0; intermediate water levels were assigned

corresponding gray-levels by interpolation. This water surface to gray-level corre-

spondence was used to generate an image on paper, and then the video system was

used to image and transform it. A comparison between the known, input wavevec-

tors and the wavenumber plane as calculated from the transformed image showed

that the system was functioning well.


Finally, as with most remote sensing techniques, a question arises as to what is

actually being measured, in this case, by the imaging and computing systems (i.e.

amplitude, amplitude squared, or some other quantity). Verification that we are, in






38

fact, measuring amplitude is as follows. For low values of k0s, that is, wavenumber

magnitude times paddle stroke, when the energy is nearly all confined to the input

wavenumber vector, if the stroke is doubled, say, the amplitude of the peak con-

centrated at that wavenumber also approximately doubles. This is true as long as

the energy is concentrated at the input wavenumber vector and it verifies qualita-

tively that we are measuring the amplitude (or the wave steepness, ku) and not

another quantity.















CHAPTER 5


RESULTS AND DISCUSSION OF PART 1


The results of the experimental program are presented and discussed in this

chapter. The first section gives experimental evidence as to the quasi-steadiness of

the wavefield, and a time-series of wavenumber contour maps during temporal

evolution preceding quasi-steadiness. In the second section, we present the experi-

mental results and compare them to the sNLS theoretical predictions. A complete

list of these experiments and important parameters are shown in Table 5.1.


5.1 Evolution of Ouasi-steadv Wavefields

In order to show that the wavefield is quasi-steady, figure 5.1 presents 2 sec-

onds (-50 waves) of a typical time series for a moderate amplitude. 25 Hz experi-

ment. (See Table 5.1, experiment GC2504. for the important parameters.) Modula-

tions of the wavetrain are present; however, the frequency spectrum of this experi-

ment's time series (see figure 5.6(c)), clearly shows that identifiable peaks exist

only for the 25-Hz input wavetrain and its super harmonics. This wavetrain had the

largest stroke of the 25 Hz experiments presented and. therefore, it is not expected

to be well-represented by the theory. In fact. figure 5.5(c) shows a photograph of

the image as seen on the video monitor, and one can see 1800 phase shifts present

in the wavefield. In spite of this strongly, nonlinear wavefield, the four contour

maps of wavenumber spectra presented in figure 5.2 are remarkably similar. These

spectra were calculated from images 1/25, 1/5, 1/2 and 1 second, respectively,

after the spectrum of figure 5.9. Very little difference exists between the contour


























(x 0.00075cm)
0

-25

-50 1

48.00 48.50 49.00 49.50 50.00
48.25 48.75 49.25 49.75


time (sec)


Two seconds of a time series from experiment GC2504.


Figure 5.1











8.

m'

(cm )
4.




Al


8.
m'
-1
(cm)
4.




0.


/ Z
/


V; I-


-+ 31'


8. 0.


-1
1' (cm )


-1
l' (cm )


Figure 5.2


Demonstration of the quasi-steadiness of the 25 Hz wavefield of
experiment GC2504. These contour maps of wavenumber spectra
represent the wavefield (a) 1/25 sec, (b) 1/5 sec, (c) 1/2 sec, and (d)
1 sec after the contour map shown in figure 5.9.


O


O









maps. It is reasonable, then to state that the experiment is quasi-steady. We have

thus shown that our experiments are not modulated in a "regular" fashion. How-

ever, modulations, in time, do exist but do not have a unique frequency at which

energy appears in the frequency spectrum. Hence, the modulations appear in a

random manner, and are partially attributed to "noise" in the experimental facility

(e.g. air currents).

We now further define what is meant by quasi-steady. The time series for a 24

Hz wavetrain which precedes a quasi-steady wavefield is shown in figure 5.3. The

length of this time series is 65.532 sec. Digitization of the wave gauge data began

simultaneously with the start of the wavemaker. A time-sequence (3, 9. 21, 33, 45,

46, 47, and 57 sec) of contour maps of wavenumber spectra are presented in figure

5.4. Clearly, the wavenumber spectra are evolving as is evidenced by the large

changes in the contours. Figures 5.4(c) and 5.4(d) show that numerical smearing

of the data is present (i.e. the "criss-cross" effect). The evolution has reached a

quasi-steady state at approximately 45 sec into the experiment. One second (-24

waves) has elapsed between recording of the images used to produce figures 5.4(e)

and 5.4(f), yet the contour maps are very similar. A comparison of figures 5.4(f)

and 5.4(g) shows little change, also. The wavefield imaged to produce figure 5.4(h)

shows that slight evolution was still occurring, but on a longer time-scale.


5.2 Experimental Results and Comparison with sNLS Theory


In order to show that the sNLS equation predicts the directional spreading of

amplitude for G-C waves of frequencies greater than 9.8 Hz. and that its prediction

of stability for frequencies less than 9.8 Hz (but greater than 6.4 Hz) for deep,

clean water is correct, five sets of experiments are presented. Paddle-driven

wavetrains with frequencies of 25, 17, 13.6, 9.8, and 8 Hz are presented for three,


























(x 0.00075cm


0.0 16.38 32.77 49.15 65.54


8.192


24.57


40.96


57.34


time (sec)


A time series of a 24 Hz wavetrain from wavemaker startup.


Figure 5.3












8.

m

(cmi)

4.
-L

m








8.

m'
-1
(cm)
4.





0.


N -1


O~4


8. 0.


-1
1' (cm )


-1
I' (cm )


Figure 5.4


A time-sequence of contour maps of wavenumber spectra for a 24
Hz wavetrain approaching quasi-steadiness: (a) 3 sec, (b) 9 sec, (c)
21 sec, (d) 33 sec, (e) 45 sec, (f) 46 sec, (g) 47 sec, and (h) 57 sec
(theoretical (- -) prediction of the sNLS eq.) into the experiment.


I __ _






45





8. (e) (f

m'

(cm






0.

I -



8. (g) (h)

m'
-I
(cm)

S4.

(\

0. ---- -

0. 4. 8. 0. 4. 8.



-1 -l
1' (cm ) I' (cm )


Figure 5.4--continued















Table 5.1. Summary of experiments



All experiments used the following:
Paddle immersion = 0.24 mm, Surface tension = 73 dyn/cm,
Channel width = 30.5 cm, water depth = 4.92 cm, and y = -4.7 cm.


Exper. Input
freq.
(Hz)


GC2502

GC2503

GC2504

GC1702

GC1704

CC1706

GC13601

CC13605

GC13606

GC9801

GC9802

GC9804

GC9810

GC0802

GC0804

GC0806


25

25

25

17

17

17

13.6

13.6

13.6

9.8

9.8

9.8

9.8

8

8

8


kos x koa


(cm)


0.13

0.19

0.25

0.09

0.18

0.27

0.04

0.18

0.22

0.03

0.05

0.10

0.26

0.04

0.08

0.12


0.08

0.16

0.18

0.11

0.11

0.34

0.04

0.09

0.14

0.03

0.04

0.07

0.26

0.05

0.11

0.16


i -X

sec"
(cm) (s )
cm5


0.079

0.079

0.079

0.110

0.110

0.110

0.135

0.135

0.135


0.119

0.119

0.119

0.105

0.105

0.105

0.125

0.125

0.125


2r.
Qm,
(cm)




31.7

16.5

14.2

22.8

22.0

7.3

56.6

24.6

14.9


+

(")




2.5

4.9

5.6

4.9

5.1

14.9

2.4

5.6

9.2


(W-1

(cm)




333.3

90.9

66.7

125.0

111.1

12.3

500.0

111.1

41.7


( -


(cm)




11.0

11.0

11.0

15.4

15.4

15.4

19.0

19.0

19.0

27.3

27.3

27.3

27.3

36.2

36.2

36.2









small to moderate amplitude experiments. For the interesting case of Wilton's

ripples. 9.8 Hz, we present a fourth, additional experiment. The 25, 17, 9.8, and 8

Hz experiments were conducted with the imager positioned such that the first re-

corded gray-level was 23 cm from the wave generator (see Table 5.1). Recall that

the imager axis was oriented about 450 from the primary wavevector (see figure

4.1). In the 13.6 Hz experiment, the imager was located such that the first imaged

pixel used in the spatial transform was 37 cm downstream. This was done because

the wavetrain extended farther downstream, evolving spatially more slowly. The

wave-gauge frequency spectra were all recorded at the same location as the first

gray-level used from the image (see Table 5.1). For each frequency presented, all

wavenumber spectra are normalized by the largest amplitude obtained by the two-

dimensional FFT of all the cases for that particular frequency. In other words, all

25 Hz experiments, for example, are contoured using the same maximum-ampli-

tude value chosen from among the three. 25 Hz experiments.


5.2.1 The 25 Hz Wavetrain


Resonant three-wave (triad) interactions are studied in Part 2. Therein, it is

shown that in the absence of "seed" waves. no distinct triad interactions occur.

The 25 Hz wave is in the admissible triad regime: however, no seed waves of

sufficient amplitude to initiate triad growth are present in the experiments reported

in Part 1. And yet, the water surface is still spatially disordered as is shown in the

three photographs of figure 5.5. Note that as wavemaker stroke is increased, the

wavefield becomes increasingly more disordered. The three amplitude-frequency

spectra which were computed from experiments GC2502. GC2503 and GC2504

(shown in figure 5.5) are presented in figure 5.6. The corresponding contour maps

of the wavenumber spectra, and an inset perspective view of the wavenumber spec-

tra, are presented in figures 5.7 through 5.9. As mentioned previously, note the



























K.


I


__.LA I


Figure 5.5


Overhead images of experiments
(c) GC2504.


(a) GC2502, (b) GC2503, and


__






















1 3 20 SO 100
2 3 10 30 70






10
120






I 3 7 20 50 100
2 5 10 30 70


5 10 30 70


frequency (Hz)


Figure 5.6


The amplitude-frequency spectra for experiments (a) GC2502,
(b) GC2503, and (c) GC2504.


amplitude

(db)











































I,


8.

1' (cm-1)


Figure 5.7


Contour map of the amplitude-wavenumber spectrum (see inset for
a perspective view) for the 25 Hz wavetrain of experiment GC2502.
Dispersion relation ( -) of the sNLS equation; the region of
spreading (- -) and the most unstable sidebands (X) as
predicted by the linear stability analysis.


12.


(cmf1)

8.






4.


m


ll


'


12.























12.


(cm-1)

8.






4.






0.


0. 4. 8. 12.


1' (cm-1)


Figure 5.8


Contour map of the amplitude-wavenumber spectrum (see inset for
a perspective view) for the 25 Hz wavetrain of experiment GC2503.
Dispersion relation (- -) of the sNLS equation; the region of
spreading (- -) and the most unstable sidebands (X) as
predicted by the linear stability analysis.























12.

m '

(cnim


Figure 5.9


0. 4. 8. 12.

1' (cm-1)
Contour map of the amplitude-wavenumber spectrum (see inset for
a perspective view) for the 25 Hz wavetrain of experiment GC2504.
Dispersion relation (- -) of the sNLS equation; the region of
spreading (- -) and the most unstable sidebands (X) as
predicted by the linear stability analysis.









rotation of the coordinate system. Superposed on the contour maps of the

wavenumber spectra are the dispersion relations of the sNLS equation as predicted

by (3.30). The maximum angle of directional spreading, as predicted by the linear

stability analysis, is depicted by radial lines. Also, the most unstable sidebands,

represented by X's, are shown. The values of the most unstable sidebands, along

with a complete summary of the experiments, are presented in Table 5.1.

In figure 5.7, the numerical smearing of the "spike" at the input wavevector is

present. In accordance with figure 4.4(c), where the angle, H. is greater than 450,

the smearing here is along the m -axis. Although little directional spreading of

amplitude can be seen in the contour map along the dispersion relation, it is clearly

visible in the perspective view. Also visible in the inset view is the second spatial

harmonic; however, an insufficient number of contours has been used to identify

it, and so it is not visible in the contour map. This is consistent with the amplitude-

frequency spectrum of figure 5.6(a) which has a very small, but discernible peak at

50 Hz. In all of the contour maps presented, herein, 10 contours were used based

on the largest peak from each of the sets of different frequency experiments.

As mentioned previously, as spreading of wave amplitude occurs, the numeri-

cal smearing which causes the "criss-cross" appears to diminish as is seen in fig-

ures 5.8 and 5.9. As the wavemaker stroke is increased, so is the directional

spreading. However, the spreading of amplitude to adjacent wavevectors is not

monotonic nor smooth, but rather occurs as local extrema. In figures 5.8 and 5.9,

it is seen that the most unstable sidebands are in the vicinity of the local extrema.

Note that the second-harmonic wa\e amplitudes have also spread directionally.

These "forced" waves (i.e. waves that do not satisfy the linear dispersion relation,

(3.4), are termed forced, while waves that do satisfy the linear dispersion relation









are called "free") travel with the primary waves and are thus forced to spread

directionally. *

An inspection of the three frequency spectra of figure 5.6, reveals that the

width of the peak present at 25 Hz is increasing as wavemaker nonlinearity (kOs) is

increased. This effect is due to the presence of free waves (sidebands) at slightly

different wavenumber magnitudes than the input wave which are necessarily of

slightly different frequencies. Recall that in figure 3.2, the dispersion relation of

the sNLS equation and the circle (representing waves of constant wavenumber

magnitude) are not perfectly matched, so that very slight frequency differences are

expected. This phenomenon is essentially a spatial, Benjamin-Feir sideband.

Rather than causing modulations of a one-dimensional wavetrain, the spatial insta-

bility causes directional spreading of energy of a two-dimensional wavetrain. In the

first two frequency spectra of figure 5.6. two harmonics are distinct, while in the

third spectrum, the third harmonic is now visible. In all of the amplitude-frequency

spectra of figure 5.6. the superharmonics have significantly less amplitude than the

primary wave. The amplitudes are ordered corresponding to the O(E") at which

they occur (i.e. the third harmonic has less amplitude than the second harmonic,

etc.).

The directional spreading, as predicted by the dispersion relation, (3.30), of

linear oscillations of the sNLS equation, have been superposed on the contour

maps. Clearly, the energy spreading is occurring along these curves. The direc-

tional spreading seems to have occurred along the dispersion relation even when

there were 1800 phase shifts present in the wavefield as in experiment GC2504.

The most unstable sideband predictions are in reasonable agreement with local

extrema along the curve. It is also evident that the maximum directional spreading

as predicted by the linear analysis is somewhat underpredicting the total spreading








occurring in the experiments. In the perspective inset of figure 5.9, the direction-

ally-spread disturbance has nearly traversed the entire quadrant: the disturbance

lies along the dispersion curve. There is also spreading of the wavevectors trans-

verse to the dispersion parabola.

5.2.2 The 17 Hz Wavetrain

The 17 Hz wavetrain lies below the 19.6 Hz parametric boundary for admissi-

ble triads, and therefore, none are expected to occur in a significant manner. In

figures 5.10 (a) through (c), we present the amplitude-frequency spectra for ex-

periments GC1702 through GC1706, respectively. As occurred with the 25 Hz ex-

periments, the widening of the peak at 17 Hz has increased with increased

wavemaker stroke. As with the previous set of experiments, this is attributed to the

presence of wavevector sidebands which are free waves and thus of slightly differ-

ent frequencies. Some identifiable, low-frequency energy was present in the first

two spectra at between 2 Hz and 3 Hz, but for the GC1706 experiment, it has

shifted to between 3 Hz and 4 Hz. These amplitude-peaks were never as large as

the second harmonic. (Recall that the frequency spectra have log-scales.) in the

GC1702 experiment, aside from the primary wave, only the second harmonic (34

Hz) is identifiable. Three harmonics are distinct in the GC1704 frequency spec-

trum, and, at least, four harmonics are discernible in figure 5.10(c).

In figures 5.11, 5.12, and 5.13, the wavenumber spectra are presented for the

17 Hz experiments. Here, the relative sizes of the second harmonics (as compared

to the primary waves) were larger than in the 25 Hz experiments. The maximum

amplitude peak occurs in experiment GC1704 which has a smaller wavemaker

motion than experiment GC1706. Presumably. this is due to the significant amount

of directional spreading occurring in the GC1706 (largest stroke) experiment.

There were also more low-wavenumber amplitudes present in the 17 Hz experi-





















2 5 10 30 70


amplitude

(db)


t1 30 70


2 5 10 30 70


frequency (Hz)


Figure 5.10 The amplitude-frequency spectra for experiments (a) GC1702,
(b) GC1704, and (c) GC1706.























12.


m '

(cm-1)

8.






4.






0.


0. 4. 8. 12.


1' (cm-1)


Figure 5.11


Contour map of the amplitude-wavenumber spectrum (see inset for
a perspective view) for the 17 Hz wavetrain of experiment GC1702.
Dispersion relation (- -) of the sNLS equation; the region of
spreading (- -) and the most unstable sidebands (X) as
predicted by the linear stability analysis.























12.

m '

(cm-1)

8.






4.






0.


Figure 5.12


0. 4. 8. 12.

1' (cm 1)

Contour map of the amplitude-wavenumber spectrum (see inset for
a perspective view) for the 17 Hz wavetrain of experiment GC1704.
Dispersion relation (- -) of the sNLS equation; the region of
spreading (- -) and the most unstable sidebands (X) as
predicted by the linear stability analysis.























12.

m '

(cm-1)

8.






4.






0.


0. 4. 8. 12.


S' (cm-1)


Figure 5.13


Contour map of the amplitude-wavenumber spectrum (see inset for
a perspective view) for the 17 Hz wavetrain of experiment GC1706.
Dispersion relation (- -) of the sNLS equation; the region of
spreading (-- ) and the most unstable sidebands (X) as
predicted by the linear stability analysis.









ments than there were in the 25 Hz experiments. Some of the low-wavenumber

amplitudes were due to viscous effects as discussed in Chapter 3. Again, the am-

plitude-wavenumber contour maps show that the most unstable sidebands are pre-

dicted reasonably well, and, that the directional spectra spread in accord with the

prediction of the sNLS equation. As with the 25 Hz experiments, these wavetrains'

second harmonics were directionally spread. The numerical smearing of the large-

amplitude peaks is also visible. Also, notice the asymmetry in the contour map of

the wavenumber spectrum of figure 5.13. In the perspective view. it is seen that the

wavenumbers, along the dispersion curve clockwise from the /-axis, are present;

however, they have less amplitude than those in the counter-clockwise direction.

Possibly, this is due to lighting effects.

Upon comparison of figures 5.10(c) and 5.13, we see that there are some

contradictions. The local extremum shown in figure 5.13, which has a wavenumber

magnitude of about k = 3.6 (less than the input wavevector). but has an identical

direction, has a free-wave frequency of 13.3 Hz. It is not identifiable in the fre-

quency spectrum. However, there is energy present at 3.7 Hz in the frequency

spectrum which corresponds to a free-wave wavenumber of 0.54. There is no en-

ergy in the wavenumber spectrum at about 0.5. We assume that the waves appear-

ing in either of the two measurement techniques which do not correspond to free-

waves are thus forced-waves (or possibly spurious noise).


5.2.3 The 13.6 Hz Wavetrain


At approximately 13.6 Hz. the phase velocity and the group velocity of G-C

waves are equal. Although this does not represent a frequency for which anything

significant occurs as relates to the NLS equation (see figure 3.2), it is a unique

frequency as regards ripples and so it is investigated. (Benney, 1976 investigated








the non-collinear case where phase velocity equals the projection of the group ve-

locity in the context of resonant interaction theory.)

Experimental results for GC13601, GC13605 and GC13606 are shown in fig-

ures 5.14(a) through 5.14(c), 5.15, 5.16, and 5.17. The results are very similar to

the 25 Hz and 17 Hz experiments with the exception that there is almost no identi-

fiable second-harmonic amplitude in the frequency spectrum, nor in the wavenum-

ber spectrum for the GC13601 experiment. In fact, this is the first perspective view

of the wavenumber spectrum in which almost no directional energy spreading is

present. (Experiment GC13601 had the smallest value of kos of the wavetrains for

which the sNLS equation is valid.) For the 13.6 Hz experiments, the most unstable

sidebands and the maximum spreading of the energy as predicted by the linear

stability analysis are in reasonable agreement. Note that in the wavenumber per-

spective views of figures 5.16 and 5.17, the sidebands grow considerably. In fact,

their magnitude, in figure 5.17 is about one-third that of the primary wavevector.

The "criss-cross" effect is present, also. Recall that these spectra were computed

from images taken 37 cm downstream as opposed to the other experiments which

were recorded at 23 cm. This apparent slower spatial development is consistent

with the 25 and 17 Hz cases, which we expect should occur more quickly in space

(and do); and the 9.8 Hz wavetrain, which we know will occur more quickly be-

cause its evolution (resonance) occurs at second order.

5.2.4 The 9.8 Hz Wavetrain, Wilton's Ripples

As shown in figure 3.2, 9.8 Hz lies on a boundary in parameter space of the

NLS equation. For frequencies greater than 9.8 Hz. the NLS equation predicts an

unstable region (for sufficiently deep water). Wavetrains. with a frequency less

than 9.8 Hz, and greater than about 6.4 Hz. are stable as regards the NLS equa-

tion. Wilton's ripples (a 9.8 Hz wavetrain for deep-water with a surface tension




















5 10 30 70


2 5 10 30 70


2 5 10 30 70


Figure 5.14


frequency (Hz)






The amplitude-frequency spectra for experiments (a) GC13601,
(b) GC13605, and (c) GC13606.


amplitude

(db)








































12.


1 (cm-1)


Figure 5.15


Contour map of the amplitude-wavenumber spectrum (see inset for
a perspective view) for the 13.6 Hz wavetrain of experiment
GC13601. Dispersion relation (- -) of the sNLS equation; the re-
gion of spreading (- -) and the most unstable sidebands (X)
as predicted by the linear stability analysis.


12.

m'
(cm-')
8.





4.





0.


A/R
\ t )_ \\


















m'
t,


-0" / /




// \
Iz_ \______.


12.


1 (cm-1)


Figure 5.16


Contour map of the amplitude-wavenumber spectrum (see inset for
a perspective view) for the 13.6 Hz wavetrain of experiment
GC13605. Dispersion relation (- -) of the sNLS equation; the re-
gion of spreading (-- ) and the most unstable sidebands (X)
as predicted by the linear stability analysis.


12.


(cm-1)

8.






4.






0.























12.


(cm-1)

8.





4.





0.


Figure 5.17


0. 4. 8. 12.
S' (cm-1)

Contour map of the amplitude-wavenumber spectrum (see inset for
a perspective view) for the 13.6 Hz wavetrain of experiment
GC13606. Dispersion relation (- -) of the sNLS equation; the re-
gion of spreading (- -) and the most unstable sidebands (X)
as predicted by the linear stability analysis.









coefficient of 73 dyn/cm), is a degenerate, three-wave (internal) resonance which

occurs at second order. In addition, the second harmonic (19.6 Hz) represents a

parametric boundary in ripple dynamics. It is the low-frequency boundary at which

resonant interaction theory no longer admits triads. Therefore, Wilton's ripples are

a very unstable wavetrain. The parabolic-shaped, directional spreading as predicted

by the sNLS equation is no longer applicable. It is expected that second harmonic

(19.6 Hz) resonance will dominate the evolution.

Four experiments were conducted for Wilton's ripples, and the amplitude-fre-

quency spectra are presented in figure 5.18. The second harmonic was present in

all of the spectra, and adjacent frequencies were also excited as the energy spread

directionally. As in the previous experiments, the width of the peak at 9.8 Hz (in

the frequency spectra) increased as wavemaker stroke was increased. For Wilton's

ripples, this phenomenon also occurred for the 19.6 Hz second harmonic. In the

GC9810 experiment shown in figure 5.18(d). at least 6 harmonics are identifiable.

Also in figure 5.18(d), background waves, which were "flat" in the other spectra,

are seen to have been excited, but at a significantly less energetic level.

In figures 5.19 through 5.22, we present the contour maps and perspective

views of the wavenumber spectra. First, note that parabolic-shaped spreading does

not occur (although significant spreading does occur in experiment GC9810). In

figure 5.20, notice the magnitude of the second harmonic in the perspective view.

It is already approaching the size of the input wave. This is in disagreement with

the frequency spectrum for this experiment, figure 5.18(b). which shows an in-

crease in amplitude of the second harmonic, but not to this extent. In fact. none of

the frequency spectra show that the magnitude of the 19.6 Hz waves grow to the

size of the 9.8 Hz waves. Remember, however, that the frequency spectra are from

point measurements at 23 cm, while the spatial information covers an area about































amplitude

(db)


2 5 10 30 70


10 30 70


2 5 10 30 70


frequency (Hz)




Figure 5.18 The amplitude-frequency spectra for experiments (a) GC9801,
(b) GC9802, (c) GC9804, and (d) GC9810.













































8.


1' (cm-1)


Figure 5.19


Contour map of the amplitude-wavenumber spectrum (see inset for
a perspective view) for the 9.8 Hz wavetrain of experiment
GC9801.


12.


m

(ci


F,


m


/


12.



















12.

m'
(cm-1)
8.









n-


4. 8. 12.


1' (cm1)


Figure 5.20


Contour map of the amplitude-wavenumber spectrum (see inset for
a perspective view) for the 9.8 Hz wavetrain of experiment
GC9802.

















m' /1'




12.

m '

(cm-')

8.










m 1
4. -






0.

0. 4. 8. 12.

1' (cm1)


Figure 5.21 Contour map of the amplitude-wavenumber spectrum (see inset for
a perspective view) for the 9.8 Hz wavetrain of experiment
GC9804.






















12.

m '

(cm-1)

8.





4.





0.


0. 4. 8. 12.


1' (cm-1)


Figure 5.22


Contour map of the amplitude-wavenumber spectrum (see inset for
a perspective view) for the 9.8 Hz wavetrain of experiment
GC9810.









22 cm by 22 cm beginning at 23 cm downstream. Thus, the frequency spectra do

not contain any spatial information, and the wavenumber spectra contain no tem-

poral information. So, it is not surprising that they do not agree in this respect,

especially since these wavetrains are evolving in space (and more slowly in time).

In figure 5.21, the first and second harmonics are shown to be of comparable

magnitude by the wavenumber spectrum. Numerical smearing is present, again, as

is seen in figures 5.20 through 5.22. In figure 5.22, the directional spreading has

occurred for the first three harmonics along an apparently. almost straight line

(perpendicular to the input wavevector).

5.2.5 The 8 Hz Wavetrain

According to the NLS equation, an input frequency of 8 Hz is a stable

wavetrain. Note that for clean, deep water, frequencies less than 9.8 Hz through

6.4 Hz are stable. Referring again to figure 3.2, along curve C1, A changes sign,

and upon crossing this curve toward lower wavenumbers, the region is again Ben-

jamin-Feir unstable. In figure 5.23. the amplitude-frequency spectra are presented

for experiments GC0802, GC0804, and GC0806. In figures 5.23a and 5.23b, the

second harmonic and the second, third, and fourth harmonics have distinct, or-

dered peak-magnitudes, respectively. However, in figure 5.23(c), the magnitude of

the third harmonic has surpassed the magnitude of the second harmonic!

In figures 5.24 through 5.26. we present the wavenumber spectra for these

three experiments. Notice the presence of a significant "criss-cross" effect reminis-

cent of the numerical smearing of figure 4.4(d). The 8 Hz wavevectors in figures

5.24 and 5.25 show no other directional spreading of energy in agreement with the

theory. However, in figure 5.26, we see the dramatic emergence of a directionally

spread third harmonic whose central peak is about one-half the peak of the pri-

mary wave. In fact, the spreading is predicted fairly well by the dispersion relation













150 -

120






S 3 7 20 50 10
2 5 10 30 70


amplitude

(db)


2 5 10 30 70


10 30 70


frequency (Hz)







Figure 5.23 The amplitude-frequency spectra for experiments (a) GC0802,
(b) GC0804, and (c) GC0806.























12.


i '

(cm-1)

8.





4.





0.


4. 8. 12.


l' (cm-1)


Figure 5.24


Contour map of the amplitude-wavenumber spectrum (see inset for
a perspective view) for the 8 Hz wavetrain of experiment GC0802.






















12.

m '
(cm-1)

8.





4.





0.


0. 4. 8. 12.


I' (cm-1)


Figure 5.25


Contour map of the amplitude-wavenumber spectrum (see inset for
a perspective view) for the 8 Hz wavetrain of experiment GC0804.























12.


i '

(cm-1)


0. 4. 8. 12.


1' (cm-)


Figure 5.26


Contour map of the amplitude-wavenumber spectrum (see inset for
a perspective view) for the 8 Hz wavetrain of experiment GC0806.
Theoretical (- -) prediction of the sNLS equation for a 24 Hz
wavetrain.









of the sNLS equation which has been superposed on the figure. It appears that the

24 Hz (third 'harmonic) is resonating with the primary wave by third-harmonic

resonance. Thus, the explanation of the large third harmonic frequency peak in

figure 5.23c. In fact, the photograph, shown in figure 4.3, clearly shows the pres-

ence of the third harmonic. Theoretically, third-harmonic resonance should occur

for an 8.38 Hz wavetrain and a 25.14 Hz wavetrain. Apparently, this is a case of

detuned, third-harmonic resonance. McGoldrick (1972) presented results which

show that the ratio of an 8 Hz response to an 8.55 Hz response (his peak response)

is about 8%. His results give a ratio of the 8 Hz response to the 8.38 Hz response

(the theoretical third-harmonic resonance) of about 13%.. Therefore, it is somewhat

surprising that the magnitude of the 24 Hz wavetrain grew so large.















CHAPTER 6


SUMMARY AND CONCLUSIONS OF PART 1


In the preceding sections, we have investigated a previously unexplored insta-

bility of G-C waves. It represents a plausible model of the instabilities of ripple

wavetrains generated mechanically in a wide. laboratory channel. The stationary,

NLS model predicts the directional spreading of amplitude reasonably well. Also,

it provides a mechanism by which energy is available at various angles, so that

non-collinear triad and quartet resonances can occur.

Three sets of wavetrains with frequencies of 25. 17 and 13.6 Hz. which are in

the sNLS unstable regime, are shown to display this instability. Using wavenumber

spectra, all of these wavetrains are shown to directionally spread their amplitude

along a dispersion curve as predicted by assuming linear modulations of the sNLS

equation. Energy appears in the vicinity of the most unstable sidebands. As non-

linearity in the wavefield in increased, this spreading of energy increases.

Wilton's ripples (9.8 Hz for clean, deep water), for which a degenerate, three-

wave (internal) resonance exists, resonates with its second harmonic and both

spread their energy directionally: however, the transfer of energy is along a path

perpendicular to the input wavevector in the wavenumber plane. Experiments with

8 Hz wavetrains which are theoretically stable according to the NLS equation, are

shown to remain stable until third harmonic resonance occurs. Then, the third

harmonic, which is now a free wave, exhibits the sNLS instability and spreads its

amplitude accordingly. One possible explanation of why the 8 and 24 Hz waves









resonate when the theory predicts 8.38 Hz and 25.14 Hz is that of detuned reso-

nance.

As mentioned earlier, one very important phenomenon neglected entirely by

the derivation of the NLS equation is viscosity. Herein, viscous effects preclude

any attempt to see whether recurrence is present in our experiments. Also, viscous

effects hamper analytic comparisons between experiment and theory.

Further experiments are required in which spatial disorder is investigated in

the presence of triads (forced to occur by wave seeding). Other related experi-

ments include modulating the wavemaker in time (problems arise because of the

difference waves generated by the wavemaker, especially if two waves are super-

posed to drive the wavemaker and thus the modulation) and the case in which

finite-water depth effects are more significant (i.e. mean, long-wave motion is sig-

nificant).

Finally, quantitative measurements of the phenomenon for use in comparison

to numerical simulations with linear viscosity included. would be useful. However,

before quantitative measurements can be made, two improvements would be bene-

ficial. First, the wave gauge probe should be replaced by a nonintrusive, laser

gauge (this is currently being developed). Second, a better understanding of the

lighting used to remotely sense the wavefield, along with a calibration of the entire

method of obtaining the wavenumber spectrum, is required.















CHAPTER 7


INTRODUCTION TO THREE-WAVE RESONANCE: PART 2


(Chapters 7 through 14 are essentially the research paper by Hammack, Perlin,

and Henderson (to appear, 1989). This material was prepared prior to the installa-

tion of the high-speed imaging system used in the study reported in Chapters 2

through 6.)

Waves for which both gravitation and surface tension are important are ubiqui-

tous on the ocean surface. These gravity-capillary waves (or simply "ripples") have

typical frequencies in the range of approximately 5-50 Hz and lengths of approxi-

mately 50-5 mm. Relative to longer gravity-induced waves, ripples have received

little attention in oceanography. However, recent interest in remote sensing of the

sea surface with satellite-based radar has focused new attention on ripples. Ripple

wavelengths are comparable to those of high-frequency radar waves, causing them

to play a major role in the backscatter of radar waves by the ocean surface. It is

somewhat paradoxical that oceanographic measurements on global scales is im-

pacted so fundamentally by surface waves with the smallest of scales.

Ripple dynamics exhibit a variety of interesting phenomena that are generic to

many nonlinear dynamical systems. One-dimensional periodic wavetrains are

fraught with instabilites which occur quickly when ripples with sensible amplitudes

are generated mechanically in a laboratory channel. The one-dimensional

wavetrains evolve rapidly into two-dimensional surface patterns which can appear

very irregular. (Herein, we refer to waves as one- or two-dimensional according to









whether their surface patterns are one- or two-dimensional. Their corresponding

velocity fields' are one dimension higher.) In Part 1, we investigated one of these

instabilities which had been unexplored previously. These instabilities are exam-

ined, herein, experimentally and analytically from the point of view of resonant

interaction theory (RIT). Theoretical results indicate that ripple wavetrains with

frequencies greater than 19.6 Hz (on a perfectly clean surface) can excite resonant

triad interactions; other than the spatial instability discussed in Part 1, quartic

resonances are the first indicated for wavetrains with frequencies less than

19.6 Hz. Experiments are presented for wavetrains in the resonant-triad regime

since these data are more definitive and analytical results are more readily avail-

able. These wavetrains are generated with and without background perturbations,

which include smaller waves with distinct frequencies and smaller waves with a

broad band of frequencies (random waves). The major conclusions from these

experiments are as follows. First, we find that a selection mechanism exists which

causes certain triads from a possible continuum to dominate the evolution of a

ripple wavetrain. Second, we show that this selection results from a "seeding" of

low-frequency wave(s) in the triad continuum by difference interactions between

the generated wavetrain and high-frequency perturbations. These perturbations

must have distinct frequencies; their amplitudes may be as small as 250 that of

the generated wave. Although a theorem by Hasselmann (1967) is not violated, the

existence of this selection mechanism is contrary to expectations because the dif-

ference interactions occur at higher-order than the triad interactions, and yet. they

determine the evolution of the wavefield. Third. we demonstrate a cycling of wave

energy among the excited members of a wave triad during their spatial evolution.

(Indeed, this is the primary mechanism we use to distinguish resonant from non-

resonant interactions.) Finally, when a ripple wavetrain is generated along with a

wide spectrum of random-wave perturbations, there is no selection and the entire









continuum of admissible triads is excited in a manner consistent with the form of

the interaction coefficients for a single triad.

An outline of the remainder of Part 2 is as follows. In Chapter 8 we present a

general summary of RIT, followed in Chapter 9 with its specific application to

ripples for resonant triad interactions. In Chapter 10 a brief description of the

laboratory facilities is presented. Previous experimental results on triad selection

are reviewed in Chapter 11. In Chapter 12 this selection process is explained and

the previous experiments are reinterpreted. Finally, in Chapter 13 we demonstrate

the evolution of a ripple wavetrain in the presence of white noise.















CHAPTER 8


RESONANT INTERACTION THEORY


The fundamental notion of RIT was formulated by Phillips (1960) for gravity

waves on deep water. Benney (1962) inter alios realized the general nature of its

application to dynamical systems. The basic idea is that the effects of weak non-

linearity may be dominated by certain wave-wave interactions which satisfy "reso-

nance" conditions. A general outline of RIT is as follows (also, see Segur, 1984).

Consider a nonlinear energy-conserving dynamical system which we represent by


N(q) = 0. (8.1)


where N is a nonlinear operator. 0(x, t) is a solution of (8.1), x = (.,y,z) is a posi-

tion vector and t is time. Suppose 0 = 0 is an equilibrium solution of (8.1) that is

neutrally stable; then infinitesimal deformations from this equilibrium state are

found by linearizing (8.1) to find


L(0) = 0. (8.2)


We then seek normal mode solutions to the linear operator L of (8.2), and, if L has

constant coefficients, this amounts to seeking solutions with the form:


01- expli(k x t +a) (8.3)


where k is a wavevector, o is the wave frequency, and a is an arbitrary and

constant phase angle. Substituting (8.3) into (8.2) yields a dispersion relation for









the linear problem:


w = W(k). (8.4)


We require that o exists in the sense that for fixed k there is a countable set of

possible values for o and that W(k) is real-valued for real-valued k. Now return to

(8.1) and seek small-but-finite deformations from the equilibrium state by means

of a formal power series:
00
(x, t; E) = en(x, t) 0 < E 1. (8.5)
n= I

At leading order in e, 01 of (8.5) satisfies (8.2). so 0 can be represented by a

superposition of M linear waves, i.e.

Al
1i= (Ae'am + Ame-io'() (8.6a)
m=

where Am is the complex wave amplitude, Am is its complex conjugate, and


O m = k,, w,,t. (8.6b)


The first nonlinear effects appear at order E: where


L(Oz) = Q(01). (8.7)


and Q is a quadratic operator. Substituting (8.6) into Q produces terms with the

form:



AA;A + Aje'2 2 + AAe- + AjAe'',+2 + A.Ae'"'-:(' + (8.8)


on the right-hand side (RHS) of (8.7). The first three terms in (8.8) correspond to

self-interactions of wave m=1; the first term represents a "mean" flow for 02 while









the second and third terms correspond to its second harmonic (first superhar-

monic). The other two terms shown in (8.8), and their complex conjugates, corre-

spond to sum and difference waves between waves m=l and m=2. (There are

analogous terms for all of the M waves in (8.6).) Supposing these quadratic inter-

actions do not vanish identically, we can examine the terms of (8.8) to see if any

of the sum and difference waves also satisfy the dispersion relation of (8.4), i.e.

does

01 02 = (k, k2) x (w (2)t = k3 x ( w3)t = 03 ? (8.9)


If so, then these waves will lead to secular terms (linear growth) for 02, and the

"three waves" of (8.9) are said to form a "resonant" triad. Eventually the resonant

terms will dominate nonresonant terms due to their secular growth; thus the non-

resonant terms are neglected in further considerations of RIT. For water waves

W(k)=W(-k). so that without loss of generality we can write the cinematic resonant

conditions of (8.9) for a resonant triad as

ki = k: k3 (8.10a)

wl = w2 w3. (8.10b)

The existence of triad resonances depends on the specific dynamical system (dis-

persion relation) being considered. If there are no terms on the RHS of (8.7) which

satisfy the resonance conditions of (8.10), then we can proceed to order E3 where

cubic interactions are encountered. Here, resonant quartets occur when the

kinematic conditions:

kl k: = k3 k4 (8.11a)

w1 W = W3 w4 (8.11b)

are satisfied. Unlike triad resonances, quartic resonances are always possible for









the dynamical systems considered here. For example, a rhombus quartet always

exists in which tW = w2 = w3 = w4 and Jk, = Ik2J = jk3j = jk4|. These procedures

can be continued to higher-order to find analogous conditions for resonances at

every order in E; though one seldom goes beyond quartic resonances.


When resonance occurs, a separate analysis is required to follow wave evolu-

tion since the secular terms will eventually disorder the expansion of (8.5). Typi-

cally, the waves in a single triad (i.e. M = 3 in (8.6a)) or a single quartet (M = 4)

are studied. Two common methods of analysis are (i) the method of multiple

scales (e.g. McGoldrick, 1972) and (ii) variational techniques (e.g. Simmons,

1969). Both methods lead to a set of coupled nonlinear partial differential equa-

tions for the complex wave amplitudes. These equations are similar for all dynami-

cal systems, differing only in real-valued "interaction coefficients" ,Ym among the

M-waves. For example, the equations for a single resonant triad (3-\\ave equa-

tions) have the form:


(a,+ U, V)Am = iymAn,,iAr,+2. (8.12)



where m is interpreted modulo 3 and U, are the respective (constant) group ve-

locities. When the nonlinear system is not energy-conserving. the 3-wave equations

of (8.12) must be modified. In the case of linear damping, (8.12) takes the form

(Craik, 1986):


(i,+U,,, V)A,, + 6,A, = iymA,,Ai+Am+:. (8.13)


where 6m are real-valued decay coefficients whose form depends on the the specific

dynamical system under consideration.















CHAPTER 9


RIT AND RIPPLES



The linear dispersion relation for gravity-capillary waves on water of quiescent

depth h has been known since Kelvin (e.g. see Rayleigh, 1945, and the reference

cited there) and has the (dimensional) form:


w2 = (gk+ Tk3)tanhkh (9.1)


where w is the radian frequency, k is the magnitude of the wavevector, g is the

gravitational force per unit mass, r is the surface tension, h is the quiescent water

depth, and Q is the mass density of the water. The kinematic possibility of reso-

nance is most easily found by a graphical procedure outlined by Simmons (1969)

for ripples. Figure 9.1 (after Simmons. 1969) shows the graphical solution of

(8.10) using (9.1) when the wavevector of each triad is collinear. (We will general-

ize to the non-collinear case from this simpler case.) The figure is constructed and

interpreted as follows. First, the dispersion relation of (9.1) is shown in (w,k)

space with an origin at 01. In the calculations for figure 9.1 we have taken

h = 4.9 cm and T = 73 dyn/cm which corresponds to values in the experiments to

be described later, and, for clarity, only one branch of (9.1) is presented. Second,

a point on the dispersion curve corresponding to a "test" wave is chosen as the

origin 02 for a graph of all branches of the dispersion relation. (The test wave

corresponds to the one generated by the wavemaker in the experiments to follow.)

In figure 9.1 a test wave with a cyclic frequency f= w/2n = 25 Hz is chosen









which corresponds to one of our experiments. (Herein, we will identify the test

wave as k, anld w,.) The intersection points of the two dispersion curves at A, B,

and C, represent the collinear, resonant triads that are possible for the particular

test wave. The individual members of each triad are found easily as shown for the

A-triad.


When the test-wave frequency is lowered in figure 9.1 by sliding 02 down the

dispersion curve, the triads at A and B will eventually coalesce, then disappear,

leaving only the triad at C for lower test-wave frequencies. Coalescence occurs at a

frequency of 2w, = 19.6 Hz for a clean water surface. The special role of this wave

frequency follows from a theorem by Hasselmann (1967). According to this theo-

rem, if only one member of a resonant triad is present initially with significant

energy, then it must have the highest frequency in the triad in order to excite the

other two members. Since the test waves in the experiments are input by the

wavemaker, and the other triad waves must be excited from (small) background

"noise," Hasselmann's theorem requires the test waves to have the highest fre-

luencies so that only the sum triads of (8.6) are possible. Hence, the C-triad of

figure 9.1 should not be excited and no triads should be available for test-wave

frequencies less than 19.6 Hz.


Figure 9.1 may be generalized to waves with non-collinear wavevectors by

rotation of the dispersion curves about the frequency axes. When rotation is per-

formed (see Simmons, 1969), the A- and B-triads are found to lie on a closed curve

providing a continuum of possible triads. The minimum frequency of any wave in

this continuum is that of the lowest-frequency wave in the collinear A-triad. i.e. tw3

in figure 9.1; the maximum frequency in the continuum is (uw wo). The C-triad in

figure 9.1 is found to lie on an open curve whose lowest frequency is (w, + w3).

The conclusions above regarding the 19.6 Hz parametric boundary for collinear




































8 10 12
k (cm-')


Figure 9.1


Graphical solution of the kinematic resonance conditions (after
Simmons, 1969) for collinear triads of a 25 Hz test wave.
(Henderson and Hammack, 1987).


(I)

(s-00
300


200