• TABLE OF CONTENTS
HIDE
 Cover
 Title Page
 Acknowledgement
 Table of Contents
 List of Figures
 Abstract
 Introduction
 Extension of non-resonant interaction...
 Comparison of solutions
 Analysis technique for laboratory...
 Laboratory study
 Bars on a mild slope
 Conclusions
 Bibliography






Group Title: UFL/COEL (University of Florida. Coastal and Oceanographic Engineering Laboratory) ; 89/012
Title: Resonant and non-resonant reflection of linear waves over rapidly varying bottom undulations
CITATION PAGE IMAGE ZOOMABLE
Full Citation
STANDARD VIEW MARC VIEW
Permanent Link: http://ufdc.ufl.edu/UF00076151/00001
 Material Information
Title: Resonant and non-resonant reflection of linear waves over rapidly varying bottom undulations
Series Title: UFLCOEL
Physical Description: viii, 88 leaves : ill. ; 28 cm.
Language: English
Creator: Anton, Jeffrey Philip, 1965-
University of Florida -- Coastal and Oceanographic Engineering Laboratory
Publication Date: 1989
 Subjects
Subject: Water waves   ( lcsh )
Waves   ( lcsh )
Fluid dynamics   ( lcsh )
Coastal and Oceanographic Engineering thesis M.S
Coastal and Oceanographic Engineering -- Dissertations, Academic -- UF
Genre: bibliography   ( marcgt )
theses   ( marcgt )
non-fiction   ( marcgt )
 Notes
Thesis: Thesis (M.S.)--University of Florida, 1989.
Bibliography: Includes bibliographical references (leaves 86-87).
Statement of Responsibility: by Jeffrey Philip Anton.
General Note: Typescript.
General Note: Vita.
Funding: This publication is being made available as part of the report series written by the faculty, staff, and students of the Coastal and Oceanographic Program of the Department of Civil and Coastal Engineering.
 Record Information
Bibliographic ID: UF00076151
Volume ID: VID00001
Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved, Board of Trustees of the University of Florida
Resource Identifier: oclc - 21717942

Table of Contents
    Cover
        Cover
    Title Page
        Title Page 1
        Title Page 2
    Acknowledgement
        Acknowledgement
    Table of Contents
        Table of Contents 1
        Table of Contents 2
        Table of Contents 3
    List of Figures
        List of Figures 1
        List of Figures 2
    Abstract
        Abstract
    Introduction
        Page 1
        Page 2
        Page 3
        Page 4
        Page 5
        Page 6
        Page 7
        Page 8
        Page 9
        Page 10
        Page 11
        Page 12
        Page 13
        Page 14
        Page 15
    Extension of non-resonant interaction theory
        Page 16
        Page 17
        Page 18
        Page 19
        Page 20
        Page 21
        Page 22
        Page 23
        Page 24
        Page 25
        Page 26
    Comparison of solutions
        Page 27
        Page 28
        Page 29
        Page 30
        Page 31
        Page 32
        Page 33
        Page 34
        Page 35
        Page 36
        Page 37
        Page 38
        Page 39
    Analysis technique for laboratory study
        Page 41
        Page 42
        Page 43
        Page 44
        Page 45
        Page 46
        Page 47
        Page 48
        Page 49
        Page 50
        Page 51
        Page 52
        Page 53
        Page 54
    Laboratory study
        Page 55
        Page 56
        Page 57
        Page 58
        Page 59
        Page 60
        Page 61
        Page 62
        Page 63
        Page 64
        Page 65
        Page 66
        Page 67
        Page 68
    Bars on a mild slope
        Page 70
        Page 71
        Page 72
        Page 73
        Page 74
        Page 75
        Page 76
        Page 77
        Page 78
        Page 79
        Page 80
        Page 81
        Page 82
        Page 83
        Page 84
    Conclusions
        Page 85
    Bibliography
        Page 87
        Page 89
Full Text







UFL/COEL-89/012


RESONANT AND NON-RESONANT REFLECTION OF
LINEAR WAVES OVER RAPIDLY VARYING BOTTOM
UNDULATIONS










By


Jeffrey Philip Anton





1989


Thesis



























RESONANT AND NON-RESONANT REFLECTION OF LINEAR WAVES OVER
RAPIDLY VARYING BOTTOM UNDULATIONS



By

JEFFREY PHILIP ANTON


A THESIS PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN
PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF MASTER OF SCIENCE


UNIVERSITY OF FLORIDA


1989















ACKNOWLEDGEMENTS


First and foremost, I thank God for giving me the capacity and opportunity to study,

learn and understand.

My most heartfelt gratitude is due my parents, Joseph and Mildred Anton, for their

unceasing support, prayers, encouragement and love, especially when I didn't believe in

myself. Thanks and love go to my loving brothers and sisters for their support and love as

well, especially my oldest brother and friend Joe for sparking my first love for the sea.

I thank Wendy for her love and tolerating five months away.

I would like to thank Dr. James Kirby for his guidance, help and patience throughout

the last two years. Also, I would like to express my appreciation to my committee members

for spending the time and effort to read and edit this thesis, and to the Office of Naval

Research who sponsored the study.

I would also like to recognize all the faculty at University of Florida and at The Ohio

State University who have taken part in my collegiate education, especially Dr. Keith

Bedford, Dr. Vince Ricca, and Dr. John Lyons who convinced me that an engineer with a

B.S. may be only "half-baked."

Finally, but far from least I owe my thanks to all of my collegiate colleagues who have

made this a grand six years of life, learning, insanity, sailing and friendship, including but

not limited to the best of BES, Sam Houston Institute of Tech, Rich, WSP, Peene, B.D.,

PAW, LAS, Matt and the Korean Contignent.















TABLE OF CONTENTS


ACKNOWLEDGEMENTS ................................

LIST OF FIGURES ....................................

ABSTRACT ........................................

CHAPTERS

1 INTRODUCTION ...................................

1.1 Review of Weak Reflection Theories ......................

1.1.1 Non-Resonant Interaction ........................

1.1.2 Formulation of Davies and Heathershaw for Non-Resonant Interaction

1.1.3 Miles' Oblique Surface Wave Diffraction ...............

1.2 Review of Strong Reflection Theories ......................

1.2.1 Mei's Resonant Interaction Solution ..................

1.2.2 Depth-Integrated Equation for Small Undulations on Mild Slopes .

1.3 Indirect Solution of Surface Elevation . . . . . .

2 EXTENSION OF NON-RESONANT INTERACTION THEORY ........

2.1 Introduction . . . . . . . . .

2.2 Governing Equation ...............................

2.3 Two Dimensional Wave Field ..........................

2.4 Solution for Periodic Bars ............................

2.5 Fourier Decomposition of the Bottom . . . . . .

2.5.1 The Resonant Case ............................

2.5.2 The Non-Resonant Case .........................

2.5.3 Full Solution .. .. ... .. ... .. ... ... .. .. ...









3 COMPARISON OF SOLUTIONS ......................... 27

3.1 Introduction ..................... ............... 27

3.2 Derivation of a Complete Governing Equation . . . .... 27

3.3 Numerical Approximations ................... ........ 29

3.4 Comparison to Existing Theories . . . ..... ...... 32

3.4.1 One-dimensional Wave Field . . . . . 32

3.4.2 Obliquely Incident Waves ........................ 34

3.5 Application of the Solutions ................... ........ 35

4 ANALYSIS TECHNIQUE FOR LABORATORY STUDY ... ......... 41

4.1 Introduction ....... ...... . . . ... 41

4.2 Theoretical Background ................... ......... 41

4.3 Inputs to Least Squares ................... ......... 47

4.3.1 Direct Signal Processing . . . ..... ........ 47

4.3.2 Ensemble Averaging of Cross-Correlation Spectra . . ... 47

4.4 Results of Least Square Error Fit . . . ..... ...... 49

4.5 Results ................ ...................... 53

5 LABORATORY STUDY ............................... 55

5.1 Introduction .. ... .. ... .. .. .. ... .. .. .. ... 55

5.2 Equipm ent ................... .............. .. 55

5.2.1 Wave Flume .............................. 55

5.2.2 Electronic Measurement . . . ..... ...... .. 58

5.3 Data Analysis ... .. .. .... . . . .. ... 59

5.3.1 Acquisition ................... .............. 59

5.3.2 Data Processing ................... ......... 61

5.3.3 Verification of Analysis Technique . . . ... 66

5.4 Results and Conclusions ................... .......... 67









6 BARS ON A MILD SLOPE ............................. 70

6.1 Introduction ....... ..... ........... .. ..... .... .... 70

6.2 Reformulation of the Mild Slope Equation . . ... 70

6.3 Numerical Solution .. ....... ........ ....... ...... 72

6.4 Model Tests and Examples ........................... 75

6.4.1 Response Over a Barfield in Front of a Wall. . . 75

6.4.2 Response Over a Barfield in Front of Beach. . . 78

6.5 Conclusions ... .... .. ....... .... .............. 84

7 CONCLUSIONS .......... ............ ............. 85

BIBLIOGRAPHY .... .......... ...... ............. .. 86

BIOGRAPHICAL SKETCH ................... .............. .. 88















LIST OF FIGURES


1.1 Domain definition sketch .......................... 5

2.1 Reflection of single mode bottom . . . ..... .. 21

2.2 Definition sketch of four cosine bumps on a flat bottom . ... 22

2.3 Contributions to reflection by individual Fourier components . 25

2.4 Sum total of the reflection from individual components . ... 26

3.1 Reflection coefficient vs. 2k/A for sinusoidal bottom. h(z) = 0.15m, four
cycles 6(z) = 0.05msin 27r/1.0m ...................... 33

3.2 Reflection coefficient vs. 2k/A for four cosine bumps. h(z) = 0.15m,
8(x) = 0.05mcos27/1.0m,A = 21r/Lb ........... ......... 34

3.3 Mei's resonant interaction theory reflection coefficient vs. angle of inci-
dence 0 and offshore wavenumber k . . . . . 35

3.4 Non-Resonant Extension, reflection coefficient vs. angle of incidence 0
and offshore wavenumber ko ......................... 36

3.5 Numerical Solution, reflection coefficient vs. angle of incidence 0 and
offshore wavenumber ko .................. ........ 37

3.6 Non-Resonant Extension Numerical Solution vs. angle of incidence 0
and offshore wavenumber ko .................. .... 38

3.7 Non-Resonant Extension Mei's Solution vs. angle of incidence 0 and
offshore wavenumber ko ........................... 39

3.8 Mei's Solution Numerical Solution vs. angle of incidence 0 and offshore
wavenumber ko ................... ............ 40

4.1 Effect of smoothing on a monochromatic spectrum . . ... 51

4.2 Interpolating resolved frequencies in the smoothed monochromatic spec-
trum to estimate the energy density at the true frequency . ... 52

5.1 Profile of bar field ................... .......... 56

5.2 Typical calibration curve ................... ....... 60









5.3 Improper gage spacing and virtual standing wave . . ... 62

5.4 Energy crossing system boundaries . . . ...... 62

5.5 Typical energy density spectrum for incident wave . . ... 63

5.6 Typical energy density spectrum for reflected wave . . ... 64

5.7 Typical energy density spectrum for transmitted wave . ... 64

5.8 Reflection theory vs. measured, .8 m bar spacing . . ... 68

5.9 Reflection theory vs. measured 1.2 m bar spacing . . .... 68

5.10 Energy Conserved: Theoretical and Measured .8 m bar spacing .. 69

6.1 Definition sketch of bar field in front of a wall . . .... 76

6.2 Wave envelope in front of a wall for qr directly and r) = f-1/2W numerical
scheme es, d = 4 ................... ............ 77

6.3 Wave amplitude at the wall for r7 directly and 7t = f-'1/W numerical
scheme es, d = 4 ................... ............ 77

6.4 Wave envelope in front of a wall for r directly and r7 = f-1/2W numerical
schemes, d = 4.5 ................... ........... 78

6.5 Wave amplitude at the wall for YT directly and t1 = f-1/2W numerical
schemes, d = 4.5 ................... ........... 79

6.6 Definition sketch of bar field on a sloping bottom in front of a shoreline 80

6.7 Wave envelope on a sloping beach with 4 sine shaped bumps, d = 10.0 80

6.8 Wave envelope on a sloping beach with 4 sine shaped bumps, d = 10.5 81

6.9 Wave amplitude at the shoreline vs. 2k/A, d = 10.0 . . ... 82

6.10 Wave amplitude at the shoreline vs. 2k/A, d = 10.5 . . ... 82

6.11 Wave amplitude at x= 7 m vs. 2k/A, d = 10.0 . . ... 83

6.12 Wave amplitude at x= 7 m vs. 2k/A, d = 10.5 . . ... 83















Abstract of Thesis Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Master of Science

RESONANT AND NON-RESONANT REFLECTION OF LINEAR WAVES OVER
RAPIDLY VARYING BOTTOM UNDULATIONS

By

JEFFREY PHILIP ANTON

August 1989

Chairman: Dr. James Thornton Kirby
Major Department: Coastal and Oceanographic Engineering

Recent studies have shown that waves propagating over a bottom with rapid undulala-

tions may experience reflection as a result of the wave interacting with the bottom. The

strength of reflection is dependent on the ratio of the wavenumber of the surface wave and

the wavenumber of the bottom undulations. Specifically, if the wavenumber of the surface

wave is close to being one-half of the wavenumber of the undulations, strong resonant re-

flections are indicated. Weaker non-resonant reflection takes place when this criterion is

not met.

This thesis is the culmination of investigations of both resonant and non-resonant inter-

actions of waves propagating over rapid undulations on an otherwise flat bottom. Results

are compared numerically and in a laboratory study. A theory for the case of a mildly slop-

ing beach is developed as well. The possibility for large amplitude standing waves between

a series of undulations in front of a shoreline and wall is investigated.














CHAPTER 1
INTRODUCTION



Observations of naturally occurring phenomena often give rise to ideas of how man may

alter, control or redirect the forces of nature. Observation and explanation are the purpose

of science, application of such phenomena that of engineering. Coastal geomorphologists

have observed periodic shore parallel bars formed on mild sloping beaches on which plunging

breakers occur. There has been speculation that once one such bar existed, others would

form, propagating a bar field outward. Also, of interest to coastal scientists and engineers

was the possibility that once the bar field formed, resonant and non-resonant reflections of

surface waves propagating over the bar field would occur.

Several theories have been put forth as to the evolution of the observed bar fields. Evans

(1940) suggested that the first bar is formed when a plunging breaker stirs up sediment on

the bottom and the falling crest behind the breaker deposits the sediment behind the wave.

It has been suggested by Carter, Liu, and Mei (1974) that this breakpoint bar will initiate

reflection of incident wave energy seaward, setting up a standing wave pattern. Due to

Lagrangian drift, causing sediment to converge at the nodes and diverge at the anti-nodes,

additional bars may form. In addition to seaward growth, it will be pointed out in the

present study that due to the possibility of a trapped resonant wave field shoreward of the

barfield, the field may also grow shoreward, as observed by McSherry (1989). Hypotheti-

cally, the growth of the bar field is a self maintaining process, where, as the bar field grows,

stronger reflection occurs causing addition growth. The initial phase of this growth has in

fact been observed in laboratory studies by Davies and Heathershaw (1984).

The wave length of the barfield on the bottom has a direct relationship with the relative

strength of reflection of a surface wave of a particular frequency. As a scientific problem,










2

investigators have been interested not only in the evolution of the undulations, but in their

effects as well. The evident strength of reflection has captured the attention of coastal

scientists. In this study, one of the primary discussions will be the nature of the reflection

resulting from the existence of a periodic bottom disturbance. Two domains of reflection,

those being resonant and non-resonant reflection, have been identified by workers in the

field.

Resonant reflection may be described as a strong backscatter of wave energy due to a

tuning mechanism between the surface wave and the bottom undulations. Specifically, for

regularly spaced undulations, or bars, resonant reflection will occur when the wavelength

of the bottom undulation is half that of the surface wave, or, equivalently when twice the

wave number k of the surface wave equals the wave number A of the bottom undulations,

2k
= 1 (1.1)

The similarity to Bragg resonant reflection in crystallography, where strong backscatter of

certain frequencies of x-rays has been used to determine the spacing between atoms in a

crystal lattice, has brought that name to the resonant backscatter of water waves. Non-

resonant reflection, while not as strong but of no less importance, can be described as

reflection away from the resonant peak.

Once the science of a phenomenon is well on its way to being worked out, the inevitable

progression is the application of the new knowledge the task of the engineer. The develope-

ment of wave reflection technology is of interest to coastal engineers as a shore protection

measure where it may provide a means to redirect the destructive energies back offshore. It

will be shown later that it is possible to choose an artificial bar configuration such that more

undesirable frequencies may be more strongly reflected. Naciri and Mei (1988) have been

studying the possibility of using the reflective characteristics of a doubly reflective structure

on a subsiding bottom to protect the oil rigs in the Ekofisk field of the North Sea. Yoon

and Liu (1987) have suggested the use of barfields to replace harbour resonators.

The purpose of this study is not to invent specific applications of the theories introduced










3

above, but rather to explore the existing solutions to the reflection predictions and offer some

deeper insight to potential advantages and problems with employing such a mechanism as

an engineering application. The primary goal of this study is to present solutions, through

various means, to the problem of predicting the reflective characteristics of shore parallel

bars.

The obvious embarkation point of the journey though this thesis will begin with a

review of the investigations and discoveries of previous scholars. Non-resonant interaction

was studied and quantified by Davies and Heathershaw (1984), while Mei (1985) and Yoon

and Liu (1987) neglected non-resonant cases and concentrated on resonant reflection. Kirby

(1986) presented a general equation describing the interaction of waves propagating over

rapid undulations of small amplitude on an otherwise slowly varying bottom and solved it

using a numerical method.

Next, an extension of the non-resonant interaction theory of Davies and Heathershaw

(1984) will be developed to accommodate oblique incidence and to solve for the reflec-

tion from individual Fourier components of the bottom undulations. Multiple component

barfields will be compared to single component bedforms as previously studied in non-

resonant interaction investigations. In effect, it will be shown that this method will provide

the engineer with a tool to construct a tuned barfield.

Returning to the differences between the forms of the solutions in the literature review,

a comprehensive comparison of the existing resonant interaction theory by Mei, the newly

developed Fourier extension of the non- resonant interaction, and the numerical solution

provided by Kirby will be presented.

Recently, Davies et al.(1989) solved the wave equation in the long wave limit by trans-

forming the water surface variable t7 to a variable W = f21 that will be forced to zero

at a shoreline. It will be developed in the final chapter how this uniquely allows the solu-

tion of the wave field (neglecting energy dissipation) over bars on a mild slope continuing

to the shoreline. Also, the indication of shoreward propagating bars will be theoretically











supported.

Lastly, to add physical support to the menagerie of theory, data from a laboratory inves-

tigation are presented. While monochromatic waves were used in the wave flume, spectral

analysis was employed to ensure the assumption of a single frequency. An exposition on the

wonders and idiosyncracies of analysis of monochromatic fields with spectral analysis is also

presented. The results of the laboratory experiments are plotted against and compared to

theoretical results of the models.

1.1 Review of Weak Reflection Theories

1.1.1 Non-Resonant Interaction

Davies (1982) studied the interaction between surface waves and a finite periodic ripple

patch on an otherwise flat bed in a two-dimensional domain. The problem had previously

been examined assuming a ripple patch infinite in horizontal extent. However, the solution

of the problem is valid only for small reflections of O(e) < 1, away from the Bragg resonant

condition where 2k/A -- 1. k is the wave number of the surface wave and A is the wave

number of the sinusoidal ripple patch. The reflection coefficient is given by


2kD 2k sin(2k/A)N b7r
sinh 2kh + 2kh A (2k/A)2 2)
where Nb is the number of periods in the ripple patch and D is the amplitude of the ripples.

This theory breaks down where 2k/A -- 1, since R becomes unbounded as Nb increases.

Davies and Heathershaw (1984) re-addressed the problem in an effort to combine the

effects of non-resonant reflection and the Bragg resonant condition. The study involved two

particular cases, one in which no attenuation of the incident wave occurs as it propagates

over the ripple patch, allowing the wave transmitted past the barfield to be equal in ampli-

tude to the incident wave. Thus, if any energy were reflected, the conservation of energy

would be violated. To address this problem, another solution was posed in which an ad hoc

linear attenuation of the incident wave amplitude was imposed to achieve an energy balance

between the incident, reflected and transmitted wave-energy fluxes. It was further assumed













<--------------- /k ---------------



h
I



6(x) ,<------2T/A ------>


Figure 1.1: Domain definition sketch

that the flow is non-separating from the ripples and it is irrotational, thus no provision is

made for the thin boundary layer above the impermeable bed. The ripple patch (Figure

1.1) is the same as used in Davies (1982).

1.1.2 Formulation of Davies and Heathershaw for Non-Resonant Interaction

Constant water depth -h is assumed and the ripples are defined as the departure 8(z)

from this mean. The barfield 6(x) has characteristic small amplitude D such that

D
0(e) < 1. (1.3)

Water surface elevation is defined as r(z, t) referenced to the still water level, z = 0. Since

the flow is assumed to be irrotational in two-dimensions, Laplace's equation is satisfied by

the velocity potential O(z, z, t).

VZ = 0 (1.4)

Proceeding with a perturbation expansion of Y, r and 6 in powers of a small parameter e


= oe1 + e22 + ... (1.5)

7 = 1 + c2r 2 + ... (1.6)


6 = e61 + e262 + ...


(1.7)










6
The bottom boundary condition (to first order) which requires that flow normal to the

bottom must vanish on the boundary which would be present without the ripple patch, is

now treated as new source of fluid motion (in second order) on the plane surface, z = -h.

The boundary condition on the bottom may be stated


OzSZ = 0 on z = -h + 6 (1.8)

and the free surface boundary conditions


rt + 0z Oz z = 0 on z = (1.9)

gri + t + 2 = 0 on z=r (1.10)

The boundary conditions are treated by expanding the governing equations 1.8, 1.9 and

1.10 in Taylor series about y = 0. This allows the original nonlinear problem to be reduced

to sets of linear problems, grouped in terms of powers of e. The first order problem may be

stated


V241=0 in -h
l71t+0lz=0 on z=0 (1.12)

g7rl-ilt=0 on z=0 (1.13)

1 = 0 on z=-h (1.14)

which describes waves propagating over a flat bottom. The bottom boundary condition to

second order contains the effect of the bottom undulations on the water motion. Specifically,

the second order problem is solvable in two separable parts, one the Stoke's theory second

order approximation, and the other which expresses the interaction between the first order

motion and O(c) bed undulations. Pursuing the second of these, the governing equation

and free surface boundary conditions remain in essentially the same form


V2 2=0 in -h

(1.15)










7

772t+2z-=O on z=O (1.16)

g92-2 = 0 on z =O (1.17)

while the bottom boundary condition may be expressed


2z. + S1i.zz 161z = 0 (1.18)

The bed form required for Davies solution is that depicted in Figure 1.1. The assumption

is that the bottom undulations will be sinusoidal in profile and small amplitude. Also, the

incident wave is restricted to be normally incident on the ripple patch.

The assumption by Davies and Heathershaw (1984) that all reflection taking place is

of second order or O(c) with respect to the incident wave causes the violation of energy

conservation to leading order. To account for this, an artificially imposed energy attenuation

correction was added to the solution procedure. The effort was to calculate the energy

carried by the wave incident on the ripple patch, calculate the sum of the reflection and

transmitted wave energies, which from the assumptions made would be greater than the

incident wave energy, and adjust the amplitudes of the transmitted and reflected waves to

the point where energy is conserved.

Included in the studies mentioned above were investigations into the possibiltiy of ripple

patch growth seaward as a result of Lagrangian drift below the standing wave field in front

of the ripple patch. Laboratory studies of the reflection characteristics of a ripple patch

on a movable bed on an otherwise flat bottom indicated some propagation of the ripple

patch. Shoreward growth downwave of a barfield on a sloping beach has been observed

in a three-dimensional laboratory study (McSherry 1989), probably a result of trapping a

resonant frequency between the barfield and the shore. This phenomena will be discussed

further in a later chapter.

1.1.3 Miles' Oblique Surface Wave Diffraction

Miles (1981) solved the same problem but allowed an arbitrary bottom form and in-

cident waves propagating at oblique angles to the bottom perturbations. A form of the







8

was assumed, that is, incident and reflected waves are allowed at the offshore boundary

and a transmitted wave allowed at the nearshore end. The governing equation was reduced

to a solvable form by making the assumption again that the reflection is small. Miles

then employed a finite cosine transform, solved the transformed equations, then applied the

inverse transform to obtain the solution.

The results of Miles' solution are

R = [il(h + K1 sinh kh)]-'(12 m2) f e2itzS(z)dz (1.19)
J-oo

and

T = 1 [il(h + K~1 sinh kh)]-(12 m2) f 6(x)dz (1.20)

where

k tanh kh = w2/g = K, (1.21)

12 + m2 = k2 (1.22)

and R is the reflection coeffecient, T is the transmission coefficient and 6(z) describes the

bottom deformation.

It can be seen from the approximations R = O(e) and T = 1 + iO(e). Conservation of

energy implies IT12 = 1 IRI, which is not satisfied by the solution to O(E). Also, where

I = m (45* angle of incidence), R = 0.

1.2 Review of Strong Reflection Theories

While Miles and Davies and Heathershaw were exploring weak non-resonant reflection,

Mei (1985), Yoon and Liu (1987), and Benjamin et al. (1988) explored resonant interac-

tion between the surface waves and rapidly varying bottom undulations. Of these, Mei

in particular has performed a number of studies on resonant interaction. The inadequacy

of Davies and Heathershaw's theory to handle resonant reflection prompted further study

into resonant interaction and obliquely incident waves. Resonant interaction occurs where

2k/A = 1, and for the purposes of the present discussion, A will be understood to be the

wavenumber of a sinusoidal bottom perturbation.











1.2.1 Mei's Resonant Interaction Solution

Mei (1985) solved Laplace's equation with an undulating bottom as described by Davies

and Heathershaw, but solved assuming coupling between the incident and reflected waves,

requiring that they be of the same order. The governing equations were linearized with

respect to the mean free surface and the mean sea bottom. The velocity potential was

given by

V2, + A, = 0; -h < z < 0 (1.23)

The bottom boundary condition is given by


O,= -Vhh VhO + eVh (6Vh) + 0(C2); z = -h (1.24)

where

Vh = (1.25)
ax ayJ
and 6 = 6(x) describes the bottom pertubation. Employing the ray approximation and

allowing the wave to be modulated in time and space, the first order potential is taken as


() = (+eiS++ ...) + (-e'-i + ...) (1.26)

where S is the phase of the +z or -z propagating waves and the velocity profile 0 is given

by
S ig coshk(z + h) A (1.27)
2w cosh kh
where A' are the complex wave amplitudes.

The assumptions in the solution procedure are that the small order undulations are su-

perimposed on a slowly varying depth (mild slope), the form of incident wave is constructed

such that it may be modulated in time and space, and a small parameter e characterizes

the slope of the free surface, the mean bottom, the bar amplitude, and C/Cg. Lastly, the

bottom contours are shore parallel, although this condition is relaxed in a later paper on a

doubly periodic bottom.










10
Mei introduces a frequency-like term flo which is defined as the cutoff frequency.

wkD
n, = .k (1.28)
2 sinh 2kh (

The reflected wave is formulated such that its amplitude is 0(1) in anticipation of strong

reflection. The solution is restricted in the sense that reflection must be strong, ie. resonant

or close to resonance. Therefore, wave parameters are described in terms of their deviation

from the true resonant case. This process is defined as detuning the wave from resonance

where k represents the perfectly tuned or resonant wavenumber. The solutions are worked

out in form of ratios between the tuned and detuned parameters. The incident wave is

slightly detuned from the Bragg resonant condition such that its wave number is k + EK,

where K is 0(1). The detuning implies a frequency deviation of cl, where

n = C,K (1.29)

The incident wave potential is given by 1.23 and the amplitude by

A = Aoei(Kz-nt) (1.30)

where x and t are slow variables. The governing equation of the wave outside the domain

defined by the ends of the bar field is reduced to

a a
(a+C, )A = (1.31)

Over the bars, the governing equations become nonhomogeneous and coupled

a a
( + C, )A= -in,B (1.32)
a a
( Cg )B = -infA (1.33)

where A is the +x propagating amplitude and B is the -x propagating amplitude and

gk2D
0 = gA2D (1.34)
S4w cosh kh (1.34)
Continuity of A and B at the ends of the domain gives four conditions so the solution in

all three regions may be easily found, where B = 0 if z > L, so no -z propagating waves

occur in this region.










11
The solution is split into four regions with respect to the cutoff frequency fo. The

cutoff frequency provides a quantitative point at which the resonant approximation becomes

unreliable, that is, where R becomes small. The reflection coefficients as a function of

distance into the bar field for the four regions are defined as follows

i) n > no Detuning frequency above cutoff

R) = -ilo sin P(L z)
PC, cos PL il sin PL

where the envelope wavenumber P is

= (2 n2)1/2
P (1.36)

and 0 < x < L and L is the length of the barfield.

ii) 0 < n < fo Detuning frequency is below the cutoff

Denoting

(M2 n2)1/2
Q = iP, where Q = ) (1.37)
Cs

the reflected wave amplitude is

olo sinhQ(L x)
( QC cosh QL + n sinhQL



iii) n = 0 Perfect tuning

Q reduces to K
-isinh (L x)
R(x) = ch (1.39)
cosh L


iv) 0 = no At the cutoff frequency

Q -+ 0
R(x) -in(L z)/Cg
1 inL/Cg

The reflection coefficient measured upwave of the bars is given by R(0).










12
The above presentation is for normally incident waves over a sinusoidal barfield on

an otherwise flat bottom. Mei also provided a solution extending the theory to oblique

incidence and a mild slope. The reader is directed to the original work for further details.

It will be shown in a later section that the resonant peak may be somewhat underes-

timated since the non-resonant interaction of severely detuned modes are neglected, and

thus not added to the resonant peak. This is especially apparent when additional Fourier

components are added to the bottom profile, in that only the dominant Fourier component

of the bottom perturbation is considered in the resonant interaction. As will be pointed

out later, second harmonic resonant peaks may become important for arbitrarily shaped

bottoms.

Benjamin et al. (1987) provided a similar solution to the resonant case but instead of

using a detuning variable, allowed the wavenumber and angular frequency of the incident

wave to be defined as physical parameters. The cutoff frequency feature does appear as

in Mei's solution. The solution is arrived at using a conformal mapping procedure. The

interested reader is encouraged to consult the original work.

1.2.2 Depth-Integrated Equation for Small Undulations on Mild Slopes

Kirby (1986) derived a depth-integrated mild slope equation for waves propagating

over an arbitrarily shaped bed restricted to small amplitude. The equation in its homoge-

neous form is Berkhoff's (1972) equation for waves propagating over a mildly varying slope.

The equation is solved in Chapter 3 using a finite difference scheme once the appropriate

boundary conditions have been established.

Berkhoff's equation is stated

V (CCgVr7) + k2CCgt = 0 (1.41)

where

C= k (1.42)

aw = 2( (1.43)
Cgg-- = T1 snh 2k (1.43)









13
The depth-integrated equation applies to waves propagating over small amplitude bed un-

dulations superimposed on a mild slope. The smallness of the rapid variation allows the

bottom boundary conditions to be expanded about the slowly-varying mean depth. Using

the Green's formula approach of Smith and Sprinks (1975), let h'(x, y) denote the total still

water depth where

h'(z, y) = h(x, y) 6(x,y) (1.44)

and h(x, y) is a slowly varying depth satisfying the mild slope condition

Vhh
h < 1, (1.45)

where

Vh = a, (1.46)

and 6(x, y) satisfies a small amplitude condition. Figure 1.1 illustrates the individual depth

components.

The problem is considered linear in wave amplitude but the first-order terms in bed-

undulation amplitude are retained, where


0 \(Vh s 0 (kS) < 1 (1.47)

Linearizing the free-surface boundary conditions and expanding the bottom boundary con-

dition about z = -h, yields to O(kS)


V2+ ,'=0 on -h< z < 0 (1.48)

ftt +gz = 0 on z=0 (1.49)

= -Vhh* Vh + Vh* (6Vh) on z= -h (1.50)

Equation 1.50 has been given by Mei(1985) and separately by Davies and Heathershaw

(1984). To leading order (6 -+ 0), the solution to 1.48, 1.49 and 1.50 is


(x, z, t) = f(x, z)(x, t) + E non propagating modes + O(k6)


(1.51)









14
where f = cosh k(h + z)/ cosh kh is a slowly varying function of z and y, and where

w2 = gktanh kh (1.52)

locally, with w being the fixed angular frequency and k the wavenumber. Using Green's

second identity to extract the propagating component of q


f h0 0
f .dz J Ofzdz = [fVz 1/} (1.53)

Manipulating the integrals and neglecting terms of second order in k6 yields


tt Vh (CCgVh) + (w2 k2CC,)4 + g Vh (8Vh. ) = O(ks)2 (1.54)
cosh2 kh

Here 4 is the velocity potential in the plane of the free surface, C = w/k, and Cg = 8w/ak.

Also note, neglecting the S terms yields Berkhoff's equation for the slowly varying bottom

alone.

In the absence of currents, 4 is simply related to surface displacement r through a

constant of proportionality, thus q is substituted in place of 6. In the monochromatic wave

case, the spatial surface displacement j can be described as

7(x, y, t) = ,(,y)e-iwt (1.55)

Substituting tr for 4 in equation 1.54, in reduced form is given by

V (CCV) + k2CC,9i 9 V (6SV) = 0. (1.56)
cosh2 kh

1.3 Indirect Solution of Surface Elevation

Davies et al. (1989) solved the shallow water wave equation by making the substitution

7 = f-1/2W (1.57)

where


f = g(h 6)


(1.58)











and where f is the shallow water limit of


f = CC (1.59)
cosh kh

and rearranging the equation into a solvable Mathieu equation form. The principal as-

sumption made is that the solution is valid only for long waves over an undulating bed.

As pointed out in a discussion by Kirby (1989), when the mild slope equation is restated

using the above transform, and is solved for the case of 2k/A : 1 the non-resonant solution

of Davies (1982) is recovered. Additionally, solving the case of 2k/A = 1, Mei's resonant

solution is recovered.

The advantage of using the substitution above is that it allows for a solution at the

shoreline with no restrictions on tr or its derivatives. A more complete formulation of the

equation and a numerical solution to it will be presented in a later chapter.















CHAPTER 2
EXTENSION OF NON-RESONANT INTERACTION THEORY



2.1 Introduction

Non-resonant interaction theories are extended to allow waves incident at oblique an-

gles over a one-dimensional topography. Then a bottom with regularly spaced bumps is

decomposed into individual Fourier components, the contribution to reflection is calculated

and the reflection coefficient calculated as the sum of the contributions.

2.2 Governing Equation

The solution given by Davies and Heathershaw results from a perturbation expansion

to second order of the components of the wave field propagating over a sinusoidal bottom of

finite length. The major assumption made is that all reflection takes place at O(c), or the

wave is weakly reflected. Thus the leading order component incident on the ripple patch

propagates over it unabated. As recognized previously, this assumption violates energy

conservation in the domain if any reflection were to take place. To account for this, Davies

and Heathershaw artificially impose a linear attenuation on the +x propagating wave, then

adjust the solution to match the requirements of energy conservation. Two additional

drawbacks to this solution are its inability to adequately handle the Bragg resonant case

of strong reflection in the area where 0(1) reflection occurs, and in its original form, the

inability to solve the problem allowing waves incident at oblique angles.

Miles (1981) solved the same problem for an arbitrary bottom and oblique incidence.

The solution method employed involved assuming a form of the incident, reflected and

transmitted wave fields and applying them to the problem. The solution method is very

similar to the one explained below. It should be noted here, however, that all three solutions,









17
(Miles, Davies and Heathershaw, and the present work) are in agreement in the final result.

Begin with the governing equation developed in Kirby (1986),


V [CC,V4] + k2CC, = V (6VO) (2.1)
cosh2 kh

where C and Cg are wave celerity and group velocity, and k is the wavenumber derived

from the local value of the slowly varying depth h. The total depth is given by


h'(x,y) = h(x, y) 6(x,y) (2.2)

where S(x,y) is the rapid bed undulation. Consider the case of undulations placed on an

otherwise constant depth h; let

a = g/ cosh2 kh = constant, (2.3)

then C, C, k are all constants as well. This allows 2.1 to be simplified to the equation


V2+ + k20 = V(b6V) (2.4)
cc,

Simplifying further, let
a = 4k
a' (2.5)
CC, 2kh + sinh 2kh (
Then, equation 2.4 may be rewritten as


V20 + k2A = a'V(SVO) (2.6)

2.3 Two Dimensional Wave Field

The problem will be extended to solve for wave propagation over a one dimensional

topography. Requiring 8 to be a function of z only, equation 2.6 becomes


V20 + k24 = a'S.Cz + 'S3V2' (2.7)

Assume the form of the general solution of 0 to be


0(x, y) = (x)eim'


(2.8)











where

m = ksin (2.9)

is a constant following Snell's law. Then

k' m' = k2(1 sin2 0) = k2 cos 2 = 12 (2.10)

Equation 2.7 now becomes

zz + 1 = 5a6'S + a'84) m' c'8 (2.11)

Allowing 8(x) -- 0 for a physically flat bottom, the solution would be given by

4 = Ae"2 + Be-'1i (2.12)

A and B will be allowed to have complex values to allow for relative phase shifts for the

most general solution.

Now, developing the boundary conditions at the ends of the domain for the case of a

device causing reflection but still allowing some transmission at the shoreward limit yields

-(x -- -oo) = ei'z + Re-"z (2.13)

where the incident amplitude is taken to be 1 and R is the amplitude of the reflected wave.

The reflection coefficient is then given by IRI. Likewise, with T being the transmitted

amplitude, the boundary condition is

(x -+ oo) = Tei"= (2.14)

Strictly, conservation of energy requires

RI2 + ITI2 = 1 (2.15)

if no energy attenuation occurs in the domain and the mean depth doesn't change.

For an arbitrary but finite bottom undulation in (x), assuming a'6 ~ (D/h) is small,

or la'8I < 1, where D is the amplitude of the undulation, let


c(a'S)* = (a'S)


(2.16)











and expand in powers of e

4 = o + e
R = Ro + cR + R2 2+ ..... (2.18)

T = To + eTi++ T2 + ..... (2.19)

Collecting first order terms gives

o,zz + 120o = 0 (2.20)

Assuming reflection is of O(c) at leading order, the solution of 2.20 is given by

o = ei't, R = 0, To = 1 (2.21)

Collecting O(E) terms,

l,zz + 12,1 = a'So.,z + a'S0o,zz m2c''68o (2.22)

Substituting 2.21 into 2.22 and rearranging gives

4,1.. + 121 = [ilar' o'(m 2 + 12)] eil' (2.23)

where (m2 + 12) = k2. The general form of the solution is

01 = cleilz + c2e-iL + 1,p (2.24)

where Olp is the particular solution.

To solve the non-homogenous part, let

q(z) = [ila's, a'S(m2 + 12)] ei' (2.25)

The particular solution is obtained by variation of parameters, and is given by

01 = ei [cg f q(F)e-iedt] + e-'t [c + q()Oe" da (2.26)
21 -00 21 2-1
where e is a dummy variable of integration. From the expansion at the offshore limit,

x -+ -00


4o + c<1 + ..... = eiz + Roe-itl + cRie-its + O(c2)


(2.27)









20
the homogeneous solution of the second order terms gives


C1 = 0 C2 = R1 (2.28)

At the nearshore limit, x -+ oo,


0o + e1i + ..... = Toeilz + eTleilz (2.29)

so

01 = Tiei2; z = oo (2.30)

From equation 2.25 and the two solutions for I1 above, the reflection and transmission at

0(c) are given by

R = 2 q(x)e"zdx (2.31)

= _q(x)e-itzdz (2.32)

Evaluating the expression 2.30 for R1 yields


R i (12 m2) f_ 6(z)e2itdz (2.33)
21 -oo

Equation 2.32 is an expression for the weak, or non-resonant, reflection of a wave of

wavenumber k over an arbitrary bottom at any angle of incidence. In its current form, it

provides the same results as determined by Miles (1981), and restricted to normal incidence

and a sinusoidal bottom would be identical to the result of Davies and Heathershaw (1984).

2.4 Solution for Periodic Bars

For the case of a sinusoidal perturbation (as assumed by Davies and Heathershaw)

6(x) = D sin 2x/L (2.34)

where L is the spacing between crests and A = 2r/L, it can be seen that the solution of

the non-resonant case by Davies and Heathershaw will be recovered in the case of normal

incidence.













1.0



0.8



6 0.6
o
u
S0.4
U
a@


0.2



0.0
0.5 1.0 1.5 2.0 2.5
2k/A A=2rf/L,

Figure 2.1: Reflection of single mode bottom



S 2kD m2 21 sin Nbr (2.
sinh2kh+2kh ( I () 2-1

where I = k. The result for this case is plotted in Figure 2.1. The solution at resonance

21 = A is given by

IRI = 2 ( m2) (2.36)
21 2
For the plotted case, the bottom configuration is


6(z) = 0.05msin 2rz/1.Om (2.37)




2.5 Fourier Decomposition of the Bottom

Exploring now the case where 6(z) is a field of discrete cosine shaped bumps with

arbitrary, but even spacing, such that

h(x) = 0Dsin(2r( bL2)/bL) ; (nL/2 bL/2) < x < nL/2 + bL/2 (2.38)
0.0 ; otherwise (2.38)


n = 0, 1,2,3

















YD
I I I
L<- ----. ,- -bL


z






Figure 2.2: Definition sketch of four cosine bumps on a flat bottom


where L is the spacing between crests and bL is the length of a single undulation, it can be

seen that the field may be described by a Fourier series expansion,

2nrx (2.39)
8(z) = Dcos (2.39)
n=0

The Fourier coefficients for the case of four cosine bumps shown in Figure 2.2 on a flat

bottom are given by


Do = (2.40)
-D
Di = (2.41)

Dn = D c 2 (1 +cosnr) (2.42)
7r(1 n2)

The cosine transform is used in this case since it is even about the starting point of the

domain.

R1 becomes

NbL 0[
R = (2 m2') [ D, cos 2n ]e2 rx (2.43)
n=o

where Nb is the number of bars in the field. For purposes of creating the most effective

design, it would be helpful to determine the relative contribution of each Fourier component










23
to the reflection assuming a constant wave field, that is


-'(12 m2) = constant (2.44)

Combining this constant with Dn, let

an = Dn '(2 m2) (2.45)

2.39 reduces to

R = an N cos e e 2dx (2.46)
n=0
Letting
2r
= A (2.47)

the integral part of the solution may be expressed by

I= f (cos nAx)e2"'dx (2.48)

Expressing the cosine term in its exponential form, I becomes

1= JNbL (ei(21+n)z + e(21-n)) dx (2.49)

2.5.1 The Resonant Case

In the special case of

21 = nA (2.50)

it can be seen

ei(21-n) = 1 (2.51)

so I becomes

[ei(21+nA)NbL 1 + NL (2.52)
21+ nA 2
Since
21
n (2.53)

I can be simplified to

I= i 1]+ + (2.54)
41 2










24

By factoring ei2rnN out of the bracketed term, thereby forcing it into a sine form with an

argument of 2rnNb where nNb are integer, it can easily be shown that the final form for

the case of 21 = nA is
NbL Nblr (2.55)
I= (2.55)
2 A

2.5.2 The Non-Resonant Case

In the case where the x component of the wave is away from resonance or

21 nA # 0 (2.56)

I will change considerably. The integral portion is stated

1 [einN22rei()2Nb e-inNb2r ei( )2rN (2.57)
12iA 21+ nA 21- nA

Letting
2 = (2.58)

2.58 becomes
-2-7 )(-Nb (eir'Nb e-irNb) (2.59)
z = 72)
X(n2 e I 2 (2.59)

Applying trigonometric identities and rearranging, gives
2
I= eiNbL sin INbL (2.60)
(-12 n2)

2.5.3 Full Solution

For a bottom with the positive branch of a cosine curve imposed on a flat bottom, the

reflection coefficient for a given wavenumber component of the wave field R, would be

calculated by summing the effect of all non-resonant Fourier components of the bottom

plus the effect of the resonant component. The full solution is given by


R= a1 ( 2 eilNbL sinlNbL+ |a, N In= (2.61)
E, U I(72 y n)


where n is the nth Fourier component of the bottom perturbation, A is 27r/L with L being

the bar spacing, Nb is the number of bars in the field, and I = k cos 0.


I































0.5 1.0 1.5 2.0 2.5
5



















0.5 1.0 1.5 2.0 2.5
2k/\ \=2n/Lb
Figure 2.3: Contributions to reflection by individual Fourier components

Plots of the contributions to the reflection of the first five modes of the barfield are shown

in Figure 2.3. The abscissa is marked for the mode, with the space between them allowed

for the reflection coefficient plot. The ordinate is referenced to the surface wavenumber over

the wavenumber of the fundamental mode of the barfield. Additionally, the total reflection

coefficient for a normally incident wave, O = 0, is shown in Figure 2.4. The ordinate units

again are 2k/A, where k is the surface wavenumber, and A is the wavenumber of the bar

spacing, or A = 21r/L. This unit designation will be irrelevant for mixed spaced bars, thus

the ordinate units would be changed to period, T, or some equally pertinent parameter.

The calculation of the reflective characteristics of a given bottom perturbation in this

manner allows the prediction of which components of the bottom may be most strongly

reflective for a particular wave frequency. However, the true power of this technique is that

it would easily lend itself to customizing a barfield to be more strongly reflective of certain

frequencies, simply by building the Fourier series of the bottom that would best reflect

the more undesirable frequencies. While the maximum reflection for a particular frequency

occurs when one of the primary components in the bottom has wavelength twice that of the











26

i. i


0.8-


"
0.6 --



S0.4



0.2 -



0.0
0.5 1.0 1.5 2.0 2.5
2k/A X=27T/L
Figure 2.4: Sum total of the reflection from individual components

surface wavelength, it should be noted as additional components are emphasized the peak

reflection of other de-emphasised components are reduced.














CHAPTER 3
COMPARISON OF SOLUTIONS



3.1 Introduction

The development of a finite difference scheme to solve the mild slope equation (Kirby

1986) will be presented. The numerical results will then be compared to the analytic

solutions of Mei (1985) and the extension to the non-resonant interaction solution.

3.2 Derivation of a Complete Governing Equation

Kirby (1986) developed an extension to the mild slope equation of Berkhoff (1972)

for shoaling waves to allow rapid, small-amplitude variations in depth. This equation was

presented in Chapter 1. This equation is again stated

V (CCV ) + k'CC, coshh2V (SVj) = 0. (3.1)

The undulating bottom on a mild slope is described by

h'(x, y) = h(z,y) 6(z,y) (3.2)

where h(x, y) is the slowly varying component, and 8(z, y) describes the undulations. Re-

stricting the model topography of equation 3.2 to one dimension in the z direction, equation

3.1 is reduced to

CC(V2 + k2 ) + (CC g)V ---- = 0 (3.3)
CC()+ ( cosh kh cosh2 kh

Allow the two dimensional surface j to represent a wave of arbitrary amplitude and

frequency which will refract over the slowly varying topography h(x) according to Snell's

law of refraction. For a given wave of frequency w incident at angle Oo in deep water, Snell's

law is

ksin = ko sin o (3.4)











where
W2
ko = (3.5)
9
Split the local wavenumber into x and y components


I = k cos 0 (3.6)


m = ksin 0 = ko sin o (3.7)

and

k2 = l' + m2 (3.8)

Since m I ko from equation 3.7, may be expressed as


ko
)= i](Z, m)etmudm (3.9)
J-ko

Substituting 3.9 into 3.3 then yields a second-order ODE for f(m)


(CCIh (r .). + [12CC, + gm2' ]c = 0 (3.10)

Equation 3.10 is a well posed problem for the reflection of waves incident at any angle after

specifying boundary conditions. Formally, the problem is posed on the interval -oo < x < 0,

where x = 0 is the shoreline and x = -oo is deep water. The solution over this interval

is, however, unwieldy. A simplification at the limits of integration is attained assuming the

incident wave condition at some finite distance offshore, xl, is known, and neglecting the

region of the surfzone, establish a second station, z2, between the topography in question

and the surfzone. The energy propagating past x2 is assumed to be fully dissipated in

breaking. Equation 3.10 is solved only in the domain zx < < z2 with boundary conditions

known at xz and z2; thus the reflection from shoreline conditions will not be included.

The boundary condition at the shoreward station, z2, is assumed to be a wave propa-

gating out of the solution domain in the +x direction


r& = l()4 ; x = 2 (3.11)










29

At the seaward station, xl, the boundary condition ijx is assumed to be a superposition

of an incident wave rj propagating in the +x direction from deep water, and a reflected

wave ir propagating in the -x direction out of the solution domain. The incident wave rj

is assumed to be known, and the reflected wave lr must satisfy a radiation condition for

propagation out of the domain


r -ix)r ; x= 1 (3.12)

Noting that

i (3.13)

and substituting into 3.12, the condition at zl is thus described.


i&z = il(2i 4) ; x = x1 (3.14)

The problem to be solved is fully specified by equation 3.10 and the boundary conditions

3.11 and 3.14.

3.3 Numerical Approximations

Proceeding further from Kirby (1987), the problem specified in the previous section

may be solved using a finite-difference scheme. The superscript will be dropped here and

the notations

p = CC, (3.15)

S= (3.16)
Scosh2 kh

are defined. The domain xl < x < 22 is discretized according to


xi= zX + (i + 1)A ; 1 < i < n (3.17)


where

Az = z2 (3.18)
n-1

All other coefficients and the variable 97 are affected by local conditions in the grid and

are defined in discrete form at the grid locations x'. From this, a centered finite-difference











scheme is developed and is given by
(p'+l +P')(1'' ') (P +P- _)(2 17'-1)


7l +
2Ax2

[(li)2pi + m2'qi6i] = 0 (3.19)

Equation 3.19 may be simplified to the form

A'ti-1 + B'i + C'+1 = 0 ; i =2,3,...n 1 (3.20)

where

A' = p +p i-1 (i +i-1) (3.21)

B' = -(pi+ + 2p' p'-1) + '(i+1 + 26' + -1)

+2Ax2[(l')2p' + m2'i5'] (3.22)

Ci = pi+ +pi (6i+1 + i') (3.23)

In order to simplify the application of the boundary conditions, the input topography

is restricted to a flat bottom at the edges of the domain;

h1 = h2 (3.24)

h"-1 = h" (3.25)

The bed undulations are also subject to this restriction in order that the waves radiating

at the boundary are not interacting with the rapid variations. Thus it is also required that

61 = 62 = 6"-1 = 6n = 0 (3.26)

Now to express the boundary conditions in finite difference form, equation 3.11 is

}"(1 a") = rn-1(1 + an) (3.27)

where

an= x"2 (3.28)
2


I












Defining 3.11 in the finite difference form,

B" = 1 a" (3.29)


An = -(1 + a") (3.30)

Likewise, at station xl the boundary condition may be restated as


(1 + a')t2 (1 aI) 1 = 2a(m)cl[e2al + 1] (3.31)


where

1 = 'i (3.32)
2

and it is assumed the incident wave is described by


7i(x) = a(m)eil(z-21) (3.33)

Again, putting 3.14 into the finite difference form gives

B' = -(1-a1) (3.34)

C' = 1+a1 (3.35)

D1 = 2a(m)a1[e2l + 1] (3.36)


The problem may be written in the form of a linear matrix equation

AT] = D (3.37)


where D is a column vector with D2 Dn = 0, r is a column vector with elements r1 n",

and A is a tridiagonal matrix with diagonal vectors A', B' and C'. The solution is obtained

using the double sweep algorithm as given by Carnahan, Luther and Wilkes (1969).

Reflection and transmission coefficients may be determined once the solution for r has

been calculated. Two estimates for the reflection coefficient R are obtained at zl. From

equation 3.31 rqr at xl may be written


= 1 a(m) (3.38)


I I










32

2 = 2 a(m)e2'1 (3.39)

Define the two estimates for reflection coefficients as


R 1 (3.40)
a(m)

R2 r 1 (3.41)
a(m)
and R as the average of R1 and R2. Transmission coefficients at X2 are estimated by

1-I"-'1
Ti = (3.42)
a(m)

T2 = (3.43)
a(m)
which are likewise averaged to obtain T. A test of the accuracy of the solution is obtained

by checking the conservation of energy requirement


R +T ,c,1 )= 1 (3.44)


By applying this model at a number of discrete frequencies and angles of incidence, predic-

tion of the reflection characteristics for a frequency and dimensional spectrum may be built.

The results of this full numerical solution will be used to compare against the oblique and

arbitrary bottom extension of non-resonant solution, and the resonant detuning solution.

3.4 Comparison to Existing Theories

The numerical solution to the mild slope equation offers a method to calculate the

reflection coefficient valid for all values of 2k/A. In this section, the numerical solution will

be used to compare the existing resonant and non-resonant interaction theories.

3.4.1 One-dimensional Wave Field

The initial investigations of this topic concentrated on bottoms of sinusoidal form.

Figure 3.1 is a plot of the three methods described previously, Mei's resonant interaction

(Mei) presented in Chapter 1, the extension of the non- resonant interaction (Non-Res

Extension), and the numerical solution of the mild-slope equation (Numeric). It can be


I











33

1.0


-- Non-res Extension-
0.8 .. ei
S\ ---- Numerical j


:. 4


0. 2

i- -






0.0
0.5 1.0 1.5 2.0 2.5
2k/A A=27r/Lb
Figure 3.1: Reflection coefficient vs. 2k/A for sinusoidal bottom. h(x) = 0.15m, four cycles
6(x) = 0.05m sin 2rf/1.0m


seen from figure 3.1 that the resonant interaction agrees well with the numerical solution

near the resonant peak. Conversly, the non-resonant extension solution agrees well in areas

of small reflection, except for a slight shift in phase with respect to 2k/A. It is also very

obvious that the non- resonant solution severely overpredicts the reflection at the resonant

peak.

Now a case of discrete but evenly spaced bumps will be investigated. The bedform

used in these calculations are identical to that described in chapter 2. It should be noted

that while the numerical solution will calculate the reflection coefficient using a discretized

bottom, the non-resonant extension and Mei's solution is a summation of the reflection

coefficients from the Fourier components of the bottom. From the plot of the numerical

solution and non-resonant extension in figure 3.2, it can be seen that a second peak of

substantial reflection occurs due to the interaction with the second Fourier component of

the bar field. The resonant peaks for the first two Fourier components were calculated inde-
















-- Non-res Extensioni
Mei
---- Numerical


0.2- /



0.0
0.5 1.0 1.5 2.0 2.5
2k.,,\ ,\27T/ .h
Figure 3.2: Reflection coefficient vs. 2k/A for four cosine bumps.
6(x) = 0.05mcos2r/1.0m,A = 2r/LL


h(z) = 0.15m,


pendently assuming the theory to be valid near resonance. Agreement between all theories

is poor. Surprisingly, the non-resonant theory resonant peaks show smaller reflection than

the numerical solution. It is possible that first order solutions may not adequately predict

the reflective characteristics of such a field.

3.4.2 Obliquely Incident Waves

Bars placed in the environment will have waves incident at all angles. The contour

plots to follow are reflection coefficient solutions for various wave numbers k propagating at

angles 0 from normal incidence. The bottom form assumed for these plots is a shore parallel

(8 = 6(z)) sinusoidal bar field. The solution for the extension of non-resonant theory for

obliquely incident waves is presented in Chapter 2 and the mild slope solution is presented

in previous sections of this chapter. The extension of Mei's resonant interaction theory is

achieved by allowing

2 gA Ahi.
w = cA (tanh )h (3.45)
2 cos O 2 cose 0v


1 0 .( . .


0.8
0.6



0.6 r-

( -
'V-
0 !






















S45









90
0.5 1.0 1.5 2.0 2.5
2k /A
Figure 3.3: Mei's resonant interaction theory reflection coefficient vs. angle of incidence 0
and offshore wavenumber k,

and the cutoff frequency to be defined by

cos 20
o = coos:- (3.46)
cos2 0



It can easily be seen from these plots, as the theories indicate, that the bar field is

invisible to waves incident at 45 degrees.

Figures 3.6, 3.7 and 3.8 are plots of the differences of the predicted reflection between

theories. Angles of incidence range from 0 to 450 since the large values of reflection and

slight phase shifts at higher angles may cause the differences to be as large as the peaks,

thus the additional information would be irrelevant.

3.5 Application of the Solutions

Kirby (1987) presented model test examples for directional and frequency spectra with

various bottom configurations. Of principal interest from a practical engineering standpoint,















--


a 45




0


90
0.5 1.0 1.5 2.0 2.5
2k /A
Figure 3.4: Non-Resonant Extension, reflection coefficient vs. angle of incidence 0 and
offshore wavenumber ko

are the cases of discrete artificial bars. The probable prototype design is a series of artificial
bars consisting of specifically shaped bumps whose longitudinal axes are shore parallel and
are laid on the bottom at predetermined spacings in the offshore direction. The process
of determining the optimum spacing is defined by Kirby as tuning the barfield. The study
assumes that the bed is non-movable, thus the seaward propagation of additional bars in
the form of sand waves as proposed by Davies and Heathershaw (1984) and scour between
the bars will not affect the reflection characteristics of the topography. In addition to the
directional spectrum comparison with previous solutions to the problem, the numerical
model was used to predict the reflection characteristics of a bar patch to be verified in
a laboratory study using normally incident waves. The laboratory study is presented in
Chapter 5.






























-o
C 45

C







90
0.5 1.0 1.5 2.0 2.5
2k /A





Figure 3.5: Numerical Solution, reflection coefficient vs. angle of incidence 0 and offshore
wavenumber ko


















2.5























0.5
45
0 Angle of Incidence 6



Figure 3.6: Non-Resonant Extension Numerical Solution vs. angle of incidence 0 and
offshore wavenumber ko























2.5















C\2












0.5
0 Angle of Incidence 0 45



Figure 3.7: Non-Resonant Extension Mei's Solution vs. angle of incidence 0 and offshore
wavenumber ko


I















CHAPTER 4
ANALYSIS TECHNIQUE FOR LABORATORY STUDY



4.1 Introduction

A formulation for the calculation of two wave field spectra travelling in opposite di-

rections to be used in analyzing the laboratory data from the experiments described in

Chapter 5 is developed. The method used is a three point method using a least squares

analysis for decomposing the measured spectra into incident and reflected spectra (Funke

and Mansard, 1980). This method requires a simultaneous measurement of the wave field

at three positions which are in reasonable proximity to each other and in a line parallel to

the direction of wave propagation.

4.2 Theoretical Background

Although these calculations may be made by measuring the wave field with two gages of

known distance apart and solving two linear equations directly, Funke and Mansard (1980)

put forth a method for resolving the wave train spectra using multiple gages to measure the

wave field and a least squares fit to resolve the incident and reflected wave spectra, in an

effort to improve accuracy and reduce sensitivity to signal noise and non-linearities of the

waves. The theory makes use of the axiom that an irregular sea state may be described as

the superposition of an infinite number of discrete components
0o
rl = n (4.1)
n=O

where

rn = Anei(knz-wt'), (4.2)

and the assumption that each component will travel at a unique speed in a given water

depth. The superposition will result in a time series qr(t),rl2(t), and rl3(t) of the water

41


-U










42
elevation at each gage position. The spacing between the gages is known, and wave celerity

may be determined by



C = (4.3)

where
27r
W= (4.4)
T
where T is the wave period. Solving the implicit equation


w = gk tanh kh (4.5)

for k iteratively, where g is the acceleration of gravity, k is the wave number, and h is the

water depth, it is possible to calculate the phase relationships between the wave trains as

they pass the probes.

Beginning by executing a Fourier transform on each signal,


B(w) = f rl(t)e-tdt (4.6)

the discrete Fourier components may be resolved and written in polar form as


Bp,n = Ap,ne'iap" (4.7)

or in rectangular form as


Bp,n = Ap,n cos ap,n + iAp,n sin tp,n (4.8)

where Ap,n is the amplitude of the nth component at gage p, and a is the phase relative to

the time origin of the record. The Fourier transform will enable the calculation of half as

many frequency components as data points, N.

These coexisting amplitude-phase spectra determined at the gage positions are a result,

as stated above, of the superposition of the discrete frequency components, and are in fact,

each a measurement of the same wave fields. The goal is now to separate out the two

interacting fields, those being the incident and reflected wave fields. Making use of the


I










43
dispersion relationship, it can be seen that it is possible to calculate the phase relationships

of each component as they are measured at each gage. By assuming superposition of two

wave fields travelling in opposite directions, the time series will be
N N
1(t) = Ai,e-(k.'-z-t) + AR,ne-i(kn(+2zrL)+wnt) (4.9)
n=l n=l
where AI and AR are the component amplitudes of the incident and reflected spectra, and

xzr is the distance from gage 1 to the point of reflection, arbitrarily set at the center of the

bar field. The record at the second gage will be identical in form, except that the phases

will be



PHI,12 = kn(x + x12) wnt (4.10)

for the incident wave train, and


PHR,12 = k(z + 2(zri z12)) wnt (4.11)

for the reflected wave train, where z12 is the distance between gages 1 and 2. The phases

will be likewise for the third gage record, with the obvious replacement of a 3 where 2

appears.

The phase lag between probes is preserved in the Fourier transform, and since it is only

these that are required to complete the calculation, the initial phase, or the phase at the

first gage can be factored out of each component at each gage. Thus, with phases referenced

to the phase at the first gage, the spectrum at a given gage may also be described by


Bp,n = ZI,neik'z" + ZR,ne-iknzlP + ZN,p,n (4.12)

where Z is the nth Fourier component of the wave field, k is the wave number of the nth

component, and XIp is the distance between the first gage and the gage in question.

It can be seen that, given only two gage spectra of known distance apart, the simulta-

neous equations may be solved for Z1 and ZR. However, to improve accuracy, additional

gages may be added, and Zi and ZR solved for using a least square error approach, where


M










44

ZN is the error spectrum for a particular gage. Following directly from Funke and Mansard

(1980), define

n = (4.13)
X12

kn = (4.14)
X13
Equation 4.12 may be restated for all three gages as


e1,n = ZI,n + ZR,n Bl,n (4.15)

e2,n = ZI,nei=' + ZR,ne-i'# B2,n (4.16)

e3,n = Zl,nei7" + ZR,ne-"'" B3,n (4.17)

where

p,k = -ZN,p,n + fe(ZI,n,ZR,n) (4.18)

where fe is an expression for the error associated with the entire domain, thus common to

all three gages.

Now a least squares fit may be used to find those values of ZR and Z1 for which the

sum of the squares of Ep,n, for all values of p is a minimum. This will occur at


fe(ZI,n, ZR,) = 0. (4.19)

Therefore, it is required that the sum of the squared error over each gage

3 3
S(ep,n) = E (ZI,ne',*" + ZR,ne-'""' Bp,n)2 (4.20)
p=l p=l

be minimized, where Op,n is either p or 7.

It is assumed that a minimum will be reached when both partial derivatives are zero.

Differentiating 4.20 with respect to Z1 and ZR results in,

3
E (ZI,ne"'," + ZR,ne-ip,n Bp,n)ei,'" = 0 (4.21)
p=l
and
3
S(Z,nei'- + ZR,ne-i pb Bp,n)e-iO'" = 0 (4.22)
p=l


I









45
Expanding the terms in the sum and rearranging terms results in two equations

Z1,(1 + ei2P* + ei2,7) + 3ZR,n = B1,. + B2,nei" + B3,nei" (4.23)

ZR,(1 + e-i2'f + e-i27") + 3Zi,n = Bl,n + B2,ne-i" + B3,ne-i'" (4.24)

which may be solved simultaneously in terms of P, 7, and Bp,n. Let

-3ZI,n + Bl,n + B2,ne-i' + B3,ne -in.
Z (1 + e-i2 + e-i2n) (425)

substitute 4.25 into 4.23

Z,1+ e + e.) 3(-3ZI,n + BI,n + B2,ne-iS + B3,ne-") )
ZIn(1 + +(1 + + e-i2-n )
(1 + e-i2Pn + e-i2"n)
Bl,n + B2,neio" + B3,nei'" (4.26)

The n subscript will be dropped here to ease the derivation, although it is understood
to be attached to all terms. Multiply both sides by (1 + e-i2p + e-i"2) and subtract the
gage spectra on the left hand side to get

Z1((1 + e'2p + ei2)(1 + e-i2, + e-i2) 9)

(B1 + B2eit + Bse'f)(1 + e-i26 + e-i2) 3(Bi + B2e-i' + B3e-') (4.27)

and finally,

S(B + B2e' e + B3')( + e + e + -i2') 3(B1 + B2e-' + B3e-i') .
((1 + ei2p + ei27)(1 + e-i2p + e-i'2) 9)

By rearranging terms, and applying the proper trigonometric identities, it can be shown
that the coefficients for the gage spectra that will solve this equation are as follows: The

divisor for the right hand side D is

(1 + e'i2 + ei27)(1 + ei26 + e'i2) 9) = 2(sin2 / + sin2 + sin2 (7 3) (4.29)

The B1 coefficient, R1 + iQ1


(e-i2, + e-i2 2) = sin2 3 + sin2 + i(sin 3 cosP + sin 7 cos -7)


(4.30)










46


the B2 coefficient, R2 + iQ2


(-2e-" + e', + e(-i2+i) = sin 7 sin 7 / + i(sin 7 cos (7 /3) 2sin ) (4.31)


and the Bs coefficient, R3 + iQ3


(-2e-' + e + e-i2p+i) = sin sin 7 / + i(sin/3 cos (7 /) 2 sin -) (4.32)


The incident spectrum may then be expressed

ZI = I-(BI(R + iQl)) + (B2(R2 + iQ2)) + (B3(R3 + iQ3))) (4.33)


and the reflected spectrum

1
ZR = D(Bi(Ri iQi)) + (B2(R2 iQ2)) + (B3(R3 iQ3))) (4.34)

The coefficients are all in terms of and 3 which are found by the geometry of the gage

array and the wavenumber k.

As ZI and ZR represent amplitude spectra, the energy density spectra may easily be

determined by squaring the amplitude and dividing by the increment in angular frequency,

or

E 2= A (4.35)
2Aw
where
27r
AW= NAt (4.36)

where At is the sampling rate.

In an effort to assure that the experiment was running correctly, and that the data were

being collected and analyzed properly, the energies of the reflected and transmitted wave

fields were compared by conservation of energy to the wave field incident from the paddle.

(see Chapter 5)


EI = ER + ET


(4.37)










47

4.3 Inputs to Least Squares

Two methods of calculating the amplitude spectra for each gage were used and their

results compared.

4.3.1 Direct Signal Processing

The first method used the full record of 2048 points. The Fourier Transform was per-

formed on each of the gage records, after the mean was taken out, using an FFT algorithm.

The resulting amplitude and phase spectra were used directly in the least square calcula-

tion of the incident and reflected wave field spectra. Note that there was no smoothing

performed on the data in this method. Demeaning the data

S2048
FPE = i048 F,(4.38)
t=1

S' = -P P (4.39)

SP E Bp = ap i (4.40)

where 7P is the elevation time record at gage p, FtP is the mean over all elements in the

record at gage p, S is the demeaned signal, and B is the complex amplitude spectrum.

4.3.2 Ensemble Averaging of Cross-Correlation Spectra

The second method is the same as the one proposed in Funke and Mansard (1980).

This method applies an ensemble averaging to the data by separating the time series into,

in this case, four separate realizations, in an effort to reduce noise effects. Of course, in

applying any type of averaging window, one sacrifices resolution in the resultant spectrum.

The second difference in this method, is that after each Fourier component is determined,

its amplitude is joined with the phase lag of that component relative to the first gage in the

series, the first gage having phase lag of zero. The computations of the gage spectra are as

follows:

The time series of 2048 points is split into four realizations of 512 points each. The












mean is computed by
S512
57Pm =-5 (4.41)
t=1
where p is the gage number 1,2 or 3, and m is the realization number, and rt is the water

elevation read in the time series t7 at time t. This value is then subtracted from each data

point

SP,m = 4 p,,m -p,m (4.42)

to yield the demeaned record S. The Fourier Transform is obtained for each realization in

an FFT algorithm resulting in

Sp,m KF Bp,m = apm ibp,m (4.43)

or in polar form

Bp-m = Apmei Pr' (4.44)

Next, the absolute amplitude spectrum is extracted from the Fourier series,

AP,m = nmB (4.45)
n At

where B* is the complex conjugate of B, and averaged by component n over the number of

realizations.

m= E Am (4.46)
1m=l
The phase of the cross correlation between the first gage and subsequent gages is at-

tached to the respective amplitudes at each frequency component. The cospectrum is

calculated by;

C12 = BB2* (4.47)

where B* is the conjugate of the transform. Note that the m and n notations have been

dropped merely for convenience at this point, and will reappear later. Thus,

C12 = (al ibi)(a2 + ib2) (4.48)

= ala2 + b1b2 + i(aib2 a2bi) (4.49)

= As1ei12 (4.50)












So,
atan b2 a2bl
tan 12 bb (4.51)
aja2 + b6b2
Averaging by frequency over the four realizations,


=~2 m= a=a b jb (4.52)
1 m=l l 2 1 2I

or
E =1 (alb2 a2bl)m
tan 2 --=1 (aa2 + bb2)M (4.53)

Thus the gage spectra used in the analysis are given by


B1 = A' (4.54)

B2 = Aei'',." (4.55)

B3 = A3ei'1'" (4.56)


4.4 Results of Least Square Error Fit

As stated above, the ensemble averaging was performed to reduce noise interactions at

the cost of resolution. Assuming the resolution of the full record analysis looks like


Aw, 2Aw, 3Aw, 4Aw, 5Aw... (4.57)

where
2r
Aw= N t' (4.58)

the resolution of the smoothed spectrum is


4Aw, 8Aw, 12Aw, ... (4.59)

and the number of spectral components will be one quarter the number in the full record

analysis. The range of frequencies covered will be the same for both spectrum lengths.

The resulting incident and reflected spectra from the full record transform behaved very

nicely and predictably. However, for the output from the smoothed spectrum scheme, this

was not the case. After a few trial runs of the program with artificial data at a discrete










50

frequency, it became apparent that the reflection coefficient (as would the energy densities)

would be significantly over or underestimated, unless the frequency picked was resolvable

in both the full record and the quarter length cross correlation method, ie.

Wpeak = mAw (4.60)

if m = 4, 8, 12,... in which case the reflection coefficient was identical using both resolutions.

No particular pattern in over- or underestimation was readily apparent, except that the

reflection coefficient spectrum in the vicinity of the chosen frequency would follow a general

upward or downward trend rather smoothly. It is probable that this phenomena is due to

leakage from the dominant frequency to adjacent frequencies. The mechanism by which this

is working is not yet understood, but would involve the interaction of the leaked spectral

data in the two Fourier spectra combined in estimating the cross-spectrum. See Figure 4.1.

The above explanation does not resolve the problem of obtaining a resultant reflection

coefficient spectrum (as well as the incident and reflected spectral energy densities) that is

smoothed or has the system noise and nonlinear effects filtered out, but rather compounds

the problem. However, the potential for extracting a reasonable result did present itself

when it was noticed that the reflection coefficient at the dominant frequency, as calculated

by the full record analysis was between those at the frequencies directly on either side of

the dominant frequency as calculated from the smoothed spectrum. See Figure 4.3. By

performing a linear interpolation between the values at these two frequencies, it was found,

for synthetic data, to yield a value very close to that calculated by the unsmoothed data.

More precisely stated, for values of w = mAw where m is equal to an integer multiple of

4, the reflection coefficients would be equal. But if m were other than a multiple of 4, a

reasonable estimate of the reflection may be calculated by linearly interpolating between the

nearest resolved frequencies.

Assuming a monochromatic wave field, a dominant frequency may be picked out by

finding the frequency with the greatest value of the power spectrum from a particular gage

spectrum in the full record analysis. Then finding the frequencies between which it lies


I












51















Energy
Density





I
I I !
I I
I I






I ii i
SI I I
I I I I







0 1 2 3 4 5 6 7 8 9 10 11 1213 nA
I I I I m



True Energy Density
These lines represent leaked energy into adjacent frequencies due to

cross correlation ensemble averaging. These leaked energies will
strongly manifest themselves at adjacent frequencies since no energy

exists there prior to calculation.


Figure 4.1: Effect of smoothing on a monochromatic spectrum











































I
I
I
I


1
i
1
I
I
I
I
I
I
I
I
I
I


I I I

0 1 2 3


I


I I I I I I I 1 1

4 5 6 7 8 9 10 11 12 13


I I I


Calculated by full record analysis

--- Calculated by cross correlation method

.... Linear interpolation










Figure 4.2: Interpolating resolved frequencies in the smoothed monochromatic spectrum to
estimate the energy density at the true frequency


Energy

Density


nAw

m










53
in the smoothed spectrum, a reasonable estimate for the true reflection coefficient may be

established by linear interpolation. For instance, assume the dominant frequency as resolved

by the full record is

Dominant = 10AW. (4.61)

This frequency is bounded by the frequencies

S= 8Aw (4.62)

and

w= 12Aw (4.63)

which are resolvable by the smoothed spectra. The reflection coefficient at one of these

bounding frequencies will be higher than the reflection coefficient at the dominant frequency

calculated using the full record, while the other will be lower. The linear interpolation is

simply

io = (2- 8)(12 10) + Kr (4.64)
(12 8)
where .rm is the reflection coefficient of the mth component of the full record spectrum. By

assuming a local linear relationship between frequency and energy density, this interpolation

can be easily be adapted for energy density calculation.

This method was developed by observing the interaction of a synthetically generated

monochromatic spectrum, which by its nature is very clean, ie. no energy at frequencies

other than the one defined. Therefore, it is justifiable to use this technique to analyze the

data collected in the laboratory experiments explained in Chapter 5, since that data itself

is very clean. Viewing a typical energy density spectrum shown in figure 5.8, it is apparent

that all of the significant energy is contained at one frequency.

4.4.1 Results

The result of performing the smoothing and interpolation scheme presented above on

the data collected in the laboratory experiments proved to be minimal. The change in the

resulting reflection coefficients was generally on the order of 10-'. It is therefore suggested










54

that the full record of a monochromatic wave field, transformed to the resolution allowed

by the recording apparatus, be used in determining incident and reflected energies, and

reflection coefficients in a monochromatic wave field.















CHAPTER 5
LABORATORY STUDY



5.1 Introduction

A laboratory experiment was performed to verify the predictions of the numerical so-

lution for monochromatic waves of normal incidence. The laboratory set up used was very

similar to that of Davies and Heathershaw (1984) except that, where Davies and Heather-

shaw used a sinusoidal ripple patch, a set of four discrete bars with positive amplitude only

on an otherwise flat bottom were installed in the wave flume. In the present study two bar

fields were tested, both with the same shaped bars but with different spacing between them.



5.2 Equipment

5.2.1 Wave Flume

The tests were done in a 26m x 0.6m x 1.lm wave flume at the Coastal and Oceano-

graphic Engineering Laboratory (COEL) at the University of Florida. The water depth

was 15 cm in the flume. The bar patch began 14m down wave of the wave generator and

ended 7.8m up wave of an energy absorbing beach at the end of the flume. Waves were

generated by a Seasim piston wave maker .4 m high. Although the Seasim system is capable

of generating a 16 band spectral wave field, only monochromatic waves were used in order

to achieve better resolution and accuracy. The paddle was driven by a servo-controlled sys-

tem with pneumatic hydrostatic balance. The signal generator was capable of accurately

controlling the period of the paddle stroke to one hundredth of a second. This provided

good resolution for the comparison curve at low frequencies and thus small values of 2k/A,

but resolution decreased at higher frequencies. The range of periods was from 0.6667 s to


I

















V 7


..c.0 .5n*>re


Figure 5.1: Profile of bar field


the system maximum of 2.5 s. The range of 2k/A was from 0.45 to 2.5.

The bar patch can be described by

h 0.05sin(2rnz); 0+ nlb < < 0.5 + n ,,
0.0; 0.5 + nla x < ,1,2,3


(5.1)


Ib = bar spacing

Two bar spacings were used, 0.8 m and 1.2 m on center. For purposes of comparison,


2x
AT-


(5.2)


assuming Ib is wavelength of the dominant Fourier component of the barfield. The bars

were constructed of fiberglass resin and mat in a female mold constructed of sheet metal

on a wood frame. They were trimmed to a tolerance of .005 m in length and .002 m in

height. Any holes that remained after curing were filled to yield a smooth surface, and

edges meeting the bottom of the flume were sanded to a sharp edge to allow a smooth

transition from bottom to bar. Pin holes were drilled into the tops of the bars to allow

trapped air to escape while the flume was filling to avoid buoyant forces on the bars during

the experiment. The bars were installed using a small amount of silicone caulk on each edge

of the bar where it met the flume side or bottom.

At the down wave end of the flume, a wave absorbing beach was constructed of rub-

berized horsehair and bagged stone. The horsehair was held in place with wire mesh in a


0.15m
A


--- -_"-


I I
I I
<- -- lb--
I i










57

convex up shape in order to dissipate wave energy in the most effective manner. This beach

extended 1.5 m up wave of the bagged stone. It was important to minimize reflection off of

this beach so as not to pollute the reflection of the bar patch.

Measurements prior to bar installation showed reflections from the beach of less than

15 percent, and generally less than 10 percent. During these measurements, the array that

was to be up wave of the bar field, the stationary array, was installed 12 m up wave of

the beach. The array on the cart was placed 2 m up wave of the beach in an effort to

determine the change in reflection coefficient due to position and examine energy decay of

the reflected wave. However, during this portion of the experiment, one of the gages in the

cart array malfunctioned and only a few runs were made with both arrays in operation.

For verification of each coefficient, two runs were made at each frequency and the results

averaged. An attempt to test the algorithm described in the the previous chapter was

performed using a hand held vertical wall in an attempt to achieve near total reflection.

The results of these tests are presented in Table 5.1.


Table 5.1: Percent Reflection from Flume End w/o Bars

Frequency Hz. Stationary Cart
0.4 10.6 na
0.5 8.4 na
0.6 6.9 na
0.7 4.7 na
0.8 4.6 na
0.9 5.8 na
1.0 6.6 6.6
1.2 6.4 13.7
1.3 11.4 10.4
1.4 6.7 8.2
1.5 4.0 8.4
1.6 20.2 41.9
vert board 1.3 78.5 79.1


The very high reflection at 1.6 Hz remains a mystery. However, the highest frequency

used in tests once the bars were installed was 1.5 Hz so no further investigation of this

phenomenon was deemed necessary.














5.2.2 Electronic Measurement

Two arrays of three gages each were employed, one 1.8 m up wave of the bar patch

and the other 1.2 m down wave. The gages were a standard in house design with minor

alterations to improve sensitivity. The gages operate by comparing the frequencies of two

inductive-capacitive (LC) circuits where:


/ = (5.3)

Inductance and capacitance were adjusted in a reference circuit to run at a constant fre-

quency around 1 MHz. The sampling circuit was identical to the reference circuit except

that an additional 'capacitor' was added, that being the capacitance contained between

the probes, that is to say the probes act as capacitance plates with water being a variable

dielectric. Since the wave amplitudes were to be less than 1.5 cm to maintain linear theory,

the probes were only 6 cm long. Normally these gages are used with probe lengths 0.5 m

or longer and the capacitance contained between them is of the order of 100 microfarads,

while the 6 cm probes contained less than 10 microfarads. This made tuning the sample and

reference circuits to be of greater importance than normal for these gages. The sampling

circuit was tuned to run at a slightly lower frequency than the reference circuit for the full

range of capacitance change in the probes. The two frequencies were subtracted in a chip

and the difference frequency sent in the form of an RF signal to the signal conditioner.

Therefore, as water level increased, the capacitance between the probes increased resulting

in a drop in the frequency in the sampling circuit thus increasing the difference between

the two frequencies, which, when processed in the signal conditioner, showed an increase in

voltage.

The signal conditioner worked somewhat like a radio receiver, converting a RF signal

to a voltage. The signal conditioner had adjustable gain and zero offset controls. The zero

offset control allowed for the positioning of the mean voltage output, and was set, such that

at still water the output was close to zero, (usually 0.2 volts). The gain was adjusted so










59

that immersing the length of the probe corresponded to a 10 volt change in output from

the signal conditioner. Thus, full immersion of the probe resulted in an output of +5 volts

and an output of-5 volts if just the tip of the probe was immersed.

The output voltage was then fed through an analog to digital conversion board mounted

on the back of a Digital Equipment Corporation Micro PDP-11 (PDP- 11). This board read

the voltage and converted it to an integer value corresponding to the voltage. A voltage

in the range of -5 to +5 volts corresponded to an integer between 0 and 4096. Thus, the

resolution achieved between discrete voltages was

10volts
-vos = 0.00244volts/division (5.4)
4096points

translating into a resolution of the probes of

6cm
10ot0.00244volts/division = .00122cm/division (5.5)
10volts

The PDP-11 would sample 2048 points at each gage location at a rate of 0.1 s or frequency

of 10Hz for one complete data set.



5.3 Data Analysis

5.3.1 Acquisition

The gages were set in two arrays of three each with 19 cm. between the first and second

gages and 31 cm. between the second and third. The choice of this spacing will be expanded

upon in the next section. Each gage in the array was attached to a rigid bar connected to a

rack and pinion vernier marked in millimeters. The rigid bar was stabilized by two vertical

parallel bearing tracks which in turn were connected to an aluminum frame. The array up

wave of the bar patch (reflected end or R-array) was mounted directly to the sides of the

flume, while the down wave array (transmitted end or T-array) was attached to a cart that

is capable of moving along the length of the flume.

The vernier allowed for precise static calibration of the gages. Calibration data was

taken at water elevation values of 2.60 cm, 1.50 cm, 0.00 cm, -1.00 cm, and -2.00 cm. The

















2- -

I I
S 0 i
52 _

-2

Operating Range


-4



-6 I
-4 -2 0 2 4
Voltage

Figure 5.2: Typical calibration curve


voltages for each gage at each elevation were stored in the PDP-11. The raw elevation vs.

voltage data was used to calculate the coefficients of a fourth order curve to account for any

subtle nonlinearities in the gage response. A typical calibration curve with the calibration

data points is shown in Figure 5.2.

Gages were calibrated at the beginning of a laboratory session. They were recalibrated

when the difference in voltage between two gages in the same array, at still water, drifted

0.1 volt, translating to a mean water shift of approximately 0.05 cm, from the reading at the

previous calibration. This would keep gage error to less than 5 percent for a 1 cm (typical)

wave, which is less than the amplitude of the measured electronic system noise.

The PDP-11 was programmed to simultaneously sample the voltage output of each gage

circuit for 2048 points at a frequency of 10 Hz. (Simultaneous sampling is suggested al-

though the computer can sample only one circuit at a time, the time difference between the

samples of adjacent gages being considered negligible.) A frequency of 10 Hz was adequate

since the range of wave frequencies was between 0.4 and 1.6 Hz allowing measurement of at










61
least the fifth harmonic of the base frequency of the wave in question. The time series of

integer values was stored in a data file in the PDP-11.



5.3.2 Data Processing

The first step in processing the data received from the PDP-11 was to convert the

voltages into real water elevations using the calibration constants. The data was then run

through the algorithm developed by Funke and Mansard (1980) described in chapter 4.

Briefly, this algorithm employs the use of the Fourier transformed data of three gages, of

known spacing, to calculate the incident and reflected spectra of a wave field passing through

the array. Any arbitrary choice of spacings will work well for this algorithm except those

spacings where the distance between the second and third gages are integer multiples of the

spacing between the first and second gages, especially in ratios of 1:2, 1:3, and 2:3. If the

gages are spaced at such a ratio, the algorithm will 'see' a virtual standing wave that will fit

inside the gages with nodal points at the gages. Thus it will calculate a fully reflected wave,

at a wave length equal to twice the distance of the spacing between the first two gages. (See

Figure 5.3.)

For each data set, four energy density spectra were calculated, viz., incident (flux to the

right) and reflected (flux to the left) spectra at the reflection end (R-array) of the barfield,

and the incident (right) and reflected (left) spectra at the transmission end (T-array). Since

the model was one dimensional, this has the effect of measuring the wave field energy passing

the boundaries established by the arrays at either end of the bar field into or out of the

control volume. Thus the incident and reflected spectra at the T-array represent the energy

transmitted over the bar field and the reflected energy off of the beach at the end of the

flume. (See Figure 5.4) During the laboratory trials, the dominant frequency, or that with

the highest energy, in each spectrum was picked out and displayed for ease in calculation

of the reflected energy ratio and reflection coefficient. A listing of the processing program

is presented in the appendix.






















GAGE 1 GAGE2 GAGE3














-- 1.0 L --1.5 L




Figure 5.3: Improper gage spacing and virtual standing wave









INCIDENT ENERGY SPECTRUM

REFLECTED ENERGY SPECTRUM TRANSMITTED ENERGY SPECTRUM











II
1 2 4 5 6Bo rie.. ......


". .. "**... ....... Boundaries..... ..*----..*** .


Figure 5.4: Energy crossing system boundaries













1.0

0.0




-1.0

-1.0


I

-5.0








FREO HZ

Figure 5.5: Typical energy density spectrum for incident wave


Since the spectra passing each array were referenced to a physical scale, that is water

elevation, one could assume by the law of conservation of energy, that the energy measured

going into the system (incident energy at the R-array only, assuming the energy re-entering

the system from the reflection of the beach at the T-array is negligible) should equal the

energy leaving the system in either direction (reflected and transmitted energy). Again, the

reader is reminded that although the energy density spectrum is used in these calculations,

the assumption that all of the energy is contained in a single frequency band and that all

spectra used in the calculation are of the same frequency resolution, this description is valid.

After the spectra had been calculated, the peak value of each spectrum was found and

displayed along with its corresponding frequency. This frequency was usually in very good

agreement with the expected peak frequency as established by the output of the signal

generator control of the wave paddle. The spectra were plotted (Figures 5.5, 5.6, 5.7)

and higher harmonics of the base frequency are evident in these plots. However, only the

characteristics of the base frequencies were investigated. With the incident, reflected, and
















64











-1.0


*2.0
U
(.0




-5.0 -





-.0 ;0







-6.0
o 0.00 O O 1.00 1 o0 .00 2.S0 1.00 3.o 0 6.00 4.SO
FREQUENCY HZ




Figure 5.6: Typical energy density spectrum for reflected wave



1.0 --


0.0


.k.0


-2.0

U
a -s.0
















0.00 O.SO 1.00 I.SO 2.00 2.O 3.0 00S

FREQ HZ


Figure 5.7: Typical energy density spectrum for transmitted wave










65

transmitted energies established, the transmitted and reflected energy ratios and reflection

coefficient could be determined. Also, total energy of the system could be tested against

the incident energy by adding reflected and transmitted energy and dividing by incident

energy. Since by linear theory

E= pgH2 (5.6)
8

and reflected and transmitted energy ratios

Rr = f (5.7)
Ej
Tt = E (5.8)
Ej
r = Reflected at R-array

t = Incident at T-array

i = Incident at R-array

(5.9)


Then the reflection coefficient is,

S E=(5.10)

By conservation of energy

E, = Er+ Et (5.11)

Or, since h(zl) = h(z2),

1 = R + T( (5.12)

As will be seen in section 5.4, for most runs only about 75 percent of the energy

measured entering the system was measured exiting the system through reflection and

transmission. Again using linear theory, estimates of the energy attenuation due to bottom

and side boundary friction were calculated. In a channel of uniform width b and depth h,

the damping of a linear wave propagating over a distance I may be estimated to be


a = aoe-Af


(5.13)











where
2k rv kb + sinh 2kh
b V 2w2kh + sinh2kh (5 )
ao is the wave amplitude incident on the barfield, a is the attenuated wave amplitude (Hunt

1957) and v is the kinematic viscosity at 1.005E-6 m2/s. For shallow water where kh -- 0

this reduces to
_f k 2v(b + 2h) (5.15)
Af-= (5.15)
4h w b
Length of travel of the waves was assumed to be the distance between the arrays for

both the transmitted and reflected waves. The approximation that the distance the reflected

wave travel equals to the total distance between the arrays is chosen by assuming that all

of the reflection would take place exactly in the middle of the bar field. Over the range of

frequencies used in the experiment, theoretical predictions showed a 10 percent attenuation

to total energy due to friction.



5.3.3 Verification of Analysis Technique

The analysis technique, being relatively new, was verified using established technique of

measuring wave envelopes using a moving wave gage. Actually, three gages were used simul-

taneously on the same cart since they were already mounted on the cart for the technique

described above. Data were collected in the same manner as with the six stationary gages.

However, since measurement of envelopes is a relative maximum amplitude measurement

of with respect to position in the wave field, and is dependent only on what the individual

gage measures, the data were not converted into true elevation but left in the form of a volt-

age reading. The data were then processed in a routine that picked out the local maxima

(crests) and minima (troughs) which were subtracted from each other and stored as a wave

height. (This data was smoothed since, often, small peaks in the raw data due to tank and

system noise resulted in gaps in the resulting envelope.) The maxima and minima of the

resulting envelope wave were then picked out in the same manner, representing maximum

(Hma) and minimum (Hmin) wave heights. A reflection coefficient was then calculated
















0.8 I










I-
0.6







0.2 -~
i-/



0.0
0.5


2.0


Figure 5.8: Reflection theory vs. measured, .8 m bar spacing

1.0 r



0.8



0.6 o \

o./l-
U-t

0.4



0. -
0.2 -\
L \ / / o / -


0.0
0.5


1.0 1.5 2.0
2k/X / \ 27' 1 .'


Figure 5.9: Reflection theory vs. measured 1.2 m bar spacing


1.0


I \ / o \ ," \ ,

1.0 1.5
2k/A AX 2T/ .8 i


r- --! --- -r----
-i











69

1.0
0
0

S0.8 0 o 0 0



w 0.6 4
0



S 0. -



S0.2 -



0.0
0.5 1.0 1.5 2.0 2.5
2k/A X=22n/.8 m

Figure 5.10: Energy Conserved: Theoretical and Measured .8 m bar spacing


the system, less the energy dissipated inside the system. Theoretical predictions show this

value of dissipation to be around 10 percent, or rather 90 percent of the energy should be

leaving. However, the sum total of energies usually measured less than the 90th percentile

(See Figure 5.10). Significant scatter is apparent in this measurement. No relationship

between the deviation in total conserved energy, and deviation in the reflection coefficient

is apparent. Scatter of the energy attenuation seems to be independent of frequency as

well. Looking at the energy levels did aid in quickly determining if a given sample was

giving unreasonable results. The energy plotted in Figure 5.10 is the sum of the ratios of

the transmitted and reflected energy of the single band being tested.

The ratios of energy contained in the full spectra were also determined and viewed.

These were slightly higher than the single band but were usually very close to the single

band ratios.


I _














CHAPTER 6
BARS ON A MILD SLOPE



6.1 Introduction

In this chapter, the effects of a series of undulations placed on a mildly sloping beach are

investigated. The equations presented remain in linear theory, thus no energy is dissipated

in breaking. Also no damping due to bottom friction is applied. It will be shown that for

this case, reflection in steady state will be complete for the whole system. The interest, then,

lies in the displacement response at the shoreward boundary and between the shoreline and

barfield.

6.2 Reformulation of the Mild Slope Equation

Davies et al. (1989) developed a linearized equation for non-dispersive long waves

propagating over sinusoidal undulations on an otherwise flat bottom. The analytic solution

was obtained by transforming the surface displacement variable and solving the resulting

Mathieu equation form. The form of the solution is restricted to two forms, or cases, those

being where 2k/A = 1 and where 2k/A # 1.- The governing equation presented in Chapter

2 is restated as the well known Mathieu equation

2W 2
a2 + r -2W 1 2) cos2z =0 (6.1)
8z2 2

where c2 = 4w2/gHoA2, Ho is the amplitude of the bed undulations and e < 1. The

transform variable is

W = H1/2Cz) (6.2)

where H(x) is the total depth according to


H(x)= Ho(1+ cosAx) and rl(x,t)= (x)e-iwt (6.3)


I










71

This form of the governing equation was then solved analytically. Kirby (1989) develops

essentially the same form but for intermediate depth, dispersive waves. Also, the form of

the governing equation is extended to accommodate obliquely incident waves. Since the

derivatives of cosh kh in the extended mild slope equation (Kirby, 1986) given in Chapter

2 are of 0(c) and using the same variable definitions, it may be rewritten as


V (fVi7) + k2p, = 0 (6.4)

where


p=CC, f =P- -g- (6.5)
cosh2 kh

Introducing the variable transformation


r1 = f-1/2W, (6.6)


6.4 becomes
V2S
V2W + [k2 + A(k26 + 2)]W = 0 (6.7)
2

where

A= 4k =c (6.8)
CC, cosh2 kh =2kh + sinh 2kh

as in Chapter 2.

If (x,y) is the horizontal plane and h = h(x),S = S(x), /ay 0, equation 6.7 becomes

6zz
W,, + [k2 + A(k26 + )]W = 0 (6.9)

Allowing oblique incidence, let


m = ksin 0 = constant (6.10)


equation 6.9 becomes


S+ [(k2 m2) + A(k26 + )]V = 0 W = lei' (6.11)
2i










72

It becomes apparent, however, that as h -+ 0, 1/f 1/2 = 1/Vg7 -- oo explicitly violating

the restriction that tr remain bounded at the shoreline. The only recourse to remedy this

situation is to require the boundary condition at the shore to be


W(x2) = 0 (6.12)

The seaward boundary conditions are again specified in the form of radiation conditions

with and incident wave propagating in the +z direction and a reflected wave propagating

in the -z direction. The incident wave ji is assumed to be known, and the reflected wave

ir, must satisfy a radiation condition for propagation out of the domain


Vr(x) = -il(x)r ; x = (6.13)

Similar to the boundary condition as in chapter 4,


W, = W W. (6.14)

and substituting, the boundary condition at xz is thus described.


*(x) = il(2, ) ; x = xi (6.15)

6.3 Numerical Solution

The problem specified in the previous section may be solved using a finite- difference

scheme very similar to the one used in the previous chapter. The full form of the transformed

mild slope equation will used in the scheme, which when expanded, becomes


W. + [(k2 2)p f=0 (6.16)
f 4f2 2f

The domain xz < x < X2 is discretized according to


X= + (i+l)Ax ; 1< i< n (6.17)

where
A 2 21
AX 2n-1 (6.18)
n I


U










73

All other coefficients and the variable W are affected by local conditions in the grid and

are defined in discrete form at the grid locations z'. From this, a centered finite-difference

scheme is developed and given by

W'-' 2Wi + W' + (k2 2p W = 0 (6.19)
Ax2 + P 2
Izt fi 4f1' 2f

where

fi+l fi- fi+l 2f' + f'-1
fc = 2A (6.20)

The scheme may be simplified by

A'W'1 + B'W' + C'W'+1 =0 ; i = 2,3,...n 1 (6.21)


where

A' = 1 (6.22)
Bi -2 -- A -2 [(k2 m')p f 2 fIn
B' = --2+AX2 2 2 f + (6.23)
fP 4fi2 2f
Ci = 1 (6.24)


The bed undulations are subject to the restriction that they do not affect the wave in

the locale of either boundary in order that the waves radiating at the boundary are not

interacting with the rapid variations. Thus it is required that

61 = 62 = 6"-1 = 6n 0 (6.25)


Expressing the boundary conditions in finite difference form, equation 6.15 is

W = 0 (6.26)


Defining 6.29 in the finite difference form

Bn= 1 (6.27)


A"=0 and C"=O


__


(6.28)










74
At station xz the boundary condition may be restated as

(1 + ac)W2 (1 a)W1 = 2f1/2a(m)al[e2a' + 1] (6.29)

where

1 = 1 (6.30)
2
and it is assumed the incident wave is described by

ti(x) = a(m)ei'(z-z1) (6.31)

Again, putting 6.34 into the finite difference form, gives

B' = -(1- a) (6.32)

C1 = 1+ 1 (6.33)

D1 = 2f /2a(m)a[e2a' + 1] (6.34)

The problem may be written in the form of a linear matrix equation

AW = D (6.35)

where D is a column vector with D2-DW = 0, tr is a column vector with elements W1 -W",

and A is a tridiagonal matrix with diagonal vectors A', B' and C'. The solution is again

obtained using the double sweep algorithm as given by Carnahan, Luther and Wilkes (1969).

The reflection coefficient may be determined once the solution for W is obtained. The

reflection coefficient may be extracted directly since W is a propagating term and f will

be the same for the +z and -z propagating components at any particular point. Two

estimates for the reflection coefficient R are obtained at xl. From equation 6.34 Wr at xz

may be written

W1 = W1 a(m) (6.36)

Wr = W2 a(m)e21a (6.37)

Define the two estimates for reflection coefficients as

R (m) (6.38)
a(m)










75

R2 = W (6.39)
a(m)
and R as the average of R1 and R2. This calculation, however, is trivial for this solution,

since no damping due to bottom friction exists in the present solution, no energy is dissap-

ated in breaking in linear theory, and no energy is transmitted past the shoreline. Therefore,

in order that energy be conserved, the energy carried by the reflected wave must be equal

and in opposite direction to the energy carried by the incident wave in steady state.

Since the boundary condition at the shoreward end of the domain is restricted to keep

t) bounded, the response at the shoreline may be estimated by extrapolating the surface

displacement directly seaward of the shoreline by
n = -1 1n-2
Al = l1 + 'zAx (6.40)

which can easily be seen reduces to

,n" = 2n"-1 "n-2 (6.41)

where
W'
I7'1 = I 1 (6.42)

Additionally, from 6.45 the magnitude of the surface displacement in the domain may

be calculated.

6.4 Model Tests and Examples

6.4.1 Response Over a Barfield in Front of a Wall

In order to verify the validity of the model formulated above, it will first be compared

to a case that the model in the form of equation 6.4 can easily handle. This would be the

case where the restriction

J =0 (6.43)

is valid. This boundary condition is required for a wave field at a vertical wall. In the finite

difference scheme, the boundary condition is applied by


1tn 17n-1 = 0


(6.44)





















S------ Lb -------
I I


Figure 6.1: Definition sketch of bar field in front of a wall


An= -1 and B"=1


(6.45)


The results for a wave at normal incidence where 2k/A = 1 with A being the fundamental

mode (A = 22r/Lb) of the barfield described by


h= 0.05sin(2irx); 0 + nl < z < 0.5 + nlb
S 0.0; 0.5 + nlb < x < nlb


n = 0, 1,2, 3


Ib = bar spacing

The bottom otherwise is assumed flat.

Also, the length Lb, between the barfield and wall is defined as a function of Lb


Lb = Lbd


where d is any real value. The domain is depicted in Figure 6.1.

Let d = 4, an integer value, such that the wall rises 2 surface wavelengths (for the

resonant case) past the barfield. The incident wave is arbitrarily set at 1 since linear theory

is being used. Also, the absolute value of the wave field is being plotted below, thus no

phase shifting or time dependence is evident. The results plotted in Figure 6.2 show the

wave envelope as calculated by both schemes for the resonant condition. The amplitude of

the wave at the wall boundary is plotted in Figure 6.3 as a function of 2k/A.


(6.46)


(6.47)


-< -------- Lb ---------





































0 2 4 6 8 10
X


Figure 6.2: Wave
schemes, d = 4


envelope in front of a wall for q directly and q = f- /2W numerical


4




3




-t


I








0.5



Figure 6.3: Wave amplitude
d=4


I 1.5
2k/X X=2n/1.0 m


at the wall for j directly and 1T = f-1/2W numerical schemes,


M














3






2
o


CO


a, 1







0 2 4 6 8 10
X

Figure 6.4: Wave envelope in front of a wall for q directly and t = f I/2W numerical
schemes, d = 4.5


Now let d = 4.5, such that barfield lies 2.25 surface wavelengths in front of the wall.

The wave envelope for this case is plotted in Figure 6.4 and the maximum displacement at

the wall in Figure 6.5.

It can clearly be seen that the choice of Lb, can have drastic effects on the wave field

between the barfield and the wall. What seems to be happening is if the spacing is an integer

multiple of half of a surface wave length, the wave field becomes trapped between the bars

and the wall. This would be due to tertiary reflection of the wave field, primary being

offshore reflection, secondary being reflection off the wall and tertiary being the reflection

by the barfield of the wave reflected off the wall. The final effect is the standing wave in

front of the wall is resonated, and potentially quite violent oscillations may occur.

6.4.2 Response Over a Barfield in Front of Beach

From the above section, it is seen that the new form of the mild slope equation is valid.

Now, attention is restricted to the case where a barfield is placed on a mild slope and waves













4 II I I '





-II
3 -









1-


o . I 1 I .

0.5 1.0 1.5 2.0 2.5
2k/A A=2n/1.0 m

Figure 6.5: Wave amplitude at the wall for r directly and 1 = f-1/2W numerical schemes,
d=4.5

are allowed to propagate to the shoreline. It is assumed that no breaking occurs as the wave

shoals and no attenuation due to bottom friction is present. The solution of the equation

in the form of 6.4 cannot be used since unrealistic restrictions on tW at the shoreline are

required. The finite difference scheme developed in section 6.3 is now employed with its

pertinent boundary condition at the shoreline. The adjusted domain can be seen in Figure

6.6.

Since the bottom is sloping, and the wavelength changes as the wave shoals, the exact

value of Lb, that would result in strong tertiary reflection is not as easily determined.

However, wave envelopes are plotted for the same frequency near resonance for two choices

of d. The bottom is plotted below the envelopes. The bottom parameters, except d, are

identical for both cases where, h = .15m, the bumps, shaped (zx) = 0.05m sin 2;r/0.5m for

the positive branch only, are spaced 1.0m apart. For Figure 6.7 d = 10.0, and d = 10.5 for

figure 6.8.
























I I
h I I
I I

,< .... L"-







Figure 6.6: Definition sketch of bar field on a sloping bottom in front of a shoreline













6 -



o 4-




> .








-2
0 5 10 15



Figure 6.7: Wave envelope on a sloping beach with 4 sine shaped bumps, d = 10.0











81
15




10




S5









-5
0 5 10 15
x

Figure 6.8: Wave envelope on a sloping beach with 4 sine shaped bumps, d = 10.5

Lastly, for the two cases above, the amplitude at the shoreline is plotted against 2k/A

in Figures 6.9 and 6.10.

Looking at the last two plots, it can be seen that very small changes in frequency

will change the resonant response between the barfield and shoreline drastically. Since the

response at the shoreline is unreasonably large,the envelope amplitude at a point (x=7 m)

midway between the barfield and shoreline will be plotted in Figures 6.11 and 6.12.

The wave envelope again is sensitive to small changes in frequency, yet the trapped

wave amplitudes are a bit more beliveable at this point in the wave field. However, it is

also clearly evident that the presence of the barfield can cause large standing waves in the

nearshore zone. Interpreting the plots, any surface displacement above 2 (the incident wave

amplitude of 1 superimposed on the reflected wave amplitude) would be identified as a

trapped mode.












82


40 j



:30 i

[ i ,i 1; ( i





io L '"" N ll
1o--


oi_, _^_ LI


N 4


Figure 6.9: Wave amplitude at the shoreline vs. 2k/A, d = 10.0

50 m r-r-- 1



40



o
CF
1. 30




lOo l

I.
101-


Figure 6.10: Wave amplitude at the shoreline vs. 2k/A, d = 10.5


I I












83

) r


1.0 1.5
2 1< /,'


0 25

2.0 2.5


Figure 6.11: Wave amplitude at x= 7 m vs. 2k/A, d = 10.0


-i


1 I 1 J
I () I F '2 o :2)


Figure 6.12: Wave amplitude at x= 7 m vs. 2k/A, d = 10.5


0F--


T F I


I

I



"L
SI-



1!-






() L
() F1


" '


' i










84

6.5 Conclusions

In the previous section, it was pointed out that the installation of a bar field described

may cause the trapping of waves in the nearshore zone. The calculations made were done

assuming no energy dissipation due to breaking or bottom friction, which may reduce the

resulting large amplitudes considerably. It may be surmised from the results that the

standing wave between the bars and shoreline may cause shoreward growth of the barfield,

just as it has been hypothesised that the standing wave seaward of the bars may cause

seaward growth of the barfield. This phenomena has been observed in the laboratory using

a barfield placed on a sand bottom by McSherry (1989).















CHAPTER 7
CONCLUSIONS



In this thesis, an extension to non-resonant interaction theories was developed to ac-

commodate oblique incidence and seabeds of other than sinusoidal shape, specifically, a

series of cosine bumps on a flat bottom. The new theory was compared to existing theories

for normal and oblique incidence. Agreement between all theories for arbitrary bottoms

was fair.

Additionally, the numerical solution of a complete governing equation for undulations

on a mild slope was compared to laboratory data. The comparison shows a slight shift

in frequency at resonant peaks. This may due to inadequate description of the bottom

boundary condition used in the solution. Included in the laboratory study was an applica-

tion of using spectral analysis to determine the incident and reflected wave energies for a

monochromatic wave field.

Finally, an investigation of wave fields between a barfield and beach was performed.

The numerical predictions, although neglecting wave damping and nonlinearities, show

the potential for large amplitude trapped modes between the barfield and shoreline. It is

apparent additional work in this application is necessary before implementing prototypes

in the environment.















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