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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