Wave transformation by bathymetric anomalies with gradual transitions in depth and resulting shoreline response

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Title:
Wave transformation by bathymetric anomalies with gradual transitions in depth and resulting shoreline response
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xx, 236 p. : ill. ; 29 cm.
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Bender, Christopher, 1976-
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Civil and Coastal Engineering thesis, Ph.D   ( lcsh )
Dissertations, Academic -- Civil and Coastal Engineering -- UF   ( lcsh )
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theses   ( marcgt )
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Thesis:
Thesis (Ph.D.)--University of Florida, 2003.
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Includes bibliographical references.
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by Christopher J. Bender.
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Printout.
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Also issued as Technical report 132.
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Vita.

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University of Florida
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WAVE TRANSFORMATION BY BATHYMETRIC ANOMALIES WITH GRADUAL
TRANSITIONS IN DEPTH AND RESULTING SHORELINE RESPONSE















By

CHRISTOPHER J. BENDER


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA


2003















ACKNOWLEDGMENTS

I am truly blessed to have my wife and best friend, Kathryn, in my life. This

completion of this work was made possible by her support, encouragement, and love. I

am fortunate beyond measure to have her next to me as I walk life's journey.

Many individuals at the University of Florida have made the completion of this

work possible. Dr. Robert G. Dean has not only been my major advisor but also my

mentor and friend during my studies at the University of Florida. I have gained many

valuable lessons from his work ethic, his character, and his tireless quest for

understanding the coastal environment. I am grateful to the other members of my

committee (Dr. Daniel M. Hanes, Dr. Andrew Kennedy, Dr. Ulrich H. Kurzweg, Dr.

Robert J. Thieke) for their instruction and involvement during the course of my doctoral

studies.

I wish to thank my family (especially Mom and Dad B, Mom and Dad Z, Caryn,

John, and Kristin) and friends for their support in all my endeavors. Each of them has

contributed to who I am today.

An Alumni Fellowship granted by the University of Florida sponsored this study

with partial support from the Bureau of Beaches and Wetland Resources of the State of

Florida.















TABLE OF CONTENTS
page

ACKNOW LEDGM ENTS ............................................................................................. ii

LIST OF TABLES ....................................................................................................... vi

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

ABSTRACT......................................................................................................................... xix

CHAPTER

1 INTRODUCTION AND M OTIVATION .......................................... ...............

1.1 M otivation................ ..................................................................................... 2
1.2 M odels Developed and Applications............................................. .............. 4

2 LITERATURE REVIEW ...................................................................................... 7

2.1 Case Studies .................................................................................................. 8
2.1.1 Grand Isle, Louisiana (1984).............................................. ............... 8
2.1.2 Anna M aria Key, Florida (1993)........................................ .......... ... 11
2.1.3 M artin County, Florida (1996) ........................................... .......... ... 15
2.2 Field Experiments ........................................................................................ 16
2.2.1 Price et al. (1978) .............................................................................. 17
2.2.2 Kojima et al. (1986)............................................................................. 17
2.3 Laboratory Experiments..... ............................. ...................................... 19
2.3.1 Horikawa et al. (1977)........................................................................ 19
2.3.2 W illiams (2002).................................................................................... 20
2.4 W ave Transformation .................................................................................. 23
2.4.1 Analytic M ethods .............................................................................. 23
2.4.1.1 2-Dimensional methods ................................................ .....24
2.4.1.2 3-Dimensional methods ........................................................34
2.4.2 Numerical M ethods ........................................................................... 40
2.5 Shoreline Response........................................................................................ 46
2.5.1 Longshore Transport Considerations .................................................. 46
2.5.2 Refraction M odels ............................................................................. 47
2.5.2.1 M otyka and W illis (1974)........................................................47
2.5.2.2 Horikawa et al. (1977) ...........................................................49
2.5.3 Refraction and Diffraction M odels................................................... 51
2.5.3.1 Gravens and Rosati (1994)....................................................51









2.5.3.2 Tang (2002)...................................................... ...................54
2.5.4 Refraction, Diffraction, and Reflection Models.................................. 54
2.5.4.1 Bender (2001) .............................................. ....................... 54

3 2-DIMENSIONAL MODEL THEORY AND FORMULATION .........................59

3.1 Introduction .................................................................................................. 59
3.2 Step Method: Formulation and Solution..................................... .......... .... 60
3.2.1 Abrupt Transition ................................................... ........................ 63
3.2.2 Gradual Transition.............................................. ............................. 65
3.3 Slope Method: Formulation and Solution................................... ........... ... 67
3.3.1 Single Transition .................................................... ......................... 69
3.3.2 Trench or Shoal ..................................................... .......................... 72
3.4 Numerical Method: Formulation and Solution.............................................. 74

4 3-DIMENSIONAL MODEL THEORY AND FORMULATION ..........................77

4.1 Introduction .................................................................................................. 77
4.2 Step Method: Formulation and Solution.................................... ........... ... 77
4.2.1 Abrupt Transition .................................................... ........................ 81
4.2.2 Gradual Transition............................................................................. 85
4.3 Exact Shallow Water Solution Method: Formulation and Solution ............... 88

5 2-DIMENSIONAL MODEL RESULTS AND COMPARISONS.........................92

5.1 Introduction .................................................................................................. 92
5.2 Matching Condition Evaluation................................................................ 92
5.3 Wave Transformation ................................................................................ 95
5.3.1 Comparison of 2-D Step Model to Numerical Model FUNWAVE 1.0. 97
5.4 Energy Reflection ........................................................................................ 99
5.4.1 Comparison to Previous Results....................................... ........... .... 99
5.4.2 Arbitrary Water Depth............... ........................................................... 101
5.4.3 L ong W aves........................................................................................... 116

6 3-DIMENSIONAL MODEL RESULTS AND COMPARISONS.........................125

6.1 Introduction...................... ......................................................................... 125
6.2 Matching Condition Evaluation....................................................................... 126
6.3 Wave Height Modification .............................................................................. 127
6.3.1 Comparison of 3-D Step Model and Analytic Shallow Water Exact
M odel................................................. ................................................. 14 1
6.4 W ave Angle M odification.......................................................................... 144
6.5 Comparison of 3-D Step Model to Numerical Models.................................... 147
6.5.1 3-D Step Model Versus REF/DIF-1................................................... 148
6.5.2 3-D Step Model Versus 2-D Fully Nonlinear Boussinesq Model......... 154
6.6 Comparison to Laboratory Data of Chawla and Kirby (1996) ...................... 156
6.7 Direction Averaged Wave Field Modification ........................................... 161









6.8 Energy R election ....................................................................................... 166
6.8.1 Comparison to Prior Results............................................................. 166
6.8.2 Effect of Transition Slope on Reflection........................................... 167
6.9 Analytic Nearshore Shoaling and Refraction Method................................... 171
6.9.1 Comparison of Analytic Method to REF/DIF-1.................................... 176
6.9.2 W ave Averaged Results ......................................................................... 183
6.10 Shoreline Evolution M odel......................................................................... 185
6.10.1 Shoreline Change Estimates Shoreward of Bathymetric Anomalies. 186
6.10.2 Effect of Nearshore Form on Shoreline Change ................................. 197
6.10.3 Investigation of Boundary Conditions ........................................... 200
6.10.4 Investigation of Transition Slope on Shoreline Evolution................ 202

7 CONCLUSIONS AND DIRECTIONS FOR FUTURE STUDY ........................208

7.1 C onclusions................................................................................................ 208
7.2 Future W ork ................................................................................................. 2 11

APPENDIX

A ANALYTIC WAVE ANGLE CALCULATION ........................................ ...213

B ANALYTIC FAR-FIELD APPROXIMATION OF ENERGY REFLECTION....215

C ANALYTIC NEARSHORE SHOALING AND REFRACTION METHOD........218

D ANALYTIC SHORELINE CHANGE THEORY AND CALCULATION...........222

R EFEREN C ES ..........................................................................................................228

BIOGRAPHICAL SKETCH ..................................................................................... 236















LIST OF TABLES


Table page

2-1 Capabilities of selected nearshore wave models................................. ........... 42

4-1 Specifications for two bathymetries for exact solution method.............................91















LIST OF FIGURES


Figure page

2-1 Aerial photograph showing salients shoreward of borrow area looking East
to West along Grand Isle, Louisiana, in August, 1985..........................................9

2-2 Aerial photograph showing salients shoreward of borrow area along Grand
Isle, Louisiana, in 1998 ...................................................... ............................ 10

2-3 Bathymetry off Anna Maria Key, Florida, showing location of borrow pit
following beach nourishment project........................... .......................................13

2-4 Beach profile through borrow area at R-26 in Anna Maria Key, Florida ..............14

2-5 Shoreline position for Anna Maria Key Project for different periods relative
to A ugust, 1993 ............................................................................................... 14

2-6 Project area for Martin County beach nourishment project .................................15

2-7 Four-year shoreline change for Martin County beach nourishment project:
predicted versus survey data. ................................... .........................................16

2-8 Setup for laboratory experiment.............................................. ....................... 19

2-9 Results from laboratory experiment showing plan shape after two hours. ............20

2-10 Experiment sequence timeline for Williams laboratory experiments .................21

2-11 Volume change per unit length for first experiment..........................................22

2-12 Shifted even component of shoreline change for first experiment......................23

2-13 Reflection and transmission coefficients for linearly varying depth [hi/hill]
and linearly varying breadth [b2/b 2]. ................................................... ..........26

2-14 Approximate reflection and transmission coefficients for the rectangular
parallelepiped of length 8.86h0 in infinitely deep water......................................27

2-15 Reflection coefficient for a submerged obstacle. ................................................28









2-16 Transmission coefficient as a function of relative wavelength (h=10.1 cm,
d=67.3 cm trench width =161.6 cm).................................................................30

2-17 Transmission coefficient as a function of relative wavelength for
trapezoidal trench; setup shown in inset diagram....................................... ..31

2-18 Reflection coefficient for asymmetric trench and normally incident waves
as a function ofKhl: h2/hl=2, h3/hl=0.5, L/hi=5; L = trench width .....................32

2-19 Transmission coefficient for symmetric trench, two angles of incidence:
L/h1=10, h2/hi=2; L = trench width. ....................................................................32

2-20 Transmission coefficient as a function of relative trench depth; normal
incidence: kIhi= 0.2: (a) L/hi=2; (b) L/hi=8, L = trench width...........................33

2-21 Total scattering cross section of vertical circular cylinder on bottom...................35

2-22 Contour plot of relative amplitude in and around pit for normal incidence;
kl/d = 7/10, k2/h=7i/102, h/d=0.5, b/a =1, a/d=2, a = cross-shore pit length,
b = longshore pit length, h = water depth outside pit, d = depth inside pit, L2
= w avelength outside pit. ...................................... ..............................................37

2-23 Contour plot of diffraction coefficient in and around pit for normal
incidence; a/L=l, b/L=0.5, d/h =3, kh=0.167 ...................................................39

2-24 Contour plot of diffraction coefficient around surface-piercing breakwater
for normal incidence; a/L=l, b/L=0.5, kh=0.167. ...............................................39

2-25 Maximum and minimum relative amplitudes for different koa, for normal
incidence, a/b=6, a/d=nt, and d/h=2......................................................................40

2-26 Comparison of wave height profiles for selected models along transect
parallel to shore located 9 m shoreward of shoal apex[*=experimental data].......44

2-27 Comparison of wave height profiles for selected models along transect
perpendicular to shore and through shoal apex [*=experimental data]. ...............44

2-28 Calculated beach planform due to refraction after two years of prototype
w aves for tw o pit depths. ...................................... ..............................................48

2-29 Calculated beach planform due to refraction over dredged hole after two
years of prototype w aves. ................................... ............................. ...........50

2-30 Comparison of changes in beach plan shape for laboratory experiment and
numerical model after two years of prototype waves..........................................51








2-31 Nearshore wave height transformation coefficients near borrow pit from
R CPW A V E study............................................................................................... 53

2-32 Nearshore wave angles near borrow pit from RCPWAVE study; wave
angles are relative to shore normal and are positive for westerly transport...........53

2-33 Reflection coefficients versus dimensionless pit diameter divided by
wavelength inside and outside the pit; water depth = 2 m, pit depth = 4 m.........55

2-34 Shoreline evolution resulting from each transport term individually for
transect located 80 m shoreward of a pit with a radius = 6 m, last time step
indicated w ith [+] .............................................................................................57

2-35 Shoreline evolution using full transport equation and analytic solution
model for transect located 80 m shoreward of a pit with a radius = 6 m, last
tim e step indicated w ith [+]. ............................................................................ 58

3-1 Definition sketch for trench with vertical transitions...........................................63

3-2 Definition sketch for trench with stepped transitions...........................................66

3-3 Definition sketch for linear transition.................................................................70

3-4 Definition sketch for trench with sloped transitions.......................................73

4-1 Definition sketch for circular pit with abrupt depth transitions............................78

4-2 Definition sketch for boundary of abrupt depth transition...................................82

4-3 Definition sketch for boundaries of gradual depth transitions .............................85

4-4 Definition sketch for boundaries of exact shallow water solution method............90

5-1 Matching conditions with depth for magnitude of the horizontal velocity and
velocity potential for trench with abrupt transitions and 16 evanescent
m odes............................................................................................. .............. 94

5-2 Matching conditions with depth for phase of the horizontal velocity and
velocity potential for trench with abrupt transitions and 16 evanescent
m odes........................................................... ........................... .. ......................94

5-3 Relative amplitude for cross-trench transect for k1hl = 0.13; trench
bathymetry included with slope = 0.1...............................................................95

5-4 Relative amplitude for cross-shoal transect for k1h, = 0.22; shoal bathymetry
included w ith slope = 0.05................................................................................ 96









5-5 Relative amplitude along cross-trench transect for Analytic Model and
FUNWAVE 1-D for klhi = 0.24; trench bathymetry included with slope
0 .1 ................................................................................................................. 9 8

5-6 Relative amplitude along cross-shoal transect for Analytic Model and
FUNWAVE 1-D for k1hi = 0.24; shoal bathymetry included with upwave
slope of 0.2 and downwave slope of 0.05.........................................................99

5-7 Comparison of reflection coefficients from step method and Kirby and
Dalrymple (1983a Table 1) for symmetric trench with abrupt transitions and
normal wave incidence: h3 = hi, h2/h1 = 3, W/hi = 10 ........................................ 100

5-8 Comparison of transmission coefficients from step method and Kirby and
Dalrymple (1983a Table 1) for symmetric trench with abrupt transition and
normal wave incidence: h3= hi, h2/h = 3, W/hi = 10 ........................................ 101

5-9 Setup for symmetric trenches with same depth and different bottom widths
and transition slopes .............................................................................................102

5-10 Reflection coefficients versus k1hi for trenches with same depth and
different bottom widths and transition slopes. Only one-half of the
symmetric trench cross-section is shown with slopes of 5000, 1, 0.2 and 0.1....103

5-11 Transmission coefficients versus klhl for trenches with same depth and
different bottom widths and transition slopes. Only one-half of the
symmetric trench cross-section is shown with slopes of 5000, 1, 0.2 and 0.1. ... 104

5-12 Reflection coefficient versus the number of evanescent modes used for
trenches with same depth and transition slopes of 5000, 1, and 0.1..................105

5-13 Reflection coefficient versus the number of steps for trenches with same
depth and transition slopes of 5000, 1, and 0.1..........................................106

5-14 Reflection coefficients versus klhi for trenches with same bottom width and
different depths and transition slopes. Only one-half of the symmetric
trench cross-section is shown with slopes of 5000, 1, 0.2 and 0.05 ..................107

5-15 Reflection coefficients versus k1ih for trenches with same top width and
different depths and transition slopes. Only one-half of the symmetric
trench cross-section is shown with slopes of 5000, 5, 2, and 1. ........................108

5-16 Reflection coefficients versus k1hi for trenches with same depth and bottom
width and different transition slopes. Only one-half of the symmetric trench
cross-section is shown with slopes of 5000, 0.2, 0.1, and 0.05 .........................109









5-17 Reflection coefficients versus klhl for shoals with same depth and different
top widths and transition slopes. Only one-half of the symmetric shoal
cross-section is shown with slopes of 5000, 0.5, 0.2 and 0.05..........................110

5-18 Reflection coefficients versus k1hi for Gaussian trench with C1 = 2 m and
C2 = 12 m and ho = 2 m. Only one-half of the symmetric trench cross-
section is shown with 43 steps approximating the non-planar slope .................11

5-19 Reflection coefficients versus k1hi for Gaussian shoal with C1 = 1 m and C2
= 8 m and ho = 2 m. Only one-half of the symmetric shoal cross-section is
shown with 23 steps approximating the non-planar slope.................................112

5-20 Reflection coefficients versus k1hi for a symmetric abrupt transition trench,
an asymmetric trench with sl = 1 and S2 = 0.1 and a mirror image of the
asym m etric trench............................................................................................. 113

5-21 Reflection coefficients versus k1hi for an asymmetric abrupt transition
trench (hi # h5), an asymmetric trench with gradual depth transitions (hi # h5
and sl = S2 = 0.2) and a mirror image of the asymmetric trench with s, = s2. ......114

5-22 Reflection coefficients versus k1hi for an asymmetric abrupt transition
trench (h, # h5), an asymmetric trench with gradual depth transitions (hi # h5
and sl = 1 and s2 = 0.2) and a mirror image of the asymmetric trench with si
S2 S .................................................................................................... ........... 115

5-23 Reflection coefficient versus the space step, dx, for trenches with same
depth and different bottom width and transition slopes. Only one-half of the
symmetric trench cross-section is shown with slopes of 5000, 1 and 0.1...........117

5-24 Reflection coefficients versus k3h3 for three solution methods for the same
depth trench case with transition slope equal to 5000. Only one-half of the
symmetric trench cross-section is shown........................................................... 118

5-25 Reflection coefficients versus k3h3 for three solution methods for same
depth trench case with transition slope equal to 1. Only one-half of the
symmetric trench cross-section is shown....................................... ............... 19

5-26 Reflection coefficients versus k3h3 for three solution methods for same
depth trench case with transition slope equal to 0.1. Only one-half of the
symmetric trench cross-section is shown...................................... ............ ... 119

5-27 Conservation of energy parameter versus k3h3 for three solution methods
for same depth trench case with transition slope equal to 1. Only one-half
of the symmetric trench cross-section is shown. ..............................................120

5-28 Reflection coefficients versus klhi for step and numerical methods for
Gaussian shoal (ho = 2 m, C1 = 1 m, C2 = 8 m). Only one-half of the








symmetric shoal cross-section is shown with 23 steps approximating the
non-planar slope ...................................................................................................12 1

5-29 Reflection coefficients versus k3h3 for step and numerical methods for
Gaussian trench in shallow water (ho = 0.25 m, C1 = 0.2 m, C2 = 3 m). Only
one-half of the symmetric trench cross-section is shown with 23 steps
approximating the non-planar slope................................................... ............. 122

5-30 Reflection coefficients versus k3h3 for three solution methods for same
depth trench case with symmetric abrupt transition trench and asymmetric
trench with unequal transition slopes equal to 1 and 0.1.....................................123

5-31 Reflection coefficients versus k3h3 for three solution methods for same
depth trench case with asymmetric abrupt transition trench and asymmetric
trench with unequal transition slopes equal to 1 and 0.2...................................124

6-1 Matching conditions with depth for magnitude of the horizontal velocity and
velocity potential for pit with abrupt transitions and 10 evanescent modes........127

6-2 Matching conditions with depth for phase of the horizontal velocity and
velocity potential for pit with abrupt transitions and 10 evanescent modes........128

6-3 Contour plot of relative amplitude with k1h, = 0.24 for pit with transition
slope = 0.1; cross-section of pit bathymetry through centerline included..........129

6-4 Relative amplitude for cross-shore transect at Y = 0 with k1hi = 0.24 for pit
with transition slope = 0.1; cross-section of pit bathymetry through
centerline included. Note small reflection. ................................................130

6-5 Relative amplitude for longshore transect at X = 300 m with k1lh = 0.24 for
pit with transition slope = 0.1; cross-section of pit bathymetry through
centerline included............................................................................................... 130

6-6 Relative amplitude versus number of evanescent modes included for
different Bessel function summations at two locations directly shoreward of
a pit with abrupt transitions in depth...............................................................132

6-7 Relative amplitude versus number of steps approximating slope for different
Bessel function summations for two pits with gradual transitions in depth........133

6-8 Contour plot of relative amplitude for shoal with k1hi = 0.29 and transition
slope = 0.1; cross-section of shoal bathymetry through centerline included.......134

6-9 Relative amplitude for cross-shore transect at Y = 0 for same depth pits for
k1hi = 0.15; cross-section of pit bathymetries through centerline included
with slopes of abrupt, 1, 0.2 and 0.07. ..............................................................135








6-10 Relative amplitude for cross-shore transect at Y = 0 for same depth pits for
klh, = 0.3; cross-section of pit bathymetries through centerline included
with slopes of abrupt, 1, 0.2 and 0.07. ................................................. .........136

6-11 Relative amplitude for alongshore transect at X = 200 m for same depth pits
for k1hi = 0.15; cross-section of pit bathymetries through centerline
included with slopes of abrupt, 1, 0.2 and 0.07. ..................................... ...137

6-12 Relative amplitude for alongshore transect at X = 200 m for same depth pits
for k1h- = 0.3; cross-section of pit bathymetries through centerline included
with slopes of abrupt, 1, 0.2 and 0.07. ............................................................... 138

6-13 Relative amplitude for cross-shore transect located at Y = 0 for same depth
shoals with k1hi = 0.15; cross-section of shoal bathymetries through
centerline included with slopes abrupt, 1, 0.2 and 0.05.......................................139

6-14 Relative amplitude for alongshore transect located at X = 300 m for same
depth shoals with khi = 0.15; cross-section of shoal bathymetries through
centerline included abrupt, 1, 0.2 and 0.05.................................................140

6-15 Relative amplitude for cross-shore transect at Y = 0 for pit with Gaussian
transition slope for k1hi = 0.24; cross-section of pit bathymetry through
centerline included. ..............................................................................................141

6-16 Relative amplitude for cross-shore transect at Y = 0 for pit with for h = C/r
in region of transition slope; cross-section of pit bathymetry through
centerline included............................................................................................ 142

6-17 Relative amplitude for longshore transect at X = 100 m for pit with for h =
C/r in region of transition slope; cross-section of pit bathymetry through
centerline included ............................................................................................ 143

6-18 Relative amplitude for cross-shore transect at Y = 0 for shoal with for h =
C*r in region of transition slope; cross-section of shoal bathymetry through
centerline included............................................................................................ 143

6-19 Relative amplitude for longshore transect at X = 100 m for shoal with for h
= C*r in region of transition slope; cross-section of shoal bathymetry
through centerline included............................................................................. 144

6-20 Contour plot of wave angles in degrees for pit with k1hi = 0.24 and
transition slope = 0.1; cross-section of pit bathymetry through centerline
included ................................................ ........... ................................................. 145

6-21 Contour plot of wave angles for Gaussian shoal with C1 =1 and C2 = 10 for
kihi = 0.22; cross-section of shoal bathymetry through centerline included
with 23 steps approximating slope.................................................................. 146








6-22 Wave angle for alongshore transect at X = 300 m for same depth pits for
klhi = 0.15; cross-section of pit bathymetries through centerline included
with slopes of abrupt, 1, 0.2 and 0.07. Negative angles indicate divergence
of w ave rays ......................... .......................................................................... 147

6-23 Relative amplitude using 3-D Step Model and REF/DIF-1 for cross-shore
transect at Y = 0 for Gaussian pit with k1hi = 0.24; cross-section of pit
bathymetry through centerline included. ........................................................149

6-24 Relative amplitude using 3-D Step Model and REF/DIF-1 for longshore
transect at X = 100 m for Gaussian pit with klh1 = 0.24; cross-section of pit
bathymetry through centerline included. .....................................................150

6-25 Relative amplitude using 3-D Step Model and REF/DIF-1 for longshore
transect at X = 400 m for Gaussian pit with k1hi = 0.24; cross-section of pit
bathymetry through centerline included. .................................................. ......151

6-26 Wave angle using 3-D Step Model and REF/DIF-1 for longshore transect at
X = 100 m for Gaussian pit with k1lh = 0.24; cross-section of pit
bathymetry through centerline included. .................................................. ......152

6-27 Relative amplitude using 3-D Step Model and REF/DIF-1 for cross-shore
transect at Y = 0 for pit with linear transitions in depth with k1ih = 0.24;
cross-section of pit bathymetry through centerline included with slope =
0 .1 ............................................................................................................... .153

6-28 Relative amplitude using 3-D Step Model and REF/DIF-1 for longshore
transect at X = 350 m for pit with linear transitions in depth with k1hi =
0.24; cross-section of pit bathymetry through centerline included with slope
= 0.1. ........................................................................................................... 154

6-29 Relative amplitude using 3-D Step Model and 2-D fully nonlinear
Boussinesq model for cross-shore transect at Y = 0 for shoal with k hi =
0.32; cross-section of pit bathymetry through centerline included....................55

6-30 Experimental setup of Chawla and Kirby (1986) for shoal centered at (0,0)
with data transects used in comparison shown ............................................... 157

6-31 Relative amplitude using 3-D Step Model, FUNWAVE 2-D and data from
Chawla and Kirby (1996) for cross-shore transect A-A with k1hi = 1.89;
cross-section of shoal bathymetry through centerline included.........................158

6-32 Relative amplitude using 3-D Step Model, FUNWAVE 2-D and data from
Chawla and Kirby (1996) for longshore transect E-E with kihl = 1.89;
cross-section of shoal bathymetry through centerline included.........................159








6-33 Relative amplitude using 3-D Step Model, FUNWAVE 2-D and data from
Chawla and Kirby (1996) for longshore transect D-D with k1hi = 1.89;
cross-section of shoal bathymetry through centerline included.........................160

6-34 Relative amplitude using 3-D Step Model, FUNWAVE 2-D and data from
Chawla and Kirby (1996) for longshore transect B-B with k1hi = 1.89;
cross-section of shoal bathymetry through centerline included......................... 161

6-35 Relative amplitude averaged over incident direction (centered at 0 deg) for
alongshore transect at X = 300 m for pit with klhl = 0.24; cross-section of
pit bathymetries through centerline included with slope = 0.1.......................... 162

6-36 Wave angle averaged over incident direction (centered at 0 deg) for
alongshore transect at X = 300 m for pit with klhl = 0.24 with bathymetry
indicated in inset diagram of previous figure. ...........................................163

6-37 Relative amplitude averaged over incident direction (centered at 20 deg) for
alongshore transect at X = 300 m for pit with k1hl = 0.24; cross-section of
pit bathymetries through centerline included with slope = 0.1.......................... 165

6-38 Wave angle averaged over incident direction (centered at 20 deg) for
alongshore transect at X = 300 m for pit with k1hl = 0.24 with bathymetry
indicated in inset diagram of previous figure. ...........................................165

6-39 Reflection coefficient versus non-dimensional diameter; comparison
between shallow water transect method and far-field approximation method. ...167

6-40 Reflection coefficient versus k1hl based on far-field approximation and
constant volume and depth pits; cross-section of pit bathymetries through
centerline included with slopes of abrupt, 1, 0.2 and 0.07.................................168

6-41 Reflection coefficient versus kihl based on far-field approximation and
constant volume bottom width pits; cross-section of pit bathymetries
through centerline included with slopes of abrupt, 1, 0.2 and 0.05...................169

6-42 Reflection coefficient versus k1hi based on far-field approximation and
constant volume and depth shoals; cross-section of shoal bathymetry
through centerline included with slopes of abrupt, 1, 0.2 and 0.07....................170

6-43 Bathymetry for two nearshore regions used in analytic shoaling and
refraction model: Pit 1; linear transition slopes with linear nearshore slope
and Pit 2; Gaussian transition slopes with Equilibrium Beach form for
nearshore slope................................................................................................ 172

6-44 Wave height and wave angle values at start of nearshore region and at
breaking for longshore transect with bathymetry with H = 1 m and T = 12 s
for Pit 1 ................................................ ....................................................... 173









6-45 Contour plot of wave height for Pit 1 in nearshore region with breaking
location indicated with H = 1 m and T = 12 s.................................................174

6-46 Wave height and wave angle values at start of nearshore region and at
breaking for longshore transect with bathymetry with H = 1 m and T = 12 s
for Pit 2 bathym etry. ........................................ ................. ................ ...........175

6-47 Contour plot of wave height for Pit 2 in nearshore region with breaking
location indicated with H = 1 m and T = 12 s.................................................176

6-48 Wave height and wave angle values at start of nearshore region from 3-D
Step Model and REF/DIF-1 (linear) with H = 1 m and T = 12 s for Pit 1..........177

6-49 Wave height and wave angle values at h = 1.68 m (X = 616 m) from 3-D
Step Model and REF/DIF-1 (linear) with H = 1 m and T = 12 s for Pit 1...........178

6-50 Wave height and wave angle values at h = 1.44 m (X = 628 m) from 3-D
Step Model and REF/DIF-1 (linear) with H = 1 m and T = 12 s for Pit 1...........179

6-51 Wave height and wave angle values at start of nearshore region from 3-D
Step Model and REF/DIF-1 (linear) with H = 1 m and T = 12 s for Pit 2...........180

6-52 Wave height and wave angle values at h = 2.04 m (X = 672 m) from 3-D
Step Model and REF/DIF-1 (linear) with H = 1 m and T = 12 s for Pit 2...........181

6-53 Wave height and wave angle values at h = 1.6 m (X = 704 m) from 3-D
Step Model and REF/DIF-1 (linear) with H = 1 m and T = 12 s for Pit 2...........182

6-54 Weighted wave averaged values of wave height and wave angle at start of
nearshore region and at breaking for longshore transect with H = 1 m and T
= 12 s and bathym etry for Pit 1.........................................................................184

6-55 Weighted wave averaged values of wave height and wave angle at start of
nearshore region and at breaking for longshore transect with H = 1 m and T
= 12 s and bathym etry for Pit 2.........................................................................184

6-56 Shoreline evolution for case Pit 1 with K1 = 0.77 and K2 = 0.4 for incident
wave height of 1 m, wave period of 12 s, and time step of 50 s with
shoreline advancement in the negative X direction; Pit 1 bathymetry along
cross-shore transect included. ...........................................................................187

6-57 Parameters for shoreline change after 1st time step showing shoreline
position, and longshore transport terms for case Pit 1 with Ki = 0.77 and K2
= 0................................................................................................ .............. 188

6-58 Final shoreline planform for case Pit 1 with K1 = 0.77 and K2 = 0, 0.2, 0.4,
and 0.77 for incident wave height of 1 m and T = 12 s, with shoreline








advancement in the negative X direction; Pit 1 bathymetry along cross-
shore transect included.........................................................................................189

6-59 Change in shoreline position with modeling time at 4 longshore locations
(Yp = 0, 100, 200, 300 m) for case Pit 1 with K1 = 0.77 and K2 = 0.4 with
shoreline advancement in the negative X direction; Pit 1 bathymetry along
cross-shore transect included. ...........................................................................191

6-60 Comparison of wave height and wave angle values at 600 m and 600 m +/-
20 m for case of Pit 1 for T = 12 s with bathymetry and transect locations
indicated in bottom plot. ........................................................... ...................192

6-61 Final shoreline planform for case Pit 2 with K1 = 0.77 and K2 = 0, 0.2, 0.4,
and 0.77 for incident wave height of 1 m, wave period of 12 s, and time step
of 50 s with shoreline advancement in the negative X direction; Pit 2
bathymetry along cross-shore transect included.............................................. 193

6-62 Shoreline evolution for case Pit 2 with K1 = 0.77 and K2 = 0 for incident
wave height of 1 m, wave period of 12 s, and time step of 50 s with
shoreline advancement in the negative X direction; Pit 2 bathymetry along
cross-shore transect included. ...........................................................................194

6-63 Change in shoreline position with modeling time at 4 longshore locations
(Yp = 0, 100, 200, 300 m) for case Pit 2 with Ki = 0.77 and K2 = 0 with
shoreline advancement in the negative X direction; Pit 2 bathymetry along
cross-shore transect included. ........................................................................... 195

6-64 Final shoreline planform for case Shoal 1 with K1 = 0.77 and K2 = 0, 0.2,
0.4, and 0.77 for incident wave height of 1 m, wave period of 12 s, and time
step of 50 s with shoreline advancement in the negative X direction; Shoal 1
bathymetry along cross-shore transect included.............................................. 196

6-65 Shoreline evolution for case Shoal 1 with Ki = 0.77 and K2 = 0.2 for
incident wave height of 1 m, wave period of 12 s, and time step of 50 s with
shoreline advancement in the negative X direction; Shoal 1 bathymetry
along cross-shore transect included. .................................................................197

6-66 Final shoreline planform for case Pit 1 with linear nearshore slope and EBP
form for T = 12 s and with Ki = 0.77 and K2 = 0 and 0.77 with shoreline
advancement in the negative X direction; Pit 1 and Pit lb bathymetry along
cross-shore transect included. ......................................................................... 198

6-67 Final shoreline planform for case Shoal 1 (linear nearshore slope) and Shoal
lb (Equilibrium Beach Profile) for T = 12 s and with K1 = 0.77 and K2 = 0
and 0.77 with shoreline advancement in the negative X direction; Shoal 1
and Shoal lb bathymetry along cross-shore transect included..........................199


xvii








6-68 Final shoreline planform for case Pit 1 for T = 12 s and with K1 = 0.77 and
K2 = 0.4 for two boundary conditions with shoreline advancement in the
negative X direction; Pit 1 bathymetry along cross-shore transect included.......201

6-69 Final shoreline planform for case Shoal 1 for T = 12 s and with K1 = 0.77
and K2 = 0.77 for two boundary conditions with shoreline advancement in
the negative X direction; Shoal 1 bathymetry along cross-shore transect
include ed ................................................................................................................202

6-70 Final shoreline planform for constant volume pits for T = 12 s and with K1
= 0.77 and K2 = 0 with shoreline advancement in the negative X direction;
cross-section of pit bathymetries through centerline included with slopes of
abrupt, 1, 0.2, 0.07. ..............................................................................................203

6-71 Final shoreline planform for constant volume pits for T = 12 s and with K1
= 0.77 and K2 = 0 and 0.4 with shoreline advancement in the negative X
direction; cross-section of pit bathymetries through centerline included with
slopes of abrupt, 1, 0.2, 0.07. ............................................ ....... ...............205

6-72 Final shoreline planform for 5 periods for constant volume pits with K1 =
0.77 and K2 = 0 with shoreline advancement in the negative X direction;
cross-section of pit bathymetry through centerline included with slope 0.2.......205

6-73 Maximum shoreline advancement and retreat versus period for constant
volume pits with K1 = 0.77 and K2 = 0; cross-section of pit bathymetries
through centerline included with slopes of abrupt, 1, 0.2, 0.07.........................206

6-74 Maximum shoreline advancement and retreat versus period for constant
volume pits with K1 = 0.77 and K2 = 0.4; cross-section of pit bathymetries
through centerline included with slopes of abrupt, 1, 0.2, 0.07.........................207

C-1 Setup for analytic nearshore shoaling and refraction method.............................218

D-1 Definition sketch for analytic shoreline change method showing shoreline
and contours for initial location and after shoreline change..............................223


xviii















Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

WAVE TRANSFORMATION BY BATHYMETRIC ANOMALIES WITH GRADUAL
TRANSITIONS IN DEPTH AND RESULTING SHORELINE RESPONSE

By

Christopher J. Bender

May 2003

Chair: Robert G. Dean
Major Department: Civil and Coastal Engineering

Analytic models for the propagation of linear water waves over a trench or shoal

of finite width and infinite length (2-D Step Model) and over an axisymmetric

bathymetric anomaly (3-D Step Model) are developed. The models are valid in a region

of uniform depth containing a bathymetric anomaly of uniform depth with gradual

transitions in depth allowed as a series of uniform depth steps approximating linear or

non-linear slopes. The velocity potential obtained determines the wave field in the

domain.

The 2-D Step Model provides the complete wave field and, therefore, the

reflection and transmission characteristics for depth anomalies of infinite length that are

either symmetric or asymmetric. The 3-D Step Model determines the wave

transformation caused by the processes of wave refraction, diffraction and reflection.

Using the known velocity potential an energy flux approach is used to calculate the wave

angle and reflected energy at large distances from the anomaly. The 3-D Step Model is








joined with an analytic shoaling and refraction model (Analytic S/R Model) to extend the

solution into the nearshore region. The Analytic S/R Model is employed to propagate

the wave field up to the point of depth limited breaking. The breaking wave conditions

are used to drive the longshore sediment transport and associated shoreline change using

the Shoreline Change Model, which provides estimates of the equilibrium shoreline

planform located landward of a bathymetric anomaly.

Comparisons of the 2-D Step Model to shallow water models developed in the

study, previous studies, and a numerical model are made with good agreement

demonstrated. The reflection coefficients found for symmetric and asymmetric trenches

and shoals indicate multiple and no instances of complete transmission, respectively, with

the solution independent of the incident wave direction for both cases.

Comparisons to a shallow water model, numerical models, and experimental data

verify the results of the 3-D Step Model for several bathymetries, with the Analytic S/R

Model verified by a numerical model for breaking wave conditions. Modeled

equilibrium planforms landward of bathymetric anomalies indicate the importance of the

longshore transport coefficients with either erosion or shoreline advancement possible for

several cases presented.













CHAPTER 1
INTRODUCTION AND MOTIVATION



Irregular and unexpected shoreline planforms adjacent to nearshore borrow areas

have increased awareness of the wave field modification caused by bathymetric

anomalies such as borrow pits or shoals. When a wave field in a region of generally

uniform depth encounters a bathymetric anomaly the wave field is modified through

wave refraction, wave diffraction, wave reflection and wave dissipation, with the first

three known collectively as scattering. These four wave transformation processes result

in a modified wave field that propagates towards, and eventually impacts, the shoreline.

The modified wave field will alter the longshore transport leading to a shoreline planform

that is held to an artificial equilibrium.

Case studies such as the beach nourishment projects at Grand Isle, Louisiana, and

Anna Maria Key, Florida, have shown the possible effects of a nearshore borrow area on

the adjacent shoreline planform. The ability to predict, and possibly design for, the

equilibrium planform in the vicinity of a bathymetric anomaly requires a better

understanding of both the wave and sediment processes near the anomaly. The focus of

this report is to better understand the wave field modifications caused by bathymetric

anomalies, of both two-dimensional and three-dimensional forms, with results providing

the foundation for study on the sediment transport processes and shoreline changes

induced.








Several studies including field and laboratory scales have been conducted to

investigate this issue. These studies examined the wave transformation over a

bathymetric anomaly with the shoreline changes caused by the altered wave field.

Earlier, dating back to the early 1900's, the focus was on the modification of a wave train

encountering a change in bathymetry, which was solved using analytic methods. This

early research included development of analytical solutions for bathymetric changes in

the form of a step, or a pit, first of infinite length (in one horizontal dimension; 2-D

models), and, more recently, of finite dimensions (in two horizontal dimensions; 3-D

models). The complexity of the 3-D models has advanced from a pit/shoal with vertical

sidewalls and uniform depth surrounded by water of uniform depth, which can be solved

analytically, to domains with arbitrary bathymetry that are solved using complex

numerical schemes. Some models combine the calculation of the wave transformation

and resulting shoreline change, whereas others perform the wave calculations separately

and rely on a different program for shoreline evolution.

1.1 Motivation

Changes in offshore bathymetry modify the local wave field, thus causing an

equilibrium planform that may be altered significantly from the previous, relatively

straight shoreline. Not only can a bathymetric change cause wave transformation, but

also may change the sediment transport dynamics by drawing sediment into it from the

nearshore or by intercepting the onshore movement of sediment. Knowledge of wave

field modifications and the resulting effects on sediment transport and shoreline evolution

is essential in the design of beach nourishment projects and other engineering activities

that alter offshore bathymetry.








Beach nourishment has become the preferred technique to address shoreline

erosion. In most beach nourishment projects, the fill placed on the eroded beach is

obtained from borrow areas located offshore of the nourishment site. The removal of

large quantities of fill needed for most projects can result in substantial changes to the

offshore bathymetry through the creation of borrow pits or by modifying existing shoals.

The effect of the modified bathymetry in the borrow area on the wave field and the

influence of the modified wave field on the shoreline can depend on the incident wave

conditions, the nourishment sediment characteristics and some features of the borrow

area including the location, size, shape and orientation.

The large quantities of sediment used in beach nourishment projects combined

with the increase in the number of projects constructed, and an increased industrial need

for quality sediment have, in many areas, led to a shortage of quality offshore fill material

located relatively near to the shore. This shortage has increased interest in the mining of

sediment deposits located in Federal waters, which fall under the jurisdiction of the

Minerals Management Service (MMS). Questions have been raised by the MMS

regarding the potential effects on the shoreline of removing large quantities of sediment

from borrow pits lying in Federal waters (Minerals Management Service, 2003).

A better understanding of the effects of altering the offshore bathymetry is

currently needed. The scattering processes of wave refraction, diffraction, and reflection

modify the wave field in a complex manner dependent on the local wave and nearshore

conditions. A more complete understanding and predictive capability of the effect of

bathymetric changes to the wave field and the resulting shoreline modification leading to

less impactive design of dredge pit geometries should be the goal of current research.








1.2 Models Developed and Applications

A better understanding of the wave field near bathymetric anomalies can be

obtained through models that more accurately represent their shapes and the local wave

transformation processes. The models developed in this study extend previous analytic

methods to better approximate the natural domain and extend the problem from the

offshore region to the shoreline. By meshing a model constrained by a uniform depth

requirement outside of the bathymetric anomaly to a nearshore model with a sloped

bottom, linear waves can be propagated from the offshore over the anomaly and into the

nearshore where shoaling and refraction lead to wave breaking and sediment transport. A

longshore sediment transport model can then predict the shoreline changes resulting from

the wave field modified by the bathymetric anomaly.

Previous analytic 3-D models and all of the previous 2-D models, with the

exception of Dean (1964) and Lee et al. (1981), have domains that contain abrupt

transitions in depth (vertical sidewalls) for the bathymetric anomaly. A more realistic

representation of natural bathymetric anomalies should allow for gradual transitions

(sloped sidewalls). The focus of the present study is the propagation of water waves over

a 2-D (trench or shoal) or 3-D (pit or shoal) bathymetric anomaly of more realistic

geometry and the wave transformation they induce.

Three solution methods are developed for a 2-D domain with linear water waves

and normal wave incidence: (1) the 2-D step method, (2) the slope method and (3) a

numerical method. The 2-D step method is valid in arbitrary water depth while the slope

method and the numerical method are valid only for shallow water conditions. The step

method is an extension of the Takano (1960) solution as modified by Kirby and

Dalrymple (1983a) that allows for a trench or shoal with "stepped" transitions that






5


approximate a specific slope or shape. The slope method is an extension of the Dean

(1964) solution that allows for linear transitions between the changes in bathymetry for a

trench or shoal creating regular or irregular trapezoids. The numerical method employs a

backward space-stepping procedure for arbitrary (but shallow water) bathymetry with the

transmitted wave specified. The 2-D models are compared against each other, with the

results of Kirby and Dalrymple (1983a) and with the numerical model FUNWAVE 1.0

[1-D] (Kirby et al., 1998).

For a 3-D domain, an analytic solution to the wave field modification caused by

bathymetric anomalies with sloped transitions in depth is developed. This solution is an

extension of previous work for anomalies with abrupt depth transitions in regions of

otherwise uniform depth that employs steps to approximate a gradual transition in depth.

A shallow water analytic solution is also developed, which is valid for specific sidewall

slope and pit size combinations. The 3-D models are validated with the laboratory data

of Chawla and Kirby (1996), the numerical models REF/DIF-1 (Kirby and Dalrymple,

1994) and FUNWAVE 1.0 [2-D] (Kirby et al., 1998), and the numerical model of

Kennedy et al. (2000) and through direct comparisons.

The application of the different models to real-world problems depends on the

situation of interest. The study of 2-D models can demonstrate the reflection caused by

long trenches or shoals of finite width such as navigation channels and underwater

breakwaters, respectively. The 3-D models can be employed to study problems with

variation in the longshore and non-oblique incidence. Wave related quantities such as

energy flux and wave direction are calculated and indicate the influence of the

bathymetric anomaly. The wave heights and directions for transects located shoreward of






6


the anomaly are combined with an analytic shoaling and refraction method to determine

the wave propagation in a nearshore zone of arbitrary slope. The longshore transport and

shoreline evolution are also calculated for the nearshore shoaledd and refracted) wave

field that occurs shoreward of the anomaly. Through the methods developed in this study

the wave transformation, energy reflection, longshore transport, and shoreline evolution

induced by a 3-D bathymetric anomaly with gradual transitions in depth can be

investigated.














CHAPTER 2
LITERATURE REVIEW



Several methods have been employed to quantify the impact on the shoreline

caused by changes in the offshore bathymetry including case studies, field experiments,

analytical developments, numerical models, and laboratory studies. The intriguing

behavior of the shoreline following beach nourishment projects at Grand Isle, Louisiana,

Anna Maria Key, Florida, and Martin County, Florida, have led to questions and

investigations regarding the impact of the significant offshore borrow areas present in

each case.

Field studies have been used to investigate the impact of offshore dredging in

relatively deeper water to attempt to define a depth at which bathymetric changes will not

induce significant wave transformation. Laboratory experiments have documented wave

transformations caused by changes in the bathymetry and the resulting effects on the

shoreline in controlled settings possible only in the laboratory.

Solutions for wave transformation by changes in the bathymetry are outlined

primarily in chronological order following the development from analytical solutions for

long waves in one horizontal dimension (2-D) through numerical models for arbitrary

bathymetry that include many wave-related nearshore processes in 3 dimensions.

Modeling of shoreline responses due to wave field modification from changes in offshore

bathymetry is examined with models that include both wave field and shoreline changes

and by coupling models that evaluate these processes independently. The wave








transformation processes included in nearshore models are important factors in the

capability to predict a salient leeward of a pit, the shoreline responses observed in the

limited laboratory experiments and at Grand Isle, Louisiana.

2.1 Case Studies

2.1.1 Grand Isle, Louisiana (1984)

The beach nourishment project at Grand Isle, Louisiana, provides one of the most

interesting, and well publicized examples of an irregular planform resulting from the

effects of a large borrow area lying directly offshore. One year after the nourishment

project was completed, two large salients, flanked by areas of increased erosion,

developed immediately shoreward of the offshore borrow area. Combe and Solieau

(1987) provide a detailed account of the shoreline maintenance history at Grand Isle,

Louisiana, specifications of the beach nourishment project that was completed in 1984,

and details of the shoreline evolution in the two years following completion.

The project required 2.1x106 m3 of sediment with approximately twice this

amount dredged from an area lying 800 m from the shore (Combe and Soileau, 1987) in

4.6 m of water (Gravens and Rosati, 1994). The dredging resulted in a borrow pit that

was "dumbbell" shaped in the planform with two outer lobes dredged to a depth of 6.1 m

below the bed, connected by a channel of approximate 1,370 m length dredged to 3.1 m

below the bed (Combe and Soileau, 1987). The salients seen in Figure 2-1 started to

form during storm events that occurred during the winter and spring of 1984/85 (Combe

and Soileau, 1987).

By August 1985 the salients and associated areas of increased erosion were

prominent features on the shoreline. An aerial survey of the area that was completed by

the New Orleans District of the Army Corps of Engineers and the Coastal Engineering








Research Center concluded that the size and location of the borrow area were such that its

presence could affect the local wave climate (Combe and Soileau, 1987). Oblique aerial

photography identified the diffraction of the wave field as a result of the borrow area

(Combe and Soileau, 1987). The area of increased erosion near the salients was found to

"affect 25% of the project length and amounted to about 8% of the neat project volume"

(Combe and Soileau, 1987, pg. 1236).




















Figure 2-1: Aerial photograph showing salients shoreward of borrow area looking East
to West along Grand Isle, Louisiana, in August, 1985 (Combe and Solieau, 1987).

Three major hurricanes impacted the project area in the hurricane season

following the project's completion, the first time that three hurricanes struck the

Louisiana coastline in the same season (Combe and Soileau, 1987). While these storms

did tremendous damage to the newly formed berm and caused large sediment losses, the

location and size of the salients remained relatively unchanged. The salients have

remained on the Grand Isle shoreline as shown by an aerial photograph from 1998








(Figure 2-2). It appears that the eastern salient has decreased in size while the western

salient has remained the same size or even become larger.


Figure 2-2: Aerial photograph showing salients shoreward of borrow area along Grand
Isle, Louisiana, in 1998 (modified from Louisiana Oil Spill Coordinator's Office
(LOSCO), 1999).

A series of detached offshore breakwaters was constructed along the eastern part of

Grand Isle in the 1990's, which terminate at the eastern salient and may have affected its

shape.

Bathymetric surveys taken through the borrow area in February 1985 and August

1986 revealed that the outer lobes had filled to about half their original depth and the

channel connecting the lobes had reached the sea bed elevation (Combe and Soileau,

1987). Currently, the borrow area is reported to be completely filled by fine material








(Combe, personal correspondence) which would have required the same approximate

volume of sediment that was dredged for the initial placement. Although the origin of the

sediment that has refilled the borrow pit is not known, it is reported to be finer than the

sediment dredged for the nourishment project, indicating that the material did not

originate from the project. While no longer a bathymetric anomaly, the borrow areas are

reported to continue to modify the wave field as local shrimpers use the waters shoreward

of the pit as a harbor to weather storms. The reason for the sheltering effect of the filled

pit may be due to the energy-dissipating characteristic of the finer material that has filled

the pit.

2.1.2 Anna Maria Key, Florida (1993)

The 1993 beach nourishment project at Anna Maria Key, Florida, is another

example of a project with a large borrow area lying offshore in relatively shallow water.

The project placed 1.6x106 m3 of sediment along a 6.8 km segment (DNR Monuments R-

12 to R-35*) of the 11.6 km long barrier island (Dean et al., 1999). The borrow area for

the project was approximately 3,050 m long and ranged from 490 to 790 m offshore in

approximately 6 m of water (Dean et al., 1999). A planview of the bathymetry near the

project including the borrow area is shown in Figure 2-3. A transect through the borrow

area, indicated in the previous figure at Monument R-26, is shown in Figure 2-4 and

shows dredging to a depth of 3.1 m below the local seabed. This figure shows one pre-

project transect, a transect immediately following completion, and two post-nourishment

transects. The post-nourishment transects indicate minimal infilling of the borrow pit.




The "DNR Monuments" are permanent markers spaced at approximately 300 m along
the Florida sandy beaches for surveying purposes








The shoreline planform was found to show the greatest losses shoreward of the

borrow area. Figure 2-5 shows the shoreline position relative to the August, 1993, data

for seven different periods. A large area of negative shoreline change indicating erosion

is found from DNR Monument numbers 25 to 34 for the July, 1997, and February, 1998,

data. This area lies directly shoreward of the borrow area shown in Figure 2-3. The

behavior of the shoreline directly leeward of the borrow area is seen to be the opposite of

the Grand Isle, Louisiana, response where shoreline advancement occurred.

Volume changes determined from profiles in the project area did not show large

negative values near the southern end of the project. The difference between the

shoreline and volume changes at the southern end of the project implies that the

constructed profiles may have been steeper near the southern end of the project as

compared to those near the northern end (Wang and Dean, 2001).

The proximity of the borrow area to the shoreline is one possible contribution to

the local erosion. Although the reason for the increased erosion in this area is not clear, it

is interesting that the anomalous shoreline recession did not occur until the passage of

Hurricanes Erin and Opal in August and October, 1995, respectively. Hurricane Opal

was a category 4 hurricane with sustained winds of 67 m/s when it passed 600 km west of

Manatee County (Liotta, 1999). A reported storm surge of 0.3 to 1.0 m, combined with

the increased wind and wave action, resulted in overtopping of the beach berm, flooding

of the back area of the project and transport of sediment to the back beach or offshore

(Liotta, 1999). The average shoreline retreat for the project area was approximately 9.1

to 15.2 m, based on observations (Liotta, 1999).












Passage Key Inlet


-13
o R-14

*R-15
eR-18
-17
-R-18
-19
*R-20

OR-21
03 -22
q -23
-24
-25

*R-26
-27
-28


-T-30

*R-31
,R-32
.R-33A
*R-34
S,,R-35
-36
*R-37
-*R-38
\ R&R-39
-R-40
-41
Longboat Pass


Figure 2-3: Bathymetry off Anna Maria Key, Florida, showing location of borrow pit
following beach nourishment project (modified from Dean et al., 1999).










9.15


6.10


3.05


0


-3.05


-6.10


-9.15 -
0 152.4 304.8 457.2 609.6
Distance from Monument (m


Figure 2-4: Beach profile through borrow area at R-26 in /
(modified from Wang and Dean, 2001).



20
.. ........ ........ ..... ............ 7. ,
Longshre Exi
ofBorriPit
Sii








1 0 ............ ...... ....




-40 .... ^ ..... ........ .. ... ... ......i....... -
P Inist





5 10 15 20 25 30
DNR Moumrent No.

Figure 2-5: Shoreline position for Anna Maria Key Project
to August, 1993 (modified from Dean et al. 1999).


762.0 914.4

a Maria Key, Florida
Mnna Maria Key, Florida


for different periods relative









2.1.3 Martin County, Florida (1996)

The Hutchinson Island beach nourishment project in Martin County, Florida was

constructed in 1996 with the placement of approximately 1.1x106 m3 of sediment along

6.4 km of shoreline, between DNR Monuments R-1 and R-25 (Sumerell, 2000). The

borrow area for this project was a shoal rising 4.9 m above the adjacent bed and lying 910

m offshore in 12.8 m of water (Sumerell, 2000). Figure 2-6 shows the borrow area

location offshore of the southern end of the project area. An average of 3 m of sediment

was dredged from the central portion of the shoal.


N~il~w PHOJ( I I


ArtANiIC
OCEAN


BORROW
AREA


Figure 2-6: Project area for Martin County beach nourishment project (Applied
Technology and Management, 1998).

The 3-year and 4-year post-nourishment shoreline surveys show reasonable

agreement with modeling conducted for the project, except at the southern end, near the

borrow area (Sumerell, 2002). Figure 2-7 shows the predicted shoreline and the survey









data for the 4-year shoreline change. This case differs from the previous two as the

borrow area did not create a pit but reduced the height of an offshore shoal. By lowering

the height of the shoal the shoreline leeward of the borrow area was exposed to greater

wave action, which is the opposite of the sheltering (through reflection) effect of an

offshore pit. The borrow area, with its large extent and proximity to the project, is a

possible reason for the higher than expected erosion at the southern end of the project.


30.5

m Predicted
15.2 Survey Data (December 99)



E 0


3 -15.2


-30.5


-45.7


Figure 2-7: Four-year shoreline change for Martin County beach nourishment project:
predicted versus survey data (modified from Sumerell, 2000).

2.2 Field Experiments

Field studies have been conducted to examine the effects of offshore dredging on

the coastal environment. The purposes of the these studies have varied and include the

tendency of a dredged pit to induce sediment flows into it from the nearshore, the

interception of sediment transport, and wave transformation effects of a newly dredged

pit on the shoreline.








2.2.1 Price et al. (1978)

Price et al. (1978) investigated the effect of offshore dredging on the coastline of

England. The tendency of a dredge pit to cause a drawdown of sediment and to prevent

the onshore movement of sediment was investigated. The study by Inman and Rusnak

(1956) on the onshore-offshore interchange of sand off La Jolla, California, was cited.

This three-year study found vertical bed elevation changes of only +/- 0.03 m at depths

greater than 9 m. Based on the consideration that the wave conditions off the southern

coast of England would be less energetic than off La Jolla, California, Price et al. (1978)

concluded that beach drawdown at a depth greater than 10 m would not occur.

A radioactive tracer experiment off Worthing, on the south coast of England, was

performed to investigate the mobility of sediment at depths of 9, 12, 15, and 18 m. The

20-month study found that at the 9 and 12 m contours there was a slight onshore

movement of sediment and it was concluded that the movement of sediment beyond a

depth contour of 18 m on the south coast of England would be negligible. Therefore, at

these locations and in instances when the onshore movement of sediment seaward of the

dredge area is a concern, dredging in water beyond 18 m depth below low water level

was considered acceptable (Price et al., 1978).

A numerical model of the shoreline change due to wave refraction over dredged

holes was also employed in the study, the details of which will be examined later in

Section 2.5.2.1. The model found that minimal wave refraction occurred for pits in

depths greater than 14 m for wave conditions typical off the coast of England.

2.2.2 Koiima et al. (1986)

The impact of dredging on the coastline of Japan was studied by Kojima et al.

(1986). The wave climate as well as human activities (dredging, construction of








structures) for areas with significant beach erosion and/or accretion was studied in an

attempt to determine a link between offshore dredging and beach erosion. The study area

was located offshore of the northern part of Kyushu Island. The wave climate study

correlated yearly fluctuations in the beach erosion with the occurrence of both storm

winds and severe waves and found that years with high frequencies of storm winds were

likely to have high erosion rates. A second study component compared annual variations

in offshore dredging with annual beach erosion rates and found strong correlation at some

locations between erosion and the initiation of dredging although no consistent

correlation was identified.

Hydrographic surveys documented profile changes of dredged holes over a four-

year period. At depths less than 30 m, significant infilling of the holes was found, mainly

from the shoreward side, indicating a possible interruption in the longshore and offshore

sediment transport. This active zone extends to a much larger depth than found by Price

et al. (1978) and by Inman and Rusnak (1956). The explanation by Kojima et al. is that

although the active onshore/offshore region does not extend to 30 m, sediment from the

ambient bed will fill the pit causing a change in the supply to the upper portion of the

beach and an increase in the beach slope. Changes in the beach profiles at depths of 35

and 40 m were small, and the holes were not filled significantly.

Another component of the study involved tracers and seabed level measurements

to determine the depths at which sediment movement ceases. Underwater photographs

and seabed elevation changes at fixed rods were taken at 5 m depth intervals over a

period of 3 months during the winter season for two sites. The results demonstrated that

sediment movement at depths up to 35 m could be significant. This depth was found to









be slightly less than the average depth (maximum 49 m, minimum 20 m) for five

proposed depth of closure equations using wave inputs with the highest energy

(H = 4.58 m, T = 9.20 s) for the 3-month study period.

2.3 Laboratory Experiments

2.3.1 Horikawa et al. (1977)

Laboratory studies have been carried out to quantify wave field and nearshore

modifications due to the presence of offshore pits. Horikawa et al. (1977) performed

wave basin tests with a model of fixed offshore bathymetry and uniform depth containing

a rectangular pit of uniform depth and a nearshore region composed of moveable

lightweight sediments. The experimental arrangement is shown in Figure 2-8. The

incident wave period and height were 0.41 s and 1.3 cm, respectively. With the pit

covered, waves were run for 5.5 hours to obtain an equilibrium planform followed by

wave exposure for three hours with the pit present. Shoreline measurements were

conducted at 1 hour intervals to determine the pit induced changes.

U
400 300 200 100 0 c
,-,--120 1
5 1o -100

30 C BEACH .60
1 SHORE INEni-- tO LS
0

(unit:cm) 20





400 300 200 100 0
Offshore distance


Figure 2-8: Setup for laboratory experiment (Horikawa et al., 1977).









The results of the experiment are presented in Figure 2-9. Almost all of the

shoreline changes with the pit present occurred in the first two hours. At the still water

level, a salient formed shoreward of the pit, flanked by two areas of erosion that generally

extend to the sidewalls of the experiment; however, the depth contour at a water depth, h

= 0.85 cm, also shown in Figure 2-9, shows only a slightly seaward displacement at the

pit centerline.


HOLE

"" 30emc-. or
20 -
h 0.5 cm

0
C.


0 Initial shoreline (h-0)
h- 0 cm

0 20 40 6C 80 100 120
Longshore distance, X (cm)


Figure 2-9: Results from laboratory experiment showing plan shape after two hours
(Horikawa et al., 1977).

2.3.2 Williams (2002)

Williams (2002) performed wave basin experiments similar to those of Horikawa

et al. (1977). The experimental setup of a fixed bed model containing a pit with a

moveable sand shoreline was constructed for similar trials by Bender (2001) and was a

larger scale version of the Horikawa et al. (1977) arrangement. The Williams

experimental procedure consisted of shoreline, bathymetric, and profile measurements

after specified time intervals that comprised a complete experiment. Figure 2-10 shows

the experiment progression sequence that was used. For analysis, the shoreline and









volume measurements were made relative to the last measurements of the previous 6-

hour phase. The conditions for the experiments were 6 cm waves with 1.35 s period and

a depth of 15 cm in the constant depth region surrounding the pit. The pit was 80 cm

long in the cross-shore direction, 60 cm in the longshore direction and 12 cm deep

relative to the adjacent bottom.



CompLete Experiment


Current Control Next Control
Phases Phases
Previous Test (covered pit) Current Test (covered pit)
Phases Phases
(open pit) (open pit)



I I I II I I
6.0 7.5 9.0 0.0 1.5 3.0 6.0 7.5 9.0 12.0 1.5 3,0 6.0
(12.0) (0.0)

Time Step (Hours)



Figure 2-10: Experiment sequence timeline for Williams laboratory experiments
(Williams, 2002)

Shoreline and volume change results were obtained for three experiments. Figure

2-11 shows the volume change per unit length versus longshore distance for results with

pit covered (control phase) and uncovered (test phase). The dashed line represents

volume changes for a covered pit relative to the time step zero which concluded 6 hours

of waves with the pit uncovered. The solid line shows the change with the pit uncovered

relative to time step six when six hours of wave exposure with the pit covered ended.

The volume change results show the model beach landward of the pit lost volume at

almost every survey location during the period with the pit covered and experienced a

gain in volume with the pit uncovered.













S- -- -150 -- --- -
It ------~..-...... v-Q.. ---------; "^-.....
'E 50
0-
ScE -1 O -120 -100 -80 -60 -40 >20 0 20 40 60 A'0 100 120 1 0
1 20 40 -6018e )100 12010

150
~ r -- ---- ---------- 4-100 ---' --------------

Longshore Position (cm)

S- 0 to 6 Hours 6 to 12 Hours

(pit covered) (pit present)
Figure 2-11: Volume change per unit length tor thirst expenment (Williams, 2002).

The net volume change for the first complete experiment (control and test phase)

was approximately 2500 cm3. Different net volume changes were found for the three

experiments. However, similar volume change per unit length results were found in all

three experiments indicating a positive volumetric relationship between the presence of

the pit and the landward beach.

The shoreline change results showed shoreline retreat, relative to Time Step 0.0,

in the lee of the borrow pit during the control phase (pit covered) for all three

experiments with the greatest retreat at or near the centerline of the borrow pit. All three

experiments showed shoreline advancement in the lee of the borrow pit with the pit

uncovered (test phase). With the magnitude of the largest advancement being almost

equal to the largest retreat in each experiment, it was concluded that, under the conditions

tested, the presence of the borrow pit resulted in shoreline advancement for the area

shoreward of the borrow pit (Williams, 2002).

An even-odd analysis was applied to the shoreline and volume change results in

an attempt to isolate the effect of the borrow pit. The even function was assumed to

represent the changes due solely to the presence of the borrow pit. The even components










were adjusted to obtain equal positive and negative areas, which were not obtained using

the laboratory data. For each experiment, the shifted even results shoreward of the pit

showed positive values during the test phase for both the shoreline and volume changes

with negative values during the control phase. The shifted even component of shoreline

change for the first experiment is shown in Figure 2-12. These results further verify the

earlier findings concerning the effect of the pit.


0
0



Ur-



U)
w


Longshore Position (cm)

-* -0 to 6 Hours -6 to 12 Hours
(pit covered) (pit present)
Figure 2-12: Shifted even component of shoreline change for first experiment (Williams,
2002).

2.4 Wave Transformation

2.4.1 Analytic Methods

There is a long history of the application of analytic methods to determine wave

field modifications by bathymetric changes. Early research centered on the effect on

normally incident long waves of an infinite step, trench or shoal of uniform depth in an

otherwise uniform depth domain. More complex models were later developed to remove

the long wave restriction, add oblique incident waves and allow for the presence of a


---------------. ..... .... .. .. t 5 -- -- ---





----------------------9---- -1-0

t0 -120 -100 -; -60 -20 0 20 60 100 120 1




-1 ----------- -








current. More recently, many different techniques have been developed to obtain

solutions for domains containing pits or shoals of finite extent. Some of these models

focused solely on the wave field modifications, while others of varying complexity

examined both the wave field modifications and the resulting shoreline impact.

2.4.1.1 2-Dimensional methods

By matching surface displacement and mass flux normal to the change in

bathymetry Lamb (1932) was one of the first to develop a long wave approximation for

the reflection and transmission of a normally incident wave at a finite step.

Bartholomeauz (1958) performed a more thorough analysis of the finite depth step

problem and found that the Lamb solution gave correct results for the reflection and

transmission coefficients for lowest order (kh) where k is the wave number and h is the

water depth prior to the step. Sretenskii (1950) investigated oblique waves over a step

between finite and infinite water depths assuming the wavelength to be large compared to

the finite depth. An extensive survey of early theoretical work on surface waves

including obstacle problems is found in Wehausen and Laitone (1960).

Kreisel (1949) developed a method that conformally mapped a domain containing

certain obstacles of finite dimensions into a rectangular strip. The reflection from

obstacles at the surface and on the bottom were considered. Kreisel (1949) presents a

proof demonstrating that the reflection coefficient is independent of the incident wave

direction for a symmetric (upwave and downwave depth equal) or asymmetric obstacle

(trench or shoal) in arbitrary water depth.

Jolas (1960) studied the reflection and transmission of water waves of arbitrary

relative depth over a long submerged rectangular parallelepiped (sill) and performed an

experiment to document the wave transformation. To solve the case of normal wave









incidence and arbitrary relative depth over a sill or a fixed obstacle at the surface Takano

(1960) used an eigenfunction expansion of the velocity potentials in each constant depth

region and matched them at the region boundaries. The set of linear integral equations

was solved for a truncated series. A laboratory experiment was also conducted in this

study.

Dean (1964) investigated long wave modification by linear transitions. The linear

transitions included both horizontal and vertical changes. The formulation allowed for

many domains including a step, either up or down, and converging or diverging linear

transitions with a sloped wall. A proposed solution was defined with plane-waves of

unknown amplitude and phase for the incident and reflected waves with the transmitted

wave specified. Wave forms, both transmitted and reflected, were represented by Bessel

functions in the region of linear variation in depth and/or width. The unknown

coefficients were obtained through matching the values and gradients of the water

surfaces at the ends of the transitions. Analytic expressions were found for the reflection

and transmission coefficients. The results indicate that the reflection and transmission

coefficients depend on the relative depth and/or width and a dimensionless parameter

containing the transition slope, the wavelength and the depth or width (Figure 2-13). In


Figure 2-13 the parameter Z, = 4' for the case of linearly varying depth and
L, S,


Z, = for linearly varying breadth where I indicates the region upwave of the
LS,

transition, Sv is equal to the depth gradient, and SH is equal to one-half the breadth

gradient. These solutions were shown to converge to those of Lamb (1932) for the case

of an abrupt transition (ZI=0).



















]/ -,-^-

I
JI __/I t L'
44' "fc!t i -







001 002 005 0.1 02 05 I 2 5 10 20 50 100 200 S00 100

Value of hi/hiI b2 /b III

Figure 2-13: Reflection and transmission coefficients for linearly varying depth [hi/hill]
and linearly varying breadth [bi2/ b,12] (modified from Dean, 1964).

Newman (1965a) studied wave transformation due to normally incident waves on

a single step between regions of finite and infinite water depth with an integral-equation

approach. This problem was also examined by Miles (1967) who developed a plane-

wave solution for unrestricted kh values using a variational approach (Schwinger &

Saxon 1968), which for this case essentially solves a single equation instead of a series of

equations (up to 80 in Newman's solution) as in the integral equation approach. The

difference between the results for the two solution methods was within 5 percent for all

kh values (Miles, 1967).









Newman (1965b) examined the propagation of water waves past long obstacles.

The problem was solved by constructing a domain with two steps placed "back to back"

and applying the solutions of Newman (1965a). Complete transmission was found for

certain water depth and pit length combinations; a result proved by Kriesel (1949).

Figure 2-14 shows the reflection coefficient, Kr, and the transmission coefficient, Kt,

versus Kooho where K, is the wave number in the infinitely deep portion before the

obstacle and ho is the depth over the obstacle. The experimental results of Takano (1960)

are included for comparison. It is evident that the Takano experimental data included

energy losses.








Numerical results of Newman (1965b)
0'7 o Takano (1960) experimental results

0.6

i0-5





Kr






0 0.2 0 0-6 0-8 1-0 1-2 14 1- 1.8 24,


Figure 2-14: Approximate reflection and transmission coefficients for the rectangular
parallelepiped of length 8.86ho in infinitely deep water (Newman, 1965b).








The variational approach was applied by Mei and Black (1969) to investigate the

scattering of surface waves by rectangular obstacles. For a submerged obstacle, complete

transmission was found for certain kho values where ho is the depth over the obstacle. A

comparison of the results of Mei and Black (1969) and those of Newman (1965b) is

shown in Figure 2-15, which presents the reflection coefficient versus kho for a

submerged obstacle. Data from the Jolas (1960) experiment are also included on the plot

and compared to the results of Mei and Black (1969) for a specific i / ho, where is the

half-length of the obstacle.



[Mei and Black (1969)], o [data from Jolas (1960)
experiment]
O'S ( / ho =4.43, h/ho= 2.78)
[Mei and Black (1969)], --- [Newman


i i \




ii 0

02

0 0-2 0-4 0 6 0-8 1-0 1-2 1-4 1-6
kho

Figure 2-15: Reflection coefficient for a submerged obstacle (Mei and Black, 1969).

Black and Mei (1970) applied the variational approach to examine the radiation

caused by oscillating bodies and the disturbance caused by an object in a wave field.

Two domains were used for both submerged and semi-immersed (surface) bodies: the

first domain was in Cartesian coordinates, with one vertical and one horizontal








dimension, for horizontal cylinders of rectangular cross section and the second domain

was in cylindrical coordinates, for vertical cylinders of circular section. The second

domain allowed for objects with two horizontal dimensions to be studied for the first time

(see Section 2.4.1.2.). Black et al. (1971) used the variational formulation to study the

radiation due to the oscillation of small bodies and the scattering induced by fixed bodies.

Black et al. demonstrated the scattering caused by a fixed object in a single figure; see

Black and Mei (1970) for further results.

Lassiter (1972) used complementary variational integrals to solve the problem of

normally incident waves on an infinite trench where the depth on the two sides of the

trench may be different (the asymmetric case). The symmetric infinite trench problem

was studied by Lee and Ayer (1981), who employed a transform method. The fluid

domain was divided into two regions, one an infinite uniform depth domain and the other

a rectangular region representing the trench below the uniform seabed level. The

transmission coefficient for the trench is shown in Figure 2-16 with the theoretical results

plotted along with data from a laboratory experiment conducted as part of the study.

Results from a boundary integral method used to compare with the theoretical results are

also plotted. The results show six of an infinite number of relative wavelengths where

complete transmission (Kt=I) will occur, a result that had been found in prior studies

(Newman (1965b), Mei and Black (1969)). The laboratory data show the general trend

of the theoretical results, with some variation due to energy losses and reflections from

the tank walls and ends.

Lee et al. (1981) proposed a boundary integral method for the propagation of

waves over a prismatic trench of arbitrary shape, which was used for comparison to









selected results in Lee and Ayer (1981). The solution was found by matching the

unknown normal derivative of the potential at the boundary of the two regions. A

comparison to previous results for trenches with vertical sidewalls was conducted with

good agreement. A case with bathymetry containing gradual transitions in depth was

shown in a plot of the transmission coefficient for a trapezoidal trench (Figure 2-17).

Note that the complete dimensions of the trapezoidal trench are not specified in the inset

diagram, making direct comparison to the results impossible.



So00



095 -
0.90
S-- [Numerical Solution]
0-85
S- [Experimental Results]
i- x [Boundary Integral Method]
S0-80


0.75

0.70 .----.
0 005 0-10 0 15 020 025
Depth to wavelength ratio (h/X)


Figure 2-16: Transmission coefficient as a function of relative wavelength (h=l0.O cm,
d=67.3 cm, trench width =161.6 cm) (modified from Lee and Ayer, 1981).

Miles (1982) solved for the diffraction by an infinite trench for obliquely incident

long waves. The solution method for normally incident waves used a procedure

developed by Kreisel (1949) that conformally mapped a domain containing certain

obstacles of finite dimensions into a rectangular strip. To add the capability of solving









for obliquely incident waves, Miles used the variational formulation of Mei and Black

(1969).


C cOr o'.. .0. 0 0. 0 .
00 so

C C





U.90 --* t

r 15.2 cm
0 03."




C,8,5 i- ..........- 161.6 cm -- -



0.5 1 0. 15 0.20 C. 2



Figure 2-17: Transmission coefficient as a function of relative wavelength for
trapezoidal trench; setup shown in inset diagram (modified from Lee et al., 1981).

The problem of obliquely incident waves over an asymmetric trench was solved

by Kirby and Dalrymple (1983a) using a modified form of Takano's (1960) method.

Figure 2-18 compares the reflection coefficient for the numerical solution for normally

incident waves and the results of Lassiter (1972). The results from a boundary integral

method used to provide verification are included. Differences in the results of Kirby and

Dalrymple and those of Lassiter are evident. Lee and Ayer (1981,[see their Figure 2])

also demonstrated differences in their results and those ofLassiter (1972). The effect of


oblique incidence is shown in Figure 2-19 where the reflection and transmission

coefficients for two angles of incidence are plotted.









0.5


04-


0.7 0.8


0. 0. 0.3 0.4 0.5 0.6
Kh,


Figure 2-18: Reflection coefficient for asymmetric trench and normally incident waves
as a function of KhI: h2/hl=2, h3/h1=0.5, L/hi=5; L = trench width (Kirby and Dalrymple,
1983a).




/- -- [Numerical solution, 01 = 0 deg]
K0.9\ / [Numerical solution, 1 =45 deg]


khi


Figure 2-19: Transmission coefficient for symmetric trench, two angles of incidence:
L/hi=10, h2/hl=2; L = trench width (modified from Kirby and Dalrymple, 1983a).

This study also investigated the plane-wave approximation and the long-wave

limit, which allowed for comparison to Miles (1982). Figure 2-20 shows transmission

coefficients with the results of the numerical solution, the long wave solution, and values

from the Miles (1982) solution, which is only valid for small kh values in each region.

For the first case, with a small relative trench width, the numerical results from Kirby and


[Kirby and Dalrymple (1983a)]
-, [Lassiter (1972, Fig. 7)]
[Boundary Integral Method]






'
--...

"xx.,, *~
\': ,,, -'


0Ii









Dalrymple compare well with the results using the Miles (1982) method and the plane-

wave solution is seen to deviate from these. For the case of a relative trench length equal

to eight, the numerical results differ from the plane-wave solution, which diverge from

the values using Miles (1982) for this case where the assumptions are violated.

1.000-__


DO, Q)'


0.994

1.00


[Long Wave Solution]
* [Numerical Solution]

-[Miles (1982) Solution]


h2/hi


Figure 2-20: Transmission coefficient as a function of relative trench depth; normal
incidence: klhl= 0.2: (a) L/hi=2; (b) L/hi=8, L = trench width. (Kirby and Dalrymple,
1983a).

The difference in scales between the two plots is noted. An extension of this

study is found in Kirby et al. (1987) where the effects of currents flowing along the

trench are included. The presence of an ambient current was found to significantly alter








the reflection and transmission coefficients for waves over a trench compared to the no

current case. Adverse currents and following currents made a trench less reflective and

more reflective, respectively (Kirby et al., 1987).

2.4.1.2 3-Dimensional methods

Extending the infinite trench and step solutions (one horizontal dimension) to a

domain with variation in the longshore (two-horizontal dimensions) is a natural

progression allowing for the more realistic case of wave transformation by a finite object

or depth anomaly to be studied. Changes in bathymetry can cause changes in wave

height and direction through the four wave transformation processes noted earlier. Some

of the two-dimensional models study only the wave transformation, while others use the

modified wave field to determine the impact of a pit or shoal on the shoreline. Several

models use only a few equations or matching conditions on the boundary of the pit or

shoal to determine the wave field and in some cases the impact on the shoreline in a

simple domain containing a pit or shoal. Other, much more complex and complete

models and program packages have been developed to solve numerically for the wave

field over a complex bathymetry, which may contain pits and/or shoals. Both types of

models can provide insight into the effect of a pit or shoal on the local wave field and the

resulting impact on the shoreline.

The wave transformation in a three-dimensional domain was investigated in a

study by Black and Mei (1970), which solved for the radially symmetric case of a

submerged or floating circular cylinder in cylindrical coordinates. A series of Bessel

functions was used for the incident and reflected waves, as well as for the solution over

the shoal with modified Bessel functions representing the evanescent modes. As

mentioned previously in the 2-D section, a variational approach was used and both the








radiation by oscillating bodies and the disturbance caused by a fixed body were studied.

The focus of the fixed body component of the study was the total scattering cross section,

Q, which is equal to the width between two wave rays within which the normally incident

wave energy flux would be equal to that scattered by the obstacle and the differential

scattering cross-section, which shows the angular distribution of the scattered energy

(Black and Mei, 1970). Figure 2-21 shows the total scattering cross section for a circular

cylinder at the seabed for three ratios of cylinder radius (a) to depth over the cylinder (h).

20


a/h=3
1-6



12



0.8 a/h=2



0-4 -

a/h=l

0 1 2 3 4 5 6
ka

Figure 2-21: Total scattering cross section of vertical circular cylinder on bottom
(modified from Black et al., 1971).

Williams (1990) developed a numerical solution for the modification of long

waves by a rectangular pit using Green's second identity and appropriate Green's

functions in each region that comprise the domain. This formulation accounts for the

diffraction, refraction and reflection caused by the pit. The domain for this method








consists of a uniform depth region containing a rectangular pit of uniform depth with

vertical sides. The solution requires discretizing the pit boundary into a finite number of

points at which the velocity potential and the derivative of the velocity potential normal

to the boundary must be determined. Applying matching conditions for the pressure and

mass flux across the boundary results in a system of equations amenable to matrix

solution techniques. Knowledge of the potential and derivative of the potential at each

point on the pit boundary allows determination of the velocity potential solution

anywhere in the fluid domain. The effect of a pit on the wave field is shown in a contour

plot of the relative amplitude in Figure 2-22. A partial standing wave pattern of increased

and decreased relative amplitude is seen seaward of the pit with a shadow zone of

decreased wave amplitude landward of the pit flanked by two areas of increased relative

amplitude.

McDougal et al. (1996) applied the method of Williams (1990) to the case of a

domain with multiple pits. The first part of the study reinvestigated the influence of a

single pit on the wave field for various pit geometries. A comparison of the wave field in

the presence of a pit versus a surface piercing structure is presented in Figures 2-23 and

2-24, which present contour plots of the transformation coefficient, K, (equal to relative

amplitude) that contain the characteristics discussed in the last paragraph. For this case

with the pit depth equal to 3 times the water depth a greater sheltering effect is found (K

= 0.4) landward of the pit than for the case of the full depth breakwater.

An analysis of the effect of various pit characteristics on the minimum value of K

found in the domain was also performed. The dimensionless pit width, a/L, (a = cross-

shore dimension, L = wave length outside pit) was found to increase the distance to the









region where K < 0.5 behind the pit and the value of K was found to decrease and then

become approximately constant as a/L increases. The minimum values of the

transformation coefficient for a wide pit are much lower than those values found in Lee

and Ayer (1981) and Kirby and Dalrymple (1983a), which may be explained by the

refraction divergence that occurs behind the pit in the 2-D case (McDougal et al., 1996).


:4 1 I. .T i i i I K i I 1 i i .. i i .i. :...:.. .
,., .' : : .





..... .......
S. ..1 107*. .

.. ..:. . .- -. .: -.-.. .
(

.... : ,. ... : .


.... ...........:... ........ :..-................. 1.09786





_- 'i -" : P
S= n 1::,::::::::1 M::::::: 5, = 1, 064.







a-j-i -j. j. i ii !: i : I: I.i i: i i i i tLLI i.jL _. I.I I'^i^': ^':!'^ ;:i:.;^^




longshore pit length, h = water depth outside pit, d = depth inside pit, L2 = wavelength
outside pit. (Williams, 1990).








The effect of the dimensionless pit length, b/L, indicates that K decreases as b/L

increases to 1 with a change in the trend, and an increase in K, from b/L = 0.55 to b/L =

0.65. Increasing the dimensionless pit depth, d/L, was found to decrease the minimum

value of K with a decreasing rate. The incident wave angle was not found to significantly

alter the magnitude or the location of the minimum K value, although the width of the

shadow zone changes with incident angle. For the case of multiple pits, it was found that

placement of one pit in the shadow zone of a more seaward pit was most effective in

reducing the wave height. However, adding a third pit did not produce significant wave

height reduction as compared to the two-pit results.

Williams and Vazquez (1991) removed the long wave restriction of Williams

(1990) and applied the Green's function solution method outside of the pit. This solution

was matched to a Fourier expansion solution inside the pit with matching conditions at

the pit boundary. Once again the pit boundary must be discretized into a finite number of

points and a matrix solution for the resulting series of equations was used. Removing the

shallow water restriction allowed for many new cases to be studied, as the wave

conditions approach deep water, the influence of the pit diminishes. A plot of the

minimum and maximum relative amplitude found in the domain versus the dimensionless

pit length (the wave number outside of the pit times the cross-shore pit dimension, koa) is

shown in Figure 2-25. The maximum and minimum relative amplitudes in Figure 2-25

are seen to occur near koa = 2n or when L = a and then approach unity as the

dimensionless pit length increases. The reason that the extreme values do not occur

exactly at koa = 27r is explained by Williams and Vazquez (1991) as due to diffraction

effects near the pit modifying the wave characteristics.































Figure 2-23: Contour plot of diffraction coefficient in and around pit for normal
incidence; a/L=l, b/L=0.5, d/h =3, kh=0.167 (McDougal et al., 1996).


Figure 2-24: Contour plot of diffraction coefficient around surface-piercing breakwater
for normal incidence; a/L=l, b/L=0.5, kh=0.167 (McDougal et al., 1996).









1K,

,n -M Min





"4



0.4






0 2 4 6 0 10 12 14 1s 1i 20

koa

Figure 2-25: Maximum and minimum relative amplitudes for different koa, for normal
incidence, a/b=6, a/d=7r, and d/h=2. (modified from Williams and Vasquez, 1991).

2.4.2 Numerical Methods

The previous three-dimensional solutions, while accounting for most of the wave

transformation processes caused by a pit, are simple in their representation of the

bathymetry and their neglect of many wave-related processes including energy

dissipation. Berkhoff (1972) developed a formulation for the 3-dimensional propagation

of waves over an arbitrary bottom in a vertically integrated form that reduced the problem

to two-dimensions. This solution is known as the mild slope equation and different forms

of the solution have been developed into parabolic (Radder, 1979), hyperbolic, and

elliptic (Berkhoff et al., 1982) models of wave propagation, which vary in their

approximations and solution techniques. Numerical methods allow solution for wave

propagation over an arbitrary bathymetry. Some examples of the parabolic and elliptic








models are RCPWAVE (Ebersole et al., 1986), REF/DIF-1 (Kirby and Dalrymple, 1994),

and MIKE 21's EMS Module (Danish Hydraulics Institute, 1998). Other models such as

SWAN (Holthuijsen et al., 2000) and STWAVE (Smith et al., 2001) model wave

transformation in the nearshore zone using the wave-action balance equation. These

models provide the capability to model wave transformation over complicated

bathymetries and may include processes such as bottom friction, nonlinear interaction,

breaking, wave-current interaction, wind-wave growth, and white capping to better

simulate the nearshore zone. An extensive review of any of the models is beyond the

scope of this paper; however, a brief outline of the capabilities of some of the models is

presented in Table 1.

Maa et al. (2000) provides a comparison of six numerical models. Two parabolic

models are examined: RCPWAVE and REF/DIF-1. RCPWAVE employs a parabolic

approximation of the elliptic mild slope equation and assumes irrotationality of the wave

phase gradient. REF/DIF-1 extends the mild slope equation by including nonlinearity

and wave-current interaction (Kirby and Dalrymple, 1983b; Kirby, 1986). Of the four

other models included, two are defined by Maa et al. (2000) as based on the transient

mild slope equation (Copeland, 1985; Madsen and Larson, 1987) and two are classified

as elliptic mild slope equations (Berkhoff et al., 1982) models.

The transient mild slope equation models presented are Mike 21's EMS Module

and the PMH Model (Hsu and Wen, in review). The elliptic mild slope equation models

use different solution techniques with the RDE Model (Maa and Hwung, 1997; Maa et al.

1998a) applying a special Gaussian elimination method and the PBCG Model employing

a Preconditioned Bi-conjugate Gradient method (Maa et al., 1998b).









Table 2-1: Capabilities of selected nearshore wave models.
RCPWAVE REF/DIF-1 Mike 21 EMS STWAVE SWAN- 3rd Gen.

Elliptic Mild
Parabolic Parabolic Elliptic Mild Conservation Conservation
Solution Slope Equation
Mild Slope Mild Slope of Wave of Wave
Method (Berkhoff et al.
Equation Equation Action Action
(1972)
Phase Averaged Resolved Resolved Averaged Averaged

No No
Spectral No Yes Yes
Use REF/DIF-S Use NSW unit

Shoaling Yes Yes Yes Yes Yes
Refraction Yes Yes Yes Yes Yes

Yes Yes Yes No
Diffraction No
(Small-Angle) (Wide-Angle) (Total) (Smoothing)

Yes Yes Yes
Reflection No No
(Forward only) (Total) (Specular)

Stable Energy Stable Energy Bore Model: Depth limited: Bore Model:
Breaking Flux: Dally et Flux: Dally et al. Battjes & Miche (1951) Battjes & Janssen
al. (1985) (1985) Janssen (1978) criterion (1978)

White- Komen et al. (1984),
No No No Resio (1987) Janssen (1991),
capping
Komen et al. (1994)
Dalrymple et al. Quadratic Hasselmann et al.
Bottom (1984) both Friction Law, (1973), Collins
No No
Friction laminar and Dingemans (1972), Madsen et
turbulent BBL (1983) al. (1988)
Currents No Yes No Yes Yes

Cavaleri &
Malanotte-Rizzoli
Wind No No No Resio (1988) (1981), Snyder et al.,

(1981), Janssen et
al. (1989, 1991)

Availability Commercial Free Commercial Free Free








A table in Maa et al. (2000) provides a comparison of the capabilities of the six

models. A second table summarizes the computation time, memory required and, where

required, the number of iterations for a test case of monochromatic waves over a shoal on

an incline; the Berkhoffet al. (1982) shoal. The parabolic approximation solutions of

REF/DIF and RCPWAVE required significantly less memory (up to 10 times less) and

computation time (up to 70 times less) than the elliptic models, which is expected due to

the solution techniques and approximations contained in the parabolic models. The

required computation times and memory requirements for the transient mild slope

equation models were found to be intermediate to the other two methods.

Wave height and direction were calculated in the test case domain for each model.

The models based on the transient mild slope equation and the elliptic mild slope

equation were found to produce almost equivalent values of the wave height and

direction. The parabolic approximation models were found to have different values, with

RCPWAVE showing different wave heights and directions behind the shoal and

REF/DIF showing good wave height agreement with the other methods, but no change in

the wave direction behind the shoal. Plots of the computed wave heights for the six

models and experimental data along one transect taken perpendicular to the shoreline and

one transect parallel to the shoreline are shown in Figures 2-26 and 2-27. Only four

results are plotted because the RDE model, the PMH model and PBCG model produced

almost identical results.

The wave directions found with REF/DIF-1 in Maa et al. (2000) were found to be

in error by Grassa and Flores (2001), who demonstrated that a second order parabolic






44


model, equivalent to REF/DIF-1 was able to reproduce the wave direction field behind a

shoal such as in the Berkhoff et al. (1982) experiment.


5 10


Longshore (m)

Figure 2-26: Comparison of wave height profiles for selected models along transect
parallel to shore located 9 m shoreward of shoal apex [=experimental data] (Maa et al.,
2000).


15---















4 8 10 12 14
Cross Shore (m)


Figure 2-27: Comparison of wave height profiles for selected models along transect
perpendicular to shore and through shoal apex [*=experimental data] (Maa et al., 2000).


3, 1- -- --------------~----








Application of numerical models to the problem of potential impact on the

shoreline caused by changes to the offshore bathymetry was conducted by Maa and

Hobbs (1998) and Maa et al. (2001). In Maa and Hobbs (1998) the impact on the coast

due to the dredging of an offshore shoal near Sandbridge, Virginia was investigated using

RCPWAVE. National Data Buoy Center (NDBC) data from an offshore station and

bathymetric data for the area were used to examine several cases with different wave

events and directions. The resulting wave heights, directions, and sediment transport at

the shoreline were compared. The sediment transport was calculated using the

formulation of Gourlay (1982), which contains two terms, one driven by the breaking

wave angle and one driven by the gradient in the breaking wave height in the longshore

direction. Section 2.5.4 provides a more detailed examination of the longshore transport

equation with two terms. The study found that the proposed dredging would have little

impact on the shoreline for the cases investigated.

Later, Maa et al. (2001) revisited the problem of dredging at the Sandbridge Shoal

by examining the impact on the shoreline caused by three different borrow pit

configurations. RCPWAVE was used to model the wave transformation over the shoal

and in the nearshore zone. The focus was on the breaking wave height; wave direction at

breaking was not considered. The changes in the breaking wave height modulation

(BHM) along the shore after three dredging phases were compared to the results found

for the original bathymetry and favorable or unfavorable assessments were provided for

ensuing impact on the shoreline. The study concluded that there could be significant

differences in the wave conditions, revealed by variations in the BHM along the shoreline

depending on the location and extent of the offshore dredging.








Regions outside the inner surf zone have also been studied through application of

nearshore wave models. Jachec and Bosma (2001) used the numerical model REF/DIF-S

(a spectral version of REF/DIF-1) to study borrow pit recovery time for seven borrow

areas located on the inner continental shelf off New Jersey. The input wave conditions

were obtained from Wave Information Study (WIS) data with nearshore bathymetry for

the existing conditions and also different dredging scenarios. Changes in the wave-

induced bottom velocity were obtained from the wave height and direction changes

determined by REF/DIF-S. The wave-induced bottom velocities were coupled with

ambient near-bottom currents to determine the sediment transport and then recovery

times of the borrow areas. The recovery times from the numerical modeling were the

same order of magnitude as recovery times estimated from two independent data sets of

seafloor change rates offshore of New Jersey.

2.5 Shoreline Response

2.5.1 Longshore Transport Considerations

The previous discussion on one and two-dimensional models focused first on

simple and complex methods of determining the wave transformation caused by changes

in the offshore bathymetry and then applications that determined the changes to the wave

height, direction and even longshore transport at the shoreline. However, none of the

applications were intended to determine the change in shoreline planforms due to an

anomaly or a change in the offshore bathymetry. With wave heights and directions

specified along the shoreline, sediment transport can be calculated and, based on the

gradients in longshore transport, the changes in shoreline position can be determined.

The longshore transport can be driven by two terms as was discussed previously

in the review of Maa and Hobbs (1998). In most situations where the offshore








bathymetry is somewhat uniform, the magnitude and direction of the longshore transport

will depend mostly on the wave height and angle at breaking as the longshore gradient in

the breaking wave height will be small. In areas with irregular bathymetry or in the

presence of structures, the transformation of the wave field can lead to areas of wave

focusing and defocusing resulting in considerable longshore gradients in the wave height.

Longshore transport equations containing a transport term driven by the breaking wave

angle and another driven by the longshore gradient in the wave height can be found in

Bakker (1971), Ozasa and Brampton (1980), who cite the formulation of Bakker (1971),

Gourlay (1982), Kraus and Harikai(1983), and Kraus (1983). While the value of the

coefficient for the transport term driven by the gradient in the wave height is not well

established, the potential contribution of this term is significant. It is shown later that

under steady conditions the diffusive nature of the angle-driven transport term is required

to modify the wave height gradient transport term in order to generate an equilibrium

planform when the two terms are both active.

2.5.2 Refraction Models

2.5.2.1 Motyka and Willis (1974)

Motyka and Willis (1974) were one of the first to apply a numerical model to

predict shoreline changes due to altered offshore bathymetry. The model only included

the effect of refraction caused by offshore pits for idealized sand beaches representative

of those found on the English Channel or North Sea coast of England. A simplified

version of the Abemethy and Gilbert (1975) wave refraction model was used to

determine the transformation of uniform deep water waves over the nearshore

bathymetry. The breaking wave height and direction were calculated and used to

determine the sediment transport and combined with the continuity equation to predict






48


shoreline change. The longshore transport was calculated using the Scripps Equation as

modified by Komar (1969):

0.045
Q= pg H C, sin(2ab) (2-1)
Ys

where Q is the volume rate of longshore transport, ys is the submerged unit weight of the

beach material, p is the density of the fluid, Hb is the breaking wave height, Cg is the

group velocity at breaking, and ab is the angle of the breaking wave relative to the

shoreline. This form of the Scripps Equation combines the transport and porosity

coefficients into one term; the values used for either parameter was not stated. This

process was repeated to account for shoreline evolution with time.

Figure 2-28 shows a comparison of the predicted shorelines for the equivalent of

two years of waves over 1 m and 4 m deep pits with a longshore extent of 880 m and a

cross-shore extent of 305 m. The detailed pit geometries were not specified.


40 -

S30 -WATER DEPTH (mi DISTANCE OFFSHORE (m)
S17 08 2740
20 1762 3050
0
ro
4J 10 -
0O





S-30 ------ Pit depth = 4 m

-0o 50 100 1500 26O 20oo 3600 3o LO400 5o00
DISTANCE ALONG SHORE-m

PLANSHAPE OF BEACH
DUE TO REFRACTION OVER DREOGED HOLE, 2740m OFFSHORE


Figure 2-28: Calculated beach planform due to refraction after two years of prototype
waves for two pit depths (modified from Motyka and Willis, 1974).








The model determined that erosion occurs shoreward of a pit, with adjacent areas

of accretion. For the wave conditions used, stability was found after an equivalent period

of two years. During the runs, "storm" waves (short period and large wave height) were

found to cause larger shoreline changes than the "normal" waves with longer periods and

smaller heights, which actually reduced the erosion caused by the storm waves. The

erosion shoreward of the pits is shown in Figure 2-28 with more erosion occurring for the

deeper pit.

2.5.2.2 Horikawa et al. (1977)

Horikawa et al. (1977) developed a mathematical model for shoreline changes due

to offshore pits. The model applies a refraction program and the following equation for

the longshore sediment transport:

0.77pg
Q 77g H 2 Cg sin(2ab) (2-2)
16(p, p)(1 b)

where X is the porosity of the sediment. Equation 2-2 is identified as the Scripps

Equation in Horikawa et al. (1977); however, to match the Scripps Equation and for a

dimensionally correct expression, the g term in the numerator should be removed. A

model by Sasaki (1975) for diffraction behind breakwaters was modified to account for

refraction only. The model computes successive points along the wave ray paths.

Interpolation for the depth and slope is used along the ray path with an iteration

procedure to calculate each successive point. The wave conditions were selected to be

typical of the Eastern Japan coast facing the Pacific Ocean. Several pit dimensions and

pit locations were used with the longshore dimension of the pit ranging from 2 km to 4

km, a cross-shore length of 2 km, pit depth of 3 m and water depths at the pit from 20 m

to 50 m.









For the configurations modeled, accretion was found directly shoreward of the pit,

flanked by areas of erosion. The magnitude of the accretion behind the pit and the

erosion in the adjacent areas were found to increase with increasing longshore pit length

and for pits located closer to shore. The shoreline planform for a model after the

equivalent of 2 years of waves is shown in Figure 2-29 with a salient directly shoreward

of the pit.



2 k hole w"lef depth: 40 m
S!(depth:3m
E AFTER 2 YEARS I 2 km I

0

0 7.3 m
2 accr tion

0 -05m
O-S
-5 -4 -3 -2 -t 0 1 2 3 4 5
Longshore distance from center of dredged hole (km)


Figure 2-29: Calculated beach planform due to refraction over dredged hole after two
years of prototype waves (Horikawa et al., 1977).

Although Horikawa et al. state that good qualitative agreement was found with

Motyka and Willis (1974), the results were the opposite with Horikawa et al. and Motyka

and Willis (1974) predicting accretion and erosion shoreward of a pit, respectively. The

proposed reason for the accretion given in Horikawa et al. was that sand accumulates

behind the pit due to the quiet water caused by the decrease in wave action behind the pit.

However, a model that considers only refraction caused by a pit and only includes a

transport term dependent on the breaking wave angle would have wave rays that diverge

over the pit and cause sand to be transported away from the area behind the pit, resulting









in erosion. The two models used different refraction programs and basically the same

transport equation with Horikawa et al. having a pit that was 2 or 3 times as large, lying

in deeper water and with longer period incoming waves. The refraction grid was 250 m

square in the Horikawa et al. model and 176 m square in Motyka and Willis.

Regardless of the differing results from Motyka and Willis, the mathematical

model results of Horikawa et al. follow the trend of the lab results contained in that study

showing accretion behind a pit (Figure 2-30); however, the aforementioned anomalous

prediction of accretion considering only wave refraction remains.









2
Predicted






0 20 40 60
Longshore distance from center
of dredged hole ( cm )


Figure 2-30: Comparison of changes in beach plan shape for laboratory experiment and
numerical model after two years of prototype waves (Horikawa et al., 1977).

2.5.3 Refraction and Diffraction Models

2.5.3.1 Gravens and Rosati (1994)

Gravens and Rosati (1994) performed a numerical study of the salients and a set

of offshore breakwaters at Grand Isle, Louisiana (Figures 2-1 and 2-2). Of particular

interest is the analysis and interpretation of the impact on the wave field and the resulting








influence on the shoreline, of the "dumbbell" shaped planform borrow area located close

to shore. The report employs two numerical models to determine the change in the

shoreline caused by the presence of the offshore pits: a wave transformation numerical

model (RCPWAVE) and a shoreline change model (GENESIS (Hanson, 1987, Hanson,

1989)) using the wave heights from the wave transformation model. RCPWAVE was

used to calculate the wave heights and directions from the nominal 12.8 m contour to the

nominal 4.3 m contour along the entire length of the island for 3 different input

conditions. Figures 2-31 and 2-32 show the wave height transformation coefficients and

wave angles near the pit (centered about alongshore coordinate 130). Significant changes

in the wave height and direction are found near the offshore borrow area. The shadow

zone centered at Cell 130 suggests the presence of one large offshore pit as opposed to

the "dumbbell" shaped borrow pit for the project described in Combe and Soileau (1987).

The shoreline changes were calculated using a longshore transport equation with

two terms; one driven by the breaking wave angle, and one driven by the longshore

gradient in the breaking wave height. Each of these terms includes a dimensionless

transport coefficient. In order for GENESIS to produce a salient leeward of the borrow

pit, an unrealistically large value for the transport coefficient associated with the gradient

in the breaking wave height (K2 = 2.4) was needed, whereas 0.77 is the normal upper

limit. While a single salient was modeled after applying the large K2 value, the

development of two salients leeward of the borrow pit, as shown in Figures 2-1 and 2-2,

did not occur. The nearshore bathymetric data used in the modeling was from surveys

taken in 1990 and 1992. Significant infilling of the borrow pit occurred prior to the










surveys in 1990 and 1992; however, details of how the pit filled over this time period are

not known.


100 110 120 130 140
Alakmpml Cordii (lld sp g 100 ft)


150 160


Figure 2-31: Nearshore wave height transformation coefficients near borrow pit from
RCPWAVE study (modified from Gravens and Rosati, 1994).


120 130 140
Alkgulbrm Coordma"r (c0ll "pciu 100 ft)


Figure 2-32: Nearshore wave angles near borrow pit from RCPWAVE study; wave
angles are relative to shore normal and are positive for westerly transport (modified from
Gravens and Rosati, 1994).








The authors proposed that the salient was formed by the refractive divergence of the

wave field created by the borrow pit that resulted in a region of low energy directly

shoreward of the borrow area and regions of increased energy bordering the area. The

gradient in the wave energy will result in a circulation pattern where sediment mobilized

in the high-energy zone is carried into the low energy zone. For GENESIS to recreate

this circulation pattern K2 must be large enough to allow the second transport term to

dominate over the first transport term.

2.5.3.2 Tang (2002)

Tang (2002) employed RCPWAVE and a shoreline modeling program to evaluate

the shoreline evolution leeward of an offshore pit. The modeling was only able to

generate embayments in the lee of the offshore pits using accepted values for the

transport coefficients. This indicates that wave reflection and/or dissipation are

important wave transformation processes that must be included when modeling shoreline

evolution in areas with bathymetric anomalies.

2.5.4 Refraction, Diffraction, and Reflection Models

2.5.4.1 Bender (2001)

A study by Bender (2001) extended the numerical solution of Williams (1990) for

the transformation of long waves by a pit to determine the energy reflection and shoreline

changes caused by offshore pits and shoals. An analytic solution was also developed for

the radially symmetric case of a pit following the form of Black and Mei (1970). The

processes of wave refraction, wave diffraction, and wave reflection are included in the

model formulations, however, wave dissipation is not. Both the numerical and analytic

solutions provide values of the complex velocity potential at any point, which allows

determination of quantities such as velocity and pressure.










The amount of reflected energy was calculated by comparing the energy flux

through a transect perpendicular to the incident wave field extending to the pit center to

the energy flux through the same transect with no pit present (Figure 2-33). The amount

of energy reflected was found to be significant and dependent on the dimensionless pit

diameter and other parameters. Subsequently a new method has been developed which

allows the reflected energy to be calculated using a far-field approximation with good

agreement between the two methods.

Pit Diameter/Wavelength(inside pit,d)
0.- 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

+ (radius = 6 m)
0.24
o o (radius = 12m)
(radius = 25 m)
0.22 x (radius = 30 m)
<> (radius = 75 m)
0.2 o

50.18- x
-
S0.16 +
I +o o
A0.14 + 0


S
0.1 0




0 0.5 1 1.5 2 2.5 3
Pit Diameter/Wavelength(outside pit,h)


Figure 2-33: Reflection coefficients versus dimensionless pit diameter divided by
wavelength inside and outside the pit; water depth = 2 m, pit depth = 4 m (Bender, 2001).

The shoreline changes caused by the pit were calculated using a simple model that

considers continuity principles and the longshore transport equation with values of the

wave height and direction determined along a transect representing the shoreline. A

nearshore slope and no nearshore refraction were assumed. The impact on the shoreline








was modeled by determining the wave heights and directions along an initially straight

shoreline, then calculating the transport and resulting shoreline changes. After updating

the shoreline positions, the transport, resulting shoreline changes, and updated shoreline

positions were recalculated for a set number of iterations after which the wave

transformation was recalculated with the new bathymetry and values of the wave height

and direction were updated at the modified breaker line.

The impact on the shoreline was found to be highly dependent on the transport

coefficients. Considering transport driven only by the breaking wave angle and wave

height, erosion was found to occur directly leeward of the pit flanked by two areas of

accretion as in Motyka and Willis (1974). Following an initial advancement directly

shoreward of the pit, erosion occurs and an equilibrium shape was reached. Examining

only the effect of the second transport term (driven by the longshore gradient in the wave

height) accretion was found directly shoreward of the pit, with no equilibrium planform

achieved. Figure 2-34 shows the shoreline evolution for each transport term. Including

both transport terms with the same transport coefficients resulted in a shoreline with

accretion directly shoreward of the pit that was able to reach an equilibrium state

(Figure 2-35). The two-term transport equation used to determine the shoreline in Figure

2-35 is


KIH2.5 sin(O-ab)cos(O-ab) K2H cos(9-ab)dH
Q (2-3)
8(s 1X- p) 8(s 1X1- p)tan(m) dy

where Hb is the breaking wave height, g is gravity, K is the breaking index, 0is the

shoreline orientation, a is the breaking wave angle, m is the beach slope, s andp are the

specific gravity and porosity of the sediment, respectively, and K1 and K2 are sediment










transport coefficients, which were set equal to 0.77 for the results presented here. A

review of the development of Eq. 2-3 is contained in Appendix D.


First Transport Term (wave angle)
200

150

100

50


0
S-50

-100


-150
Erosion
-200
79.5 80 80.5 81
Shoreline Position (m)


Second Transport Term (dHbldx)




















40 60 80 100 120
Shoreline Position (m)


Figure 2-34: Shoreline evolution resulting from each transport term individually for
transect located 80 m shoreward of a pit with a radius = 6 m, last time step indicated with
[+] (modified from Bender and Dean, 2001).

In application the wave height and angle calculations were limited to a uniform depth

region; therefore no nearshore shoaling and refraction was performed to resolve the depth

limited breaking values. The wave height and angle values at the transect location

representing the shoreline were assumed to be the values at breaking; this limitation is

removed in the model developed in Section 6.10.

In these figures the water depth and pit depth were 2 m and 4 m, respectively, the

period was 10 s, the incident wave height was 1 m, and averaging over 5 wave directions

was used to smooth out the longshore variation in the wave height at large distances from

the pit. The time step was 120 s and 10 iterations of shoreline change were calculated

between wave height and direction updates for a total modeling time of 48 hours. The










diffusive nature of the angle-driven transport term is seen to modify the much larger


wave height gradient transport term in order to generate an equilibrium planform when


the two terms are used together. Comparison of these results with those described earlier


establishes the significance of wave reflection and the second transport term on the


equilibrium planform.


50

0
Q 0


S -50so
-10
-100


Full Transport Equation (both terms)




















Erosion


-150 F


-200
78


78.5 79 79.5 80 81
Shoreline Position (m)


0.5 81 81.5


Figure 2-35: Shoreline evolution using full transport equation and analytic solution model
for transect located 80 m shoreward of a pit with a radius = 6 m, last time step indicated
with [+] (modified from Bender and Dean, 2001).


I '













CHAPTER 3
2-DIMENSIONAL MODEL THEORY AND FORMULATION



3.1 Introduction

The reflection and transmission of normally incident waves by two-dimensional

trenches and shoals of finite width with sloped transitions between the depth changes are

studied. Prior two-dimensional studies, outlined in Chapter 2, have all investigated the

interaction of water waves with changes in bathymetry that have featured domains with

an abrupt transition, with the exception of Dean (1964) and Lee et al. (1981). A more

realistic representation of natural trenches and shoals should allow for gradual transitions

(sloped sidewalls).

The focus of the 2-dimensional study is the propagation of water waves over a

2-D trench or shoal of more realistic geometry. This will extend the study of Dean

(1964) that investigated long wave modification by a sloped step and Lee et al. (1981),

which did not directly address the effect of the transition slope on the reflection and

transmission coefficients. Three solution methods are developed for linear water waves:

(1) the step method, (2) the slope method and (3) a numerical method. The step method

is valid in arbitrary water depth while the slope method and the numerical method are

valid only for shallow water conditions. The step method is an extension of the Takano

(1960) eigenfunction expansion solution as modified by Kirby and Dalrymple (1983a)

that allows for a trench or shoal in arbitrary depth with "stepped" transitions that

approximate a specific slope or shape. The slope method extends the long wave solution








of Dean (1964) that allows for linear transitions between the changes in bathymetry for a

trench or shoal creating regular or irregular trapezoids. The numerical method employs a

backward space-stepping procedure for arbitrary (but shallow water) bathymetry with the

transmitted wave specified.

3.2 Step Method: Formulation and Solution

The two-dimensional motion of monochromatic, small-amplitude water waves in

an inviscid and irrotational fluid of arbitrary depth is investigated. The waves are

normally incident and propagate in an infinitely long channel containing a two-

dimensional obstacle (trench or shoal) of finite width. Details of the fluid domain and the

formulation of the solution vary depending on the case studied: abrupt transition or

gradual transition with the slope approximated by the step method.

The step method is an extension of the Takano (1960) formulation for the

propagation of waves over a rectangular sill. The eigenfunction expansion method of

Takano (1960) was extended in Kirby and Dalrymple (1983a) to allow for oblique wave

incidence and again in 1987 by Kirby et al. to include the effects of currents along the

trench. In the present formulation the method of Takano, as formulated in Kirby and

Dalrymple (1983a), was generally followed for normal wave incidence.

The solution starts with the definition of a velocity potential:

j (x,z,t)= qj (x,z)e' (j= 1 J) (3-1)

where j indicates the region, J is the total number of regions (3 for the case of a trench or

sill with an abrupt transition), and a is the angular frequency. The velocity potential

must satisfy the Laplace Equation:








a + a b(XZ) = 0 (3-2)
a9x 2 Z 2

the free-surface boundary condition:

84(x,z) 2
a )+ = 0 (3-3)
az g

and the condition of no flow normal to any solid boundary:

9<^(x,z)
a (x 0 (3-4)
on

The velocity potential must also satisfy radiation conditions at large Ix|.

The boundary value problem defined by Eq. 3-2, the boundary conditions of Eqs.

3-3 and 3-4, and the radiation condition can be solved with a solution in each region of

the form

+ cosh[k1(h1 + z) eiL +B cos[, (h ,)]e (Xx)
(x,z) = A cosh[kjh ] e-ik(x) + B,, cos (=l (h + e
coshkhTc e n=i

(j = -J), (n = -oo) (3-5)

In the previous equation A1+ is the incident wave amplitude coefficient, A( is the

reflected wave amplitude coefficient and AJ+ is the transmitted wave amplitude

coefficient. The coefficient B is an amplitude function for the evanescent modes

(n = 1 -+ oo) at the boundaries, which are standing waves that decay exponentially with

distance from the boundary. The values of the wave numbers of the propagating modes,

kj, are determined from the dispersion relation:

-2 = gkj tanh(khj) j = 1 J) (3-6)


and the wave numbers for the evanescent modes, K ,n are found from








a-2 = -gK,, tan(K,h) (j= 1 -J), (n = 1 oo) (3-7)

In each region a complete set of orthogonal equations over the depth is formed by Eqs. 3-

5 to 3-7.

To gain the full solution, matching conditions are applied at each boundary

between adjacent regions. The matching conditions ensure continuity of pressure:

S= j (x= xJ), (j = 1-- J-1) (3-8)

and continuity of horizontal velocity normal to the vertical boundaries:

-= (x = x), j= J-1) (3-9)
ax ax

The matching conditions are applied over the vertical plane between the two regions:

(-hi z < 0) if h, < hi+, or (- hi+, < z < 0) if hj > h+1.

In order to form a solution, one wave form in the domain must be specified,

usually the incident or the transmitted wave. Knowing the value of the incident, reflected

and transmitted wave amplitudes, the reflection and transmission coefficients can be

calculated from

KR = a (3-10)
a!

a, cosh(k,h,)
K, = c (3-11)
a, cosh(k,h,)

where the cosh terms account for the change in depth at the upwave and downwave ends

of the trench/shoal for the asymmetric case. A convenient check of the solution is to

apply conservation of energy considerations:


K' + a-] =1 (3-12)
R a, nk, 1






63

where nj is the ratio of the group velocity to the wave celerity:

2k.h
n si =-(- 1+j (3-13)
7 2 sinh(2k h.)J

3.2.1 Abrupt Transition

The solution of Takano (1960) for an elevated sill and that of Kirby and

Dalrymple (1983a) for a trench are valid for abrupt transitions (vertical walls) between

the regions of different depth. For these cases the domain is divided into three regions (J

= 3) and the matching conditions are applied over the two boundaries between the

regions. The definition sketch for the case of a trench with vertical transitions is shown

in Figure 3-1 where W is the width of the trench.

z Nl


X Xi X2


Region 1 Region 2 1 Region 3



*c--- W -



Figure 3-1: Definition sketch for trench with vertical transitions.

Takano constructed a solution to the elevated sill problem by applying the

matching conditions [Eqs. 3-8 and 3-9] for a truncated series (n =1 -> N) of

eigenfunction expansions of the form in Eq. 3-5. Applying the matching conditions

results in a truncated set of independent integral equations each of which is multiplied by

the appropriate eigenfunction; cosh[kj(hj+z)] or cos[Kj,n(hj+z)]. The proper eigenfunction

to use depends on whether the boundary results in a "step down" or a "step up"; thereby








making the form of the solution for an elevated sill different than that of a trench. With

one wave form specified, the orthogonal properties of the eigenfunctions result in 4N+4

unknown coefficients and a closed problem.

By applying the matching conditions at the boundary between Regions 1 and 2

(x = x ), 2N+2 integral equations are constructed. For the case of a trench with vertical

transitions (Figure 3-1) the resulting equations are of the form

o 0
S((x1,, z) cosh[k,(h, + z)]dz = 2(I, ) ,cosh[k, (h, + z)]dz (3-14)
-h, -h,

0 0
fo,(x,,z)cos[K,,,(h, + z)]dz = f2 (x1,z)cos[KI,,(h + z)]dz (n = 1 N) (3-15)
-hA -hA

0J^ (x,,z)cosh[k2(h2 + z)dz = (x,, z) cosh[k2(h + z)]dz
Ax Aax
-hi -h,
(3-16)

= (x,,z) cosh[k2 (h2 + z)]dz
-h2


J (x z) cosK2, (h2+ z)]dz = (, z)cos[K2, (h2 + z)ldz

-h, (n = 1 N)(3-17)
(x z) cos K2,n (h2 + z)]dz
-h2

The limits of integration for the right hand side in Eqs. 3-16 and 3-17 are shifted from -hi

to -h2 as there is no contribution to the horizontal velocity for (- h2 < z < -h,) at x = xl

and (- h < z < -h3) at x = x2, for this case. In Eqs. 3-14 and 3-15 the limits of

integration for the pressure considerations are (- h, < z < 0) at x = x, and (- h3 z < 0)

at x = x2.

At the boundary between Regions 2 and 3 the remaining 2N+2 equations are

developed. For the case of a trench the downwave boundary is a "step up", which








requires different eigenfunctions to be used and changes the limits of integration from

the case of the "step down" at the upwave boundary [Eqs. 3-14 to 3-17].

0 0
J2 (x2,z)cosh[k3(h + z)]dz = (x2, z) cosh[k3(h, + z)]dz (3-18)
-h3 -h3

0 0
(x2 2,z)cos[K3,n(h3 + z)]dz= 3(x2,z)cos[K3,,(h3 + z)]dz (n=1- N) (3-19)
-h3 -h3


z) cosh[k2(h2 + z)]dz = (x2, z) cosh[k2 (h + z)]dz (3-20)
-h2I -h3


S (x2, z) cos[K2, (h2 + z)]dz = (2 (x,z)cos[K2,,(h, + z)dz (n =1 N)(3-21)
-ax -h ax

At each boundary the appropriate evanescent mode contributions from the other

boundary must be included in the matching conditions. The resulting set of simultaneous

equations may be solved as a linear matrix equation. The value of N (number of non-

propagating modes) must be large enough to ensure convergence of the solution. Kirby

and Dalrymple (1983a) found that N = 16 provided adequate convergence for most

values of klh.

3.2.2 Gradual Transition

The step method is an extension of the work by Takano (1960) and Kirby and

Dalrymple (1983a) that allows for a domain with a trench or sill with gradual transitions

(sloped sidewalls) between regions. Instead of having a "step down" and then "step up"

as in the Kirby and Dalrymple solution for a trench or the reverse for Takano's solution

for an elevated sill, in the step method a series of steps either up or down are connected

by a constant depth region followed by a series of steps in the other direction. A sketch








of a domain with a stepped trench is shown in Figure 3-2. In this method, as in the case

of a trench or a sill, a domain with J regions will contain J-1 steps and boundaries.


-- lR ---
X XI X2 X3 X4 X5
I I
h, h2 z4 5 h








Region 1 (R1)| R2 R3 Region 4 RS Region 6

Figure 3-2: Definition sketch for trench with stepped transitions.

Each region will have a specified depth and each boundary between regions will have a

specified x location where the matching conditions must be applied.

At each boundary the matching conditions are applied and depend on whether the

boundary is a "step up" or a "step down." With the incident wave specified, a set of

equations with 2(J-1)N+2(J-1) unknown coefficients is formed.

The resulting integral equations are of the form: for(j =1 -> J -1)

if(- hj > -hj ) at x = xj then the boundary is a "step down";

0 0
fJ(x,,z)cosh[k h +z)] dz = ft (x ,z)cosh[k,(hi +z)]dz (3-22)
-h, -h,

0 0
f (xj,z)cos[Kj, (hj + z)]dz = Jj+,(xj,z)cos[K,, (hj + z)]dz (n=1 -) N)(3-23)
-h, -hi

o a< 0 (, \ + d
J (x. z) cosh[kj,(,,h + z)dz = al(x z)cosh[k+,,(h+,, +z dz (3-24)
-h -hi








0i Q ji I / \1
J C (x ,z)cos[Ki,,, +(h + + z)]dz = J (xz)cos[K+,n (h+, +z)dz
-h, -AI

(n = 1 -+ N) (3-25)

if (- hi < -h ) at x = xj then the boundary is a "step up";

o 0
i (x, z) cosh[kj, (ih, + z)]dz = i,+, (xj, z) cosh[k ,, (hj + z)]dz (3-26)
-h+) -hA+I

0 0
Ji (xj, z) cos[Kj+,n (h+,, + z)dz = J+ (x., z) cos[Kj+.,, (h, + z)]dz
-hj+l -hl+h

(n =1 -- N) (3-27)


Sx(x j, z)cosh[k (h, + z)]dz = J ( z) cosh[kj (h. + z)]dz (3-28)
ax ax


J --(x, z) cos[K ,,(hj + z] dz = J l(xj,z)cos[K,, (j z) dz
-h -hj+

(n = 1 N) (3-29)

At each boundary (xj) the appropriate evanescent mode contributions from the

adjacent boundaries (xjli, xj+1) must be included in the matching conditions. The

resulting set of simultaneous equations is solved as a linear matrix equation with the

value of N large enough to ensure convergence of the solution.

3.3 Slope Method: Formulation and Solution
The slope method is an extension of the analytic solution by Dean (1964) for long

wave modification by linear transitions. Linear transitions in the channel width, depth,

and both width and depth were studied. The solution of Dean (1964) is valid for one

linear transition in depth and/or width, which in the case of a change in depth allowed for








an infinite step, either up or down, to be studied. In the slope method a domain with two

linear transitions allows the study of obstacles of finite width with sloped transitions.

The long wave formulation of Dean (1964) for a linear transition in depth was

followed. By combining the equations of continuity and motion the governing equation

of the water surface for long wave motion in a channel of variable cross-section can be

developed. The continuity equation is a conservation of mass statement requiring that the

net influx of fluid into a region during a time, At, must be equal to a related rise in the

water surface, rl. For a channel of uniform width, b, this can be expressed as

[Q(x) Q(x + Ax)]At = bAx[rq(t + At) r(t)] (3-30)

where Q(x) and Q(x+Ax) are the volume rates of flow into and out of the control volume,

respectively. The volume flow rate for the uniform channel can be expressed as the

product of the cross sectional width, A, and the horizontal velocity, u, in the channel:

Q=Au (3-31)

By substituting Eq. 3-31 into Eq. 3-30 and expanding the appropriate terms in

their Taylor series while neglecting higher order terms, Eq. 3-30 can be rewritten as

(Au) = b (3-32)
ax Ot

The hydrostatic pressure equation is combined with the linearized form of Euler's

equation of motion to develop the equation of motion for small amplitude, long waves.

The pressure field, p(x,y,t), for the hydrostatic conditions under long waves is

p(x,z,t) = pg[i(x,t)- z] (3-33)

Euler's equation of motion in the x direction for no body forces and linearized motion is

1 9p 6u
Sap = u (3-34)
p Ox at








The equation of motion for small amplitude, long waves follows from combining Eqs. 3-

33 and 3-34:

-0 -g Ou (3-35)
ax at

The governing equation is developed by differentiating the continuity equation [Eq. 3-32]

with respect to t:


---(Au)= b -- -- A = b (3-36)
at 8x at ax at at2

and inserting the equation of motion [Eq. 3-35] into the resulting equation, Eq. 3-36

yields the result


g A = b (3-37)
ax r ax t2

Eq. 3-37 is valid for any small amplitude, long wave form and expresses r1 as a function

of distance and time. Eq. 3-37 can be further simplified under the assumption of simple

harmonic motion:

r7(x,t) = 77, (x)ei(o+a) (3-38)

where a is the phase angle. Eq. 3-37 can now be written as


g abha7 +x r =O0 (3-39)
b 9x ax

where the subscript r l(x) has been dropped and the substitution, A = bh, was made.

3.3.1 Single Transition

The case of a channel of uniform width with an infinitely long step either up or

down was a specific case solved in Dean (1964). The definition sketch for a "step down"

is shown in Figure 3-3. The three regions in Figure 3-3 have the following depths:









Region 1, x < xj; h = h, (3-40)

x
Region 2, x < x < x2; h = h3 (3-41)
x2

Region 3, x > x,; h = h3 (3-42)



Region 1 Region 2 Region 3
x-] x x2









Figure 3-3: Definition sketch for linear transition.

For the regions of uniform depth, Eq. 3-39 simplifies to



2X2

which has the solution for rl of cos(kx) and sin(kx) where k = and X is the wave
A

length. The most general solution ofrl(x,t) from Eq. 3-43 is

r(x, t) = B, cos(kx t + a ) + B, cos(kx + at + a2) (3-44)

The wave form of Eq. 3-44 consists of two progressive waves of unknown amplitude and

phase: an incident wave traveling in the positive x direction and a reflected wave

traveling in the negative x direction.

For the region of linearly varying depth, Eq. 3-41 is inserted in Eq. 3-39 resulting

in a Bessel equation of zero order:









2x + r+ 7 = 0 (3-45)
x ax
Ox2 Ox

where


S= x2 (3-46)
gh,

The solutions ofrl(x) for Eq. 3-45 are

q(x) = Jo(2p 12x1'2) and Yo(2P1/2x' 2) (3-47)

where Jo and Yo are zero-order Bessel functions of the first and second kind, respectively.

From Eq. 3-47 the solutions for r(x,t) in Region 2 follow

q(x,t)= BJo(2fl/2x1/2)cos(crt+a3)+ BYo(21/2x1/2)sin(at + a3)
(3-48)
+ B4J(2fl12x1/2)cos(t + a4)- B4 Y(2P/212x'2)sin(Tt + a4)

The wave system of Eq. 3-48 consists of two waves of unknown amplitude and phase;

one wave propagating in the positive x direction (B3) and the other in the negative x

direction (B4).

The problem described by Figure 3-3 and Eqs. 3-44 and 3-48 contains eight

unknowns: Bl-+4 and a(X14. Solution to the problem is obtained by applying matching

conditions at the two boundaries between the three regions. The conditions match the

water surface and the gradient of the water surface:

77 = 77,, at x = x (j = 1,2) (3-49)


atx = x (j = 1,2) (3-50)
Ox Ox








Eqs. 3-49 and 3-50 result in eight equations (four complex equations), four from

7rC
setting a t = 0 and four from setting at = which can be solved for the eight


unknowns as a linear matrix equation.

3.3.2 Trench or Shoal

The slope method is an extension of the Dean (1964) solution that allows for a

domain with a trench or a sill with sloped transitions. Two linear transitions are

connected by a constant depth region by placing two solutions from Dean (1964) "back to

back." A trench/sill with sloped side walls can be formed by placing a "step down"

upwave/downwave of a "step up." The definition sketch for the case of a trench is shown

in Figure 3-4.

In the slope method the depths are defined as follows

Region 1, x < x,; h = h, (3-51)

Region 2, x < x < x2; h = h + s, (x x,) (3-52)

Region 3, x < x < x3; h = h3 (3-53)

Region 4, x3 < x < x4; h=h3-s (x-x3) (3-54)

Region 5, x > x4; h = h5 (3-55)

where hi, h3, hs, si, s2, and W are specified. With the new definition for the depth in


regions 2 and 4, the definition of the coefficient P in Eq. 3-45 changes to f = and
gsI
2
p = -- in regions 2 and 4, respectively.
gs2





















W-- ----


Figure 3-4: Definition sketch for trench with sloped transitions.

The matching conditions of Eqs. 3-49 and 3-50 are applied at the four boundaries

between the regions. With the transmitted wave specified and by setting a t = 0 and


a t = for each matching condition a set of 16 independent equations is developed.
2

Using standard matrix techniques the eight unknown amplitudes and eight unknown

phases can then be determined.

The reflection and transmission coefficients can be determined from


KR = aRand KT = a' (3-56), (3-57)
a, a,

Conservation of energy arguments in the shallow water region require


K'+ K[ ] = 1 (3-58)


This method can be extended to the representation of long wave interaction with

any depth transition form represented by a series of line segments.








3.4 Numerical Method: Formulation and Solution

A numerical method was developed to determine the long wave transformation

caused by a trench or shoal of arbitrary, but shallow water bathymetry. A transmitted

wave form in a region of constant depth downwave of the depth anomaly is the specified

input to the problem. Numerical methods are used to space step the wave form

backwards over the trench or shoal and then into a region of constant depth upwave of the

depth anomaly where two wave forms exist; an incident wave and a reflected wave.

As in the long wave solution of Section 3, the continuity equation and the

equation of motion are employed to develop the governing equation for the problem.

The continuity equation and the equation of motion in the x direction are written in a

slightly different form than in Eqs. 3-32 and 3-35 of Section 3.3:

-= (3-59)
at ax


gh aq (3-60)
ax at

Taking the derivative of Eq. 3-59 with respect to t and the derivative of Eq. 3-60

with respect to x results in the governing equation for this method:

S-gh a27g dh-= 0 (3-61)
9at2 x2 dx ax

where the depth, h, is a function of x and ri may be written as a function of x and t:

77 = qr(x)e"' (3-62)

Inserting the form of r in Eq. 3-62 into the governing equation of Eq. 3-61 casts the

equation in a different form (equivalent to Eq. 3-39)

gh a28r(x) g dh ar(x)
Cr(x)+ + 2 -2 d=0 (3-63)
a2 x2 2 d ax








Central differences are used to perform the backward space stepping of the

numerical method.

(x) F(x + Ax) 2F(x)+ F(x x) (364)
Ax2

F F(x + Ax) F(x x)
FA'(x)= (3-65)
2Ax

Inserting the forms of the central differences into Eq. 3-63 for rJ results in

(x)+gh 77(x+ Ax)- 27()+ (xAx) g dh 77(x+Ax)-(x-Ax) (366)
a 2 Ax2 a2 dx 2Ax

For the backward space stepping calculation, Eq. 3-66 can be rearranged
) gh g dh 1 [ 2gh
Ax)2 2 dx 2Ax 1 2x2
77(x -Ax) = d2Ax (3-67)
gh g dh 1
C.2Ax2 O2 dx 2Ax

To initiate the calculation, values of r(x) and rl(x+Ax) must be specified in the

constant depth region downwave from the depth anomaly. If the starting point of the

calculation is taken as x = 0 then the initial values may be written as

H
r7(0) = (3-68)


77(Ax) = H [cos(kAx) i sin(kAx)] (3-69)

The solution upwave of the depth anomaly comprises of an incident and reflected

wave. The form of the incident and reflected waves are specified as

H1
r], = cos(kx ot ,) (3-70)
2

HR
qR = cs(kx + o't eR) (3-71)
2








where the E's are arbitrary phases. At each location upwave of the depth anomaly the

total water surface elevation will be the sum of the two individual components:

H1
77T = /i + 77R = cos(kx E,) cos(ot) + sin(kx e,) sin(ot)
2
+ HR cos(k s ) cos(Ot) sin(kx eR ) sin(Ot)
2

= cos(ot)[ cos(hk ,) + cos(k c8)
2 2
I

+ sin(ot) sin(kx e,) Hr sin(kx R)
2 2


= VI2 +2 cos(O C) 6 = tan-'-j (3-72)


Using several trigonometric identities, Eq. 3-72 can be reduced further to the form

7Tr = JH + H +2HH, cos(2kx- e, ) cos(ot ) (3-73)

which is found to have maximum and minimum values of


rTrax = (H, +H,) (3-74)
2


r7m = H, -HR (3-75)

Eqs. 3-74 and 3-75 are used to determine the values of Hi and HR upwave of the

trench/shoal, and allow calculation of the reflection and transmission coefficients.













CHAPTER 4
3-DIMENSIONAL MODEL THEORY AND FORMULATION

4.1 Introduction

The three-dimensional motion of monochromatic, small-amplitude water waves in

an inviscid and irrotational fluid of arbitrary depth is investigated. The waves propagate

in an infinitely long, uniform depth domain containing a three-dimensional axisymmetric

anomaly (pit or shoal) of finite extent. The addition of the second horizontal dimension

provides many new, and more practical, possibilities for study compared to the 2-D

model domains, which excluded longshore variation.

Two different models are developed for the 3-D domains that contain linear

transitions in depth. The analytic step method is an extension of Bender (2001) that

determines the wave transformation in arbitrary water depth for domains with gradual

transitions in depth that are approximated by a series of steps of uniform depth. The

exact analytic model solves the wave transformation in shallow water for specific

bathymetries that reduce the governing equations to known forms.

4.2 Step Method: Formulation and Solution

The step method for a three-dimensional domain is an extension of the Bender

(2001) formulation for the propagation of waves past a circular anomaly with abrupt

transitions. This method allowed oblique wave incidence, but was limited to the shallow

water region. Following Bender (2001) with significant changes in notation the

governing equations for the three-dimensional models are developed.








Details of the fluid domain and the formulation of the solution vary depending on

the case studied: abrupt transition in depth between regions or gradual transition with the

slope approximated by the step method. The definition sketch for the case of a circular

pit with an abrupt depth transition is shown in Figure 4-1.


r (r,0)
Region 2
^^_ {^ ^<. -


Region 1



f ,


Figure 4-1: Definition sketch for circular pit with abrupt depth transitions.

The domain is divided into regions with the bathymetric anomaly and its

projection comprising Regions 2-+Ns+l where Ns is the number of steps approximating

the depth transition slope and the rest of the domain, of depth hi, in Region 1. For the

case of an abrupt transition in depth the bathymetric anomaly occurs in Region 2 of

uniform depth h2, where abrupt is defined as one step either down or up. For the case of

a gradual depth transition the bathymetric anomaly will be divided into subregions with

the depth in each subregion equal to hj for each step j = 2 -> N, +1.








The solution starts with the definition of a velocity potential in cylindrical

coordinates that is valid in each Region j:

0j = Re( j(r,0,z)e-') (j = -+ N,) (4-1)

where w is the wave frequency.

Linear wave theory is employed and Laplace's solution in cylindrical coordinates is

valid:


V 2 o+ + + = 0 (4-2)
r 2 r ar r2a02 (z2

where the free surface boundary condition is


a2 = 0 (4-3)
az g

and the bottom boundary condition is taken as

S= 0 (4-4)
az

at z = -hi in Region 1 or z = -hj in Region 2.

Separation of variables is used to solve the equations with the velocity potential

given the form

Q(r,O,z) = R(r)O(O)Z(z) (4-5)

A valid solution for the dependency with depth is

Z(z) = cosh(kj(hj+z)) (4-6)

where kj is the wave number in the appropriate region and hj is constant within each

region. Inserting Eqs. 4-5 and 4-6 into the Laplace equation gives

1 1
R"O + -R' + 2E"R + ROk2 = 0 (4-7)
r r








If the form of the dependence with 0 is assumed to be

(O)= cos(mO) (4-8)

then Eq. 4-7 may be reduced to

r2R" +rR' +R(k2r2 -m2)= 0 (4-9)

which is a standard Bessel equation with solutions Jm(kr), Ym(kr), and Hm(kr). The

dependency of the solution on 0 cancels out of Eq. 4-9; a result of the separation of

variables approach. The wave number is determined from the dispersion relation:

a2 = gkj tanh(khj) (4-10)

The standard Bessel solutions and the wave numbers of Eq. 4-10 represent the plane

progressive wave component of the solution.

Another solution is found when the dependence with depth is defined as

Z(z) = cos(Kj(hj+z)) (4-11)

Inserting Eqs. 4-5, 4-8 and 4-11 into the Laplace Equation gives

r2R" +rR'- R(K2r2 + m2)= 0 (4-12)

which is a modified Bessel equation with solutions Km(Kr) and Im(Kr) where K is the

wave number for the evanescent modes obtained using

c = -gKt,, tan(Kcih) (n = 1 -+ oo) (4-13)

where n indicates the number of the evanescent mode. The modified Bessel solutions

and the wave numbers of Eq. 4-13 represent the evanescent mode solutions, which decay

with distance from each interface between regions. The evanescent terms are included to

account for the distortion of the plane wave near the interface (Black and Mei, 1970) and

to extend the range of the solution into the arbitrary depth region.




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