UFL/COEL 2001/005
WAVE FIELD MODIFICATIONS AND SHORELINE RESPONSE
DUE TO OFFSHORE BORROW AREAS
by
Christopher J. Bender
Thesis
2001
WAVE FIELD MODIFICATIONS AND SHORELINE RESPONSE DUE TO
OFFSHORE BORROW AREAS
By
CHRISTOPHER J. BENDER
A THESIS PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
MASTER OF SCIENCE
UNIVERSITY OF FLORIDA
2001
LIST OF TABLES
Table Page
5.1: Coefficients and constants used in the shoreline change models.... ............75
6.1: Values of parameters used in laboratory model for different trials in shoreline
change experiment............ .............................. ......... 106
6.2: Dimensions of different pits used in shoreline change
exp erim ents............ ................... ................................. 107
ACKNOWLEDGMENTS
Many individuals at the University of Florida made the completion of this work
possible. I first must thank my advisor and committee chairman Dr. Robert G. Dean for
his abundant help and tireless work during the course of this study. His ideas and
suggestions assisted me in countless ways. I would like to thank the other members of
my committee, Dr. Daniel M. Hanes and Dr. Robert J. Thieke, for their help throughout
the study. The staff at the Coastal Engineering Laboratory including Sidney Schofield,
Jim Joiner, Vic Adams and Vernon Sparkman solved many of the problems I experienced
during the laboratory experiments and made my research possible. I would like to
acknowledge the Coastal Engineering Archives staff at the University of Florida
including Helen Twedell and Kimberly Hunt for their help during the last two years. I
thank Becky Hudson for answering my questions covering many topics in the graduate
catalog.
The Office of Beaches and Coastal Systems of the Florida Department of
Environmental Protection provided partial funding for this study through a contract on
the causes of erosional hot spots.
I would like to thank my wife for her love, constant support and confidence in me
and my endeavors. In addition I would like to thank my parents, family and friends for
their encouragement and all that they have done to make me who I am.
TABLE OF CONTENTS
page
A C K N O W L E D G M E N T S ..................................................................... .....................ii
LIST OF TABLES ..................................................................... vi
LIST O F FIG U RE S ...................... ................ .. ................................vii
A B ST R A C T ............................................................................................ xiv
1 INTRODUCTION ............................................. ............... 1
1.1 Problem Statement............................... ................ 1
1.2 O objectives and Scope ........................................................................................2
2 LITERATURE REVIEW ............... ........................... ............................. 4
2 .1 Intro du ctio n ............................................................................... 4
2.2 D discussion of P previous Studies ......................................................................... 4
2.2.1 Rectangular Pits of Infinite Length................................................... 4
2.2.2 Pits of A rbitrary Shape ........................................................................ .......... 6
2.2.3 Shoreline Change by Offshore Pits ............................................7
2.2.4 Laboratory and Field Studies .................................. ....... .. ............ 8
3 T H E O R Y ................................................................... ................................. . . 9
3 .1 Introdu action ............................................................................. 9
3.2 G governing E equations ......................................... ............... ............. .. 9
3.2.1 Flow Over A Pit .................................. ........................ ...... ... 9
3.2.2 Energy Considerations ............................................ ............................ 11
3.2.3 Shoreline C change ......................................... ................. ................ 12
4 M O D E L S ................................. ................... ...................................... 17
4 .1 Intro du ctio n .............................................................................. ...................17
4.2. Numerical Solution Model of Williams (1990) ............................... .................. 17
4.2.1 Green's Function Solution of Williams (1990) ............................................ 17
4.2.2 Wave Height Reduction for Pit of Arbitrary Shape ........................................20
4 .2 .3 R election from P it ......... .................................................... ..................... 22
4.2.4 Shoreline Change Induced by Pit............................................... ............... 24
4.2.5 Models for Solid Structure of Finite Dimensions...........................................27
4.3 Analytical Solution Model for Circular Pit....................................................28
4.3.1 W ave Height Reduction for Circular Pit ................................... ................. 28
4.3.2 Reflection from Circular Pit ..... .......................................30
4.3.3 Shoreline Change Induced by Pit .... ............. ................................... 31
4.4 MacCamy and Fuchs Solution Model for Cylinder .............................................32
5 RESULTS AND DISCU SSION .......................................................... ..... ........... 34
5.1 Introduction ............................................................... 34
5.2 W ave H eight R education M models ........................................ ...... ...................... 34
5.2.1 Comparison of Numerical Model Results with Results of Williams (1990) ...34
5.2.2 Comparison of Numerical Model with Analytic Solution Model................... 39
5.2.3 Results for M odels w ith Solid Cylinder ....................................................... 50
5.2.4 Plots of W ave Fronts in Fluid D om ain ................................ .................... 54
5.2.5 M odel Sensitivity ................................. ............... ........ 56
5.3 Energy Reflection Caused by Pit............ ................. ............... 59
5.3.1 A nalytic Solution M odel .......................................................... ... ........ 59
5.3.2 N um erical M odel ....................... ......... ................................. ......... 68
5.3.3 MacCamy and Fuchs Solution Model for Solid Cylinder............ ...............72
5.4 Shoreline Change Induced by Pit ...............................................74
5.4.1 Comparison of M odel Results ............................................... ............... 74
5.4 2 Shoreline Evolution with Time................. ............. ........... .................... 85
5.4.3 C ase H stories .................................................................. .. .... ........ 90
5.4.3.1 G rand Isle, L ouisiana ............................................... ....... ................... 90
5.4.3.2 M artin County, Florida.............. .......................... .............. ............... 96
6 LABORATORY RESULTS AND DISCUSSION.................................................. 102
6.1 Introduction ......... ............................ ........................ ......... 102
6.2 Experimental Setup and Equipment ................ ... ..................................... 102
6.3 Experim ents...................................................................................................... 104
6.3.1 W ave H eight R education ........................................................ ..... ........... 104
6.3.2 Shoreline C change ..................................... ............................................. 104
6.4 Experim ent Results and D discussion ............... .............................. ................ 105
6.4.1 Wave Height Reduction. ............................... .............. 105
6.4.2 Shoreline C change .............................................. ........ ..... .. ............ 106
6.5 Comparison with Numerical Models.................... ...................... 115
7 SUMMARY AND CONCLUSIONS......................................................... 122
LIST OF REFERENCES ........................................... 127
BIO GR APH ICAL SK ETCH ......... ..................................................... ............... 129
LIST OF FIGURES
Figure Page
3.1: Definition sketch for flow over pit............ ............................... ................9
3.2: Definition sketch for shoreline change problem ...........................................13
5.1: Contour plot of relative amplitude from numerical model for 8 by 8 m pit and
base conditions with location of pit drawn... ........................35
5.2: Contour plot of relative amplitude for p = 0, kid = 7/10, k2d = 7/10/2,
h/d = 0.5, b/a = 1.0 and a/d = 2.0 equal to 8 by 8 m pit with base
conditions; from W illiams (1990) ....................... .................36
5.3: Contour plot of relative amplitude from numerical model for 8 by 24 m pit
with p = 450 and other base conditions with location of pit drawn.............37
5.4: Contour plot of relative amplitude for P = 450, kid = d/10, k2d = rll02,
h/d = 0.5, b/a = 3.0 and a/d = 2.0 equal to 8 by 24 m pit with 0 = 450
and other base conditions; from Williams (1990) ................................38
5.5: Contour plot of relative amplitude from numerical model for circular pit
with r = 12 m, 200 points defining the pit boundary and base conditions
w ith location of pit draw n ................... ......................... ............. 40
5.6: Contour plot of relative amplitude from analytic solution model for circular
pit with r = 12 m, 80 terms taken in the series summation and base
conditions with location of pit drawn .................. ...... .................41
5.7: Contour plot of percent difference for numerical model and analytic solution
model for circular pit with r = 12 m, 200 points defining the pit boundary
in the numerical model, 80 terms taken in the series summation in the
analytic solution and base conditions with location of pit drawn ............41
5.8: Plot of transect taken parallel to the xaxis at y = 0 for pit of radius 12 m
showing wave direction (p = 0) ...............................................42
5.9: Relative amplitude for numerical model and analytic solution model for
transect show in Figure 5.8 for pit of radius = 12 m with 200 points
defining the pit boundary in the numerical model and 80 terms taken
in the series summation for analytic solution with pit drawn ....................43
5.10: Relative amplitude for numerical model and analytic solution model for
3 transects parallel to the yaxis at X = 0, 24 and 100 m for pit of
radius = 12 m with 200 points defining the pit boundary in the
numerical model and 80 terms taken in the series summation for
analytic solution w ith pit drawn ............ .......................................44
5.11: Contour plot of relative amplitude from analytic solution model for
circular pit with r = 75 m, 110 terms taken in the series summation
and base conditions w ith pit draw n .................................................45
5.12: Contour plot of percent difference for numerical model and analytic
solution model for circular pit with r = 75 m, 600 points defining
the pit boundary in the numerical model, 110 terms taken in the series
summation in the analytic solution and base conditions with pit drawn.......46
5.13: Relative amplitude for numerical model and analytic solution model for
transect parallel to the xaxis at y = 0 m for pit of radius = 75 m+ with
600 points defining the pit boundary in the numerical model and 100
terms taken in the series summation for analytic solution with pit drawn......47
5.14: Relative amplitude for numerical model and analytic solution model for
transect parallel to the yaxis at X = 100 m for pit of radius = 75 m with
600 points defining the pit boundary in the numerical model and 100
terms taken in the series summation for analytic solution with pit drawn......48
5.15: Relative amplitude for numerical model and analytic solution model for
transect parallel to the yaxis at X = 500 m for pit of radius = 75 m with
600 points defining the pit boundary in the numerical model and 100
terms taken in the series summation for analytic solution with pit drawn......49
5.16: Contour plot of relative amplitude from MacCamy and Fuchs solution
model for a circular cylinder with r = 12 m, 80 terms taken in the
series summation and base conditions with pit drawn ...........................50
5.17: Contour plot of percent from from solid numerical model and MacCamy
and Fuchs solution model for a circular cylinder with r = 12 m, 200
points defining the pit boundary, 80 terms taken in the series
summation and base conditions with pit drawn................................. 52
5.18: Relative amplitude for solid numerical model and for MacCamy and
Fuchs solution model for transect parallel to the xaxis at y = 0 m for
pit of radius = 12 m with 200 points defining the pit boundary in the
numerical model and 80 terms taken in the series summation for
analytic solution with analytic solution model for circular pit shown..........52
5.19: Relative amplitude for solid numerical model and for MacCamy and Fuchs
solution model for transect parallel to the yaxis at x = 100 m for pit
of radius = 12 m with 200 points defining the pit boundary in the
numerical model and 80 terms taken in the series summation for analytic
solution with analytic solution model for circular pit of the same radius
shown and with pit drawn .................. ............... ... ............... 53
5.20: Contour plot showing wave fronts as lines of constant phase from analytic
solution model for circular pit with r = 12 m, 80 terms taken in the series
summation base conditions with pit drawn .........................................54
5.21: Contour plot of showing wave fronts as lines of constant phase from
MacCamy and Fuchs solution model for circular pit with r = 12 m,
80 terms taken in the series summation base conditions with pit drawn.......55
5.22: Plot of numerical model values of maximum and minimum relative amplitude
versus the number of points defining the pit boundary for a grid of points
defining the fluid domain with a circular pit of radius 12 m and other
base conditions with analytic model solution for same conditions drawn......56
5.23: Plot of analytic solution model values of relative amplitude versus the number
terms taken in the series summation for five points on the xaxis with a
circular pit of radius 12 m and other base conditions............................57
5.24: Plot of numerical model values of relative amplitude for two sections of a
transect parallel to the yaxis at x equal 100 m for five different
numbers of points defining the pit boundary for a circular pit of
radius 12 m and other base conditions with the analytic solution
model results for the same conditions drawn................................... 61
5.25: Plot of transect taken parallel to the yaxis for energy flux calculations with
pit of radius 12 m showing wave direction ................... ...............60
5.26: The four energy flux terms from the analytical solution model for transect
shown in Figure 5.25 with base conditions and 90 points taken in the
summ ation............................................ ................... . 61
5.27: Energy flux at each point in transect shown in Figure 5.25 with incident value
(dotted) for base conditions and 90 terms taken in the series summation......62
5.28: Reflection coefficients determined as the average over one energy flux
oscillation for each upcrossing in the transect................. ...............63
5.29: Reflection coefficients for different transect locations with several incident
wave periods for the transect shown in Figure 5.25 taking the
reflection coefficient as the average of the last energy flux oscillation
in the transect............. .................. ..................... .......... .64
5.30: Reflection coefficients versus pit diameter divided by wavelength outside
the pit and pit diameter divided by wavelength inside the pit for
different pit radii for the transect shown in Figure 5.25 taking the
reflection coefficient as the average of the last energy flux oscillation
in the transect .............. ... .............. ............. .... ......... 65
5.31: Reflection coefficients versus water depth divided by pit depth for different
pit radii for the transect shown in Figure 5.25 ..................... .............66
5.32: Reflection coefficients versus pit diameter divided by wavelength inside
pit with different pit radii for a pit of radius equal to 12 m and the
transect shown in Figure 5.25 ................. ........ ...... ......... 67
5.33: Reflection coefficients versus the number of points defining the pit
boundary for the numerical model with a pit of radius equal to 12 m,
a period of 12 s and the other base conditions with the analytic solution
model result shown ................................. .. ... ............ 69
5.34: Percent difference in relative amplitude, percent difference in energy flux
values and difference in energy flux values for analytic solution model
and numerical model for a pit of radius equal to 12 m, 500 points on
the pit boundary in the numerical model, and the other base conditions
along a transect at x equal to 100 m ................................................ 70
5.35: Reflection coefficients versus the pit diameter divided by the wavelength
outside the pit for the numerical model for different pit radii along a
transect at x equal to 100 m and the other base conditions with the
analytic solution model results shown..........................................72
5.36: Reflection coefficients versus the cylinder diameter divided by the wavelength
outside the pit with the MacCamy and Fuchs solution model for different
pit radii along a transect at x equal to 0 m and the other base conditions......73
5.37: Relative Amplitude along the transect for the numerical model with a pit of
radius equal to 50 m, at x = 1500 m with 400 points defining the pit
boundary and the other base conditions for this shoreline change model......76
5.38: Longshore transport value along the transect for the numerical model with
a pit of radius equal to 50 m, at x = 1500 m with 400 points defining
the pit boundary................... ............................... .. ........... 77
5.39: Shoreline change from the full transport equation for a 300 s time step
along the transect for the numerical model with a pit of radius equal to
50 m, at x = 1500 m with 400 points defining the pit boundary and the
other base conditions for the shoreline change model ............................78
5.40: Shoreline change for a 300 s time step along the transect for the numerical
model with a pit of radius equal to 50 m, at x = 1500 m with 400 points
defining the pit boundary and the other base conditions for the shoreline
change model ............. ........ ................. . ..... ........ .. 79
5.41: Shoreline change for first and second transport terms for a 300 s time step
along the transect for the numerical model with a pit of radius equal to
50 m, at x = 1500 m with 400 points defining the pit boundary and the
other base conditions for the shoreline change model ............................80
5.42: Longshore transport along the transect for the analytic solution model and
the numerical model with a pit of radius equal to 50 m, at x = 1500 m
with 400 points defining the pit boundary and the other base conditions
for the shoreline change model............ ................... .................81
5.43: Shoreline change for a 300 s time step along the transect for the analytic
solution model and the numerical model with a pit of radius equal to
50 m, at x = 1500 m with 400 points defining the pit boundary and the
other base conditions for the shoreline change model......................... 82
5.44: Relative amplitude along transect for numerical model with a rectangular
pit (40 x 200 m) at x = 1500 m with 800 points defining the pit boundary
and the other base conditions for this shoreline change model .................. 83
5.45: Shoreline change for 300 second time step along transect for numerical
model with a rectangular pit (40 x 200 m) at x = 1500 m with 800
points defining the pit boundary and the other base conditions for
this shoreline change model ............. ...................... ....... ........... 84
5.46: Total shoreline change for 300 second time step along transect for numerical
model with a rectangular pit (40 x 200 m) at x = 1500 m with 800 points
defining the pit boundary and the other base conditions for this
shoreline change model ........... ............. ........... ............... 85
5.47: Shoreline evolution for analytic solution model for 2 hour time step with
pit radius of 6 m, water depth of 2 m, pit depth of 3 m and 50 terms
taken in the series summation using full transport equation....................87
5.48: Shoreline evolution for analytic solution model for 2 hour time step with
pit radius of 6 m, water depth of 2 m, pit depth of 3 m and 50 terms
taken in the series summation using each transport term separately............88
5.49: Shoreline change over each time step for analytic solution model for 2 hour
time steps with pit radius of 6 m, water depth of 2 m, pit depth of 3 m
and 50 terms taken in the series summation ................. ................ 89
5.50: Aerial photograph showing salients shoreward of borrow area looking
East to West in August, 1985 (Combe and Soileau, 1987) ....................91
5.51: Relative amplitude along the transect for beta equal to 6.5 degrees and
averaged over five betas from numerical model for the Grand Isle, LA
case study ........................................... ............ .......... . 93
5.52: Shoreline change for one 300 s time step and longshore transport (filtered
and unfiltered) for Grand Isle, LA case study with 50 terms taken in
the moving average ............... . ................ .............. .... ....... 94
5.53: Longshore transport (filtered and unfiltered) from numerical model for Grand
Isle, LA case study with 25 and 75 terms taken in the moving average........95
5.54: Shoreline change for one 300 s time step from numerical model for Grand
Isle, LA case study with 25 and 75 terms taken in the moving average........96
5.55: Shape of borrow area and transect location for Martin County, FL
numerical model ............. ... ......................... .......... 98
5.56: Relative amplitude along transect for bathymetry predredging and
postdredging for Martin County, FL numerical model ......................... 98
5.57: Change in shoreline position during one time step for original bathymetry
and postdredging for Martin County, FL numerical model ......................99
5.58: Difference in shoreline change during one 300 second time step from original
bathymetry to shoreline change after dredging of shoal for Martin
County, FL num erical m odel ........... ................................. ..... 100
6.1: Schematic layout of fixedbed model used in the laboratory experiments .........103
6.2: Change in dry beach width with time for Pit 1 for Trial 1 from initial
equilibrium without pit, to shoreline after 3 hours with pit ..................... 108
6.3: Change in dry beach width with time for Pit 2 for Trial 3 from initial
equilibrium without pit, to shoreline after 3 hours with pit ..................... 109
6.4: Change in shoreline position from equilibrium without a pit to after 1 hour
with a pit present for 3 pit sizes and wave conditions of Trial 1............... 109
6.5: Change in shoreline position from equilibrium without a pit to after 3 hours
with a pit present for 3 pit sizes and wave conditions of Trial 1..............10
6.6: Change in shoreline position from equilibrium without a pit to after 1 hour
with a pit present for 3 pit sizes and wave conditions of Trial 2..............111
6.7: Change in shoreline position from equilibrium without a pit to after 3 hours
with a pit present for 3 pit sizes and wave conditions of Trial 2.............. 112
6.8: Change in shoreline position from equilibrium without a pit to after 1 hour
with a pit present for 2 pit sizes and wave conditions of Trial 3...............13
6.9: Change in shoreline position from equilibrium without a pit to after 3 hours
with a pit present for 2 pit sizes and wave conditions of Trial 3...............113
6.10: Change in shoreline position from equilibrium with no cylinder to after
3 hours with a solid cylinder of diameter equal to 32 cm present for
the wave conditions of Trial 2 with the pit results drawn ..................... 114
6.11: Contour plot of relative amplitude from numerical model for Pit 1 and
incident wave conditions of Trial 1 with wave guide drawn and x = 0
representing the baseline for the experiment ............... ................. 115
6.12: Relative amplitudes determined from the numerical model for experiment
Trials 1 and 3 with 3 pits at x equal to 0.4 m with 150 points on the
pit boundary and the location of the waveguides used in the
experiment drawn.............. ...................... ... .......... 116
6.13: Change in shoreline position from equilibrium from the numerical model
for experiment Trials 1 and 3 with 3 pits at x equal to 0.4 m with
150 points on the pit boundary, one 300 second time step, and the
location of the waveguides used in the experiment drawn.................. 117
6.14: Change in shoreline position by each transport term from the numerical
model for experiment Trials 1 and 3 with and Pit 1 at x equal to 0.4 m
with 150 points on the pit boundary, one 300 second time step.............119
6.15: Change in shoreline position by each transport term from the numerical
model for parameters from Horikawa et al. (1977) at x equal to 0.4 m
with 150 points on the pit boundary, one 300 second time step............ 120
Abstract of Thesis Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Master of Science
WAVE FIELD MODIFICATIONS AND SHORELINE RESPONSE DUE TO
OFFSHORE BORROW AREAS
By
Christopher J. Bender
May 2001
Chairman: Dr. Robert G. Dean
Major Department: Civil and Coastal Engineering
This study was undertaken to form a better understanding of the wave field
modifications and resulting shoreline changes caused by offshore borrow areas. These
offshore areas can cause wave refraction, diffraction, reflection and dissipation resulting
in a shoreline that is modified to a new equilibrium by the pit altered wave field. In the
context of a beach nourishment project, this new equilibrium can lead to the presence of
erosional hot spots that reflect negatively on the overall project performance.
Numerical and analytic solutions are used to model the wave conditions in and
around the pit using shallow water wave theory. The numerical solution employs the
formulation by Neil Williams in his paper on wave diffraction by a pit, published in
1990. This method uses Green's Theorem and suitable Green's functions on a pit
boundary of arbitrary shape. The analytic solution uses a series expansion of Bessel
functions at each point in the defined fluid domain for a circular pit. These models
determine the complex velocity potential found anywhere in the fluid domain, allowing
quantities such as the velocity and pressure to be solved. This allows modeling of the
energy reflected and shoreline change induced by a pit. The wave field modification
caused by a pit is compared to that due to a solid structure using numerical and analytic
solutions. A laboratory experiment was conducted to determine the wave field
modification and shoreline change caused by pits of various sizes under several incident
wave conditions.
The wave field modification caused by a pit of finite dimensions is found to be
significant, with a partial standing wave pattern of increased and decreased wave
amplitude in front of the pit, and an area of reduced relative amplitude flanked by two
lobes of increased amplitude behind the pit. The numerical and analytic solutions for a
circular pit compare well, with larger pit radii leading to greater differences in the results.
The energy reflected by a pit is found to be considerable and to depend on the
dimensionless pit diameter and other parameters. The shoreline change shoreward of the
pit shows accretion directly behind the pit bordered by areas of erosion, with both
transport terms of importance. The laboratory results for the shoreline change behind a
pit show either accretion or erosion directly behind the pit; trends that vary in their
correlation with the computer models. In summary, an offshore pit is shown to modify
the incident wave field, which can lead to energy being reflected and diffracted, and
associated changes to the shoreline near the pit.
CHAPTER 1
INTRODUCTION
1.1 Problem Statement
Beach erosion is a primary concern for many coastal communities. The beach
system provides natural storm protection, recreation area and wildlife habitat. When
significant erosion occurs, the benefits provided by the beach system are diminished.
Beach nourishment has emerged as a "natural" technique for restoring the equilibrium of
the beach system. By adding sand to the dynamic sediment system in the nearshore zone,
beach nourishment replaces the material lost to erosion; an advantage over "hard"
solutions such as seawalls, breakwaters and groins.
Beach nourishment projects require sediment for the beach fill that is placed at the
nourishment location. The fill material needs to be carefully chosen to have sediment
characteristics that combine with the native sediment to provide successful performance.
The fill sediment can be dredged from local offshore locations or nearby inlets,
transported from distant offshore locations or mined from inland areas. When the fill
sediment is dredged from offshore, borrow pits are usually created. These borrow pits
range in size and location offshore depending on the amount of material needed and the
location where suitable fill material is found.
Recent cases of erosional hot spots (EHS) associated with beach nourishment
projects have increased interest in the prediction of the mechanisms by which borrow pits
alter the local wave field. An offshore borrow pit can cause four wave transformation
processes: wave refraction, wave diffraction, wave reflection and wave dissipation. A
better understanding of the interaction of the incident wave field and borrow pit
characteristics such as size and depth is needed to determine the effect of the pits on the
nourished beach and to anticipate the effects of various borrow pit designs.
The problem of waves propagating over a pit of uniform depth has been studied in
the past by various methods. Some studies have investigated the reflection and
transmission characteristics of a constant depth trench of infinite length located in water
of constant depth. More recently the diffraction caused by a rectangular pit of finite
shape and constant depth has been studied numerically. Laboratory as well as field
studies have been performed to determine the shoreline response caused by offshore
dredge pits.
1.2 Objectives and Scope
The present study employs numerical, theoretical and laboratory approaches. The
numerical analysis uses a boundary element approach for a pit of finite dimensions in
uniform depth as in Williams (1990). The fluid is considered incompressible and the
flow irrotational allowing the application of potential flow and linearized shallow water
wave theory. The fluid regime is divided into two regions; one representing the pit area
and the other, the region outside the pit. Using Green's second identity and appropriate
Green's functions, the velocity potential and derivative of the potential normal to the pit
are found at the interface of the two regions using standard matrix techniques. The
velocity potential at any location in the fluid domain can then be determined after a
reapplication of Green's Law with the values determined on the pit boundary.
An analytical solution to the pit problem is obtained using the general method of
MacCamy and Fuchs (1954) with a circular pit instead of a solid cylinder by allowing a
solution inside the pit with appropriate boundary conditions. The solution inside the pit
is of the same general form as the incident solution. As in the numerical solution the
boundary conditions match the pressure and flow across the pit boundary. This analytical
solution allows for the determination of the potential at any location in the fluid domain
containing a circular pit.
For both the numerical and analytical solutions, the free surface elevation is
obtained based on the known velocity potential at any point. This provides the basis for
the development of a diffraction diagram of relative amplitude over the entire field of
interest. The velocity potential can also be used with sediment transport and sediment
conservation equations to calculate the shoreline change landward of the pit, as well as
the energy reflection caused by the presence of the pit. The numerical solutions to these
cases are compared to the analytic solutions for both a pit and a solid structure.
Laboratory experiments were carried out to study the wave field modification
around a pit and the shoreline evolution behind pits of several planform shapes in
uniform water depth. The wave height alteration and shoreline change were compared to
the computer model results for different pit depths and sizes as well as different incident
wave characteristics. In addition to comparison with the laboratory results, the
methodology is illustrated by application to two beach nourishment projects: Grand Isle,
LA and Martin County, FL.
CHAPTER 2
LITERATURE REVIEW
2.1 Introduction
Wave field modification by an offshore pit has been studied numerically,
analytically, and in laboratory and field studies. The purpose of this chapter is to present
a broad review of the studies undertaken on the wave transformation caused by pits and
the resulting nearshore changes. As stated before, the pit can induce four transformation
processes: wave refraction, wave diffraction, wave reflection and wave dissipation. The
studies differed in their methodology, the transformation processes) studied and their
conclusions.
2.2 Discussion of Previous Studies
2.2.1 Rectangular Pits of Infinite Length
A general analysis of wave propagation over variable depth geometries was
performed by Kreisel (1949). The procedure involved mapping the fluid domain to a
rectangular strip, which for suitable geometries allows a linear integral equation to
resolve the velocity potential through iteration. Matching the solution along the
geometrical boundary separating the regions of different depths is a method used by
Bartholomeitz (1958) and Miles (1967).
The reflection and transmission coefficients for a trench of uniform depth with
water depth upwave and downwave from the trench that is uniform, although not
necessarily equal, was studied by Lassiter (1972) for the case of monochromatic plane
progressive waves. Complementary variational integrals were used to solve for the
4
velocity potential after applying matching conditions on the trench boundaries. Results
show reflection coefficients demonstrating a dependency on the dimensionless wave
frequency, k*h. The analysis is performed for different wave and trench conditions and
shows complete transmission at certain k*h values.
Lee and Ayer (1981) developed an analytical solution for normally incident
waves over a rectangular trench of infinite length. The analytic solutions for the region
outside the pit and inside the pit are matched and solved numerically. The region outside
the pit is considered one continuous strip from negative to positive infinity with the pit
region occurring below this for a specified distance and depth. Reflection and
transmission coefficients found in the study show an infinite number of discrete
frequencies at which waves are completely transmitted, while at other frequencies
reflection coefficients of 0.45 can be obtained. The reflection coefficient was defined as
the ratio of the reflected wave amplitude to the incident wave amplitude. The results
compare reasonably well with those of Lassiter (1972) and show that for a deep trench,
the transmission coefficient is a minimum when the pit length divided by the incident
wavelength is equal to onehalf The transmission coefficient can reach one when the pit
length is equal to the wavelength.
Miles (1982) studied the diffraction of normally incident, long waves over a
trench of infinite length using the mapping procedure of Kreisel (1949). A correction
was applied to determine the reflection and transmission coefficients for oblique waves.
The ability to determine results for long waves is a feature that Lassiter (1972) and Lee
and Ayer (1981) do not include.
The diffraction of waves with large incident angles was studied by Kirby and
Dalrymple (1983) using the boundary matching conditions discussed previously and a
modified form of the solution used by Takano (1960). For normally incident waves the
reflection coefficient results nearly match those of Lee and Ayer (1981), but show some
difference from Lassiter (1972). Much lower values of transmission coefficients are
found for an incident angle value of 45" as opposed to normal incidence. This study was
extended to include a current flowing along the trench in Kirby et al. (1987).
Ting and Raichlen (1986) applied the theory of Lee and Ayer (1981) to examine
the dependence of the kinetic energy in the pit region to the incident frequency and pit
characteristics. The goal was to determine the frequencywise dependence of the induced
motion in the trench relating to navigational channels.
2.2.2 Pits of Arbitrary Shape
A Green's function approach was used by Willliams (1990) to determine the
diffraction of long waves by a rectangular pit. Long waves were studied rendering the
problem two dimensional in two horizontal coordinates, where the previous studies were
two dimensional in one horizontal and one vertical coordinate. Using the potential and
derivative of the potential on the pit boundary and applying Green's second identity
allows for the solution of the velocity potential at any point in the fluid domain. The
results show a shadow zone created behind the pit with two bands of increased wave
height flanking the shadow zone. A partial standing wave system develops seaward of
the pit defined by alternating bands of increased and decreased wave amplitudes. For the
cases presented in the paper a reduction often percent in the wave amplitude is found
landward of the pit and increases of 10 % occur seaward of the pit. The long wave
restriction of the previous study was removed in Williams and Vasquez (1991) using a
threedimension Green's function and a double Fourier series expansion as the solution
inside the pit boundary.
The method of Williams (1990) was expanded to include diffraction by multiple
pits in McDougal et al. (1996). The effect of pit width, pit depth, pit length, and angle of
incidence on the relative amplitude and location of the shadow zone are investigated for a
single pit. The minimum diffraction coefficient is much lower than for the infinite length
trench studied by Lee and Ayer (1981) and Kirby and Dalrymple (1983) with the possible
explanation that the finite width pit results in a refraction divergence in the lee of the pit.
2.2.3 Shoreline Change by Offshore Pits
The shoreline response due to wave refraction over dredged holes was
investigated by Motyka and Willis (1974) with a mathematical model. The model used
wave characteristics typical of the English Channel coasts of Britain with deepwater
angles ranging from 20" to 10". The beach was found to erode directly behind the
dredged hole with increasing erosion for an increasing pit depth and decreasing original
water depth.
This finding is opposite to that of Horikawa et al. (1977) who found shoreline
advancement in the lee of a dredge pit in their model. It is interesting to note that in
Horikawa et al. it is stated that the results concur with those of Motyka and Willis (1974),
but they clearly do not. The model uses wave data defined for different seasons with six
different pit configurations. The results show a salient forming behind the pit with the
size dependent on the incident wave values, pit characteristics and duration of simulation.
It is suggested that the sand accumulates behind the pit due to the relatively calm water
occurring there as a result of the reduced wave action.
A study by Gravens and Rosati (1994) examined the EHSs at the Grand Isle,
Louisiana Nourishment Project and concluded that refraction around an offshore borrow
pit can cause low energy immediately landward of the pit and higher energy at the ends
of the pit. This situation results in a salient forming behind the pit as a current, and
therefore sediment flow, is created by the difference in wave setup at the edge of the pit
and behind the pit. Their approach used the full CERC transport equation with an
unusually high value of 2.4 for the coefficient K2 needed to achieve the salient behind the
pit.
2.2.4 Laboratory and Field Studies
As part ofHorikawa et al. (1977) a laboratory experiment was conducted to see if
a salient formed behind an offshore borrow pit as in the numerical model of the same
paper. Using a fixedbed model in a small wave basin, an equilibrium beach profile was
created under monochromatic, shorenormal waves with no pit and then with the
presence of a rectangular pit. The results show the formation of a salient behind the pit
for the conditions tested. The shoreline change results compare fairly well to those of
their numerical model with larger values of accretion and erosion occurring in the
laboratory tests. The reasons for this difference in offshore displacement are theorized as
resonant waves particular to the wave tank and wave refraction over the inshore seabed,
which was not considered in the model.
Two other studies that investigated the effects of offshore dredge pits on the
shoreline are Price (1978) and Kojima et al. (1986). Price (1978) examined locations off
the coast of England through rods driven into the seabed, tracer studies, and numerical
modeling and concluded that sand mining in water depths greater than 46 ft (14 m) to
59 ft (18 m) caused little effect to the coast. Kojima et al. (1986) concluded that offshore
mining likely played a role in observed beach erosion off the coast of Japan. This
analysis was based on examining wind and wave data, shoreline histories, offshore
mining activities, sand tracers, and monitoring of dredge pits.
CHAPTER 3
THEORY
3.1 Introduction
This chapter presents the fundamental equations used as the basis for this study.
These governing equations are used in their respective sections of the study as the
foundation of the models for the numerical and analytic solutions. The models employ
these fundamental equations to solve the problems of flow over a pit, reflection from a pit
and shoreline change induced by a pit.
3.2 Governing Equations
3.2.1 Flow Over A Pit
A uniform wave field encounters a region of constant water depth containing a pit
of finite dimensions and uniform depth as shown in Figure 3.1.
S Y3
x
ii ItRcaion 2
Figure 3.1: Definition sketch for flow over pit
10
Long waves will be studied as the greatest influence of the pit occurs in the
presence of these waves. The area of study is divided into two regions. The pit, of depth
d, and its projection comprise Region 1 while the rest of the fluid domain, of depth h, is
contained in Region 2. The fluid is taken to be inviscid and incompressible, and the flow
is taken to be irrotational. Linear shallow water wave theory is applicable and Laplace's
solution is valid:
V2 = + (3.1)
x2 Y2 a2
with the bottom boundary condition:
o= 0 (3.2)
az
on z = h in Region 2 or on z = d in Region 1.
Solving, using separation of variables and O(x, y, z, t) = X(x)Y(y)Z(z)T(t), a
valid solution is Z(z) = cosh(k(h+z)). Inserting this into the Laplace equation leads to:
a +k2a0 (3.3)
aX 2 ay2
The depth averaged velocity potential can be defined as:
D = Re( (x, y)e "') (3.4)
where (o is the wave frequency. The wave numbers in the two fluid regions are:
k = (3.5)
2 g= d
k2 = (3.6)
vgh
The boundary conditions for the interface, F, between the two regions are:
1 = 02 (3.7)
on F, which equates pressure across the boundary and:
d h a2 (3.8)
On On
on F, which equates discharge across the boundary. The solution must also meet the
radiation condition for large r:
limr ,( ik2 )( ) (3.9)
where the complex incident potential is defined as 0('.
The value of instantaneous free surface elevation may be found using the
equations:
r= 1= Ae(Ot) (3.10)
g t g
A = + ,2.ag (3.11)
tan = ag (3.12)
Treat
for the 0 values found inside and outside of the pit.
3.2.2 Energy Considerations
The amount of energy reflected by a pit is determined using a timeaveraged
energy flux approach. At each point along a transect running parallel to the incident wave
fronts, the depth integrated, timeaveraged energy flux and incident energy flux are
determined.
The time averaging is obtained by taking the conjugate of one of the complex
variables:
EFluxT = pTu = pT, (* means conjugate) (3.13)
where p is the pressure and u is the velocity in the direction perpendicular to the transect:
PT = P (3.14)
at
u, (3.15)
Ox
v, (3.16)
ay
The total potential is defined as the sum of the incident potential and the reflected
potential due to the presence of the pit:
S= 0 + R (3.17)
Therefore the equation to determine the time average energy flux, in the x direction, at a
single point due to the incident potential and the reflected potential is:
EFluxr = PTU = (p, + PR)(UI + UR) = pu + puR + pRUI + pUR (3.18)
The EFluxx r values at each transect point are multiplied by the spacing between the
transect points and the water depth to determine the depth integrated, timeaveraged
energy flux in the x direction through the transect. The timeaveraged energy flux in the
y direction can be determined using Eq. 3.16 instead of Eq. 3.15 with a transect parallel
to the xaxis.
3.2.3 Shoreline Change
The principle of an equilibrium beach profile is the foundation for all planform
evolution concepts. The continuity equation is used to allow for a relationship between
the shoreline position and the change in volume for the profile. The assumptions of
straight and parallel bottom contours and no refraction are made. These concepts are
used to determine the longshore transport and resulting shoreline change. The wave
angle incident on the shore must be known to use the transport equation. A sketch of
shoreline change problem is shown in Figure 3.2.
N Y
4 I
the transect the timeaveraged energy flux in the x and y directions are determined using
/ 0 .
/ \
Eqs. 3.13 to 3.18. The angle of wave approach at each point is then solved in this way:
a = tan' EFluxy (3.19)
SEFlux )
14
Using the definition of the depth average velocity in Equation 3.4 the wave angle
is found in an alternative way using the following methodology and equations:
0 = 0(real + ioimag = A + iB
B
A + iB = C cos(a) + iC sin(a) A = C cos(a) & B = C sin(a) tan(a) =
A
where a is the angle of wave approach. To track a constant phase:
da da
Da = 0 = ax + dy
9x Sy
Cy Dax
ax Da/
/ dy
Taking the derivative of a defined in Equation (3.21) with respect to x and y:
(3.20)
(3.21)
(3.22)
(3.23)
OB DA
A B
Sda _x Ox
sec2 a= a
x A2
AB aA
A B
oa Dy Dy
and sec2 a
Gy A2
and combining:
A aB B A
Oy (A B B
A = 1 real
DA ,
= L real
Ox Ox
9A a
= Z real
Cy cDy
B = 1qimag
Ox= imag
8x 8x
aB 8 9
B= = imag
9y Dy
(3.24,3.25)
where:
(3.26)
(3.27,3.28)
(3.29,3.30)
(3.31,3.32)
The wave angle is then determined using the formula:
a= tan ( (3.33)
dx
The value of the longshore transport angle is equal to a +7t for the previous derivation.
From the coordinate system in Figure 3.2 it is seen that when a is greater than 0, the
longshore transport is positive and the transport is negative when a is less than 0. This
method provides an angle, but the quadrant of the angle is not determined due to the
negative sign in Eq. 3.26. Due to this limitation, only the magnitude of the angle from
the first method can be checked.
The wave angle and wave amplitude at each point of a transect representing the
shoreline are used to determine the longshore transport with the full CERC transport
equation:
KH 25 sin(ab)cos(Oab) K2H 25 cos(Oab)dHb
Q= +4 b (3.34)
8(s 1X1 p) 8(s1X1 p)tan(r) dy
where Hb equals the wave height at breaking, ab equals the wave angle at breaking, 0
equals the shoreline orientation, s equals the specific gravity ratio of sand to water, p
equals the porosity of the sediment, K equals the breaking criteria, KI and K2 equal the
sediment transport coefficients, and tan(y) equals the slope from nearshore to depth of
closure. The first transport term is driven by the waves approaching the shore at an angle
and the second term, which is sometimes not included, is due to the gradient in the wave
heights or setup. Bakker (1971) developed a transport term based on the variation in
longshore wave height. This method is outlined in Ozasa and Brampton (1979).
The shoreline change is calculated using the formula:
&x 8Q 1 oQ 1
= aQ  Ax = Q At (3.35)
at oy h, + B oy h, + B
where h* is the closure depth, B is the berm height and a positive change in xdirection
indicates erosion. The time step At needed for model stability is determined using:
Ay2
Atm < (3.36)
2G
where:
K H 2.5
G =1 (3.37)
8(s IX p)h. +B
CHAPTER 4
MODELS
4.1 Introduction
In this study the problem of long wave propagation over a pit is solved with a
numerical solution and an analytic solution. Both the numerical and analytic solution
models are compared to solutions for a solid structure. This chapter provides details of
the solution method for each model. The numerical model uses the solution of Williams
(1990), which applies Green's second identity to points defined on a pit boundary of
arbitrary shape. The analytic solution model solves the problem in a manner similar to
the MacCamy and Fuchs solution for diffraction by a cylinder, but with a circular pit
instead of a cylinder.
4.2. Numerical Solution Model of Williams (1990)
The solution model of Williams (1990) determines the velocity potential at any
point inside or outside of a pit of arbitrary shape in the fluid domain. The ability to
determine the velocity potential at different points is utilized in the models to determine
the reflection from the pit and the shoreline change induced by the pit. Also, the
Williams solution is modified to allow for the study of solid barriers.
4.2.1 Green's Function Solution of Williams (1990)
The governing equations and definition sketch of Section 3.1 provide the basis for
this solution method. The complex incident potential is defined as:
q( (x,y,t)= igH ek2(xcosp+ysin pwt) (4.1)
2w
where H is the incident wave height and p is the incident wave angle. Suitable Green's
functions for the two regions are defined as:
G = H(kR) and G, = Ho(k2R) (4.2,4.3)
2 2
for Region 1 and Region 2, respectively where H' is the Hankel function of the first kind,
zero order equal to Jo (kR) + iYo (kR).
The distance between any point in the fluid domain and one on the pit boundary is
defined as:
R= (xx')2 (yy (4.4)
with (x,y) defining the location of a point on the pit boundary and (x',y') defining the
location of a point not on the pit boundary. At locations of r = r' the Green's functions
each have a singularity resulting from Yo(0) = oo, which must be accounted for. The
second Green's function is seen to satisfy the radiation condition.
The divergence theorem and Green's theorem are used to achieve the desired
solution allowing for determination of the potential at any point in Region 1 or Region 2.
The divergence theorem is:
fVj" VdV = I V ndS (4.5)
and is true ifV and its partial derivatives are continuous in v and on S. To apply Green's
theorem we substitute:
V = ,V ,2 (4.6)
where 41 has been defined previously and 2 is equal to G, both of which are scalar
functions of position. Using the first form of Green's theorem:
fIIff4V22 + (V21) (V02)]dV = S n1V2dS (4.7)
By interchanging and 2 :
fJ [2V2 0 + (V2). (V41)]dV = fn.402V 1dS (4.8)
Subtracting the Eq. 4.8 from Eq. 4.7 gives the second form of Green's theorem:
IJJJV V202 2 V21 ]dV = (n (1V2 2 2V1)dS (4.9)
If 4 and 02 are both solutions of (V2 + k2 ) = 0 then the left hand side of the previous
equation vanishes and we are left with:
Jsn" (0 V02 02V4 )dS= 0 (4.10)
Accounting for the residue found at the singularity on the pit boundary, F, over
Region 1 and using the above equation with 4, and G1 leads to:
al, (r') + ii(r) (r,r') G, (r, r') (r) =0 (4.11)
with a, = 2 ifr' is inside F, a, = 1 ifr' is on a smooth portion ofF and a( = 1/2 ifr' is on a
corner point ofF. Applying 2 and G2 to the same equation gives the following
equation:
a2 2 ()= 2(r) (r, r) G2(r, ) o (r) + 2S (r') (4.12)
with 2 = 2 ifr' is inside 2 = 1 if'in a smooth portion ofand2 = 3/2 ifr' is on a
with a2 = 2 ifr' is inside F, a2 = 1 ifr' is on a smooth portion ofF and a2 = 3/2 ifr' is on a
corner point of In these equations r is the location of a point on the pit boundary and r'
is a point inside or outside of the pit.
In order to solve numerically for the potential inside and outside of the pit, 0 and
Son the pit must be determined. This is achieved by setting r' on the pit boundary and
On
using the boundary conditions stated earlier. This procedure leads to two equations that
can be solved numerically with standard matrix procedures after discretizing the pit
boundary into a specified number of segments:
z(r f (r)1 (r,r')h G(r,r') (r) l =0 (4.13)
)'J an d oan
a22 2 r, )G(rr') (r)" = 22 (r') (4.14)
rc an an
where the integrals are replaced by summations. The above equations lead to values of
and on F that are assumed constant for the length of each segment. Knowing the
On
values of 02 and 2 on F allows for the potential at any point in the fluid to be
an
determined with the previous solutions. The singularities inherent in the Green's
functions are accounted for in the a values which must be determined for each point, r', in
the fluid domain.
The value of instantaneous free surface elevation may be found using the Eqs.
3.10 through 3.12.
4.2.2 Wave Height Reduction for Pit of Arbitrary Shape
A numerical model was developed to solve the problem of diffraction around a pit
of uniform depth, in water depth which is otherwise uniform assuming linearized shallow
water wave theory. The model follows the solution of Williams (1990) and obtains
results that are very similar. The basic program results in plots of relative amplitude in
and around the pit. The user defines values for:
beta angle of incident wave approach in radians
h water depth around pit in meters
d depth of pit in meters
T period of incident waves in seconds
H uniform height of incident waves in meters
x x values defining pit in meters, entered in counterclockwise motion around pit
y y values defining pit in meters, entered in counterclockwise motion around pit
nump number of points defining pit in calculations, greater number gives better
accuracy
totalpoints number of points in grid where potential is found in fluid domain,
must be a square number
C multiplier for grid boundary, grid size is C*wavelength (out of pit) on each
side of pit
The program calculates values of wavelength and wavenumber inside and outside
of the pit using linearized shallow water theory. Four subprograms are used in the main
MATLAB program during the execution of the models; these programs can be obtained
by contacting the author.
The program outputs the velocity potential at each point defined in the fluid
domain. Knowing the complex value of the potential at each point in the grid allows for
the calculation of the instantaneous free surface elevation at each location using Eqs. 3.10
through 3.12. The relative amplitude (diffraction coefficient) of the grid locations is
taken as:
H
A, =R (4.15)
HI
Several checks were performed on the numerical model. The values obtained are
very similar to those of Williams (1990) and McDougal, et al (1996), both of which use
the same theory and primary equations as this model. The values of 0 and on the pit
9n
boundary and the diffraction pattern are found to be symmetric with the x axis for a
symmetric pit with normal incident waves, as would be expected. A pit depth, d, equal to
the surrounding water depth, h, is found to create free surface elevations that are found
with no pit present. The program results check with the boundary conditions.
4.2.3 Reflection from Pit
This model uses the values of potential along a transect parallel to the yaxis to
calculate the timeaveraged energy flux in the x direction at each point along the transect
and the sum of the energy flux along the transect.
The methodology of Section 3.2.2 is followed for these calculations. Using the
definition for the velocity potential in Eq. 4.1 the real values of the incident pressure and
velocities in the x and y directions are determined with:
P = pgH cos[kxcos(0)+kysin(O)ot] (4.16)
at 2
Using the identity cos(a f) = cos(a)cos(f) + sin(a)sin( f),
P = pgH {cos[kx cos(f) + ky sin(, )]cos(ct) + sin [k cos(f) + ky sin( f)]sin(Wt)} (4.17)
2
The final expression for the real value of the incident pressure is:
P, = P cos(a t ,) (4.18)
with P, = S + S and 6, = tan' 1 where:
2 2
2 2
The real value of the incident velocity in the x direction obtained using:
U, = gH k cos(0)cos[kxcos(0) +kysin(0) wt] (4.21)
8x 2 o
Using the same methodology as in the pressure calculation the final expression for the
real value of the incident velocity is:
U, = U Icos(wt eU) (4.22)
with UI = JS_ +S and s, = tan' 2 where SI and S2 are determined using
Eqs. 4.19 and 4.20 with a coefficient of gHk. Taking the derivative of the velocity
2w
potential at each point with y instead of x and following the procedure just described
determines the value of velocity in the y direction.
The model returns values of complex velocity potential at each point along the
transect parallel to the yaxis. The reflected pressure and velocity at each location are
determined using the value of the total velocity potential at each point.
Rota, = A, + iZ B,, (4.23)
where An and Bn are the values determined from each point on the pit boundary plus the
incident potential if the point is outside the pit.
PR = PO = iop A,, + i B,, = p( B,, + i A,, (4.24)
The final expression for the real value of the reflected pressure is:
PR = CWp cos(Wt ERP) (4.25)
where C = B + A and RP = tan A
n"
The real value of the reflected velocity in the x direction is determined as:
UR (real)+ image ) e (4.26)
8 Ox xx O)x
where the summation of the complex derivative of the velocity potential with x at each
point is taken. The equations used to determine are shown in Section 4.2.4. The
Dx
final expression for the real value of the reflected velocity is:
UR = cos(t ERU) (4.27)
where C = (real) + 42 (imag) and RU =tan1
O a (real)
The total velocity potential is determined from Eq 3.17 and the timeaveraged energy
flux for each of the four quantities in Eq. 3.18 is determined from:
UP = Pcos(c, ,) (4.28)
2
4.2.4 Shoreline Change Induced by Pit
This model uses the values of potential along a transect parallel to the y axis to
calculate the relative amplitude, wave angle, longshore transport, and shoreline change
along the transect. The methodology and inputs from the main numerical model are used
except that there is no grid, just a transect of points, and instead of only calculating the
potential, 4, values of O and are determined for each point along the transect.
ax ay
The calculation of 0, and are performed in a new program. This program
CX ay'
aOx ay
calculates and at each point along the transect. The same method as the main
8x ay
program is used for calculating 0, with the same results. The values of are found
ax
starting with the Eq. 4.12:
2 ) 2 (r, r') G2 (r, 2 (r)]dF + 2 '' (r) (4.29)
a21r J on 'n
Taking the partial derivative with respect to x' gives:
8 1 8 aDG 8 p'"
2 ( L 2 (r) 2 (r,r')G(rr') 2 (r)]dF + 27r (r')
ax a( o o
a 1 a aG a 8 1
0 2(rP2 [[02(r)  (r,r') G2(r,') (r)]dF +27 (r')
Ox a2 ax2 n x an Ox
(4.30)
This equation can be broken into three parts, with part A being the partial derivative with
respect to x' of the incident potential:
A: a 0(r')=ik2 cos()(f (r') (4.31)
ax
The second part is the partial derivative with respect to x' of the Green's function:
aG i (x X')
B: (k)[J, (kR)+ iY, (kR)] (4.32)
ax 2 R
The last part is the partial derivative with respect to x' of the derivative of the Green's
function normal to the pit boundary. This derivative must be expanded to accommodate
any pit orientation:
SaG a 9G 9G 8 aG a 9 aG
= a +b a +b (4.33)
x' an ox ax' y' &x' x ax' y'
where a and b are direction coefficients dependent on the angle the pit boundary makes
with the x axis at that point:
a QG 9 9 R 8 99G R 9 '9R
a a and b  b a (4.34,4.35)
ax ax aR x' ax ax y y x
a(a = a k( )[J (kR)+ Y(kR)] (4.36)
aR ax) 2 R2
9R (x x')
=(4.37)
x' R
b ) = b2 (kR)+ iY2(kR)] (4.38)
aR ay 2 R2
The resulting equation for last part is:
C: a k2 2 [2 (kR) + iY2 (kR)][a(x x') + b(y y')] (4.39)
9x 'n 2 R2
The values for are found in the same manner with the resulting equations being:
_ 1 G 2a
S (r)= [[ (r) (r,r') G2(r,r') (r)] l+2 (r') (4.40)
ay' ay ny n
A: a (r')= k2 in(,p) (r') (4.41)
ay'
B: =G (k)[J (kR)+ iYI(kR)] (4.42)
y' 2 R
C: caG ir k (Y2 )[J2(kR)+iY2(kR)Ia(x x')+b(y y')] (4.43)
by 9n 2 R
With these equations the program computes the integral as the summation of the
contribution from each point on the pit times the spacing between the points. The
program returns the value of 0 the potential at each point in the transect, , the
Ox
derivative of the potential with x' at each point, and the derivative of the potential
Oy
with y' at each point. The shoreline change is determined using these values and the
equations in Section 3.2.3.
4.2.5 Models for Solid Structure of Finite Dimensions
A solid structure is modeled by changing the boundary conditions at the pit to
have a no flow condition at the pit border. By making = 0 on the border the equation
on
to solve for 2 and the matrix solution from the original numerical model is changed.
The equation used to solve for 2 is,
)= J 2r) 2 (rr') d + 27 (r') (4.44)
an 2
In order to solve numerically for the potential outside of the pit, 0 on the pit must
be determined. This is achieved by setting r' on the pit boundary and using the boundary
conditions stated earlier. This procedure leads to two equations that can be solved
numerically with standard matrix procedures after discretizing the pit boundary into a
specified number of segments:
a 2 (r')+ 2(r) (r,r') = 0 (4.45)
a2 2 (r') f2(r) 20= ((r') (4.46)
),'L on
This leads to values of 0 on F that are assumed uniform for the length of each segment.
Knowing the values of 2 on F allows for the potential at any point in the fluid to be
determined with the previous solutions in the same procedure as in the main numerical
model for the pit.
4.3 Analytical Solution Model for Circular Pit
An analytical solution was developed for the problem of diffraction by a circular
pit of finite dimensions. This problem and solution are similar in form to that of
diffraction around a cylinder solved by MacCamy and Fuchs (1954). The solution
method involves defining an incident and reflected velocity potential outside the pit as
well as one inside the pit. An analytical solution model is developed that determines the
velocity potential at any point inside or outside of the pit. Using these values of velocity
potential, the reflection from the pit and the shoreline change induced by the pit are
determined using other models.
4.3.1 Wave Height Reduction for Circular Pit
The setup for the problem is the same as shown in Figure 1, but with a circular
pit. The methodology and inputs from the numerical model are used except that there is
no real pit boundary with points defined on it and therefore no calculations of 0 and
 on the pit. The value of 0 is calculated directly from the final equation for each
an
point on the grid and the free surface elevation is then determined. This solution is for a
circular pit of radius, a, and uniform depth, d, in water of otherwise uniform depth, h.
The velocity potential outside the pit is the addition of the incident wave, 1,, and the
reflected wave, R :
01 = MI 1 m cos(mO)J.(kr) e (4.47)
_m=0
igH
where Pm = 1 for m = 0 and 2im otherwise, M r is the distance from the point in
2wo
the fluid domain to the center of the pit and 0 is the angle between the two points
measured clockwise from the positive x axis. The reflected velocity potential is:
0R = [Am cos(mO)[J.(kr)+iYm(kr)] eicot (4.48)
_m=0
The solution inside the pit is defined as:
Oi = B cos(mO)JM(kr) e' (4.49)
The boundary conditions that must be met are the same as Equations 3.7 and 3.8,
except on r equal to the pit radius, a.
=, = out I r= (3.7)
ha d =d = (3.8)
Using this boundary condition, the solutions for Am and Bm for m equals 0 to oo:
k2h J' h (k2a)Jm(k,a)
MI) ~ m r 2 Jm (k2)
Akd J, (k,a) (4.50)
Jm (k, a) k,d
B, = J(k2a)Mi,, +Am[J. (k2a)+iY, (k2a)] (4.51)
J, (k, a)
The values for Am and Bm are inserted into the equations for JO and 0,,t equal to
01 + O Knowing the complex value of the potential at each point in the grid allows for
the calculation of the instantaneous free surface elevation at each location using equations
3.10 through 3.12. The relative amplitude of the grid locations with the pit and without
the pit is taken as:
HI
4.3.2 Reflection from Circular Pit
The energy reflection from the circular pit follows the methodology of Section
3.2.2 and equations 3.13 through 3.18. From the definition of the incident and reflected
velocity potentials, the pressure at any point outside the pit can be defined as:
p = PI ( O) I,,, cos(mO)J,,(k2r) e(t) (4.52)
PR = p(iW) Am cos(mO)(J (k2r)+iY '(k2r) e(t) (4.53)
_m=0
A polar coordinate system is used in this solution, which makes some algebra
necessary in order to determine the velocity in the xdirection. The velocity in the x
direction is obtained using the following equations:
u = vel, cos(O) velo sin(O) (4.54)
where velr = and velo = 0'a
ar r a0
Using these equations the equation for the incident and reflected velocities in the
udirection are determined as:
u, = Mk2 cos(0) ,m cos(mS0) J, (k2r) + m (k2r) e(lt)
m=0 2
+sin(0)MI [ sin(mO)J,,(k2r) e(it) (4.55)
r Lm=o
u, =k2 cos(O) Am cos(mO) Jm (k2r) + )+ ( (k2)+ i ( (r)) e(
_m=0 , _
+ sin(,) mAm sin(mO)(Jm(k2r) + iY(kr)) e(i"t) (4.56)
r m=0
The depth integrated, timeaveraged energy flux in the x direction at any point along the
transect is obtained from:
EFluxT = pTu = (p + Ps)(u, +us)* = p, u + pUs+ psU* + psUs (3.17)
The EFlux, values at each transect point are multiplied by the spacing between the
transect points and the water depth to determine the xdirected energy flux through the
transect.
A different solution for the energy flux is found inside the pit due to the lack of an
incident velocity potential there. The energy flux inside the pit is obtained using:
p. = p(iWCO B cos(mO)Jm(kr) e (a) (4.57)
m=0
u,i = k cos(O) Bm cos(mO) JM+kr)k + J.(,r) e
sin(O)L[i_, J k jc (4.58)
+sin() mBm sin(mO)J(kr) e (4.58)
r _m=0
The value of the velocity in the y direction is found using:
v = vel, sin(O) + velo cos(O) (4.59)
4.3.3 Shoreline Change Induced by Pit
The shoreline change induced by a circular pit follows the methodology of section
3.2.3 and equations 3.19 through 3.37. Values of L and are needed for this model
ax ay
for equations 3.29 through 3.32. In the preceding section the equations for ui and UR are
given in equations 4.54 through 4.56. Equation 3.15 is used to determine from u.
Ox
The following equations are used to determine the velocity in the y direction:
v1 = Mk2 sin(O) tlm cos(mO) JM, (k2r) Jm (k2 r)J e ()
cos(O) tim sin(mO)J(k2r) e(t) (4.60)
r [.m=0
R k2 sin(0) FA cos(mO) JM 1(k2)iY1(k2r)+ (Jm(k2r i(k2 )) e
cos(0) mA sin(ml)(J rk2 k2r) (i) (4.61)
r "1m=0
The value of is determined as:
9y
=V (3.16)
ay
4.4 MacCamy and Fuchs Solution Model for Cylinder
This model uses the solution for the diffraction around a cylindrical object first
solved by MacCamy and Fuchs (1954) for general depths. The setup for the problem is
the same as shown in Figure 1, but with a solid cylinder. The methodology and inputs
from the analytic solution model are used. The value of 0 is calculated directly from the
final equation for each point on the grid and the free surface elevation is then determined.
This solution is for a circular cylinder of radius, a, in water of otherwise uniform depth, h.
= M /, co0[ (r) J' (k2r)(J. (k2r) + iYm (k2r))) (4.62)
S+ Or = = =M ,. cos(m) J.(k2iY(ka)) (4.62)
M=o (.(ka)+iym(k2a))
where r, 0, and MI have been previously defined.
Knowing the complex value of the potential at each point in the grid allows for
the calculation of the instantaneous free surface elevation at each location using equations
3.10 through 3.12. The relative amplitude of the grid locations with the pit and without
the pit is taken as:
HI
A,,, R (4.15)
Values for the reflection caused by a cylinder and the shoreline change induced by a
cylinder can be obtained using the same methodology used with the analytic model.
CHAPTER 5
RESULTS AND DISCUSSION
5.1 Introduction
Many trials were run to verify and analyze the models that were developed. The
numerical model for an arbitrary shaped pit based on the theory in Williams(1990) was
compared to the results of that paper to verify the current model. After demonstrating
good agreement between the developed numerical model and that of Williams (1990) the
analytic solution model developed for a circular pit was compared to the results found
with a circular pit using the numerical model. These results for a circular pit are then
compared to the wave field alteration by a solid cylinder of the same dimensions based
on the theory of MacCamy and Fuchs (1954). The wave field alteration by a solid
cylinder is also found using a modified form of the numerical model with a noflow
boundary condition allowing for the modeling of any arbitrary shape as a solid entity.
For some of these models the energy reflected by the pit or cylinder as well as the
shoreline change induced by the pit is determined and compared for the different models
and for different incident wave and pit considerations.
5.2 Wave Height Reduction Models
5.2.1 Comparison of Numerical Model Results with Results of Williams (1990)
The relative amplitude in the fluid domain surrounding a pit of arbitrary shape is
determined in a numerical model based on the solution of Williams (1990). The base
conditions for the following trials are a water depth, h, of 2 m, a pit depth, d, of 4 m and
35
incident waves with 13 equal to zero and a period of 12.77 s (kh = 0.22). For this trial the
pit is comprised of 120 points and a grid of 1600 points defines the fluid domain. Figure
5.1 shows a contour plot of the relative amplitude for these conditions and a square pit
(8x8 m). A partial standing wave is seen to develop in front of the pit with a significant
area of wave sheltering behind the pit. Two "lobes" of increased wave amplitude are
seen to project out behind the pit at an approximate angle of 30 degrees.
40' / 0. ,
~1 1.04
'1..0
20 '" .0 ...
1 098
20 1
00 94
S ........... . ... ...
30 20 10 0 10 20 30 40
XDirection (mn)
Figure 5.1: Contour plot of relative amplitude from numerical model for 8 by 8 m pit and
base conditions with location of pit drawn
The maximum increase in relative amplitude is found directly in front of the pit and is on
the order of 10%. The shadow zone directly behind the pit encompasses the minimum
relative amplitude, which is also on the order of 10%. Figure 5.2 contains the plot from
Williams (1990) for the same model parameters.
..3 .
Figure 5.2: Contour plot of relative amplitude for P = 0O, kid = 7/10, k2d = '/1042,
h/d = 0.5, b/a = 1.0 and a/d = 2.0 equal to 8 by 8 m pit with base conditions; from
Williams (1990)
The two figures are in close agreement, as they should be since the same theory and
formulation were used in the development of both models.
Comparing Figures 5.3 and 5.4 provides further verification of the numerical
model. Figure 5.3 is a contour plot with the base conditions except that the pit is 8 by
24 m and the incident wave angle is now 45 degrees. Once again the alternating bands of
increased and decreased relative amplitude are found seaward of the pit, although offset
by the incident angle for this trial. As in the first trial, there are two lobes of increased
wave amplitude that border the shadow zone behind the pit. The magnitudes of the
maximum and minimum relative amplitude are seen to be larger for this trial, than for the
previous one due to the increased pit size.
40 gt&i
604.
 40 20 0 20 40 60 80
XDirection (m)
Figure 5.3: Contour plot of relative amplitude from numerical model for 8 by 24 m pit
with P = 45' and other base conditions with location of pit drawn
The results from Williams (1990) for these model parameters are seen in Figure 5.4, and
once again good agreement between the results is evident.
A thorough investigation of the effect of the pit dimensions and incident wave
conditions on the diffraction coefficient (relative amplitude) is found in McDougal et al
(1996). In this paper the method of Williams (1990) and the current numerical model are
used to determine the variation of the minimum diffraction coefficient with the non
,' .__1_0_ .. _ _"I
60 ,40 ;20 0 20 40 60 80
Figure 5.3: Contour plot of relative amplitude from numerical model for 8 by 24 m pit
with = 450 and other base conditions with location of pit drawn
dimensional values of pit width, pit depth, pit length and angle of wave incidence. The
variation in the location of a defined wave shadow zone is also determined for these four
parameters. The minimum diffraction coefficient is found to generally decrease with
increasing dimensionless pit width (a/L) and dimensionless pit depth (d/L). The
minimum diffraction coefficient is found to decrease for dimensionless pit lengths (b/L)
of 0.1 to 0.55 and then increase for values of b/L near 0.6 and then fall again as the
dimensionless pit length increases to 1.
"i
'1
Figure 5.4: Contour plot of relative amplitude for P = 45, kid = rl10, k2d = X/102,
h/d = 0.5, b/a = 3.0 and a/d = 2.0 equal to 8 by 24 m pit with 1 = 450 and other base
conditions; from Williams (1990)
Only a slight variation in the diffraction coefficient is found when varying the angle
incidence between 0 and 45 degrees.
1"1:7
: 1;
'~OJ
;'J'?r;
C. ~i;]
McDougal et al (1996) point out that the diffraction coefficients found with the
numerical models are much smaller than those found in the twodimensional cases of an
infinitely long trench as in Lee and Ayer (1981), Kirby and Dalrymple (1983) and
Furukawa (1991). A possible explanation is given that the finite with pit results in a
refraction divergence in the lee of the structure; with the divergence still present as the pit
width increases.
5.2.2 Comparison of Numerical Model with Analytic Solution Model
With confidence that the numerical model accurately determined the velocity
potential and therefore wave amplitude in the presence of a pit, it was next compared the
analytic solution model for further verification. Defining a circular pit in the numerical
model allowed for the direct comparison of results for the numerical model and the
analytical solution model. For the numerical model a pit was created with 72 points at
equal arc length spacing. Increasing this to 360 points resulted in almost no difference in
the model results. These points defined the skeleton of the circular shape and next a
certain number of equally spaced points were placed between these to more accurately
represent a circular boundary. Figure 5.5 shows the results for the numerical model of a
circular pit with a radius of 12 m, 200 points defining the pit boundary and 1600 points in
the grid.
A period of 12 s is now used as the base period in the following model results. The
center of the pit is the origin of the coordinate system and the base conditions for the
water depth, pit depth and wave criteria are used. The features seen in Figure 5.1 are
repeated with the partial standing wave in front of the pit, and the lobes of increased
amplitude behind the pit bordering the shadow zone of decreased relative amplitude. The
40
wave field alteration due to the pit in Figure 5.5 is seen to be larger than in Figure 5.1
partly due to the larger pit area of the circular pit.
,'' ,.,, /7 r. f' '" '1
100 "'. i . .... 7
100 80 60 40 20 0 20 40 60 80 100
XDirection (m)
Figure 5.5: Contour plot of relative amplitude from numerical model for circular pit with
r = 12 m, 200 points defining the pit boundary and base conditions with location of pit
drawn
The analytic solution model results for this trial are found in Figure 5.6 with the
series summation taken as the first 80 terms. The relative amplitude values are found to
be very close to those of the numerical model. The percent difference between the two
. "l. ",",. RelAmp, "na,
models results is shown in Figure 5.7 as 100 he
ReAmpnumenca,
percent error is seen to vary from 1% to 6% with the error decreasing as the number of
points defining the pit in the numerical solution is increased. These low values of percent
error indicate good agreement between the two models for this trial.
mis s n i ig
seissumtontkn ste is 8 em. h eltv apiud aue r fudt
bevr loet hoeo henmrca oe.Th ecntdfeenebtee h w
moes eulsissow n iur .7a R AMnmeialRe'M.nlyi *0. h
va LRe A rnumencal
41
" . ... .....'. ...... ..... ..
Fi . .. I
40 n "IL:
.. .. . .... ... ...... ... .,
I . \. \,
XDirection (mn
Figure 5.6: Contour plot of relative amplitude from analytic solution model for circular
pit with r = 12 m, 80 terms taken in the series summation and base conditions with
location of pit drawn
NJ, .' j' "
S. '
,,, . . . .
. '' ' ..
; ,' .' ' ' , "
,, .' ,, , . ,, >..  . .
S: v.) ;'*'. J; '
   ,
100 60 30 40 ,Io 0 20 4) 6, 00.' 100
XMDrteon (m)
Figure 5.7: Contour plot of percent difference for numerical model and analytic solution
numerical model, 80 terms taken in the series summation in the analytic solution and base
conditions with location of pit drawn
conditions with location of pit drawn
Another way of comparing the two model results is by taking a transect across the
fluid domain and comparing the relative amplitude values for the different models along
the transect. Figure 5.8 shows a representative transect taken parallel to the xaxis for the
pit with a radius of 12 m. The spacing along the transect is defined and the velocity
potential and therefore free surface elevation can be determined at each point along the
transect.
10
15
150 100
50 0
XDirection (m)
50 100 150 200
Figure 5.8: Plot of transect taken parallel to the xaxis at y = 0 for pit of radius 12 m
showing wave direction (3 = 0)
A comparison of the relative amplitude for the previous case is shown in Figure
5.9 for the transect in Figure 5.8. The results of the numerical model and the analytic
solution model are found to be in good agreement with the numerical model values
 1   ;  
0
50 Wave Direction
0,
0
>o^ ___ ^ ___ _ _^l....~ ^ ___ ,.._ ___ I ._ _i._ 
200
slightly higher than those for the analytic solution. The numerical model produces
scattered values for points very near the pit boundary, due to the singularity that occurs
there in the solution method.
1 3 ..................... r .................... ........... .... ..................... I ... .. ..... ............. r .........................................
1.2
d solid=Analytic Model
n mi ,d a 8 in dashed = Numerical Model
l i i j pi
t J'\ \J \l I
0.7
0I
200 150 100 50 0 50 100 150 200
XDirection (m)
Figure 5.9: Relative amplitude for numerical model and analytic solution model for
transect show in Figure 5.8 for pit of radius = 12 m with 200 points defining the pit
boundary in the numerical model and 80 terms taken in the series summation for analytic
solution with pit drawn
The models are further compared in Figure 10 with 3 transects taken
perpendicular to the direction of wave propagation at X equal to 0, 24 and 100 m. The
relative amplitude values for the two models are very similar along all 3 transects with
the numerical model having noticeably smaller values directly behind the pit for the X =
0 m transect and also to a lesser extent for the X =100 m transect. In this plot the primary
44
lobes of increased wave amplitude are seen to spread and increase as x increases with the
oscillations of increased and decreased relative amplitude growing smaller as the y
distance from the pit increases.
X = 0 (m)
300 ..... .. ..............
Pit
100
200
100 o .
300
0.8 1 1.2
Relative Amplitude
X = 24 (m)
300 r............ .... . 
200
100
0
200
.1'
R;t
300
0.8 1 1.2
Relative Amplitude
X = 100 (m)
300 ... ................ ..................
)
/
200
100 "
100
200
300
0.8 1 1.2
Relative Amplitude
Figure 5.10: Relative amplitude for numerical model and analytic solution model for 3
transects parallel to the yaxis at X = 0, 24 and 100 m for pit of radius = 12 m with 200
points defining the pit boundary in the numerical model and 80 terms taken in the series
summation for analytic solution with pit drawn
The reduced amplitude behind the pit is seen to decrease as the distance behind the pit
increases.
With the good agreement between the models for a representative small pit, a
second trial was conducted with a pit radius of 75 m to better represent the size of a pit
found in the coastal environment. The wave field modification caused by a pit of this
size with the predefined base conditions is seen in Figure 5.11 with the summation of the
first 110 terms taken in the analytic solution. The magnitudes of the maximum and
minimum relative amplitudes are 1.71 and 0.058 respectively. The relative amplitude
increase and reduction are substantially larger than those found with a circular pit of 12 m
radius.
o 00,' \ .' ,* *
I...I, ... ,,,. ,
,.ri
1 ,! I'.: * '. f' ";, '" ." ,," "".. ,,' 
200.1;1 .. 1 .
S .( .. .'
) " '" ..
'" '".i "' " I" ."'.'', :i ."'.:
100 \ o,, ,' i, / ..
.r ; 
'" 1' I '"" '":
., ',.. ,"ljJ
200 100 0 100 200 300
XDirection (m)
Figure 5.11: Contour plot of relative amplitude from analytic solution model for circular
pit with r = 75 m, 110 terms taken in the series summation and base conditions with pit
drawn
The maximum relative amplitudes are found in the lobes of increased amplitude behind
the pit. The extremely low values of relative amplitude are found just inside these lobes
in the shadow region behind the pit where an area with relative amplitude values less than
0.2 occurs. The partial standing wave pattern that develops in front of the pit has
oscillations of large magnitude, but they are not strongly evident in the contour plot due
to the even larger values occurring behind the pit. They will be shown later through
transect plots.
The contour plot of the numerical model results with 600 points defining the pit is
similar to that in Figure 5.11. The increased number of points on the pit was used to
allow for spacing between points on the pit that was similar, although larger, to that in the
previous trial of 200 points defining a pit with a radius of 12 m.
Once again, the relative difference was calculated to compare the values from the
two models at each grid point; this can be found in Figure 5.12.
... ....300.. .
3 o    * * 
9 32
,100
.
300
300 200 100 0 100 200
XDlrectlon
300
Figure 5.12: Contour plot of percent difference for numerical model and analytic solution
model for circular pit with r = 75 m, 600 points defining the pit boundary in the
numerical model, 110 terms taken in the series summation in the analytic solution and
base conditions with pit drawn
1'
100
100
9O 3'
The most noticeable feature of this plot is the large percent error values occurring on the
edge of the shadow zone behind the pit. At one of these two points the value of the
relative amplitude from the analytic model is 0.058, while for the numerical model it is
0.029, which results in a percent error over 100%; however, the large error is due to
division by a very small number (0.058) and the validity of the value there is
questionable. For most of the plot the percent error is found to be less than 5% and only
16 points in the grid of 1600 points show a percent error great than 10%, which indicates
good agreement between the two models.
As shown before, transects through the fluid domain are another tool useful in
comparing the results from the two models. A transect of relative amplitude values taken
parallel to the xaxis is shown in Figure 5.13.
1.2   "  .r .............,.r .. .........  .... .... I ...................
0.8
S0.7 
0. 4'. .. ..... .... ..
Sv. ii i I
400 300 200 100 0 100 200 300
X Direction (m)
Figure 5.13: Relative amplitude for numerical model and analytic solution model for
transect parallel to the xaxis at y = 0 m for pit of radius = 75 m with 600 points defining
the pit boundary in the numerical model and 100 terms taken in the series summation for
analytic solution with pit drawn
The numerical model solution is seen to result in smaller values in front of the pit that are
slightly out of phase with the analytic model, larger values inside the pit, and good
agreement between the models behind the pit. Two noticeable spikes in the numerical
solution are seen at the pit border and occur due to the proximity of the point to the pit
boundary where a singularity occurs. The large size of the pit leads to waves forming
inside the pit where the relative amplitude is less than 1, but large oscillations are seen.
Taking a transect parallel to the yaxis located just behind the pit at X equal to
100 m results in Figure 5.14.
1.2
0.8e
wL 0.8
400 300 200 100 0
Y Direction (m)
100 200 300 400 500
Figure 5.14: Relative amplitude for numerical model and analytic solution model for
transect parallel to the yaxis at X = 100 m for pit of radius = 75 m with 600 points
defining the pit boundary in the numerical model and 100 terms taken in the series
summation for analytic solution with pit drawn
f It
if I
U: U
ft~ ~ '4
'S
Ui
.4
50o
0
  I ____I __ ; __ :  I i  
This plot shows good agreement between the models and highlights the large gradient in
the relative wave amplitude occurring behind the pit where the lobes of increased
amplitude border the shadow zone. The shadow zone shows two spikes of very low wave
amplitude ( 0.2) bordering the shadow zone where relative amplitudes of 0.6 are seen.
The rapid decrease in the oscillations of the relative amplitude with distance from the pit
is also noticeable.
Taking a transect at a greater distance behind the pit shows the spreading of the
lobes on increased wave amplitude concurrent with spreading in the wavelength of the
oscillations of relative amplitude. Figure 5.15 shows a transect taken parallel to the y
axis at X equal to 500 m, with almost no difference noticeable between the two models.
Si~
0
Y Direction (m)
Figure 5.15: Relative amplitude for numerical model and analytic solution model for
transect parallel to the yaxis at X = 500 m for pit of radius = 75 m with 600 points
defining the pit boundary in the numerical model and 100 terms taken in the series
summation for analytic solution with pit drawn
A decrease in the maximum and minimum values of the relative amplitude are seen in
this transect when compared to Figure 5.14.
5.2.3 Results for Models with Solid Cylinder
The wave field modification by a solid structure was analyzed by employing the
solution of MacCamy and Fuchs (1954) for the diffraction caused by a circular cylinder.
This model is similar to that for the analytic solution model for the circular pit, except
with a noflow constraint for the boundary condition. Figure 5.16 shows a contour plot
of the relative amplitude for a cylinder with a radius of 12 m and the base conditions
without a pit depth, d.
20 0 20
x Direction (m)
Figure 5.16: Contour plot of relative amplitude from MacCamy and Fuchs solution model
for a circular cylinder with r = 12 m, 80 terms taken in the series summation and base
conditions with pit drawn
A partial standing wave developed in front of the cylinder, as occurred for the case of the
pit, however these oscillations are 90 degrees out of phase with that of a pit. The contour
pattern behind the cylinder is seen to differ from the case of the pit as the areas of
minimum relative amplitude are not directed out from the pit along the xaxis, but project
out at an angle. The maximum relative amplitude shows a 60% increase in the wave
amplitude in front of the cylinder and the minimum relative amplitude indicates a 40%
reduction both behind the pit and in the first trough in front of the pit. These relative
amplitude values are much larger than those found for a pit of the same radius.
By taking a noflow boundary condition in the numerical model, a solid structure
numerical model was developed. The results of this model for the previous case compare
very well with those of the MacCamy and Fuchs solution. The percent difference
between these two models for this trial can be seen in Figure 5.17. In this numerical
model, 200 points define the cylinder boundary. The contour plots shows less than 1%
error for each point in the grid with the largest errors occurring directly behind the pit and
in two lobes of increased amplitude projecting out behind the pit.
Transects taken through the fluid domain compare the values obtained by the two
models and illustrate the effect of the cylinder on the wave field. Figure 5.18 shows the
relative amplitude for the two models for a transect taken parallel to the xaxis. Included
in this plot are the relative amplitude values for the analytic solution for a circular pit of
the same radius, seen in Figure 5.9. The results of the two cylinder models are so close
that they appear as a single line except for the numerical model values directly in front of
and behind the pit. The results show a large value of relative amplitude directly in front
of the cylinder and a reduction in relative amplitude, which diminishes rapidly behind the
iOO.. *>y.^.... . .......... ....... ., ..
00 .' ,
'30 '..'/" '
30 i '/ /' :" *'*" "".: .. i
0 :
100 80 60 40 20 0 20 4 60 80 100
,h . 
', '. ..22.
01
40.
S..0 15
C" 26 '"""^^ "
.i 0. 2
100
100 60 40 2 0 N0 40 60 80 15 2
x Direction (in)
Figure 5.17: Contour plot of percent from from solid numerical model and MacCamy and
Fuchs solution model for a circular cylinder with r = 12 m, 200 points defining the pit
boundary, 80 terms taken in the series summation and base conditions with pit drawn
  .................... " ' ............ . ........r..................
Dashed = Circular Pit
.... ........... ... ........ . .. ......
X Direction nm)
Figure 5.18: Relative amplitude for solid numerical model and for MacCamy and Fuchs
solution model for transect parallel to the xaxis at y = 0 m for pit of radius = 12 m with
200 points defining the pit boundary in the numerical model and 80 terms taken in the
series summation for analytic solution with analytic solution model for circular pit shown
pit. The plot shows the phase difference between the case of a pit and that of a cylinder.
The pit is seen to result in a greater sheltering effect directly behind the pit. The large
value right at the cylinder boundary due to the noflow condition is contrasted by the
reduction in wave amplitude found at this location for the pit.
A transect taken parallel to the yaxis at x equal to 100 meters is shown in Figure
5.19. Once again the values of the two cylinder models are so close they appear as one
line.
1,2 ............................ ............................ r ................... .... ........ ... F ............................ r ............................
1.2
solid = Solid Cylinder
1.15 dashed = Circular Pit
300 200 100 0 0 0 i300
I i j I I.. i
.Y Direction (m)
1.05 I! j \i i I /
with 200 points defining the pit boundary in the numerical model and 80 terms taken in
SI l n w alc sl n l f l p
the same radius shown and with pit drawn
0,85 l / I, U ,i
v : ii 4 \ i \
\ 1 i 1
300 200 100 0 100 200 300
Y Direction (m)
Figure 5.19: Relative amplitude for solid numerical model and for MacCamy and Fuchs
solution model for transect parallel to the yaxis at x = 100 m for pit of radius = 12 m
with 200 points defining the pit boundary in the numerical model and 80 terms taken in
the series summation for analytic solution with analytic solution model for circular pit of
the same radius shown and with pit drawn
To further compare the solution to that of a pit of the same radius, the solution of the
analytic model is also included (Figure 5.10). This plot shows that the minimum relative
amplitude for the case of a cylinder does not lie directly behind the pit, but in two areas
project out behind the pit. The sheltering in these areas is found to be greater than in the
case of the circular pit. The relative amplitude oscillations are found to die off much
more quickly for a pit.
5.2.4 Plots of Wave Fronts in Fluid Domain
Another way to consider the wave modification caused by a pit is to plot the wave
fronts as lines of constant phase. The complex value of the velocity potential was used to
determine the phase at each point in the grid and from this a contour plot of phase was
created. Figure 5.20 shows the contour line wave fronts for the case of a pit with a radius
of 12 m and the base conditions using the numerical model results, which are very close
to those of the analytic solution model.
100 80 60 40 20 0 20 40 60 80 100
40 j.. 7
X Dirction m I
Figure 5.20: Contour plot showing wave fronts as lines of constant phase from analytic
100 80 60 40 z2. 20 40 60 80 It~,
XDir~erlion ,mi
Figure 5.20: Contour plot showing wave fronts as lines of constant phase from analytic
solution model for circular pit with r = 12 m, 80 terms taken in the series summation base
conditions with pit drawn
The plot shows lines of constant phase, which illustrate the wave fronts at a point in time.
The phase was calculated as e= atan (lma The asymptotes of the tangent function
I real 2
cause the closely spaced contour lines observed in the figure. The other contour lines
indicate the wave fronts and clearly show an increase in wave speed as the front travels
through the pit, which results in a divergence of the wave front at the rear of the pit.
Further behind the pit this "bulge" is seen to spread laterally, which at large distances
directly behind the pit results in a concave shape; a result verified in the shoreline change
model.
The contour plot of phase associated with a solid cylinder is shown in Figure 5.21.
NUMERICAL contour plot of phase
20 0 21
X Direction (m)
2
Figure 5.21: Contour plot of showing wave fronts as lines of constant phase from
MacCamy and Fuchs solution model for circular pit with r = 12 m, 80 terms taken in the
series summation base conditions with pit drawn
This plot of the results of the solid numerical model shows wave crests that diffract
around the cylinder, as would be expected. A greater disturbance to the entire wave field
is seen in this plot as compared to Figure 5.20.
5.2.5 Model Sensitivity
The numerical model is based on the assumption that the pit can be represented by
a number of equally spaced points defining its boundary. The value of 0 and 0 are
b&
determined at each point and using these values, the velocity potential can be determined
at any point in the fluid domain. Increasing the number of points that define the pit
should result in more accurate results for the model, as a better representation of the pit
should result. Figure 5.22 shows the value of the maximum and minimum relative
amplitudes for the parameters used in Figure 5.5 with 900 points in the fluid domain.
1Z rS1 ^ I;r
.2 
0.8
0 .7 i" 
dotwith analytic model solutionalytic solufor same conditions drawtion
0 5 ......... ... .......... .. ..... ..... ....... .......... ....... .............. : ...... ... j ............... .............. ..............
0 100 200 300 400 500 600 700 800 900 1000 1100
Number of Points Defining Pit Boundary
Figure 5.22: Plot of numerical model values of maximum and minimum relative
amplitude versus the number of points defining the pit boundary for a grid of points
defining the fluid domain with a circular pit of radius 12 m and other base conditions
with analytic model solution for same conditions drawn.
The values are seen to approach a near constant value after the number of points
on the pit is 200. Increasing the number of points on the pit boundary from 200 to 1000
results in a 0.4% change in maximum relative amplitude for this case; 1.1555 to 1.151.
For a pit with a 12 m radius, 200 points on the boundary results in a spacing of 0.38 m
between the points on the pit.
The analytic solution model and the MacCamy and Fuchs model are based on a
series summation to obtain accurate results. The relative amplitude values found with the
analytic solution for several points are plotted against the number of terms taken in
Figure 5.23.
1.3 ..................... .................... .................... 1 ..................... .... . . . .......... ..................... .................... ....................
08h
20 40 60 80 100
Terms Taken in Series Summation
120 140 160
Figure 5.23: Plot of analytic solution model values of relative amplitude versus the
number terms taken in the series summation for five points on the xaxis with a circular
pit of radius 12 m and other base conditions
i ' \
i \ (100.00)
,", ' ' (800.0)
(300,0)
....... ........... .. .._... .. ........ .... .... .. ........ .... (150.0)
/
___
The relative amplitude is seen to become constant for each distance from the pit with less
than 140 terms taken in the summation, except for 1000 m. At 140 terms taken in the
summation, a value of NAN or notanumber was returned by the MATLAB program for
unknown reasons. The number of summation terms needed to achieve a constant relative
amplitude value (within 5 decimal places) is seen to increase as the distance from the pit
increases. The radius was not found to influence this value, but decreasing the period
was found to result in more terms required in the summation to achieve a constant value.
The same results were found for the MacCamy and Fuchs model. In each analytic model
trial, the limit used was one that ensured the value of the velocity potential had reached a
constant value.
This means that in the models run, the analytic solution results in a constant
relative amplitude value at each point, but increasing the number of points on the pit can
change the numerical solution values. Figure 5.24 compares the relative amplitude
values found along two sections for a transect taken parallel to the yaxis at x equal to
100 m. The base conditions were used with a pit radius of 12 m. The effect of increasing
the number of points on the pit boundary is seen to result in numerical model results that
approach those of the analytical solution model. The top plot in the figure is located at
the first peak of increased amplitude outside the pit and shows numerical model values
that are larger than those for the analytic solution. The bottom plot is for the section
directly behind the pit and shows numerical model values less than the analytic values.
In both figures a less than 3% change in relative amplitude is observed by increasing the
number of points on the pit from 100 to 1000.
59
1.17  
1.165 nump = 100
...........................
S... nump= 200
1. 155
S." n...ump = 1000
1.145 '.. .... ,, ..
.+'' += Analytic Solution :.
1.174 72 70 68 66 64 62 6
74 72 70 68 66 64 62 6
Y Direction (m)
0.84F r .  r  r
+= nalytic Solution
0.83 ... ...
S ... ".. .... .nump = 100
S0. ....82. . nump 4 mp=
0.2 i. .. .. nunmp= 200
o .. ......... .
8 7 6 5 4
Y Direction (m)
0
3 2 1 0
Figure 5.24: Plot of numerical model values of relative amplitude for two sections of a
transect parallel to the yaxis at x equal 100 m for five different numbers of points
defining the pit boundary for a circular pit of radius 12 m and other base conditions with
the analytic solution model results for the same conditions drawn.
5.3 Energy Reflection Caused by Pit
5.3.1 Analytic Solution Model
The amount of energy reflected by a pit was determined using the timeaveraged
energy flux through a transect parallel to the yaxis (shorenormal wave fronts). Due to
the symmetry of the problem for a symmetric pit with normally incident waves, a transect
over only one side was used as can be seen in Figure 5.25. At each point along the
transect the four energy flux terms described in Section 3.2.2 were calculated. The
amount of energy reflected was determined as the difference between the amount incident
on the transect and the amount passing through the transect; the sum of the four energy
flux terms.
wave direction
 *
................ ................... t ..................................... ..............
300 200 100 0
X Direction (m)
100 200 300 400
Figure 5.25: Plot of transect taken parallel to the yaxis for energy flux calculations with
pit of radius 12 m showing wave direction
Dividing this difference by the energy incident on half of the pit and taking the square
root resulted in a reflection coefficient.
The analytic solution with its exact, separate equation for the incident and
reflected potentials provided for a more direct approach to the reflection problem. Using
the equations shown in Section 4.3.2 the four terms of energy flux were determined at
each transect point. Figure 5.26 shows the four terms for the representative transect
shown in Figure 5.25.
1 0 0 ................... .................. ............... : ................... .................. ................... .................. ....................
1 00
It
100F
9200
6300
>
400
500
600
40(
0
The first term consisting of the incident pressure and xdirected velocity, PiUi*, is
constant at all points until the boundary of the pit is reached, as would be expected. The
PiUr* term is seen to have a large negative value just outside the pit and then oscillate
with quickly diminishing amplitude as the distance from the pit increases.
3000  ....r... ........... .. ....
S.......................... ........................... ............................. ........
S2000
1000
LU
600 500 400 300 200 100 0
O.............. .............................................. ....... ................................... .............................
0 0 0 .......... .... ................ .
I Soo \
1000 
1000 ......... L ........... .......................................... L ...........................
600 500 400 300 200 100 0
200 ............... ...............J.. .. T.  .... ..
200:
2 0 0 ............................ ............................ I............................ ............................: ............................ ...................... ..
shown in Figure 5.25 with base conditions and 90 points taken in the summation
600 500 400 300 200 100 0
2 0 0 0 ; ............................ ............................ I ............................ ............................ ............................ ............................
a 000:
600 500 400 300 200 100 0
Y Direction (m)
Figure 5.26: The four energy flux terms from the analytical solution model for transect
shown in Figure 5.25 with base conditions and 90 points taken in the summation
This negative value near the pit indicates reflected energy caused by the pit. The PrUi
term follows the same form as PiUr*, except with a smaller negative value at the pit
boundary and oscillations that do not dampen as quickly. The PrUr* term has a small
value until very near the pit boundary where it rises steadily until inside the pit where it is
the only term and shows more energy passing through the pit near the sides than at the
62
center. This term shows energy passing through the pit, which is largest near the edge of
the pit and decreases towards the center.
Summing these four terms at each transect point results in Figure 5.27, which
shows the total energy flux through the transect at each point, along with the constant
incident value.
3000:
2800 j I
2600[ v
2400
2200
82000
S1800 
1600 .
1400 
1200 ........................................................ ..... ....................... ............... .......................................... ..... ....................
600 500 400 300 200 100 0
Y Direction (m)
Figure 5.27: Energy flux at each point in transect shown in Figure 5.25 with incident
value (dotted) for base conditions and 90 terms taken in the series summation
It is clear that the energy flux oscillates as the distance from the pit increases. This
results in a fluctuating energy sum for the transect depending on where the transect ends.
Ideally, the transect would extend out far enough to where the fluctuations reduce to zero,
but the computing requirement for this limit is not practical for these models. For these
energy models, the transect was taken out to y equal to 600 m with a value taken every
meter. To solve the fluctuating energy problem, an average over 1 oscillation was made
to determine the reflection coefficient for that oscillation in the transect. By comparing
the reflection coefficients found by ending the transect at each upcrossing along an entire
transect out to y equal to 600 m it was found that a stable value was reached by the end
of the transect, and therefore 600 m was a long enough transect. Figure 5.28 shows the
reflection coefficients for the transect shown in Figure 5.25 with the energy flux shown in
Figure 5.27. The reflection coefficients is determined as the average over one oscillation
in the energy flux from each upcrossing, with 1 being closest to the pit.
0.26

0.259 i
0.258
0.257 .
0
* 0.256 
o 0.255
S0.254
0.253
0.252
0.251
A ..
+* 
0 1 2 3 4 5 6
Uperossing Number
+ +
1 +
+ T
7 8 9 10 11
Figure 5.28: Reflection coefficients determined as the average over one energy flux
oscillation for each upcrossing in the transect
.............J............. ..... ..................................................
64
The reflections coefficients are seen to reach an almost steady value as the distance from
the pit increases.
Next, the location for the transect was investigated. Figure 5.29 shows reflection
coefficient values for transects taken at different x locations; in front of, through and
behind the pit.
0.2
0.2
0.2
0.2
0
o
,0.2
n.
0.2
CM
Analytic Solution (h=2.d=4,a=12) for X=0(+) X=50(*) X=100(o) X=200(d) X=100(x)
12 13 14 15 16
Period [s]
17 18 19 20 21
Figure 5.29: Reflection coefficients for different transect locations with several incident
wave periods for the transect shown in Figure 5.25 taking the reflection coefficient as the
average of the last energy flux oscillation in the transect
Different period waves were used for a pit of radius 12 m with the other base conditions.
The reflection coefficients are seen to not depend on where the transect is taken. It
i
+ (X 0)
;5 
(X = 50)
o (X = 100)
4 o (X = 200)
x (X=100)
3.
:2
1 ................ ..... ..........
a
11
should be noted that at larger periods, fewer oscillations in the energy flux value occur
before y equals 600 m and therefore these values might be slightly less accurate.
A series of trials was made to determine how the reflection coefficient varied with
the pit diameter and incident wavelength. Figure 5.30 shows how the reflection
coefficient varies with D/L(h) and D/L(d) the wavelength outside and inside the pit,
respectively.
0
0.26
0.24
0.22 *
0,2
0 ",
U 0.18
0.14
0.12
0.14
c 
0.08
0.06
0
Pit DiameterWVavelength(inside pit.d)
0.2 0.4 0.6 0.8 1 1.2 1.4
1.6 1.8 2
1 1.5 2
Pit Diameter/Wavelength(outside pith)
Figure 5.30: Reflection coefficients versus pit diameter divided by wavelength outside
the pit and pit diameter divided by wavelength inside the pit for different pit radii for the
transect shown in Figure 5.25 taking the reflection coefficient as the average of the last
energy flux oscillation in the transect
This figure indicates that the energy flux is highly dependent upon the pit diameter to
wavelength ratios. A maximum reflection coefficient of 0.257 is seen at values of 0.5

+ (radius = 6 m)
o o (radius = 12 m)
(radius = 25 m)
x (radius = 30 m)
(radius = 75 m)
,;:
+,
<)
4
* 4. <
f
**
4,
and 0.3 for the diameter over the wavelength outside the pit and diameter over the
wavelength inside the pit, respectively. The minimum value of the reflection coefficient
shown is 0.07, but this will approach zero as the pit size approaches zero. The
oscillations in the relative amplitude have a period of D/L of approximately 0.8 outside
the pit and approximately 0.6 inside the pit for these base conditions of pit depth equal to
twice the water depth.
For the same incident wave conditions, a deeper pit should result in greater
reflection coefficients than a shallower pit of the same size. This is shown in Figure 5.31,
which is plot of the reflection coefficient versus the pit depth for pits of radius equal to 12
m and 25 m with the other base conditions and the pit depths ranging from 3 m to 12 m.
0.55
0.5
0.45
0.4
"z 0.35
0
o 0.3
I 0.25
0.2
0.15
0.1
0.05
1 0.2 0.3 0.4 0.5 0. (
Water Depth (h) / Pit Depth (d)
Figure 5.31: Reflection coefficients versus water depth divided by pit depth for different
pit radii for the transect shown in Figure 5.25
+ + (radius= 12 m)
o (radius = 25 m)
0
\
**J,
The reflection coefficient values for the pit of radius equal to 12 m are seen to approach
those of the 25 m for the larger values of pit depth.
Another way of examining the influence of the pit depth on the reflection
coefficient is shown in Figure 5.32, which shows the reflection coefficient for two
different pit depths over a range of pit diameter divided by wavelength values.
0.45
0.35
0.4
t5 0251
o.
02
0.15
0.05.
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7
Pit Diameter I Wavelength(inside pit.d)
Figure 5.32: Reflection coefficients versus pit diameter divided by wavelength inside pit
with different pit radii for a pit of radius equal to 12 m and the transect shown in Figure
5.25
D
The maximum reflection coefficient values are seen to occur for the same values
L(d)
with the deeper pit having the higher reflection coefficient.
I:'
,
i.
: 
s
o (radius = 12 m)
+ + (radius = 25 m)
0 water depth / pit depth = 0.25
water depth / pit depth = 0.5
5.3.2 Numerical Model
Determining the reflection coefficient for a pit using the numerical model solution
is not as straight forward as using the analytic solution model. The numerical model
presents some bookkeeping problems for transects through the pit due to the
perturbations in the velocity potential values around the singularity at the pit boundary.
As seen with the analytic solution in Figure 5.29, the reflection coefficient value is
independent of where the transect is taken, and this holds for the numerical solution
allowing the transect to be taken outside the pit at x equal to 100 m.
The number of points defining the pit boundary was found to have an effect on
the reflection coefficient values determined by the numerical model; as was found for the
relative amplitude values. This presents a larger problem than the singularity found at the
pit boundary. Figure 5.33 shows the energy flux value determined as the average value
from the last upcrossing versus the number of points defining the pit boundary for a pit
radius of 12 m with the other base conditions. The reflection coefficient value from the
analytic solution model for this case has been included in the figure. At first, the
numerical model values rise steadily as the number of point's increases, but then the
increase in relative amplitude from increasing the number of points diminishes rapidly.
With 1000 points on the pit boundary, a spacing of 0.075 m between the points on the pit,
the numerical solution is still around 9 percent lower than the analytic solution value.
When modeling a pit with a much larger radius, the number of points defining the
boundary needed to keep this small spacing leads to a very large computing requirement.
Figure 5.34 shows a comparison for the results of the numerical and analytic
solutions for a pit of radius equal to 12 m, 500 points defining the pit boundary in the
numerical model and the other base conditions for a transect taken at x equal to 100 m
and 500 points defining the pit boundary in the numerical solution.
0.24
0.22
0.16 F
100 200 300 400 500 600 700 800
Number of Points Defining Pit Boundary
900 1000 1100
Figure 5.33: Reflection coefficients versus the number of points defining the pit
boundary for the numerical model with a pit of radius equal to 12 m, a period of 12 s and
the other base conditions with the analytic solution model result shown
To try and identify where the difference in the energy flux values from the two
models occurred, the relative amplitude and energy flux at each point in a transect were
compared. The percent difference in the relative amplitude and energy flux energy were
determined at each transect point and plotted, where the percent difference is equal to
analtyic numerical *10
analytic
I  ........... .r .. ..
analytic solution
+
I

U.b
0.4
0.
0 '.2 L 
600 500 400 300 200 100 0
0.4
0.2
0.2 J \/ \
0.4
600 500 400 300 200 100 0
2 r\
^VvVV W\ \ 
10 i I
600 500 400 300 200 100 0
Y Direction (m)
Figure 5.34: Percent difference in relative amplitude, percent difference in energy flux
values and difference in energy flux values for analytic solution model and numerical
model for a pit of radius equal to 12 m, 500 points on the pit boundary in the numerical
model, and the other base conditions along a transect at x equal to 100 m
The percent difference found for the relative amplitude and energy flux are seen to be
less than 0.5 % with the largest error occurring behind the pit. Close examination of the
energy flux errors shows a slight skewness in the error with the negative values larger
than the positive values in the oscillations occurring as the distance from the pit
increases. These larger negative values, along with the area directly behind the pit lead to
a larger value for the total energy flux through the transect for the numerical model, and
therefore a lower reflection coefficient. These small values in percent error are shown in
another way as differences in the energy flux at each location (analytical numerical).
These values are all less than 10 ns/m2, but when multiplied by the distance along the
transect, the total difference can become significant.
For a 600 m transect with a pit of 12 m radius with a 12 s period incident wave
and the base conditions, the analytic solution model gives an incident energy flux of
1.6295*106 ns/m2 through the transect, which with the reflection coefficient of 0.255
found in Figure 5.30, equates to an energy flux passing through the transect of
1.6252*106 ns/m2 or a difference of only 4230 ns/m2. This difference in very small
when compared the sum of the incident and actual values of energy flux along the entire
transect. If a very large number of points is not used to define the pit boundary, this error
can result in the numerical model producing results with energy flux values through the
transect larger than those incident on the transect; a 'negative' reflection coefficient.
Even with a large number of points defining the pit boundary the numerical model
does not produce results the match well with those of the analytic solution model. Figure
5.35 shows the reflection coefficients from the numerical and analytic models versus the
pit diameter divided the wavelength outside of the pit. Results were obtained using four
different pit sizes all with the base conditions for pit depth, water depth and angle of
incidence. For the numerical model, the pits of radius equal to 6, 12, 25, and 30 m where
defined by 200, 400, 800, and 900 points on the pit boundary, respectively. For four
trials, two each of pit radius equal to 25 m and 30 m, the sum of the energy flux through
the transect was found to be greater than the incident sum through the transect, resulting
in a zero value for the reflection coefficient. Even though the spacing between the points
on the pit boundary were similar, the reflection coefficient values do not mesh together
72
for the different radius values with the same dimensionless diameter, as they do in the
analytic solution results.
analytic solution model
o X
o 0
0
ei
+
+

t
4
r,
*
+ (radius = 6m)
o (radius = 12 m)
* (radius = 25 m)
x (radius = 30 m)
v *
x
j numerical model
(larger symbols)
*
0.4 0.6 0.8
Pit Diameter/Wavelength(outside pit,h)
Figure 5.35: Reflection coefficients versus the pit diameter divided by the wavelength
outside the pit for the numerical model for different pit radii along a transect at x equal to
100 m and the other base conditions with the analytic solution model results shown
This suggests that when radius in the numerical model is increased, increasing the
spacing on the pit by the same factor does not result in the same accuracy when compared
to the analytical results. Dependencies on the pit dimensions, and incident wave field
characteristics must be involved.
5.3.3 MacCamy and Fuchs Solution Model for Solid Cylinder
The energy reflected by a solid cylinder was investigated using the MacCamy and
Fuchs solution. This model was used to compare the wave field changes caused by a
0.25 P
0.05 
0.2
solid cylinder to those caused by pit in Section 5.2.3. Using the same procedure for the
determining the energy flux through a transect as was previously discussed the reflection
coefficient was determined for solid cylinders of different radius (Figure 5.36).
0.9 ................................................................ ........................ .....................
0.85 F
.* 4:. "
. X I
4
) *+
+*
0; **
o (radius= 12 m)
^ (radius= 25 m)
x (radius = 50 m)
+ (radius = 75 m)
0 0.5 1 1.5 2 2.5 3
Cylinder Diameter/Wavelength(water depth,h)
Figure 5.36: Reflection coefficients versus the cylinder diameter divided by the
wavelength outside the pit with the MacCamy and Fuchs solution model for different pit
radii along a transect at x equal to 0 m and the other base conditions
The reflection coefficient values are much larger than those found for a pit of the same
dimensionless diameter, with a peak value for the cylinder of 0.85 and for the pit of
0.257. The maximum value of the reflection coefficient for the cylinder is seen to occur
once at a dimensionless diameter of 0.35 and then again as the dimensionless diameter
increases near 3. Slight oscillations in the reflection coefficient for the cylinder are
apparent, but they are small as compared to the oscillations seen in Figure 5.30 for the
pit. The large oscillations of the relative amplitude in the pit can be explained by
resonance in the reflected waves occurring for certain dimensionless diameters, where
this cannot happen for the case of a cylinder.
5.4 Shoreline Change Induced by Pit
5.4.1 Comparison of Model Results
The preceding sections have documented the wave field modification caused by a
pit, as well as quantified energy reflection by a pit. These changes to the incident wave
field can impact the shoreline and result in changes there as well. This section presents
the shoreline change induced by an offshore pit.
In these models a transect parallel to the yaxis was considered to represent the
shoreline. The models used to determine the energy flux were extended to calculate the
wave direction and relative amplitude at each point along the transect. The wave direction
(EFlux\
was determined as a = tan (Eq. 3.19). Knowing the shoreline orientation
and the values of wave direction and height at each location allowed for the longshore
transport to be calculated with the CERC longshore transport equation with both transport
terms, Eq. 3.34. The changes in shoreline position were determined using these values of
longshore transport in the continuity equation (Eq. 3.35). The values of the coefficients
and constants in these equations that were used in the models are listed in Table 5.1. The
assumptions used in these models are detailed in Section 3.2.3.
For these models a situation more representative of the coastal environment was
chosen. To compare the numerical and analytic models a circular pit was required, and a
radius of 50 m was used. The pit is 8 m deep and located in 4 m of water. The transect
was taken 1500 m behind the pit and the transect representing the shore extended 1600 m
in the longshore direction with a point every 5 m.
Table 5.1: Coefficients and constants used in the shoreline change models
K1= 0.77 K2 = 0.77 K= 0.78 s = 2.65
P = 0.5 tan(y) = 0.05 h* + B = 6 (m) At = 300 s
Incident wave heights of 1 m were used with periods of 10, 12 and 14 s. These wave
periods lead to h/L values that meet the shallow water condition of h/L less than 1/10.
The shoreline change models were exercised for monochromatic waves with one
direction resulting in changes in the shoreline position that oscillated with large
amplitude to great distances from the pit. This effect is interpreted as the result of the
diffraction pattern. This resulted in shoreline changes that were unrealistic for the coastal
environment where waves of different period and different incident angle impact the
shore. To account for this, an averaging procedure was employed where the models were
run for incident angles from 10 to 100 every 20 and for 3 different periods. By taking
the average of the relative amplitude, longshore transport, and shoreline change values
over the different incident wave angles and then averaging those values for the 3 different
periods, a better representation of a typical nearshore situation was made.
The numerical model for this trial had 400 points defining the pit boundary. The
relative amplitude for the situation described above is found in Figure 5.37 with the axes
oriented in the same manner as the transects that were presented for the energy flux with
the waves approaching from the left.
1.4 1.2 1 0.8
Relative Amplitude
0.6 0.4 0.2
Figure 5.37: Relative Amplitude along the transect for the numerical model with a pit of
radius equal to 50 m, at x = 1500 m with 400 points defining the pit boundary and the
other base conditions for this shoreline change model
This figure shows the relative amplitude for the case of shore normal waves with a period
of 12 s (dashed line), the average relative amplitude of the 11 wave directions for a
period of 12 s (dotted line), and the average of the average relative amplitude of the 11
wave directions for periods of 10, 12, and 14 s (solid line). The values for the single
wave direction and period are seen to oscillate very far from the pit while the averaged
values do not. Further averaging over the wave direction for each period was found to
200 .
400
600
16
.. . .. ..
< ' .
' .. . . i..
solid (avg of wave direction and 3 periods)
Spotted (avg of wave direction for T = 12 s)
S dashed (value for beta = deg and T = 12
/
MII'     r..T........
U'~" 
)
not change the values near the pit, but slightly reduced the oscillations far from the pit.
Ideally a large number of periods and wave directions would be used to approximate the
coastal environment, resulting in even more smoothing of the oscillations; however due
to the computational requirements this problem presented, an average over the wave
direction for only one period was deemed satisfactory. The averaged values over the
wave direction are seen to greatly reduce the maximum relative amplitude compared to
the single direction and period, but only slightly reduce the minimum value. This is due
to the larger values of wave direction moving the shadow zone from directly behind the
pit to areas alongshore where the largest relative amplitudes were found for the more
shorenormal waves; thereby reducing the average value at these locations.
The same averaging procedure was performed on the longshore transport values
determined with the numerical model for the same conditions, Figure 5.38.
8007 ' r r   . .......... : .... .... T  .
600
soiid (avg of wave direction and 3 periods ...
400 dotted (evg of wave direction for T 12 s)
dashed (value for beta = 0 deg and T= 12 s) .......
200
o4 r,...
..200. L ... .
 00 * .... ..... ..
j. '.
::00 .... .'
0.25 0.2 015 0.1 0.05 0 0.05 0.1 0.15 02 0.25
Longshore Transport (m3)
Figure 5.38: Longshore transport value along the transect for the numerical model with a
pit of radius equal to 50 m, at x = 1500 m with 400 points defining the pit boundary
Once again the averaging is seen to result in a significant reduction in the oscillations far
from the pit. The large peaks in the longshore transport behind the pit are also greatly
reduced by the averaging. The second averaging has little effect, except far from the pit
where the oscillations are slightly reduced.
The shoreline change was calculated based on these values of longshore transport
using the full transport equation for the three methods presented in the last two figures.
Figure 3.39 shows the results for a 300 s time step with a negative change value
indicating shoreline advancement. As seen previously the single direction and wave
period model leads to great fluctuations in the shoreline change, even at large distances
from the pit.
800... ,.
600 
..... .. .....................
400 .
200
200
400 ... sol d (ai v i of Woive direction anid 3 periods)
600 ..dashed (value.for .be.ta 0 .de and T = 12 s
0.4 0.3 0.2 0.1 0 0.1 0.2 0.3 0.4
Change in Shoreline Position (m)
Figure 5.39: Shoreline change from the full transport equation for a 300 s time step along
the transect for the numerical model with a pit of radius equal to 50 m, at x = 1500 m
with 400 points defining the pit boundary and the other base conditions for the shoreline
change model
The averaging procedure is seen to remove these fluctuations and shows a salient forming
behind the pit flanked by two areas of erosion and then oscillations in the shoreline
change as distance from the pit increases. The salient directly behind the pit is found to
be slightly smaller than the two immediately upshore and downshore from it. Figure 5.40
provides a closer view of these changes.
200
6001
0.2 0.15 0.1 0.05 0 0.05
Change in Shoreline Position (m)
0.1 0.15 0.2
Figure 5.40: Shoreline change for a 300 s time step along the transect for the numerical
model with a pit of radius equal to 50 m, at x = 1500 m with 400 points defining the pit
boundary and the other base conditions for the shoreline change model
This plot clearly shows the increased reduction in the oscillating shoreline change values
at large distances from the pit resulting from the extra averaging.
... .
solid (svg of wave direction and 3 periods) ,
dotted (avg of wave direction for T = 12 s) ....
..,
min ........ ..
80
The magnitude of each transport term was investigated. Figure 5.42 shows the
first and second terms of the transport equation plotted separately for the total transport
shown in the previous plot.
600 F
200
200 
400
600 
800
0.
0.1 0.05 0 0.05
Change in Shoreline Position (m)
0.1 0.15 0.2
Figure 5.41: Shoreline change for first and second transport terms for a 300 s time step
along the transect for the numerical model with a pit of radius equal to 50 m, at x = 1500
m with 400 points defining the pit boundary and the other base conditions for the
shoreline change model
The second transport term is seen to contribute greatly to the total transport, especially
behind the pit where the gradients in the wave height are the greatest. This second
transport term is also seen to oscillate at a higher frequency than the first transport term.
It is clear that the two transport terms can oppose each other in some locations, such as
behind the pit where refraction may cause the waves to diverge and result in little change
solid (1st transport term)
dashed (2nd transport term)
r "~"
..~~~
""~
I`:::
~
':Ic;. .
"
'"~
ZI
 ..,!
~: 
 
. 
,
~.
~
"'~
''
~,,c"'
I'
~ `'
on~
2
81
or erosion for the first transport term, but the gradient in the wave height of the second
term leads to accretion behind the pit.
The analytic solution model was run for the same pit and incident wave
conditions with the same averaging techniques applied. The results compare well with
those of the numerical model. Figure 5.42 shows for a wave period of 12 s, the wave
direction averaged value of the longshore transport for the two models for half of the
transect.
dii
100..
500
solid (numerical model)...
600 dotted (analytic solution model) .... .....
700
....................................... ........ ............... ...... ......
800 ...................................................................................... .:: .:. :::: :;.. .......... .. ....... .... ...
0.25 0.2 0.15 0,1 0.05 0 0.05
Longshore Transport (m3)
Figure 5.42: Longshore transport along the transect for the analytic solution model and
the numerical model with a pit of radius equal to 50 m, at x = 1500 m with 400 points
defining the pit boundary and the other base conditions for the shoreline change model
The magnitudes of the numerical values are found to be larger than those of the analytic
solution model. Increasing the number of points defining the pit should bring the values
into closer agreement. These differences in longshore transport are carried into the
shoreline change results, Figure 5.43.
.... ... ..... ....
......... ..................... ..... .
.. . . .. .. .....
.... . . . ... . . ..... .
....... '.'. ...............o n ic l mo
... . ........... dotted (.nalyt solution model)
0.08 .0.06
0.04 0.02 0 0.02
Change in Shoreline Position (m)
0.04 0.06 0.08 0.1
Figure 5.43: Shoreline change for a 300 s time step along the transect for the analytic
solution model and the numerical model with a pit of radius equal to 50 m, at x = 1500 m
with 400 points defining the pit boundary and the other base conditions for the shoreline
change model
The larger values in longshore transport for the numerical model found in the previous
figure results larger gradients and therefore larger shoreline changes in Figure 5.43.
A comparison was made of the shoreline change caused by a circular pit to that of
a rectangular pit, which more accurately represents the shape of most borrow areas. For
this test the pit dimensions were 40 m in the crossshore direction and 200 m in the
longshore direction; this results in an area almost equal to that for a circular pit with a
600
400
200
200
400
600
800
0.1
ot A
83
50 m radius. A 12 s incident period was used, with 800 points defining the pit boundary,
and the same pit depth, water depth and transect location were used as in the trial for the
circular pit of 50 m radius. The relative amplitude found for the case of the large
rectangular pit is shown in Figure 5.44.
1.4 1.2 1 0.8
Relative Amplitude
0.6 0.4 0.2
Figure 5.44: Relative amplitude along transect for numerical model with a rectangular pit
(40 x 200 m) at x = 1500 m with 800 points defining the pit boundary and the other base
conditions for this shoreline change model
The relative amplitude was found to be significantly smoothed by averaging over the five
wave directions, beta equal to 5, 2, 0, 2, 5 degrees. The shadow zone was seen to be
larger for this case of the long pit, as compared to the circular pit with the same area,
which was expected. This will result in differences in the shoreline changes as well.
200 
200
400
600
800
1.6
" / ...
Dashed (value for beta = 0, T = 12 s)
S . solid (avg of wave direction for T = 12 s)
/...I t
,
The shoreline change for each of the two transport terms are shown in Figure
5.45.
600
400
200
200
400
600
0.03 0.02 0.01 0 0.01
Change in Shoreline Position (m)
0.02 0.03 0.04
Figure 5.45: Shoreline change for 300 second time step along transect for numerical
model with a rectangular pit (40 x 200 m) at x = 1500 m with 800 points defining the pit
boundary and the other base conditions for this shoreline change model
The two transport terms are shown to result in different shoreline changes directly behind
the pit, as was seen in Figure 5.41 for the case of the circular pit. The 2nd transport term
leads to shoreline advancement directly behind the pit due to the large gradients in the
local wave height. Both plots show erosion flanking the shadow area of the pit. The total
shoreline change induced by the rectangular pit is shown in Figure 5.46 with the results
for the circular pit drawn.
t,/*
'" solid (1st transport term)
dashed (2nd transport term)
 [both averaged for 5 directions]

i N ,
800
0.0
4
"
