Wave field modifications and shoreline response due to offshore borrow areas

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 x-axis 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 y-axis 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 x-axis 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 y-axis 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 y-axis 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 x-axis 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 y-axis 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 x-axis 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 y-axis 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 y-axis 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 pre-dredging and
post-dredging for Martin County, FL numerical model ......................... 98

5.57: Change in shoreline position during one time step for original bathymetry
and post-dredging 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 fixed-bed 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 wave-guides 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 wave-guides 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 one-half 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 frequency-wise 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

three-dimension 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 fixed-bed model in a small wave basin, an equilibrium beach profile was

created under monochromatic, shore-normal 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 time-averaged

energy flux approach. At each point along a transect running parallel to the incident wave

fronts, the depth integrated, time-averaged 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, time-averaged

energy flux in the x direction through the transect. The time-averaged energy flux in the

y direction can be determined using Eq. 3.16 instead of Eq. 3.15 with a transect parallel

to the x-axis.

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 time-averaged 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(O-ab) -K2H 25 cos(O-ab)dHb
Q= +4 b (3.34)
8(s -1X1- p) 8(s-1X1- 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 x-direction

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 p-wt) (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= (x-x')2 (y-y (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 sub-programs 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 y-axis to

calculate the time-averaged 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 y-axis. 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 =tan-1
O a (real)

The total velocity potential is determined from Eq 3.17 and the time-averaged 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 (Y-2 )[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)] e-icot (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 x-direction. 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

u-direction 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, time-averaged 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 x-directed 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 /, co-0[ (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 no-flow

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
X-Direction (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
X-Direction (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 two-dimensional 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
X-Direction (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 AMnmeial-Re'M.nlyi *0. h
-va LRe A rnumencal

41

" . ... ..-...'. ...... ..... ..

Fi . .. I

40 n "IL:

.. .. . .... ... ...... ... .,
I . \. \,

X-Direction (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
X-MDrteon (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 x-axis 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
X-Direction (m)

50 100 150 200

Figure 5.8: Plot of transect taken parallel to the x-axis 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
X-Direction (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 y-axis 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
X-Direction (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
X-Dlrectlon

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 x-axis 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 x-axis 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 y-axis 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 y-axis 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 y-axis 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 no-flow 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 x-axis, 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 no-flow 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 x-axis. 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 x-axis 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 no-flow condition is contrasted by the

reduction in wave amplitude found at this location for the pit.

A transect taken parallel to the y-axis 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 y-axis 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~,
X-Dir~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 rS------1---------- ----------------------------------^--- 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 x-axis 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 not-a-number 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 y-axis 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 y-axis 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 time-averaged

energy flux through a transect parallel to the y-axis (shore-normal 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 y-axis 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-

9-200-

6-300
>-

-400

-500

-600
-40(

0

The first term consisting of the incident pressure and x-directed 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 n-s/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 n-s/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 n-s/m2 or a difference of only 4230 n-s/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
e-i

+

+
-

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 y-axis 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

shore-normal 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

o-4 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 cross-shore 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

"

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