 Front Cover 
 Title Page 
 Table of Contents 
 List of Figures 
 Introduction 
 Methodology 
 Stepbystep discussion of... 
 Case A Nourishment along an uninterrupted... 
 Case B Nourishment downdrift of... 
 Numerical procedure 
 Examples illustrating application... 
 References 
 Appendix A. Deep water wave equivalents... 
 Appendix B. Program listing and... 
 Appendix C. Numerical example... 
 Appendix D. Numerical example... 
 Appendix E. Numerical example... 
 Appendix F. Numerical example... 
 Appendix G. Numerical example... 
 Appendix H. Numerical example... 
 Appendix I. Numerical example... 

Full Citation 
Material Information 

Title: 
Development of methodology for thirtyyear shoreline projections in the vicinity of beach nourishment projects 

Series Title: 
UFLCOEL 

Physical Description: 
1 v. (various foliations) : ill. ; 28 cm. 

Language: 
English 

Creator: 
Dean, Robert G ( Robert George ), 1930 Grant, Jonathan University of Florida  Coastal and Oceanographic Engineering Dept Florida  Dept. of Natural Resources.  Division of Beaches and Shores 

Publisher: 
Coastal & Oceanographic Engineering Dept., University of Florida 

Place of Publication: 
Gainesville Fla 

Publication Date: 
1989 
Subjects 

Subject: 
Shore protection  Florida ( lcsh ) Beaches  Florida ( lcsh ) Beach erosion  Florida ( lcsh ) Coastal and Oceanographic Engineering thesis M.S Coastal and Oceangraphic Engineering  Dissertations, Academic  UF 

Genre: 
government publication (state, provincial, terriorial, dependent) ( marcgt ) bibliography ( marcgt ) nonfiction ( marcgt ) 
Notes 

Bibliography: 
Includes bibliographical references. 

Statement of Responsibility: 
by Robert G. Dean and Jonathan Grant ; prepared for Division of Beaches and Shores, Florida Dept. of Natural Resources. 

General Note: 
"December 15, 1989." 

Funding: 
This publication is being made available as part of the report series written by the faculty, staff, and students of the Coastal and Oceanographic Program of the Department of Civil and Coastal Engineering. 
Record Information 

Bibliographic ID: 
UF00076124 

Volume ID: 
VID00001 

Source Institution: 
University of Florida 

Holding Location: 
University of Florida 

Rights Management: 
All rights reserved, Board of Trustees of the University of Florida 

Resource Identifier: 
oclc  22187426 

Table of Contents 
Front Cover
Front Cover
Title Page
Title Page
Table of Contents
Table of Contents 1
Table of Contents 2
Table of Contents 3
List of Figures
List of Figures 1
List of Figures 2
List of Figures 3
Introduction
Page 1
Background
Page 1
Page 2
Page 3
Page 4
Page 5
Page 6
Page 7
Page 8
Page 9
Page 10
Page 11
Page 12
Methodology
Page 13
Page 14
Page 15
Page 16
Page 17
Page 18
Page 19
Page 20
Page 21
Page 22
Page 12
Page 23
Page 24
Page 25
Page 26
Page 27a
Page 27b
Page 27c
Page 28
Page 29
Page 30
Page 31
Page 32
Page 33
Page 34
Page 35
Page 36
Page 37
Page 38
Stepbystep discussion of methodology
Page 39
Page 40
Page 38
Case A Nourishment along an uninterrupted shoreline
Page 41
Page 42
Page 43
Case B Nourishment downdrift of a littoral barrier
Page 44
Page 43
Page 45
Page 46
Page 47
Numerical procedure
Case A Nourishment along uninterrupted shoreline
Page 47
Page 48
Page 49
Page 47
Page 50
Page 51
Page 52
Page 53
Page 54
Page 55
Page 56
Case B Nourishment with structures present
Page 57
Page 58
Examples illustrating application of methodology
Page 59
Page 60
Page 61
Page 62
Page 63
Page 64
References
Page 65
Appendix A. Deep water wave equivalents for shoreline modeling
Page 66
Page 67
Page 68
Page 69
Page 70
Page 71
Appendix B. Program listing and sample input and output
Page 72
Page 73
Page 74
Page 75
Page 76
Page 77
Page 78
Page 79
Page 80
Page 81
Page 82
Page 83
Appendix C. Numerical example 1
Page 84
Page 85
Page 86
Page 87
Page 88
Page 89
Page 90
Page 91
Page 92
Page 93
Appendix D. Numerical example 2
Page 94
Page 95
Page 96
Page 97
Page 98
Page 99
Page 100
Page 101
Page 102
Page 103
Appendix E. Numerical example 3
Page 104
Page 105
Page 106
Page 107
Page 108
Page 109
Page 110
Page 111
Page 112
Page 113
Appendix F. Numerical example 4
Page 114
Page 115
Page 116
Page 117
Page 118
Page 119
Page 120
Page 121
Page 122
Page 123
Appendix G. Numerical example 5
Page 124
Page 125
Page 126
Page 127
Page 128
Page 129
Page 130
Page 131
Page 132
Page 133
Appendix H. Numerical example 6
Page 134
Page 135
Page 136
Page 137
Page 138
Page 139
Page 140
Page 141
Page 142
Page 143
Appendix I. Numerical example 7
Page 144
Page 145
Page 146
Page 147
Page 148
Page 149
Page 150
Page 151
Page 152
Page 153

Full Text 
Development of Methodology for ThirtyYear Shoreline
Projections in the Vicinity of Beach Nourishment Projects
December 15, 1989
Prepared for:
Division of Beaches and Shores
Florida Department of Natural Resources
3900 Commonwealth Boulevard
Tallahassee, FL 32399
Prepared by:
R. G. Dean
and
Jonathan Grant
Coastal and Oceanographic Engineering Department
University of Florida
336 Well Hall
Gainesville, FL 32611
UFL/COEL89/026
DEVELOPMENT OF METHODOLOGY FOR
THIRTYYEAR SHORELINE PROJECTIONS IN THE
VICINITY OF BEACH NOURISHMENT PROJECTS
by
Robert G. Dean
and
Jonathan Grant
Prepared for:
Division of Beaches and Shores
Florida Department of Natural Resources
3900 Commonwealth Boulevard
Tallahassee, FL 32399
December 15, 1989
TABLE OF CONTENTS
INTRODUCTION 1
BACKGROUND 1
General Description of Sediment Transport Processes in the Vicinity of A Beach
Nourishment Project .................... ............ 1
Profile Equilibration .................... ............. 2
"Spreading Out" Losses .................. :. ........... 2
Background Erosion ..................... ............ 5
Role of Retention Structures ............................ 5
Role of Sediment Size on Transport Rates . . . ..... .... 5
Significance of Wave Height ................... ......... 10
W ave Direction ................... .................... 10
General Characteristics of Equilibrium Beach Profiles . . ... 10
METHODOLOGY 12
Profile Equilibration ................... ............... 12
Longshore Sediment Transport ........................... 20
Combined Linearized Equations ................... ........ 22
Rectangular Beach Nourishment Project . . . ..... 25
Erosion Adjacent to a Littoral Barrier . . . ..... ..... 30
Numerical Solution .................................. 33
Boundary Conditions ................... ............... 36
Wave and Other Parameters of Use in Applying the Methodology . ... 38
STEPBYSTEP DISCUSSION OF METHODOLOGY 38
Graphical Procedure ................... ................ 38
CASE A NOURISHMENT ALONG AN UNINTERRUPTED SHORE
LINE 41
Step 1 Specify Beach Nourishment Project Characteristics . . ... 41
Step 2 Determine the Equilibrated Project Width, Ayo . . .... 41
Step 3 Calculate Effective Alongshore Diffusivity, G . . ... 41
Step 4 Calculate Shoreline Position Due to Spreading Out Losses ...... ..43
Step 5 Calculate Background Erosion Losses . . . ... 43
Step 6 Calculate Resulting Shoreline Position . . . ... 43
CASE B NOURISHMENT DOWNDRIFT OF A LITTORAL BAR
RIER 43
Step 1 Specify Beach Nourishment Characteristics ............... 44
Step 2 Determine the Equilibrated Project Width, Ayo ............ 44
Step 3 Calculate Effective Alongshore Diffusivity, G ............... 44
Step 4 Calculate Shoreline Position Due to Spreading Out Losses ...... ..46
Step 5 Calculate Background Erosion Losses . . . . ... 46
Step 6 Calculate Resulting Shoreline Position . . . ... 46
NUMERICAL PROCEDURE 47
CASE A NOURISHMENT ALONG AN UNINTERRUPTED SHORE
LINE 47
STEP 1 Specify Beach Nourishment Project Characteristics ......... 47
STEP 2 Determine Equilibration Project Width, Ayo ............ 47
STEP 3 Develop Background Erosion Data as Piecewise Linear Segments .47
STEP 4 Develop Input File ............................ 47
STEP 5 Run Program .................... ........... 51
STEP 6 Examine Output in File DNRBS.OUT . . . ..... 51
CASE B NOURISHMENT WITH STRUCTURES PRESENT 57
STEP 4B Specify a Reference Background Transport .............. 57
STEP 5B Specify Structure Location(s) and Length(s) in Program ...... 57
EXAMPLES ILLUSTRATING APPLICATION OF METHODOLOGY 59
Graphical Examples .................... ............. 59
Numerical Examples .................... ............. 59
REFERENCES 65
APPENDIX A 66
APPENDIX B 72
APPENDIX C 84
APPENDIX D 94
APPENDIX E 104
APPENDIX F 114
APPENDIX G 125
APPENDIX H 134
APPENDIX I 144
LIST OF FIGURES
1 Effect of Nourishment Material Scale Parameter, AF, on Width of Resulting Dry
Beach. Four Examples of Decreasing A. . . . ... .... 3
2 "Spreading Out" Losses Occurring Due to Mobilization of Sediments by Waves. 4
3a Variation of Shoreline Position with Time at Various Locations Relative to a
Nourishment Project. No Background Erosion. . . . . 6
3b Variation of Shoreline Positions with Time at Various Locations Relative to a
Nourishment Project. Uniform Background Erosion of 2 ft/yr. . . 7
4 Illustration of Nourishment Stabilization by Terminal Structure. . . 8
5 Plot of K vs D. Results of Present and Previous Studies (modified from Dean,
1978). ........................................... 9
6 Shoreline Orientation Downdrift of a Complete Littoral Barrier . .... 11
7 Beach Profile Factor, A, vs Sediment Diameter, D, in Relationship h = Ay2/I
(modified from Moore, 1982). Note: A(ftl/3) = 1.5 A(m/3) . ... 13
8 Recommended Distribution of h. Along the Sandy Shoreline of Florida. . 14
9 Three Generic Types of Nourished Profiles. . . . ...... 15
10 Effect of Increasing Volume of Sand Added on Resulting Beach Profile, AF =
0.1 m1/3,AN = 0.2 m1/3,h. = 6m, B = Im. ................... 17
11 Variation of NonDimensional Shoreline Advancement Ayo/W. with A' and V'.
Results Shown for h./B = 2.0. .......................... 18
12 Variation of NonDimensional Shoreline Advancement Ayo/W. with A' and V'.
Results Shown for h./B = 4.0. .......................... 19
13 Definition Sketch ................... ............... 21
14 Variation of Ratio C,/Co vs h./Lo. ......................... 23
15 Approximate Estimates of G(ft2/s) Around the Sandy Beach Shoreline of the
State of Florida. Based on the Following Values: K = 0.77, g = 32.2 ft/sec2,
s = 2.65, p = 0.35, n = 0.78, h. From Fig. 8, B Estimates Ranging from 6
to 9 ft, Ho from Figure 23, T From Figure 24. . . . ... 24
16 Example of Evolution of Initially Rectangular Nourished Beach Planform. Ex
ample for Project Length, of 4 Miles and Effective Wave Height, H, of 2
Feet and Initial Nourished Beach Width of 100 Feet. . . ... 26
17a Evolution of an Initially Rectangular Beach Planform on an Otherwise Straight
Shoreline. Results for t' = 0, 0.1, 0.2, 0.5 and 1.0. . . . ... 27a
17b Evolution of an Initially Rectangular Beach Planform on an Otherwise Straight
Beach. Results for t' = 0, 2.0, 4.0, 6.0 and 8.0. . . . ... 27b
17c Evolution of an Initially Rectangular Beach Planform on an Otherwise Straight
Beach. Results for t' = 0, 10.0, 15.0, 20.0 and 30.0. . . . ... 27c
18 Percentage of Material Remaining in Region Placed vs. the Parameter VUGt7 29
19 Example of Shoreline Evolution in Response to Littoral Barrier. Based on
Method of PelnardConsidere. Longshore Sediment Transport Rate Used
in Example = 180,000 cubic yards per year. Littoral Barrier Length = 160 ft. 31
20 Pelnard Considere Solution for Shoreline Recession Downdrift of a Complete
Littoral Barrier .......... .. .... ................... 32
21 Two Alternative Methods for Predicting Beach Nourishment Performance Down
drift of a Littoral Barrier ............................. 34
22 Computational Scheme Used in Numerical Method. . . . ... 35
23 Recommended Values of Effective Deep Water Wave Height, Ho, Along Florida's
Sandy Shoreline. ................... .............. 39
24 Recommended Values of Effective Deep Water Wave Period, T, Along Florida's
Sandy Shoreline. ................... .............. 40
25 Form for Computation of Performance Along Uninterrupted Shoreline . 42
26 Form for Computations of Performance Downdrift of a Littoral Barrier . 45
27 Data Input Preparation Form for Numerical Procedure . . ... 48
28 Input File DNRBS.INP for Example 2 ....................... 49
29 Example of Output File DNRBS.OUT for Input File in Figure 27. Example No.
1. (Total of 11 Pages of Output. ......................... 52
30 Estimates of Net Annual Longshore Sediment Transport Along Florida's East
Coast ......................................... 58
C1 Numerical Example 1, Ayo = 112 ft, Nourishment Length = 2 miles, Zero Back
ground Erosion ................... .. ........ ..... 86
C2 Numerical Example 2, Shoreline Position Variation with Time at Locations In
dicated and Shown in Figure C1. . . . .... ........ 87
D1 Numerical Example 2, Ayo = 112 ft, Nourishment Length = 2 miles, Uniform
Background Erosion = 2 ft/yr. .......................... 96
D2 Numerical Example 2, Shoreline Position Variation with Time at Locations In
dicated and Shown in Figure D1. ........................ 97
E1 Numerical Example 3, Ayo = 112 ft, Nourishment Length = 2 miles, Variable
Background Erosion ............................... 106
E2 Numerical Example 2, Shoreline Position Variation with Time at Locations In
dicated and Shown in Figure E1 .......................... 107
F1 Numerical Example 4, Ayo = 112 ft, Nourishment Length = 1,000 ft, No Back
ground Erosion ................... ................116
F2 Numerical Example 2, Shoreline Position Variation with Time at Locations In
dicated and Shown in Figure F............. ... ............. 117
G1 Numerical Example 5, 112 ft Long Structure at North End of Project, Nourish
ment Length = 2 miles, No Background Erosion. . . . ... 126
G2 Numerical Example 2, Shoreline Position Variation with Time at Locations In
dicated and Shown in Figure G1. . . . ... ...... 127
H1 Numerical Example 6, 112 ft Long Structure at South End of Project, Nourish
ment Length = 2 miles, Uniform Background Erosion. . . ... 136
H2 Numerical Example 2, Shoreline Position Variation with Time at Locations In
dicated and Shown in Figure H1. ........................ 137
I1 Numerical Example 7, 112 ft Long Structure at South End of Project, Nourish
ment Length = 2 miles, Variable Background Erosion. . . ... 146
I2 Numerical Example 2, Shoreline Position Variation with Time at Locations In
dicated and Shown in Figure I1.......................... 147
DEVELOPMENT OF METHODOLOGY FOR
THIRTYYEAR SHORELINE PROJECTIONS IN THE
VICINITY OF BEACH NOURISHMENT PROJECTS
INTRODUCTION
The purpose of this report is to develop and illustrate with examples readily applied
methodologies for calculating the response of shorelines in the vicinity of beach nourishment
projects. The need for such methodology is a result of Florida Statutes 161.053(G) and Rule
16B33.024(3)(e) which require, with minor exceptions, coastal structures to be located
landward of a thirty year projection of the Seasonal High Water Shoreline (SHWL).
The conceptual interpretation of these Statutes and Rule is that the performance of
the beach nourishment project should be considered in projecting the Seasonal High Water
Line (SHWL) position to a time thirty years into the future. This requires consideration
of both the background erosion rate which is the normal rate in areas that have not been
nourished and the shoreline retreat component due to "spreading out" losses from the beach
nourishment project.
BACKGROUND
General Description of Sediment Transport Processes in the Vicinity of
a Beach Nourishment Project
In general, when sand is placed in conjunction with a beach nourishment project, this
project represents an "anomaly" to the shoreline planform and the natural processes will
tend to smooth out this anomaly. In addition, many times the placed profile will be steeper
than the natural profile and the profile will tend to equilibrate over time. The sections below
describe the individual processes and characteristics of the response of a beach nourishment
project.
DEVELOPMENT OF METHODOLOGY FOR
THIRTYYEAR SHORELINE PROJECTIONS IN THE
VICINITY OF BEACH NOURISHMENT PROJECTS
INTRODUCTION
The purpose of this report is to develop and illustrate with examples readily applied
methodologies for calculating the response of shorelines in the vicinity of beach nourishment
projects. The need for such methodology is a result of Florida Statutes 161.053(G) and Rule
16B33.024(3)(e) which require, with minor exceptions, coastal structures to be located
landward of a thirty year projection of the Seasonal High Water Shoreline (SHWL).
The conceptual interpretation of these Statutes and Rule is that the performance of
the beach nourishment project should be considered in projecting the Seasonal High Water
Line (SHWL) position to a time thirty years into the future. This requires consideration
of both the background erosion rate which is the normal rate in areas that have not been
nourished and the shoreline retreat component due to "spreading out" losses from the beach
nourishment project.
BACKGROUND
General Description of Sediment Transport Processes in the Vicinity of
a Beach Nourishment Project
In general, when sand is placed in conjunction with a beach nourishment project, this
project represents an "anomaly" to the shoreline planform and the natural processes will
tend to smooth out this anomaly. In addition, many times the placed profile will be steeper
than the natural profile and the profile will tend to equilibrate over time. The sections below
describe the individual processes and characteristics of the response of a beach nourishment
project.
Profile Equilibration
As noted, beach nourishment projects are generally placed with profiles which are steeper
than the natural profile for the size of sediment that is used in the beach nourishment project.
Thus over the years this profile will tend to equilibrate to its natural shape. In addition, if
the sediment size used in the beach nourishment is fine, the profile will tend to be rather
mild in slope and the shoreline advancement will be small for a given volume of beach
nourishment per unit length of beach. Figure 1 shows the qualitative effect of grain size
in terms of the dry beach width for the same added volume per unit length of beach. The
upper panel presents the profile that would result for a beach nourishment grain size which
is larger than the native sand resulting in a fairly wide dry beach width. The three lower
panels illustrate the effect of decreasing grain size maintaining the volume per unit beach
length the same. It is seen that with decreasing grain size the dry beach width progressively
decreases to a point where in the lower panel the dry beach width is zero. For this condition
all of the sand that has been placed has been moved offshore in a profile which is consistent
with the grain size used in the nourishment.
"Spreading Out" Losses
The placement of a beach nourishment project results in a planform anomaly which
interacts with the waves to result in sediment transport away from this anomaly. This
process is illustrated in Figure 2 and shows the transport occurring away from the anomaly
in a manner that will result in a smoothing or spreading out of the sediment. The term
"spreading out" losses actually refers to a redistribution of the sediment and not a total
loss to the system but rather a loss from the region in which the sediment is placed. As will
become evident later, this loss from the nourished area is manifested as a gain of sediment
volume in the nourishmentadjacent areas.
92.4m
7"r 7A B = 1.5m
45.3m
^ 1h*
b) Nonintersecting Profiles
AN= AF= 0.1m1 /3
,5.9m
c) NonIntersecting Profiles"
AN= 0.1m11/3AF = 0.09m1/3
d) Limiting Case of Nourishment Advancement, 1/3
NonIntersecting Profiles, AN= 0.1m1/3,AF = 0.085m
I I I I 1 I
100
200
300
400
500
h,= 6m
= 6m
600
OFFSHORE DISTANCE (m)
Figure 1. Effect of Nourishment Material Scale Parameter, AF,on Width of
Resulting Dry Beach. Four Examples of Decreasing AF.
:
* "Spreading Out"
Losses
SWaves
S Planform Anomaly Due to
Beach Nourishment
"Spreading Out"
Losses
(From Region Placed)
Figure 2. "Spreading Out" Losses Occurring Due to Mobilization
of Sediments by Waves.
Background Erosion
Usually the need for a beach nourishment project is due to a background erosion which,
for an ideal project, is relatively slow. With the placement of the beach nourishment
project, there will be two components of shoreline retreat. It will be assumed that the
two components of shoreline recession, i.e. background erosion and the component due to
spreading out losses, can be added linearly. The background erosion which was present
prior to the placement of the beach nourishment project will continue. Figure 3 illustrates
qualitatively the superposition of these two components for several locations within and
adjacent to a beach nourishment project. Figure 3a presents the case for no background
erosion and Figure 3b for a uniform background erosion of 2 ft/yr.
Role of Retention Structures
In some cases, especially short beach nourishment projects, it may be worthwhile to
consider the use of retention structures to extend the life of the projects. Figure 4 illustrates
qualitatively one such application. Structures must be used with great care, especially in
areas where there is a substantial longshore transport magnitude. An additional situation in
which retention structures have been used effectively to prevent loss of sediment in Florida
has been at the ends of littoral systems such as at the termini of barrier islands. Two such
locations are the north jetty at John's Inlet in Pinellas County and the two small terminal
structures at the south end of Gasparilla Island in Lee County.
Role of Sediment Size on Transport Rates
It has been noted that the dominant losses due to a beach nourishment project are due
to spreading out losses or transport away from the region where the sediment is placed.
The sediment transport is proportional to a coefficient, K, which has been found to depend
on sediment size as shown in Figure 5; thus with the use of coarser grained material, the
project will perform much more effectively. Although there has not been any substantial
documentation to illustrate adverse effects of using material which is substantially coarser
Z o
OWL
=I 100
Og
W Z
ODO
00
I c
cn 0
Figure 3a.
5 10 15 20 25
TIME (Years) AFTER NOURISHMENT
Variation of Shoreline Position with Time at Various Locations Relative to a
Nourishment Project. No Background Erosion.
100
a
w
LLJ
CI
W
3t
LI
D
z 0~
X <
Q
0
J
0
cc
UL
50
0 5 10 15 20 25 30
TIME (Years) AFTER
NOURISHMENT
Figure 3b. Variation of Shoreline Positions with Time at Various Locations Relative to a
Nourishment Project. Uniform Background Erosion of 2 ft/yr.
Figure 4. Illustration of Nourishment Stabilization
by Terminal Structure.
2.0
\ Res
u 1.0 
^\
0.5
1.0
DIAMETER, D (mm)
Figure 5. Plot of K vs. D. Results of Present and Previous Studies (modified
from Dean, 1978).
ult From This Study,
Santa Barbara
relationship Suggested
Previously
*
I I
than the native material, it has been hypothesized that if such material is used it may
effectively armor the beach in the nourishment area thereby resulting in less transport from
the area nourished and a deficit and associated erosion on the area downdrift of the project.
Significance of Wave Height
After placement of a beach nourishment project, it is evident intuitively that the mobiliz
ing effects of wave height cause profile equilibration and the spreading out losses mentioned
earlier. Thus the determination of reliable, effective wave heights is important to the pre
diction of the performance of any beach nourishment projects.
As will be described later, for two identical projects which are placed in areas where
the wave height differs by a factor of two, the longevity of these projects would differ by a
factor of 5.3.
Wave Direction
It is somewhat surprising that on a long, uninterrupted shoreline the effect of wave
direction is relatively unimportant to the performance of a beach nourishment project.
The interpretation of this insensitivity will be discussed in a later portion of this report.
However, wave direction is extremely important in the case of a beach nourishment project
located adjacent to a structure which interferes with the longshore sediment transport.
Figure 6 illustrates such a situation where sand is placed immediately downdrift of a jetty
and the orientation of the beach planform immediately adjacent to the jetty is parallel to
the incident wave crests. Thus, it will be necessary to provide estimates of wave direction
or to develop alternative methodologies which do not require accurate estimates of wave
direction.
General Characteristics of Equilibrium Beach Profiles
In general, equilibrium beach profiles tend to be concave upward and the profiles tend
to be milder in slope for the finer sediment and steeper for coarser sediment. Equilibrium
beach profiles have been found by Bruun (1954) and Dean (1977) to be reasonably well
Inlet
N. \
Shoreline Orientation Downdrift of a Complete
Littoral Barrier.
Figure 6.
represented by the form
h(y) = Ay2/3 (1)
in which h is the depth at a distance y seaward of the shoreline and A is a scale parameter.
A significant contribution to the objectives of this report was developed by Moore (1982)
in the form of a plot of the sediment scale parameter, A, in terms of the sediment size,
Figure 7.
A second important relationship to the objectives of this study is that of closure depth,
h,. Closure depth is a concept which describes the maximum depth to which sediments
will be mobilized by the waves. Although in general this closure depth is expected to be
dependent on wave height and wave period, for purposes of this study, the closure depth will
be regarded as a value dependent on position around the state of Florida. The recommended
closure depth versus location around the state is presented in Figure 8.
METHODOLOGY
Profile Equilibration
In considering the profiles resulting from beach nourishment, generically there are three
types of nourished, equilibrated profiles. These are presented in Figure 9. Referring to the
top panel in this figure of intersecting profiles, a necessary but not sufficient requirement
for intersecting profiles is that the fill material be coarser than the native material. One
can see that an advantage of such a profile is that the nourished profile "toes in" to the
native profile thereby negating the need for material to extend out to the closure depth.
The second type of profile is one that would usually occur in most beach nourishment
projects. Nonintersecting profiles occur if the nourished material grain size is equal to or
less than the native grain size. Additionally, this profile always extends out to the closure
depth, h,.
(J.) Cl)
"
ta
w
Cl)
!i
r oU
Cn
1.0
0.10
0.01
0
Relationship
From Hughes'
Field Resuts From Individual Field Profiles where a
Range of Sand Sizes was given
 Lot R
A From Swart's
AA Laboratory Results
.U1
10.0
100.0
SEDIMENT SIZE D (mm)
Figure 7. Beach Profile Factor, A, vs Sediment Diameter, D, in Relationship
h = Ay2/3 (modified from Moore, 1982). Note: A(ft1l3) = 1.5 A(ml/3).
12 16 20 24
h* (Feet)
h, (Feet)
12 16 20 24
Figure 8. Recommended Distribution of h, Along the Sandy Shoreline
of Florida.
 I U .
a) Intersecting Profile AF>AN
Added Sand ~:
b) NonIntersecting Profile
Virtual Origin of
Nourished Profile
Added Sand
c) Submerged Profile AF
Figure 9. Three Generic Types of Nourished Profiles.
The third type of profile that can occur is the submerged profile (Figure 9c) the char
acteristics of which are shown in greater detail in Figure 10. This profile type requires the
nourished material to be finer than the native. It can be shown that if only a small amount
of material is used then all of this material will be mobilized by the breaking waves and
moved offshore to form a small portion of the equilibrium profile associated with this grain
size as shown in the upper panel. With increasing amounts of fill material, the intersection
between the nourished and the original profile moves landward until the intersection point
is at the water line. For greater quantities of material, there will be an increase in the dry
beach width, Ayo, resulting in a profile of the second type described.
The next major section describes the methodology for calculating planform response to a
beach nourishment project. It is assumed that profile equilibration occurs when the material
is placed. This assumption is not important to the final thirty year projection. Actually, of
course the profile equilibration will occur gradually, but will probably be near completion
within a few years. This assumption merely allows the overall response calculations to
be carried out in two steps. Following the discussion of profile equilibration, graphical and
numerical methods are presented for predicting the shoreline (planform) evolution. As might
be expected the numerical method provides greater flexibility for representing realistically
the actual situation.
It can be shown that the initial additional dry beach width, Ayo, is related to the
placed and native sediment characteristics and the closure depth, h,, and berm height, B.
To render the results more compact, the results are cast in the following non dimensional
form
AY" h,
=, f ( /A ,V /BW,, (2)
in which W, is the width of the active surf zone on the native profile, i.e.
W = (h/AN)3/2 (3)
Figures 11 and 12 present results of Ayo/W. for h,/B values of 2 and 4, respectively.
OFFSHORE DISTANCE (m)
100
200
300
400
5
7
SJ_ B = 1.5m
h.= 6m
a) Added Volume = 120 m3/m
b) Added Volume = 490 m3 /m 
Case of Incipient Dry Beach
Figure 10. Effect of Increasing Volume of Sand Added on Resulting
Beach Profile. AF= 0.1m1/3, AN= 0.2m113, h, = 6m, B = 1m.
,+4
z
0
W 10
LJ
;00
I
I I I I I
10.
1.0
o %
W. ____ I' ____ ___ect ____
I AF'' hV
or Ay= \\e
0.01 ___ v Y 0.1
~
y w A a V' = 0.05
T   '\ ! i  
Asymptotes V' = 0.02
W, 0
0 1.0 2.0 2.8
Figure 11. Variation of NonDmensona' = 0. Advancement
Swith= 0.0nd Results Shown for h, /B = 2.0.
rtes AV' = VIBW, = 0.002
tah BY , ;"
0.001 Definition Sketch
0 1.0 2.0 2.8
A' = AF /AN
syo/W* with A' and V'. Results Shown for h, /B = 2.0.
1.0
S0 NonIntersecting
i Profilesi '
V = /BW, =5.0 
Intersecting
2_ Profiles .
I i V' 
0.1
0 0.1 .
A7  .05
W0 Asymptotes
W* for Ayo= 0
J0.02
1 
0.01 .01
w
S= 0.005
I I
S=B AN' BW,
0 0.002
0 1.0 2.0 2.8
Definition Sketch _
 ff(l AF V 
W, B AN BW
0.0001 =00
0 1.0 2.0 2.8
A' = AF/AN
Figure 12. Variation of Nondimensional Shoreline Advancement Ayo/W., with
A' and '. Results shown for h, /B = 4.0.
It is seen that for each non dimensional volume, the nondimensional additional beach
width increases with increase in ratio of sediment scale parameters; however, the increase is
relatively small for ratios greater than 1.2. Additionally, there is some lower ratio of scale
parameters for each nondimensional volume below which there will be no additional dry
beach width. This corresponds to the case presented in Figure Id. As noted previously,
the profile equilibrations will be assumed to occur instantaneously. The stage is now set for
consideration of the longshore sediment transport and planform evolution.
Longshore Sediment Transport
The equations available for representing planform evolution are a sediment transport
equation and a sand conservation (or continuity) equation. The transport equation is em
pirically based and describes the total transport in the longshore direction due to waves
arriving at a breaking angle, as to the shoreline. The continuity equation is fundamental
and simply balances sediment volume changes with transports into and out of the region
under consideration. These equations are:
K H5/2 igV sin(P Yb) cos(f ab) (4)
Transport: Q = 8(s 1)(1 (4)
Continuity: (5)
at ax
in which V is sediment volume per unit length of beach, g is gravity, C is the ratio of
breaking wave height to water depth (usually taken as 0.78), f represents the azimuth of
the outward normal to the shoreline, ab represents the azimuth of the direction from which
the breaking waves originate, s is the specific gravity of the sediment (approximately 2.65),
p is the inplace porosity of the sediment (usually taken as 0.35) and t is time. Figure 13
presents a definition sketch for ab and f/. The sign convention used in this report is that
the positive x (and Q) direction are to the right as an observer looks offshore.
For most shoreline evolution models and those that will be presented here, the model
predicts the position of one contour, such as the NGVD contour or the SHWL contour.
0
U
0
4
Reference
Base Line
Figure 13. Definition Sketch.
These models assume that as beaches erode or accrete the profile moves without change of
form in a landward or seaward direction, respectively. Thus after equilibration occurs, the
shoreline change, Ay, associated with a volumetric change, AV, can be shown to be given
by
A = B (6)
(h, + B)
The two governing equations, namely the transport and conservation equations, can
be applied directly to predict the evolution of a beach nourishment project or they can
be combined in a linearized manner. Both of these approaches will be described in the
following sections.
Combined Linearized Equations
Eq. (4) describes the sediment transport in terms of the difference between the shoreline
orientation and wave direction. Foregoing the algebra, it can be shown that the combined
and linearized equation governing the evolution of a beach system is
9y 82y
G= aG (7)
in which the parameter, G, can be interpreted as the "alongshore diffusivity" and is ex
pressed as
K HO2.4 904 cos1W (o co) cos 2(Po e.)
8(s 1)(1 p)C*c0.4(h. + B) cos(/o a,*)
where the subscript "o" denotes deep water conditions, C, is the wave celerity in the water
depth h, and Eq. (8) is derived in Appendix A. The ratio C,/Co is
C./Co = tanh ( 2 ) (9)
in which Co = gT/2r, CGo = gT/4r and C,/Co is presented vs h,/Lo in Figure 14.
Figure 15 presents approximate values of G along the sandy beach shorelines of the state
of Florida.
Equation (7) is the socalled heat conduction or diffusion equation which is wellknown
in classical physics and has many known solutions. Two solutions which are of interest
represented by the form
h(y) = Ay2/3 (1)
in which h is the depth at a distance y seaward of the shoreline and A is a scale parameter.
A significant contribution to the objectives of this report was developed by Moore (1982)
in the form of a plot of the sediment scale parameter, A, in terms of the sediment size,
Figure 7.
A second important relationship to the objectives of this study is that of closure depth,
h,. Closure depth is a concept which describes the maximum depth to which sediments
will be mobilized by the waves. Although in general this closure depth is expected to be
dependent on wave height and wave period, for purposes of this study, the closure depth will
be regarded as a value dependent on position around the state of Florida. The recommended
closure depth versus location around the state is presented in Figure 8.
METHODOLOGY
Profile Equilibration
In considering the profiles resulting from beach nourishment, generically there are three
types of nourished, equilibrated profiles. These are presented in Figure 9. Referring to the
top panel in this figure of intersecting profiles, a necessary but not sufficient requirement
for intersecting profiles is that the fill material be coarser than the native material. One
can see that an advantage of such a profile is that the nourished profile "toes in" to the
native profile thereby negating the need for material to extend out to the closure depth.
The second type of profile is one that would usually occur in most beach nourishment
projects. Nonintersecting profiles occur if the nourished material grain size is equal to or
less than the native grain size. Additionally, this profile always extends out to the closure
depth, h,.
c 0.05 o
CO
L=
0    
0
0 0.05 0.10 0.15 0.20
h*/Lo
Figure 14. Variation of Ratio C*/Co vs. h*/Lo
0.02 0.06 0.10 0.14
G(ft2/s)
G(ft2/s)
0.02 0.06.101
::/^
\^
Figure 15. Approximate Estimates of G(ft2/s) Around the Sandy
Beach Shoreline of the State of Florida. Based on
the Following Values: K = 0.77, g = 32.2 ft/sec2,
s = 2.65, p = 0.35, K= 0.78, hFrom Fig. 8., B Estimates
Ranging from 6 to 9 ft, Ho from Fig. 23, T from Fig. 24.
,,
here will be discussed below; these solutions pertain to the graphical methodology thereby
allowing a first estimate of the performance of a beach nourishment project. These solutions
and the development of the combined and linearized equation concepts are due to Pelnard
Considere (1956).
Rectangular Beach Nourishment Project
The first solution of interest is for the evolution of an initially rectangular beach nour
ishment project of length, which projects a distance Ayo from the original shoreline. The
solution is
(,t) = rf [ ( + l) erf [ (2 1)} (10)
in which the term "erf" refers to the error function described mathematically as
erf(z) = e2 du (11)
in which u is a dummy variable.
Figure 16 illustrates an example of the performance of such a beach nourishment project
and Figures 17a, b and c present the results in nondimensional form. It can be seen from
Eq. (9) that if the term is the same for two beach nourishment projects the non
dimensional performance of the two beach nourishment projects will be the same. Thus,
for two projects constructed with the same wave characteristics but with one project twice
the length of the second project, the first project will lose the same percentage of sediment
as the second project in a duration that is four times as long as that for the second project.
Similarly if two projects have the same length but the first project has a wave height one
half that of the second wave height then the first project will have a longevity which is in
excess of five times the longevity of the second project. In general this relationship may be
stated as
t2 = l ( 2 ( )24 (12)
DISTANCE FROM ORIGINAL
SHORELINE, y (ft)
Nourished Beach Planform
6 4 2 0 2 4 6 8
ALONGSHORE DISTANCE, X (miles)
Figure 16.
Example of Evolution of Initially rectangular Nourished Beach Planform.
Example for Project Length, J, of 4 Miles and Effective Wave Height, H,
of 2 feet and Initial Nourished Beach Width of 100 Feet.
1.0
0.9
0.8
0.5
O.4i
0.3
0.2
0.1
0.0
Figure 17a.
S Gt
t' = 16 G
92
 Initial Planform, t' = 0.0
I I I I I I I I I
\\
I'\
''
2.0
2.0
3.0 3.5 4.0 4.5
x/(Z/2)
I I
5.0 5.5
Evolution of an Initially Rectangular Beach Planform
on an Otherwise Straight Shoreline. Results for
t' = 0, 0.1, 0.2, 0.5 and 1.0.
I I I I
6.0 6.5 7.n 7.5
____________ 0.0
t I =0.0
......................g t 0. 1
.t'=0.2
t'=0.5
t =.1.0
I ( I )

 0.6 
'\I
0.5 
0.1
0.3
0.1 _ ___ **
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.
x/(k/2) t =0.0
t =2.0
Figure 17b. Evolution of an Initially Rectangular Beach Planform on an t'=4.0
Otherwise Straight Shoreline. Results for t' = 0,2.0,4.0,
6.0 and 8.0. =6.0
t' =8.0
10.7
0.6
0.5
0.4
.0 . 
0.3 
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5
x/(/2) t'=0.0
Figure 17c. Evolution on an Initially Rectangular Beach Planform .................. =10.0
on an Otherwise Straight Shoreline. Results for  =15.0
t' = 0,10.0, 15.0, 20.0 and 30.0. t'=20.0
t' =30.0
in which tl and t2 represent the times required for projects 1 and 2 to lose the same
percentage of sand from the region placed. Thus, the longevity of a project in terms of
the time required to lose a certain percentage of the sediment from the project area varies
directly as the square of the length of the project and inversely as the 2.4 power of the wave
height.
Equation (10) may be integrated to determine the fraction of material, M, remaining
within the area placed. This is shown formally as
1 ) /2
M(t) = /2 y(x,t)dx (13)
Ayot f/2
and upon carrying out the integration the result is
M(t) = 2 [e(/2 ) 1] + erf (14)
which is plotted in Figure 18 where the horizontal axis is the parameter encountered previ
ously in the solution for the evolution of this particular planform.
If we are interested in the time required for 50% of the nourished material to be trans
ported out of the area placed, then from Figure 17 we see that the appropriate value of
G/e is 0.46. Thus the time required to lose 50% of the sediment from the region placed
is
t5o = 0.21 (15)
G
in which all variables are in consistent units. A more readily applied form is
to = 8.7 2 (16)
where t50 is in years, e is in miles and Hb is the breaking wave height in feet. As an example
a project 2 miles in length with an effective breaking wave height of 2 ft would "lose" 50%
of the volume placed due to spreading out losses in
22
tso = 8.7 = 6.15 years (17)
It is emphasized that this solution is for a long unobstructed shoreline and includes only
spreading out losses, i.e. no background erosion.
L
LL Gtt/
SO 1.0 0.5 1.0
'Z LU t = Time After Placement
.0 0,) G= Alongshore Diffusivity Initial
cc4;CFill _
U. L. Asymptote Planform
OZz0.5 / M=1 :
2 O . I
0 4 0 :
00
CLI0.0
Ou 0 1 2 3 4 5 6
Figure 18. Percentage of Material Remaining In Region Placed vs. the Parameter V Gt1j
L
Erosion Adjacent to a Littoral Barrier
The second analytical solution of relevance to this study is that of the downdrift erosion
adjacent to a littoral barrier as shown in Figure 19. The solution for this situation is
applicable for an initial condition of a straight and uniform shoreline and a wave arriving
at a constant direction. The solution is presented as
(X, t) = V ex ( z Werf( ) t < t (18)
y(x,t) = Yerfc ( > t > t, (19)
where
erfc(z) = 1 erf(z) (20)
t 4G tan2 (21)
in which Y is the length of the structure, 0 represents the angle of the approach wave
and t, is the time at which bypassing commences. Because we are interested primarily
in the beach response downdrift of a barrier and there is usually no bypassing, Eq. (18)
would be the solution of primary interest. Figure 20 presents the non dimensional solution,
y/( \i tan 0), versus non dimensional distance, x/IV4Gt, from the downdrift jetty.
There are two approaches to predicting shoreline changes downdrift of a littoral barrier,
such as a jetty. One method, that just described, requires knowledge and specification
of an effective wave direction. Available information to define wave directions is quite
limited, especially on the west coast of Florida. Fortunately a second method, which will
be recommended here, requires data which are more readily available along the Florida
coastline.
The recommended procedure utilizes background erosion data rather than an effective
wave direction. The justification for the use of background erosion data rather than wave
3
w
J
^J
g2
0
_1
DISTANCE
LANDWARD (ft)
YEARS
100
DISTANCE
SEAWARD (ft)
Initial Shoreline
0
cc
U
U.
2
z
3 0
z
<,
(/
Example of Shoreline Evolution in Response to Littoral
Barrier. Based on Method of PelnardConsidere. Longshore
Sediment Transport Rate Used in Example =180,000 cubic
yards per year. Littoral Barrier Length = 160 ft.
Figure 19.
NONDIMENSIONAL DISTANCE DOWNDRIFT OF COMPLETE LITTORAL BARRIER
x/44i/t
0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1
OU. Littoral Barrier Waves
) 0.2 
(,j Y
Oz
ZU) 0.4 
o '
Z x0.
0
Z 0.6
Figure 20. Pelnard Consldere Solution For Shoreline Recession Downdrlft of a Complete Littoral Barrier.
direction is that the local background erosion rates in the vicinity of a littoral barrier are
due to and a manifestation of waves arriving at the shoreline. This alternate recommended
method would not be possible in the case where an inlet is to be cut because at that
time there are no a priori background erosion data. Fortunately, in Florida, quite reliable
background erosion data exist in the vicinity of most inlets.
For the recommended approach, the modifications to the graphical method described
previously for an uninterrupted shoreline are small and are illustrated diagrammatically in
Figure 21. The only changes are that (a) the effective length of the project, e', is twice the
physical length of the project, e, and (b) the waves are considered as advancing normal to
shore. This accomplishes the desired effect of a zero transport at the littoral barrier, since
the transport at the center of a project for normally incident waves is zero.
The methods described here will be illustrated by later examples.
Numerical Solution
The numerical solution that will be presented here is a so called explicit scheme in which
the equations for sediment transport and continuity are solved sequentially. In particular
referring to Figure 22, the shoreline positions are held constant for a time step, At, while
the sediment transport is computed. Following this operation, the sediment transport is
held constant for a time step and the equation of continuity is applied to these transport
values to update the shoreline positions.
This type of explicit model referred to here has a stability criterion which limits the max
imum time step, At, that can be utilized. The maximum time step is given approximately
by
1 Az2
(At)maz = (22)
2G
and G is defined in Equation (8) and approximate values presented in Figure 15. For most
purposes in Florida, a time step of 86,400 seconds (1 Day) and a grid size (Az) of 500 feet
are reasonable. From Eqs. (8) and (22) it is seen that the larger the wave height, the smaller
the allowable time step. Also, the smaller the grid size, the smaller the allowable time step.
Littoral
Barrier 
i
S  Nourishment
a) Method With Waves Approaching
at a Specific Angle. Background
Erosion Without Effect of Littoral
Barrier
Littoral
Barrier
Waves (0 o) = 0
Nourishment
b) Recommended Method With Waves
Approaching Normal to Shoreline.
Background Erosion Includes Effect
of Littoral Barrier.
Figure 21. Two Alternative Methods For Predicting Beach Nourishment
Performance Downdrift of a Littoral Barrier
Ql"A
Q \+1
Figure 22. Computational Scheme Used in Numerical Method.
As noted previously, one of the primary advantages of the numerical solution is the much
greater flexibility of specifying initial conditions and input to the model. Additionally, with
minor modifications to the program, renourishments could be represented.
To effectively utilize the greater flexibility inherent in the numerical procedure and in
particular to include structures where desired, the background erosion rates are translated
into background transport rates. Formally the background transport rates, QB(x), are
determined from the continuity equation
QB(x) = QB(xo) (h. + B) B dx (23)
in which 2 is the background shoreline change rate and xo is a reference shoreline location
at which a reference transport QB(Xo) is specified.
Boundary Conditions
The application of the sediment transport and continuity equations with initial planform
conditions require specification of boundary conditions at the two ends of the grid system
in order to complete the problem formulation. In general, there are two types of boundary
conditions. The first that will be discussed is a specified shoreline position at one or both of
the ends of the computational domain. A simple example of the specified shoreline positions
would be that the shoreline is fixed at its initial value or the value could be prescribed over
the computational time period. A second boundary condition that could be applied is a
specified discharge at one or both ends of the computational domain. Examples of situations
in which each of these boundary conditions would be applied are discussed below.
The fixed boundary condition could be applied at the ends of a computational domain
for the case of a beach nourishment project on an uninterrupted shoreline; however, if the
ends of the computational domain are too close to the changes that would occur due to the
nourishment, then these conditions can adversely affect the accuracy of the results. A useful
and direct approach to evaluating whether the fixed boundary conditions are sufficiently
distant from the point of interest is to simply double the extent of the computational domain
and to evaluate the effects on the shoreline changes in the region of interest over the period
for which the computations are carried out.
The second type of boundary condition of interest is the specified transport boundary
condition. Examples where a specified transport boundary condition would be appropriate
are immediately downdrift of a partial or complete littoral barrier. If the barrier were
a complete obstruction to the longshore sediment transport, then a specified discharge of
zero would be appropriate; however, if there was some bypassing around the littoral barrier,
then the volume per unit time of the bypassing would be the appropriate input transport
boundary condition. Obviously in this case since the discharge values are centered at the
grid lines, it would be appropriate to locate a grid line at the littoral barrier. The transport
boundary condition could also be applied at the ends of the computational grid. If this
were done, the shoreline displacement would be free to vary with time. If the transport
boundary condition is specified as zero at the ends of the computational grid, there would
be no change of volume within the computational domain. This could be the case in
which complete littoral barriers existed at the two ends of the system of interest. In the
model developed for this project, the boundary condition imposed at the two ends of the
computational domain is the transport condition with the background transport as the
imposed values.
A situation in which the boundary condition will change within the computational pe
riod might be a case where a groin of specified length was included somewhere within the
computational domain. As the shoreline advances seaward toward the groin tip, the bound
ary condition would be a zero transport condition. However after the shoreline reached the
end of the groin then the shoreline would remain fixed at that position which would in effect
then be a fixed shoreline position boundary condition. In a case where the longshore sed
iment transport direction changed with time, the boundary condition at a structure could
alternate between a fixed transport boundary condition and a specified shoreline position.
Wave and Other Parameters of Use in Applying the Methodology
Four parameters will be presented and recommended for applying the methodology de
veloped in conjunction with this study.
The first parameter of interest is the limiting depth of motion, h.. Although this quantity
is not known precisely, recommended values for h. have been presented in Figure 8. The
berm height, B, is also required and appropriate values can be determined from profiles at
the site of interest. Generally, berm heights range between 6 and 9 ft (above NGVD) in
Florida.
A third parameter of interest is the effective wave height. The recommended distribution
of wave heights around the Florida peninsula is shown in Figure 23. These wave heights
were based primarily on the Coastal Data Network results where available. It is seen that,
on the Florida east coast, the wave heights vary from the largest near the Florida/Georgia
border and decrease toward the southern portions of the state. On the Florida west coast,
the heights decrease toward the north with very low values along the Big Bend area, then
increase toward the Florida/Alabama border. Finally, estimates of effective wave period
are presented for the coast of Florida in Figure 24. Approximate values of the longshore
diffusivity parameter, G, have been presented in Figure 15 and may be used as a reasonable
approximation.
STEPBYSTEP DISCUSSION OF METHODOLOGY
In this section the limitations and the stepbystep application of the graphical and
numerical procedures will be presented.
Graphical Procedure
The graphical procedure as presented here pertains to (1) a rectangular nourishment on
an uninterrupted shoreline, and (2) a rectangular beach nourishment immediately downdrift
of a complete littoral barrier such as a jetty. In both of these cases it is considered that
the shoreline change is the linear sum of the result of the spreading out losses and the
background erosion rate as determined by historical data;
H eff2(eet)
.q3
1 3 5 8
JA
MA
SST
CC x
x CL
VE WP
MI
1 3 5 8
H eff 2(feet)
4. .
Figure 23. Recommended Values of Effective Deep Water Wave Height,
Ho, Along Florida's Sandy Shoreline.
2 6 10 14
Wave Period, T(sec)
0)
w 14
S10
0
S2
(0
Figure 24. Recommended Values of Effective Wave Period, T, Along
Florida's Sandy Shoreline.
40
Wave Period, T(sec)
2 6 10 14
!
Wave and Other Parameters of Use in Applying the Methodology
Four parameters will be presented and recommended for applying the methodology de
veloped in conjunction with this study.
The first parameter of interest is the limiting depth of motion, h.. Although this quantity
is not known precisely, recommended values for h. have been presented in Figure 8. The
berm height, B, is also required and appropriate values can be determined from profiles at
the site of interest. Generally, berm heights range between 6 and 9 ft (above NGVD) in
Florida.
A third parameter of interest is the effective wave height. The recommended distribution
of wave heights around the Florida peninsula is shown in Figure 23. These wave heights
were based primarily on the Coastal Data Network results where available. It is seen that,
on the Florida east coast, the wave heights vary from the largest near the Florida/Georgia
border and decrease toward the southern portions of the state. On the Florida west coast,
the heights decrease toward the north with very low values along the Big Bend area, then
increase toward the Florida/Alabama border. Finally, estimates of effective wave period
are presented for the coast of Florida in Figure 24. Approximate values of the longshore
diffusivity parameter, G, have been presented in Figure 15 and may be used as a reasonable
approximation.
STEPBYSTEP DISCUSSION OF METHODOLOGY
In this section the limitations and the stepbystep application of the graphical and
numerical procedures will be presented.
Graphical Procedure
The graphical procedure as presented here pertains to (1) a rectangular nourishment on
an uninterrupted shoreline, and (2) a rectangular beach nourishment immediately downdrift
of a complete littoral barrier such as a jetty. In both of these cases it is considered that
the shoreline change is the linear sum of the result of the spreading out losses and the
background erosion rate as determined by historical data;
CASE A NOURISHMENT ALONG AN UNINTERRUPTED
SHORELINE
The computation sheet presented as Figure 25 has been developed and should be refer
enced when reviewing the stepbystep procedure described below.
Step 1 Specify Beach Nourishment Project Characteristics
These include
Project Length, e
Sediment Size, D
Volume Added Per Unit Length, V
Step 2 Determine the Equilibrated Project Width, Ayo
To accomplish this
h, from Figure 8
Estimate B from local profile data berm height
Determine AF and AN from Figure 7 from sediment sizes and local profile data,
respectively
Calculate Ayo/W. from Figures 11 and 12, interpolating if necessary.
Step 3 Calculate Effective Alongshore Diffusivity, G
The alongshore diffusivity, G, is obtained as expressed by Eq. (8) and is calculated from
the wave, sediment and other local factors (G can also be estimated from Figure 15).
Determine Ho from Figure 23
Determine T from Figure 24
Determine C. from Figure 14
BEACH NOURISHMENT PROJECTION
(Graphical Computations, Uninterrupted Shoreline)
General Location:
Wave Height, Ho
Wave Period, T
Wave Direction, ao:
(Fig. 23): ft, Closure Depth, h, (Fig. 8): ft
(Fig. 24): sec, Sediment Size, D: mm
0, Transport Factor, K (Fig. 5):
Berm Height, B: ft
Alongshore Diffusivity, G (From Equation below or Figure 15).
SK HI4C~g0o4 cos(0o ao) cos 2(/o a,)
8 (s 1)(1 p)CK0.4(h. + B) cos(Po a*)
Background Erosion
= ft2/s
Equilibrated Beach Width, Ayo
AN (Fig. 7) or From Profile:
Ap (Fig. 7):
Volume Per Unit Length:
Ayo (Figs. 11 and 12):
Project Length, = __ miles =
ftl/3
ft'/3
ft3/ft
ft
ft
For 30 years
(1) (2) (3) (4) (5) (6)
Distance X y { ) /Ayo Ys YB (ft) = YN =
From Center, x(ft) (Fig. 17) (ft) 30 x ER y, Yb (ft)
Figure 25. Form for Computation of Performance Along Uninterrupted Shoreline.
x Erosion Rate (ER)
ft/yr
* Other Recommended Values:
= 0.78
s = 2.65
p = 0.35
g = 32.2
Step 4 Calculate Shoreline Position Due to Spreading Out Losses
Calculate nondimensional time for t = 30 years or other time of interest
Gt
t' = 16
where all variables are in consistent units
Calculate x/(t/2) at locations of interest (Column 2, Bottom Table in Figure 25)
Determine y/Ayo from Figure 17 (Column 3, Bottom Table in Figure 25)
Step 5 Calculate Background Erosion Losses
Estimate background erosion rate from DNR data base
Multiply rate by time (30 years) to obtain background erosion component (Column
5, Bottom Table in Figure 25)
Step 6 Calculate Resulting Shoreline Position
Add linearly the changes due to spreading out losses and background erosion to obtain
the total changes. If the area of interest is not within the project area, apply the same
methodology, however, here the spreading out losses (from the project area) will result in a
shoreline advancement (see Figure 3).
CASE B NOURISHMENT DOWNDRIFT OF A
LITTORAL BARRIER
As discussed previously, there are two methods for calculating response downdrift of a
littoral barrier. It is recommended that the method utilizing background erosion data be
applied rather than the method requiring the wave approach angle. The recommended
method is described below.
The computation sheet presented as Figure 26 for this case has been developed and
should be referenced along with the stepbystep procedure described.
Step 1 Specify Beach Nourishment Characteristics
These include (same as for Case A)
Project Length, e (Effective Length, e' = 2e)
Sediment Size, D
Volume Added Per Unit Length, V
Step 2 Determine the Equilibrated Project Width, Ayo
(Same procedure as for Case A)
h. from Figure 8
Estimate B from local profile data berm height
Determine AF and AN from Figure 7 and local profile data, respectively
Calculate Ayo/W. from Figures 11 and 12, interpolating if necessary.
Step 3 Calculate Effective Alongshore Diffusivity, G
(Same as for Case A)
The alongshore diffusivity, G, is obtained as expressed by Eq. (8) and is calculated
from the wave, sediment and other local factors (G can also be estimated from Figure 15).
Determine Ho from Figure 23
Determine T from Figure 24
Determine C, from Figure 14
* Other Recommended Values:
= 0.78
s = 2.65
p = 0.35
g = 32.2
Step 4 Calculate Shoreline Position Due to Spreading Out Losses
Calculate nondimensional time for t = 30 years or other time of interest
Gt
t' = 16
where all variables are in consistent units
Calculate x/(t/2) at locations of interest (Column 2, Bottom Table in Figure 25)
Determine y/Ayo from Figure 17 (Column 3, Bottom Table in Figure 25)
Step 5 Calculate Background Erosion Losses
Estimate background erosion rate from DNR data base
Multiply rate by time (30 years) to obtain background erosion component (Column
5, Bottom Table in Figure 25)
Step 6 Calculate Resulting Shoreline Position
Add linearly the changes due to spreading out losses and background erosion to obtain
the total changes. If the area of interest is not within the project area, apply the same
methodology, however, here the spreading out losses (from the project area) will result in a
shoreline advancement (see Figure 3).
CASE B NOURISHMENT DOWNDRIFT OF A
LITTORAL BARRIER
As discussed previously, there are two methods for calculating response downdrift of a
littoral barrier. It is recommended that the method utilizing background erosion data be
BEACH NOURISHMENT PROJECTION
(Graphical Computations, Downdrift of a Littoral Barrier)
General Location:
Wave Height, Ho (Fig. 23): ft, Closure Depth, h. (Fig. 8): ft
Wave Period, T (Fig. 24): sec, Sediment Size, D: mm
Wave Direction, ao: 0, Transport Factor, K (Fig. 5):
Berm Height, B: __ ft
Alongshore Diffusivity, G(From Equation Below or Figure 15)
K HA4CO g04 cos(0o ao) cos 2(o a,)
8 (s 1)(1 p)C,c0 4(h. + B) cos(Po a*)
Background Erosion
x Erosion Rate (ER)
ft/yr
For 30 years
= ft2/s
Equilibrated Beach Width, Ayo
AN (Fig. 7) or From Profile: ftl/3
AF (Fig. 7): ft1/3
Volume Per Unit Length: ft3/ft
Ayo (Figs. 11 and 12): ft
Project Length, = miles = ft
Effective Project
Length, e' = 2e = _miles = ft
Gt G(30x365x24x3600)
16 =)2 16 )
(1) (2) (3) (4) (5) (6)
Distance y (Z) /Ayo y, YB(ft) = YN =
From Littoral Barrier, x(ft) (Fig. 17) (ft) 30 x ER y, Yb (ft)
Figure 26. Form for Computation of Performance Downdrift of a Littoral Barrier.
* Other Recommended Values:
K = 0.78
s = 2.65
p = 0.35
g = 32.2
Step 4 Calculate Shoreline Position Due to Spreading Out Losses
Calculate nondimensional time for t = 30 years or other time of interest
16 Gt 4 Gt
where all variables are in consistent units.
(Note: Different coefficient from Case A)
Calculate x/('/2) at locations of interest where the origin of x is at the littoral barrier
Calculate y/Ayo from Figure 17 (Note in this case, the horizontal axis in Figure 17
is to be interpreted as x/('/2) or equivalently, x/e.)
Step 5 Calculate Background Erosion Losses
Estimate background erosion rate from DNR data base
Multiply rate by time to obtain background erosion component
Step 6 Calculate Resulting Shoreline Position
Add linearly the changes due to spreading out losses and background erosion to obtain
the total changes. If the area of interest is not within the project area, apply the same
methodology, however, here the spreading out losses (from the project area) will result in a
shoreline advancement (see Figure 3).
L
NUMERICAL PROCEDURE
As noted previously, the numerical procedure provides greater flexibility for representing
shoreline and beach nourishment conditions. Prior to using the program, there is a certain
amount of data preparation that is required. Some of this preparation is similar to that for
the graphical procedure as described earlier. The numerical procedure also allows input of
structures of arbitrary lengths at any location within the computational domain. At this
stage, the program is straightforward, but not overly "user friendly".
As for the case of the "Graphical Procedure", the methodology will be illustrated below
for the case of nourishment along an uninterrupted shoreline and for the case of structures
present. The preparation sheet presented as Figure 27 has been developed to assist in
data preparation and should be referenced along with the stepbystep procedure described
below.
CASE A NOURISHMENT ALONG AN UNINTERRUPTED
SHORELINE
STEP 1 Specify Beach Nourishment Project Characteristics
This is the same as described previously for the Graphical Procedure. The only difference
is that now greater flexibility is available with the numerical procedure allowing varying
volumes of nourishment along the shoreline including any number of nourishment segments.
STEP 2 Determine Equilibration Project Width, Ayo
Utilize same method as described for Graphical Procedure
STEP 3 Develop Background Erosion Data as Piecewise Linear Segments
STEP 4 Develop Input File
A description of the input file (DNRBS.INP) is given below and Figure 28 presents an
example input file.
NUMERICAL PROCEDURE
As noted previously, the numerical procedure provides greater flexibility for representing
shoreline and beach nourishment conditions. Prior to using the program, there is a certain
amount of data preparation that is required. Some of this preparation is similar to that for
the graphical procedure as described earlier. The numerical procedure also allows input of
structures of arbitrary lengths at any location within the computational domain. At this
stage, the program is straightforward, but not overly "user friendly".
As for the case of the "Graphical Procedure", the methodology will be illustrated below
for the case of nourishment along an uninterrupted shoreline and for the case of structures
present. The preparation sheet presented as Figure 27 has been developed to assist in
data preparation and should be referenced along with the stepbystep procedure described
below.
CASE A NOURISHMENT ALONG AN UNINTERRUPTED
SHORELINE
STEP 1 Specify Beach Nourishment Project Characteristics
This is the same as described previously for the Graphical Procedure. The only difference
is that now greater flexibility is available with the numerical procedure allowing varying
volumes of nourishment along the shoreline including any number of nourishment segments.
STEP 2 Determine Equilibration Project Width, Ayo
Utilize same method as described for Graphical Procedure
STEP 3 Develop Background Erosion Data as Piecewise Linear Segments
STEP 4 Develop Input File
A description of the input file (DNRBS.INP) is given below and Figure 28 presents an
example input file.
BEACH NOURISHMENT PROJECTION
(Numerical Procedure)
General Location:
Wave Height, Ho (Fig. 23):
Wave Period, T (Fig. 24):
Wave Direction, cao:
Deep Water Contour Orientation, f/o:
Longshore Axis Orientation, Mp:
Grid Dimension, Ax:
Time Increment, At:
ft.,
sec.,
o
o
o
ft
sec
Closure Depth, h* (Fig. 8):
Berm Height, B:
Sand Diameter, D:
Transport Factor, K (Fig. 5):
VFACT:
Background Transport, QREF:
IREF:
IMAX:
TIMES:
No. of Structures, NS:
___ft.
___ft.
mm
ft3/s
Structure Specificiation
Structure Structure Structure
Number Location, I Length (ft)
Equilibrated Beach Width Ayo
Background Erosion
x
Erosion Rate, ER, (ft/yr)
Nourishment Specification
I Range
Ayo
AN (Fig. 7) or From Profile:
AF (Fig. 7):
Volume Per Unit Length:
Ayo (Figs. 11 and 12):
ft1/3
ft1/3
ft3/ft
ft
to
to
to
to
to
Figure 27. Data Input Preparation Form for Numerical Procedure.
48
EXAMPLE OF INPUT FILE: DNRBS.INP
(Example No. 2)
EXAMPLE NO. 2 UNIF. BACK. EROS. NO STRUCK. 2 MILE PROJ. ] 1e+deqr*dio
r wasve peJrpiP pU0rr4 +j 1A0 Drred
e C kA* .1 r r T'*t. &V
2.00 6.0 90.0 90.0 180.0 500.0 86400.0
1 .0o4 A"fIv. M +.0i 4rI F ,, r'c,' J f 0.0 r 5s5 0 N T
/ 
17.0 6.0 0.77 1.0 0.0 1 180 10950 0
hL)3 a+ C4Ayrmh v,
0.0 2.
90000. 3.
Frr fH
80 100
80 112.0
81 112.0
82 112.0
83 112.0
84 112.0
85 112.0
86 112.0
87 112.0
88 112.0
89 112.0
90 112.0
91 112.0
92 112.0
93 112.0
94 112.0
95 112.0
96 112.0
97 112.0
98 112.0
99 112.0
100 112.0
0 90000. 2.0 49500.
0 100000. 3.0 140000.
ri a C ell /oucrtsf_.k / iorrU
I auirs
iVofe; The
t ere
2.0
2.0
60000.
tf (rrI Ce/l No: Ao V,0lueQ
lOerc. Prorv;ccd
6eL.e Vs C4V
Pu rf os c s.
Figure 28. Input File DNRBSoINP For Example 2
3.0o PaItrs of(tIJ.s+OmeS,
J ErMos ;n as)
mrs
vfur4
aw r L; ^i, s
r (An' ta v%
NUMERICAL PROCEDURE
As noted previously, the numerical procedure provides greater flexibility for representing
shoreline and beach nourishment conditions. Prior to using the program, there is a certain
amount of data preparation that is required. Some of this preparation is similar to that for
the graphical procedure as described earlier. The numerical procedure also allows input of
structures of arbitrary lengths at any location within the computational domain. At this
stage, the program is straightforward, but not overly "user friendly".
As for the case of the "Graphical Procedure", the methodology will be illustrated below
for the case of nourishment along an uninterrupted shoreline and for the case of structures
present. The preparation sheet presented as Figure 27 has been developed to assist in
data preparation and should be referenced along with the stepbystep procedure described
below.
CASE A NOURISHMENT ALONG AN UNINTERRUPTED
SHORELINE
STEP 1 Specify Beach Nourishment Project Characteristics
This is the same as described previously for the Graphical Procedure. The only difference
is that now greater flexibility is available with the numerical procedure allowing varying
volumes of nourishment along the shoreline including any number of nourishment segments.
STEP 2 Determine Equilibration Project Width, Ayo
Utilize same method as described for Graphical Procedure
STEP 3 Develop Background Erosion Data as Piecewise Linear Segments
STEP 4 Develop Input File
A description of the input file (DNRBS.INP) is given below and Figure 28 presents an
example input file.
Card 1 (Format: 20A4): Identification Card with 80 Characters of Alphanumeric
Input
Card 2 Format: 8F8.2): Contains the Following Input Parameters
First Parameter: Deep Water Effective Wave Height in Feet, Ho (From
Figure 23)
Second Parameter: Wave Period in Seconds, T (From Figure 24)
Third Parameter: Deep Water Wave Direction, ao, in Degrees
Fourth Parameter: Deep Water Contour Orientation, Po, in Degrees
Fifth Parameter: Longshore Axis Orientation, p, in Degrees
Sixth Parameter: Grid Dimension, Ax, in Feet
Seventh Parameter: Time Increment, At, in Seconds
Card 3 Format: 5F8.2,4I6): Contains the Following Input Parameters
First Parameter: Depth of Limiting Motion, h,, in Feet (From Figure 8)
Second Parameter: Berm Height, B, in Feet
Third Parameter: Sediment Transport Parameter, K (From Figure 5)
Fourth Parameter: Factor to Increase or Decrease Proportionally All Input
Beach Widths, Ayo
Fifth Parameter: Background Transport, QBKREF (cubic feet/sec) (See
Eq. (23))
Sixth Parameter: Grid Line Index, IREF, at Which QBKREF is to Apply
Seventh Parameter: Number of Grids, IMAX
Eighth Parameter: Number of Time Steps, NTIMES
Ninth Parameter: Number of Structures, NS
Card 4 Format: 5(I6,F8.3)): Note this Card (and Possibly a Subsequent Card if NS
> 5) is only Present if NS > 0 and Contains NS Pairs of Grid Lines and
Structure Lengths. At Present the Program is Dimensioned to Accommo
date Up To 10 Structures
Cards 5 and 6 Format (8F8.2): These Two Cards Contain Pairs of (x, EB(x)) where x is
in Feet and EB is the Location Background Erosion in Feet/Year. The
Program is Presently Configured for Seven Pairs; However, it is Possible to
Specify Background Erosion Conditions with as Little as Two Pairs. For
Example, if the Background Erosion is Uniform at Two Feet/Year and the
Computational Domain is 60,000 ft in Length, the Two Active Pairs Could
be: 0.0 2.0 80000.0 2.0
The Remaining Five Pairs Entered Would be Immaterial. Note it is nec
essary to provide two cards here, even if all the meaningful information is
contained in the first card.
Card 7 This Card Specified the First, NNOUS, and Last, NNOUE, Grid Indices for
the Nourished Segment
Cards 8 and Following (Format: 16, 3F8.2): Each of These Cards Specifies the Grid Index,
I, and the Associated Shoreline Advancement, Ayo (I)
This completes specification of the input File DNRBS.INP
STEP 5 Run Program
STEP 6 Examine Output in File DNRBS.OUT
A description of the output file DNRBS.OUT is presented below and Figure 29 presents
an example of this output with annotations. This output is for the input file presented in
Figure 28.
Card 1: This card is an image of the first input card which is an identification card
Cards 2,3,4,5,6: These cards simply repeat input values
Cards 7 and 8: These two cards are pairs of (x, EB(x)) specified in Input Cards 5 and 6
Next Block of Data: Presents pairs of (I,QBI) in which QBI is the background erosion
transport across the Ith grid line. The units of QBI are in ft3/sec
Next Card: This card repeats the first nourished grid index, NNOUS, and the last
nourishment grid index, NNOUE, as provided by Input Card 7
Next Block of Data: Presents three entries per grid: (I,X(I), DYO(I)), in which I is the
grid block index, x(I) is the x coordinate of the grid block and DYO(I)
is the initial nourished width at the grid block. In the example
presented, because there are 450 sets of entries, one for each grid
block.
Next Block of Data: Provides pairs of I, Y(I) for one year after nourishment for all grid
blocks
Next Card: Presents the proportion of the additional dry beach area relative to the
initial area that remains within the project area after one year. This
proportion is denoted PCT(LCUR)
Remaining Output: The remaining output consists of detailed shoreline output for 5,
10, 20 and 30 years and the proportional surface area remaining
for each of the thirty years.
This completes the description of the information in the output file DNRBS.OUT
'/3
Figure 29. Example of Output File DNRBS.OUT for Input File in
Figure 28. Example No. 2. (Total of 5 Pages of
Output).
EXAMPLE OF OUTPUT FILE: DNRBS.OUT
(Example No. 2)
EXAMPLE NO. 2 UNIF. BACK. EROS. NO STRUC. 2 MILE PROJ.
HO = 2.00 FT., T = 6.00 SEC., ALPO = 90.00 DEG., BTAO =
XMU = 180.00 DEG., DX = 500.00 FT., DT = 86400.00 SEC.
HSTR = 17.00 FT., B = 6.00 FT., XK = .77 VFACT =
QBKREF = .00 FT.**3/SEC.
IREF = 1, IMAX = 180, NTIMES = 10950, NS =
90.00 DEG.,
1.00
2.00 .90E+05
3.00 .10E+06
2.00 .50E+05
3.00 .14E+06
2.00 .60E+05
2.00
BACKGROUND EROSION TRANSPORT RATES
.001 4
.005 9
.009 14
.012 19
.016 24
.020 29
.023 34
.027 39
.031 44
.034 49
.038 54
.042 59
.045 64
.049 69
.053 74
.056 79
.060 84
.063 89
.067 94
.071 99
.074 104
.078 109
.082 114
.085 119
.089 124
no 129
52 134
139
.104 144
.107 149
.00E+00
.90E+05
3.00
1
6
11
16
21
26
31
36
41
46
51
56
61
66
71
76
81
86
91
96
101
106
111
116
121
126
131
136
141
146
.000
.004
.007
.011
.015
.018
.022
.026
.029
.033
.036
.040
.044
.047
.051
.055
.058
.062
.066
.069
.073
.077
.080
.084
.088
.091
.095
.098
.102
.106
2
7
12
17
22
27
32
37
42
47
52
57
62
67
72
77
82
87
92
97
102
107
112
117
122
127
132
137
142
147
.001
.004
.008
.012
.015
.019
.023
.026
.030
.034
.037
.041
.044
.048
.052
.055
.059
.063
.066
.070
.074
.077
.081
.085
088
.092
.096
.099
.103
.106
3
8
13
18
23
28
33
38
43
48
53
58
63
68
73
78
83
88
93
98
103
108
113
118
123
128
133
138
143
148
.002
.006
.009
.013
.017
.020
.024
.028
.031
.035
.039
.042
.046
.050
.053
.057
.061
.064
.068
.071
.075
.079
.082
.086
.090
.093
.097
.101
.104
.108
5
10
15
20
25
30
35
40
45
50
55
60
65
70
75
80
85
90
95
100
105
110
115
120
125
130
135
140
145
150
.003
.007
.010
.014
.018
.021
.025
.028
.032
.036
.039
.043
.047
.050
.054
.058
.061
.065
.069
.072
.076
.079
.083
.087
.090
.094
.098
.101
.105
.109
___ ___ ___ ___ ___ ___ ___ ___ ___
156
161
166
171
176
181
.113
.117
.120
.124
.128
.131
157
162
167
172
177
80 100
INITIAL SHORELINE
1
3
5
7
9
11
13
15
17
19
21
23
25
27
29
31
33
35
37
39
41
43
45
47
49
51
53
55
57
59
61
63
65
67
69
71
73
75
77
79
81
83
85
87
89
91
93
95
97
99
101
103
105
0.
1000.
2000.
3000.
4000.
5000.
6000.
7000.
8000.
9000.
10000.
11000.
12000.
13000.
14000.
15000.
16000.
17000.
18000.
19000.
20000.
21000.
22000.
23000.
24000.
25000.
26000.
27000.
28000.
29000.
30000.
31000.
32000.
33000.
34000.
35000.
36000.
37000.
38000.
39000.
40000.
41000.
42000.
43000.
44000.
45000.
46000.
47000.
48000.
49000.
50000.
51000.
52000.
.114
.117
.121
.125
.128
158
163
168
173
178
.115
.118
.122
.125
.129
159
164
169
174
179
(INCL. NOURISHMENT) POSITION
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
112.00
112.00
112.00
112.00
112.00
112.00
112.00
112.00
112.00
112.00
.00
.,00
.00
2
4
6
8
10
12
14
16
18
20
22
24
26
28
30
32
34
36
38
40
42
44
46
48
50
52
54
56
58
60
62
64
66
68
70
72
74
76
78
80
82
84
86
88
90
92
94
96
98
100
102
104
106
500.
1500.
2500.
3500.
4500.
5500.
6500.
7500.
8500.
9500.
10500.
11500.
12500.
13500.
14500.
15500.
16500.
17500.
18500.
19500.
20500.
21500.
22500.
23500.
24500.
25500.
26500.
27500.
28500.
29500.
30500.
31500.
32500.
33500.
34500.
35500.
36500.
37500.
38500.
39500.
40500.
41500.
42500.
43500.
44500.
45500.
46500.
A "7 en
53
50500.
51500.
52500.
'lr^
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
112.00
112.00
112.00
112.00
112.00
112.00
112.00
112.00
112.00
112.00
112.00
.00
.00
.00
.115
.119
.123
.126
.130
160
165
170
175
180
.116
.120
.123
.127
.131
111
113
115
117
119
121
123
125
127
129
131
133
135
137
139
141
143
145
147
149
151
153
155
157
159
161
163
165
167
169
171
173
175
177
179
100
1
7
13
19
25
31
37
43
49
55
61
67
73
79
85
91
97
103
109
115
121
127
133
139
145
151
157
163
55000.
56000.
57000.
58000.
59000.
60000.
61000.
62000.
63000.
64000.
65000.
66000.
67000.
68000.
69000.
70000.
71000.
72000.
73000.
74000.
75000.
76000.
77000.
78000.
79000.
80000.
81000.
82000.
83000.
84000.
85000.
86000.
87000.
88000.
89000.
116
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
000
TIME
2.00
2.00
2.00
2.00
2.00
2.00
2.00
2.00
2.00
2.00
1.93
.18
15.06
58.16
96.98
102.52
74.10
26.75
2.31
1.77
2.00
2.00
2.00
2.00
2.00
2.00
2.00
2.00
112
114
116
118
120
122
124
126
128
130
132
134
136
138
140
142
144
146
148
150
152
154
156
158
160
162
164
166
168
170
172
174
176
178
180
.084
3
9
15
21
27
33
39
45
51
57
63
69
75
81
87
93
99
105
111
117
123
129
135
141
147
153
159
165
55500.
56500.
57500.
58500.
59500.
60500.
61500.
62500.
63500.
64500.
65500.
66500.
67500.
68500.
69500.
70500.
71500.
72500.
73500.
74500.
75500.
76500.
77500.
78500.
79500.
80500.
81500.
82500.
83500.
84500.
85500.
86500.
87500.
88500.
89500.
.000 .542
1 YEARS
2.00 4 
2.00 10 
2.00 16 
2.00 22 
2.00 28 
2.00 34 
2.00 40 
2.00 46 
2.00 52 
2.00 58 
1.87 64 
.84 70
20.45 76 2
66.34 82 7
100.24 88 10
100.24 94 9
66.34 100 5
20.45 106 1
.84 112
1.87 118 
2.00 124 
2.00 130 
2.00 136 
2. 
2.
2.uu iD4 
2.00 160 
2.00 166 
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.542
.084
.000
2.00
2.00
2.00
2.00
2.00
2.00
2.00
2.00
2.00
2.00
1.96
.87
10.62
49.83
92.71
103.87
81.19
33.87
4.34
1.60
1.99
2.00
2.00
2.00
2.00
2.00
2.00
2.00
2
8
14
20
26
32
38
44
50
56
62
68
74
80
86
92
98
104
110
116
122
128
134
140
146
152
158
164
2.00
2.00
2.00
2.00
2.00
2.00
2.00
2.00
2.00
2.00
1.77
2.31
6.75
4.10
2.52
6.98
8.16
5.06
.18
1.93
2.00
2.00
2.00
2.00
2.00
2.00
2.00
2.00
5
11
17
23
29
35
41
47
53
59
65
71
77
83
89
95
101
107
113
119
125
131
137
143
149
155
161
167
2.00
2.00
2.00
2.00
2.00
2.00
2.00
2.00
2.00
1.99
1.60
4.34
33.87
81.19
103.87
92.71
49.83
10.62
.87
1.96
2.00
2.00
2.00
2.00
2.00
2.00
2.00
2.00
6
12
18
24
30
36
42
48
54
60
66
72
78
84
90
96
102
108
114
120
126
132
138
144
150
156
162
168
2.00
2.00
2.00
2.00
2.00
2.00
2.00
2.00
2.00
1.98
1.31
7.07
41.64
87.43
104.31
87.43
41.64
7.07
1.31
1.98
2.00
2.00
2.00
2.00
2.00
2.00
2.00
2.00
__ ___ __ _I_ __ ___
1
7
13
19
25
31
37
43
49
55
61
67
73
79
85
91
97
103
109
115
121
127
133
139
145
151
157
163
175
2.00 176
LCUR =
LCUR =
LCUR =
LCUR =
10.00 2
10.00 8
10.00 14
10.00 20
10.00 26
10.00 32
9.98 38
9.86 44
9.37 50
7.68 56
3.12 62
6.41 68
21.71 74
40.01 80
54.71 86
59.00 92
50.69 98
33.93 104
16.05 110
2.57 116
5.09 122
8.46 128
9.61 134
9.92 140
9.99 146
10.00 152
10.00 158
10.00 164
169 10.00
175 10.00
LCUR =
LCUR =
LCUR =
LCUR =
LCUR =
20.00 2
20.00 8
19.99 14
19.98 20
19.92 26
19.76 32
19.31 38
18.25 44
16.05 50
12.04 56
5.64 62
3.15 68
13.40 74
23.14 80
29.93 86
31.79 92
28.14 98
20.10 104
9.92 110
.01 116
8.06 122
13.62 128
16.96 134
2.00 177
. PCT(LCUR)
SPCT(LCUR)
I PCT(LCUR)
SPCT(LCUR)
TIME =
10.00 3
10.00 9
10.00 15
10.00 21
10.00 27
10.00 33
9.97 39
9.82 45
9.21 51
7.17 57
1.93 63
8.58 69
24.70 75
42.92 81
56.29 87
58.45 93
48.29 99
30.84 105
13.41 111
.91 117
5.89 123
8.76 129
9.70 135
9.94 141
9.99 147
10.00 153
10.00 159
10.00 165
10.00 171
10.00 177
PCT(LCUR)
PCT(LCUR)
PCT(LCUR)
PCT(LCUR)
PCT(LCUR)
TIME =
20.00 3
20.00 9
19.99 15
19.97 21
19.91 27
19.71 33
19.19 39
17.98 45
15.53 51
11.15 57
4.33 63
4.80 69
15.12 75
24.55 81
30.62 87
31.55 93
27.06 99
18.48 105
8.19 111
1.51 117
9.16 123
14.31 129
17.34 135
2.00 178
.78
.68
.60
.53
5
10.00
10.00
10.00
10.00
10.00
9.99
9.96
9.77
9.00
6.58
.59
10.91
27.75
45.69
57.54
57.54
45.69
27.75
10.91
.59
6.58
9.00
9.77
9.96
9.99
10.00
10.00
10.00
10.00
10.00
10
20.00 4 20.00
20.00 10 20.00
19.99 16 19.99
19.97 22 19.96
19.89 28 19.86
19.65 34 19.59
19.05 40 18.89
17.68 46 17.34
14.95 52 14.32
10.19 58 9.16
2.96 64 1.51
6.48 70 8.19
16.82 76 18.48
25.85 82 27.06
31.16 88 31.55
31.16 94 30.62
25.85 100 24.55
16.82 10i 15.12
6 4.80
2 4.33
10.1~ i4q 11.15
14.95 130 15.53
17.68 136 17.98
YEARS
2.00 179
4
10
16
22
28
34
40
46
52
58
64
70
76
82
88
94
100
106
112
118
124
130
136
142
148
154
160
166
172
178
.47
.42
.38
.33
.30
10.00
10.00
10.00
10.00
10.00
9.99
9.94
9.70
8.76
5.89
.91
13.41
30.84
48.29
58.45
56.29
42.92
24.70
8.58
1.93
7.17
9.21
9.82
9.97
10.00
10.00
10.00
10.00
10.00
10.00
5
11
17
23
29
35
41
47
53
59
65
71
77
83
89
95
101
107
113
119
125
131
137
143
149
155
161
167
173
179
5
11
17
23
29
35
41
47
53
59
65
71
77
83
89
95
101
107
113
119
125
131
137
10.00
10.00
10.00
10.00
10.00
9.99
9.92
9.61
8.46
5.09
2.57
16.05
33.93
50.69
59.00
54.71
40.01
21.71
6.41
3.12
7.68
9.37
9.86
170
176
6
12
18
24
30
36
42
48
54
60
66
72
78
84
90
96
102
108
114
120
126
132
138
10.00
10.00
10.00
10.00
10.00
9.98
9.90
9.50
8.10
4.17
4.40
18.82
37.00
52.84
59.18
52.84
37.00
18.82
4.40
4.17
8.10
9.50
9.90
9.98
10.00
10.00
10.00
10.00
10.00
10.00
20.00
20.00
19.98
19.94
19..80
19.42
18.49
16.53
12.86
6.88
1.54
11.66
21.66
29.10
31.87
29.10
21.66
11.66
1.54
6.88
12.86
16.53
18.49
YEARS
1
7
13
19
25
31
37
43
49
55
61
67
73
79
85
91
97
103
109
115
121
127
133
20.00
20.00
19.99
19.95
19.83
19.51
18.70
16.96
13.62
8.06
.01
9.92
20.10
28.14
31.79
29.93
23.14
13.40
3.15
5.64
12.04
16.05
18.25
6
12
18
24
30
36
42
48
54
60
66
72
78
84
90
96
102
108
114
120
126
132
138
L __ __ __ _^ ___ __ __ I__ C_ __ In^ I_ __ _I_ _C I_
9.98 144
10.00 150
10.00 156
10.00 162
10.00 168
10.00 174
10.00 180
2.00 180 2.00 4/
145
151
157
163
169
175
1
7
13
19
25
31
37
43
49
55
61
67
73
79
85
91
97
103
109
115
121
127
133
139
145
151
157
163
169
175
19.51
19.83
19.95
19.99
20.00
20.00
LCUR =
LCUR =
LCUR =
LCUR =
LCUR =
LCUR =
LCUR =
LCUR =
LCUR =
LCUR =
LCUR =
LCUR =
LCUR =
LCUR =
LCUR =
LCUR =
LCUR =
LCUR =
LCUR =
LCUR =
59.99
59.52
58.93
58.13
57.01
55.45
53.40
50.80
47.68
44.15
40.40
36.71
33.41
30.83
29.24
28.83
29.64
31.59
34.45
37.92
41.66
45.36
48.77
51.72
54.14
56.02
57.42
58.41
59.12
59.63
LCUR =
TIME
59.92
59.43
58.82
57.97
56.78
55.15
53.00
50.31
47.11
43.53
39.77
36.13
32.92
30.49
29.08
28.88
29.89
32.01
34.99
38.53
42.28
45.95
49.30
52.16
54.49
56.28
57.61
58.55
59.21
59.71
3
9
15
21
27
33
39
45
51
57
63
69
75
81
87
93
99
105
111
117
123
129
135
141
147
153
159
165
171
177
30 PCT(LCUR)
146 19.59 147
152 19.86 153
158 19.96 159
164 19.99 165
170 20.00 171
176 20.00 177
10 PCT(LCUR)
11 PCT(LCUR)
12 PCT(LCUR)
13 PCT(LCUR)
14 PCT(LCUR)
15 PCT(LCUR)
16 PCT(LCUR)
17 PCT(LCUR)
18 PCT(LCUR)
19 PCT(LCUR)
20 PCT(LCUR)
21 PCT(LCUR)
22 PCT(LCUR)
23 PCT(LCUR)
24 PCT(LCUR)
25 PCT(LCUR)
26 PCT(LCUR)
27 PCT(LCUR)
28 PCT(LCUR)
29 PCT(LCUR)
2
8
14
20
26
32
38
44
50
56
62
68
74
80
86
92
98
104
110
116
122
128
134
140
146
152
158
164
170
176
19.65
19.89
19.97
19.99
20.OC
20.0C
30
59.84
59.34
58.69
57.8C
56.54
54.83
52.59
49.81
46.54
42.91
39.15
35.55
32.4E
30.17
28.97
28.97
30.17
32.45
35.55
39. 15
42.93
46.54
49.81
52.59
54 .8
56.54
57.79
58.6E
59.31
59.78
148
154
160
166
172
S178
.26
.23
.19
.16
.13
.10
.08
.05
.02
.00
.03
.05
.08
.10
.13
.15
.17
.20
.22
.24
YEARS
4
10
16
22
28
34
40
46
52
58
S64
70
76
82
88
94
100
S106
i 112
118
124
130
136
142
148
154
160
166
172
178
19.71
19.91
19.97
19.99
20.00
20.00
59.76
59.24
58.56
57.62
56.29
54.49
52.17
49.30
45.95
42.28
38.53
34.99
32.01
29.89
28.88
29.08
30.49
32.92
36.13
39.77
43.53
47.11
50.31
53.00
55.15
56.77
57.96
58.80
59.39
59.85
149
155
161
167
173
179
5
11
17
23
29
35
41
47
53
59
65
71
77
83
89
95
101
107
113
119
125
131
137
143
149
155
161
167
173
179
19.76
19.92
19.98
19.99
20.00
20.00
59.68
59.14
58.43
57.42
56.02
54.14
51.72
48.77
45.36
41.66
37.92
34.45
31.59
29.64
28.83
29.24
30.83
33.41
36.71
40.40
44.15
47.68
50.80
53.40
55.45
57.00
58.12
58.91
59.48
59.92
150
156
162
168
174
180
6
12
18
24
30
36
42
48
54
60
66
72
78
84
90
96
102
108
114
120
126
132
138
144
150
156
162
168
174
180
19.80
19.94
19.98
20.00
20.00
20.00
59.60
59.04
58.28
57.22
55.75
53.78
51.27
48.23
44.76
41.03
37.31
33.92
31.19
29.42
28.81
29.42
31.19
33.92
37.31
41.03
44.76
48.23
51.27
53.78
55.74
57.21
58.27
59.02
59.56
59.99
.26
CASE B NOURISHMENT WITH STRUCTURES
PRESENT
In this case, all of the description presented for Case A is relevant with the exceptions
noted below. Because Steps 1, 2, 3, and 4 are identical, they will not be repeated here.
STEP 4B Specify a Reference Background Transport
As has been described earlier, in situations where structures are present, it is necessary to
establish the net background longshore transport rate as this will interact with the structure.
The net longshore background transport on the east coast of Florida could be estimated
from Figure 30. Since background transport rates on the west coast are so variable spatially,
no attempt will be made here to provide a recommendation. Rather, it is suggested that
each rate should be developed on a casebycase basis.
The background transport rate is specified to the program on Card 3 as QBKREF and
the grid index value associated with the background transport rate QBKREF is specified
as IREF on Card 3. Note that QREF must be specified in units of ft3/second and that the
conversion factor from cubic yards per year to cubic feet per second is
Q(cubic feet per second) = 8.56 x 107 Q(cubic yards per year)
STEP 5B Specify Structure Location(s) and Length(s) in Program
In the current version of the program, up to 10 structures can be specified including the
grid line and length. The structures interact with the background sediment transport and
the transport induced by the beach nourishment project.
Specification of the structure number, location and length is by Card 4 (this card present
only if structures are specified).
S 0 1 jHOLUEM "___
,/SANTA I HO .0C0,. 00
( r WALTO S'D  CS 600,000 yd3/yr
S ^7 BAY \ N 0. / \/ HAM LTON. ) JACKSONVILLE r"
i4 / 1, WjMADISON'wn / I ^1 r
LIBERy WAKULLA 4 BAKER (
TAYLORI I B am ,,  "
,GULF FRANKLIN R ST. AUGUSTINE
FORD 'S
I CHRIST O MARINELAND
GU. 0o ,'LEVY F',. X
LE\ I I RION AYTONA 3
S AON 'voLUSI 500,000 yd /yr
.EW SMYRNA
0 CITUS ILAKE L ...
,SUMTE'R I O"' / \
HERNANDb'
P A I I ORANGE Q
PASCO ' APE CANAVERAL
N.1* ._ CI \\OSSCEOLAv\l\l. ,
Sr, POLK oSSCEOLA 350,000 yd /yr
 i i ID
LL 4 1l: RI RO BEACH
A MANATEE HARDEE OKEE 1 230,000 yd /yr
S'" nxDu== I ,,KCHBEE 1 ST. \ l ,
S ) HIo C LCIE FiT. PIERCE
LAKE MARTIHN
OKEECHO JUPITER I.
HARLOTTL BADEE \ 230,000 yd/yr
S I I PALM BEACH
'7LEE HENRYY I PALM BEACH
Alon ...... EERF ELD
COLLIER BROWARO 120,000 yd3/yr
^ BAKERS HAULOVER
.. 47 10,000 yd 3/yr
'NRO oADE MIAMI
Figure 30. Estimates of Net Annual Longshore Sediment Transport
Along Florida's East Coast.
EXAMPLES ILLUSTRATING APPLICATION OF
METHODOLOGIES
In this section, a number of examples are presented illustrating application of the method
ologies. The purpose of these examples is to familiarize the reader thoroughly with the
methodologies and the anticipated results. As in preceding sections of this report, the
examples will be organized by "Graphical Methodology" and "Numerical Methodology".
Graphical Example
The following four examples illustrate application of the methodology to the following
situations.
Graphical Example 1: Uninterrupted Shoreline, No Background Erosion
Graphical Example 2: Uninterrupted Shoreline, Uniform Background Erosion
Graphical Example 3: Uninterrupted Shoreline; NonUniform Background Erosion
Graphical Example 4: Downdrift of a Littoral Barrier, NonUniform Background Erosion
The computations and results are presented on the following four worksheets.
Numerical Examples
A number of examples were run with the numerical methodology and are described
briefly on the following page. Because the documentation for each example is fairly exten
sive, each example is presented in an individual appendix.
Numerical Example 1:
Numerical Example 2:
Numerical Example 3:
Numerical Example 4:
Numerical Example 5:
Numerical Example 6:
Numerical Example 7:
Uninterrupted Shoreline, No Background Erosion, Nourish
ment Length = 2 Miles, Initial Added Width = 112 ft, Wave
Height = 2.0 ft, Waves Normally Incident, Results Presented
in Appendix C.
Uninterrupted Shoreline, Uniform Background Erosion of 2
ft/yr, Nourishment Length = 2 Miles, Initial Added Width
= 112 ft, Wave Height = 2.0 ft, Waves Normally Incident,
Results Presented in Appendix D.
Uninterrupted Shoreline, Variable Background Erosion,
Nourishment Length of 2 Miles, Initial Added Width = 112
ft, Wave Height = 2.0 ft, Waves Normally Incident, Results
Presented in Appendix E.
Uninterrupted Shoreline, No Background Erosion, Nourish
ment Length = 3,500 ft, Wave Height = 2.0 ft, Waves Nor
mally Incident, Results Presented in Appendix F.
One Structure 112 ft Long Located at North End of Nourish
ment Project, Nourishment Length = 2 Miles, Initial Added
Width = 112 ft, Wave Height = 2.0 ft, Waves Normally In
cident, No Background Erosion, Results Presented in Ap
pendix G.
One Structure 112 ft Long Located at South End of Nour
ishment Project, Uniform Background Erosion of
2 ft/yr, Waves Normally Incident, Nourishment Length = 2
Miles, Initial Added Width = 112 ft, Wave Height = 2.0 ft,
Results Presented in Appendix H.
One Structure 112 ft Long Located at South End of Nour
ishment Project, Waves Approaching at 100 Angle to Shore
line, Variable Background Erosion, Nourishment Length = 2
Miles, Initial Added Width = 112 ft, Wave Height = 2.0 ft,
Results Presented in Appendix I.
For each numerical example, the input file, DNRBS.INP, and output file, DNRBS.OUT,
are presented and the results are discussed and plotted.
RAPH ICAL EXtmYPILe i
(z4ero APc4lJqroC't4'1 Eyosi ")
BEACH NOURISHMENT PROJECTION
(Graphical Computations, Uninterrupted Shoreline)
General Location:
Wave Height, Ho (Fig. 23): a ft, Closure Depth, h, (Fig. 8): /7 ft
Wave Period, T (Fig. 24): 6( sec, Sediment Size, D: mm
Wave Direction, ao: 0 o, Transport Factor, K (Fig. 5): 0,77
Berm Height, B: bo ft
Alongshore Diffusivity, G (From Equation below or Figure 15).
SK H ^Cog0.4 cos(io ao) cos 2(8o a.)
8 (s 1)(1 p)C..0.4(h + B) cos(8o a.)
o0,77 olf .o i)^ ioh117 ft2/A
Background ( oErosion Equilibrated Beach Width, yo.
Background Erosion Equilibrated Beach Width, Ayo
z Erosion Rate (ER)
o ft/yr
For 30 years
AN (Fig. 7) or From Profile:
Ap (Fig. 7):
Volume Per Unit Length:
Ayo (Figs. 11 and 12):
Project Length, e, = 2 O miles =
0.Z o ft'/3
O.iO ft'/3
5"/1S ft3/ft
112. ft
/; f, /t,
(1) (2) (3) (4) (5) (6)
Distance Y ( z) /Ayo y, YB(ft) = N =
From Center, x(ft) (Fig. 17) (ft) 30 x ER y, y, (ft)
_o.o _0 O, 2z 31t1 0 3/.1
.52 o To Z..2, 2.1/1 0 2 _'r_
/OJ 5'f 0 O.22 ^ 2q.,o
6 (Pp I CpL C rAmPLE 2
CUnrform %ivLd roun^ Eros(on
BEACH NOURISHMENT PROJECTION
(Graphical Computations, Uninterrupted Shoreline)
General Location:
Wave Height, Ho (Fig. 23): .0 ft, Closure Depth, h. (Fig. 8): 17 ft
Wave Period, T (Fig. 24): (0oO sec, Sediment Size, D: mm
Wave Direction, ao: Oo __, Transport Factor, K (Fig. 5): 01'7
Berm Height, B: & ft
Alongshore Diffusivity, G (From Equation below or Figure 15).
,, K H4C g0 cos(Po ao) cos 2(fo c.)
Background Erosion
8 (s 1)(1 p)C.c0.4(h. + B) cos(/o a.)
XU L I= Dl0,f7 ft2/s
Equilibrated Beach Width, Ayo
x Erosion Rate (ER)
,52.o 3,o ft/yr
or 30 yars,
For 30 years
AN (Fig. 7) or From Profile:
AF (Fig. 7):
Volume Per Unit Length:
Ayo (Figs. 11 and 12):
Project Length, = A:' 0 miles =
O, .z !fti/3
:', 2 O ftl/3
.5// 3 ft3/ft
// ft
/o, s' ft
16Gt G(30x365x24x3600)
16= 16 2= / 5 =
2 J2
(1) (2) (3) (4) (5) (6)
Distance y )1 /Ayo y, /B(ft) = N =
From Center, x(ft) (Fig. 17) (ft) 30 x ER y, ys (ft)
o o_._ 0..28 31,/4 & 8.>
528O o ,. 2 O zq, _o oq
lo_ e _so o._z ( 1. 6 _
6i RAP I CAL. EXAMPLE ,
(%Noyi LU l 4.oy ErosiQov)
BEACH NOURISHMENT PROJECTION
(Graphical Computations, Uninterrupted Shoreline)
General Location:
Wave Height, Ho (Fig. 23): 0. ft, Closure Depth, h. (Fig. 8): oI ft
Wave Period, T (Fig. 24): 6, sec, Sediment Size, D: mm
Wave Direction, ao: 0 .0 o, Transport Factor, K (Fig. 5): 0,TT
Berm Height, B: ,o ft
Alongshore Diffusivity, G (From Equation below or Figure 15).
=f K H^C g0o4 cos(o a0) cos 2(8o a,)
8 (s 1)(1 p)C.C0.4(h. + B) cos(0o a*)
= Sa4g e9 X.?Qg/e ( ^ I 0
,Iq47 ft2 /s
Background Erosion
Equilibrated Beach Width, Ayo
x Erosion Rate (ER)
I__o ,o ft/yr
For 30 years
AN (Fig. 7) or From Profile:
AF (Fig. 7):
Volume Per Unit Length:
Ayo (Figs. 11 and 12):
Project Length, 1, = R*, ) miles =
.2. S ftl/3
0 ft1/3
5"/ 3 ft3/ft
112. ft
/(j 56 0 ft
(1) (2) (3) (4) (5) (6)
Distance 2 y (N) /Ayo y, yB(ft)= y =
From Center, x(ft) (Fig. 17) (ft) 30 x ER y, ye (ft)
._tQ52>o . 0 .Cz 2.t> 30,0 5" 4
20o o. z2.2I 3173 g.4
o o o.2< 31,cW 45., 13.o
_2_80 i o,2i. 2?,I 2, 2.3 *_
+__,_ IZr7. 10..2. :::/." 3. 4
SRPH%\CAL E4lm L LL
(NoYI ^YIqoH^ E nsso
BEACH NOURISHMENT PROJECTION
(Graphical Computations, Downdrift of a Littoral Barrier)
General Location:
Wave Height, Ho (Fig. 23): 'OD ft, Closure Depth, h, (Fig. 8): 170 ft
Wave Period, T (Fig. 24): ( .'o sec, Sediment Size, D: mm
Wave Direction, ao: 0, Transport Factor, K (Fig. 5): 0o'7
Berm Height, B: 0 ft
Alongshore Diffusivity, G(From Equation Below or Figure 15)
K HOC/g0o4 cos(#o ao) cos 2(fo a.)
8 (a 1)(1 p)C,0o4(h. + B) cos(#o a.)
Background Erosion
,= J147 ft2/s
Equilibrated Beach Width, Ayo
x Erosion Rate (ER)
0 ,o.' ft/yr
fBo )_ o
F orbs 8 .O
2 izo 1 4,0
For 30 years
AN (Fig. 7) or From Profile:
AF (Fig. 7):
Volume Per Unit Length:
Ayo (Figs. 11 and 12):
Project Length, e, = miles
Effective Project
Length, e' = 2 = __ miles =
0o,2 ftl/3
0, 2 ft'/3
1/ 1 ft3/ft
//2 ft
= /2, $i: ft
21 ; /2< ft
Gt 1G(30x365x24x3600)
16 = 16 = :: S T =
(e)2 (2)2
(1) (2) (3) (4) (5) (6)
Distance y I() /Ayo y, yB(ft) = UN =
From Littoral Barrier, X(ft) (Fig. 17) (ft) 30 x ER y, yb (ft)
0 0 OSz. 58.. Goo s z
5z80 0.5 0.50 go.o 4 o 4z."
o6s60o .o o0.q. 477o a24 1i2.
2/)20o 2,0o o,. ILo 2 5
r
REFERENCES
Balsillie, J. (1987) "Offshore Profile Description Using the Power Curve Fit, Part II: Stan
dard Florida Offshore Profile Tables", Beaches and Shores, Technical and Design
Memorandum No. 821IIa, Florida Department of Natural Resources, Tallahassee,
FL.
Bruun, P. (1954) "Coast Erosion and the Development of Beach Profiles", Beach Erosion
Board, Technical Memorandum No. 44.
Dean, R. G. (1977) "Equilibrium Beach Profiles: U.S. Atlantic and Gulf Coasts", Depart
ment of Civil Engineering, Ocean Engineering Report no. 12, University of Delaware,
Newark, DE.
Dean, R. G. (1978) "Review of Sediment Transport Relationships and the Data Base",
Proceedings of a Workshop on Coastal Sediment Transport with Emphasis on the
National Sediment Transport Study", Report DELSG1578, University of Delaware,
Newark, DE.
Dean, R.G. (1987) "Additional Sediment Input to the Nearshore Region", Shore and
Beach, Vol. 55, Nos. 34, p. 7681.
Moore, B.D. (1982) "Beach Profile Evolution in Response to Changes in Water Level
and Wave Height", Masters Thesis, Department of Civil Engineering, University of
Delaware, Newark, DE.
Pelnard Considere, R. (1956) "Essai de Theorie de l'Evolution des Formes de Rivate en
Plages de Sable et de Galets", 4th Journees de l'Hydraulique, Les Engergies de la
Mar, Question III, Rapport No. 1.
I
APPENDIX A
DEEP WATER WAVE EQUIVALENTS
FOR SHORELINE MODELING
APPENDIX A
DEEP WATER WAVE
EQUIVALENTS FOR SHORELINE MODELING
Consider the transport equation
EbCG6 cos(p8 ab) sin(#, ab)
pg(s 1)(1 p)
I I
I \l
I
I I I 1
//I/I
/ / I
/1 / I
1 / /
I I /
I I I
(A.1)
N ,^
I
I
I
1
I.
I
I
I
I
Definition Sketch
The bathymetry of concern will be considered as straight and parallel bottom contours
seaward of the effects of a beach nourishment project. This seaward depth limit is denoted as
h,. For depths smaller than h., it is assumed that all contours are parallel to the shoreline.
The azimuth, ,g, of the outward normal within the depth limit affected by the nourishment
project is related to the azimuth of the outward normal, 0o, outside the limit of the project
by
P(s) = o + A /(x) (A.2)
in which A/ is small.
Using conservation of energy and Snell's law to transform Eq. (A.1) from the breaker
line to the depth contour h,,
S E*CG. cos(W, a*) sin(1, a)(A.3)
q =K Cb (A.3)
pg ( 1)(1 p)C*
and using Eq. (A.2)
K E, CG, sin 2(Po + Af ac,) CA.4)
2 pg(s 1)(1 p) C,
and expanding
E, CG. sin 2(Po a.) C
Q = K cos 2AP
2 pg(s 1)(1 p) C.
E. CGc cos 2(1o ac ) Ob
+ K co sin 2A C (A.5)
2 pg(s 1)(1p) C,
Since Ap is small, cos 2A/3 P 1 and sin 2A/p 2Afl, and the first term is recognized as
the transport without the project present (the background transport, QB) and the second
term the transport induced by project placement, Qp.
The background transport will first be expressed in terms of deep water wave charac
teristics
iK E*CG cos(?o ac) sin(fo a*) Cb
BACKGROUND = QB = K
SK EoCGo cos(po ao) sin(flo so) Cb
pg(s 1)(1 p) Co
Eq. (A.6) contains Cb which we now wish to relate to deep water conditions. Using
energy conservation,
EbCGb cos(P, ab) = E*CG. cos(P, a,) = ECG. cos(fo + Ari a,)
Therefore
EbCG cos(/a, ab) = E.CG. [cos(Po ca.) cos Ap sin(p3, ac) sin AP]
and since Ap8 is small, the last term can be neglected and cos Ap8 P 1. Finally
EbCG, cos(f3, ab) : EoCco cos(/?o c0o)
and employing the following shallow water approximations
CGb Cb '/ "gh.
Hb ; z he ( Fas 0.78)
g92HCb cos(/, ab) = g,2HCGo cosQ(3o ao)
K'C5 cos(P ab) = g2H CGo cos(PO O)
gHb= [2 G cos(fo Co) (A.7)
in which cos(p, as) has been approximated by unity.
Returning now to the project transport and using conservation of energy considerations
and Snell's law to transform to deep water
K EoCGo cos 2(Po a.) cos(po ao) Cb
pg(s 1)(1 p)cos(Po a) C,
Employing Eq. (A.7), the project related transport can now be written without reference
to shallow water
K H2.^C12g04 cos12 (Po o) 1
QP = V cos 2(0o a.) A~ (A.9)
8(s 1)(1 p) cos(Po a*,)oc04 *CA
Using Snell's law,
?o a* = sin1 C[ sin(Po ao)] (A.10)
rio s, = sinJ o
The shore planform direction anomaly Ap is
A tan (A.11)
ax ax
Combining Eqs. (A.8) and (A.11) with the continuity equation
ay 1 aQp
at (h. + B) ax
we find
ay K H24C o^'g0.4
at 8(s 1)(1 p)Cr.04(h, + B)
Defining the longshore diffusivity,
cos 2(# ao) cos 2(/o a,) a2y
[ cos(Po a.) I x2
K H24 C12^4 [cos1.2( o ao) cos 2(/o a.) (
8(s 1)(1 p)C*.0.4(h. + B) cos(o a.) (
and it is noted that G is now expressed entirely in terms of deep water wave quantities
(with the use of Eq. (A.10)). The diffusion equation for shoreline evolution is obtained in
the usual form
ay 82y
at Ga2Y (A.14)
at 82
We now consider the equations that will be used for numerical analysis. Commencing
with Eq. (A.3) and inserting Eq. (A.2) in the cosine term
Q K ECG. [cos(Po a.) cos Ap sin(olo a.) sin A] sin( )C (A.15)
Q = (, a)C (A.15)
/o "AJ, P)*
and since Ap is small and using conservation of energy
KEoCGo cos(po o) ,
Q = sin(p, a,)C
pg(s 1)(1 p)C*
Combining Eq. (A.7) with the expression for deep water wave energy, Eo
Eo = pg
8
yields
K H2^C4g04 cos12(o ao)
Q = 8( sim(p )o
8(s 1)(1 p)C,. n.4 sin( )
(A.16)
(A.17)
(A.18)
and
a, = o sin1 [C sin(lo ao)] (A.19)
which completes the development. It is noted that with the exception of the trigonometric
term involving (V, a,) and the term C., all quantities are expressed in terms of deep
water conditions.
(A.12)
Representative Wave Conditions
To simplify input conditions it is desirable to define representative wave characteristics.
In developments here, we will consider a constant wave direction, but timevarying wave
height and period. At each time, the waves will be considered as represented by a single pe
riod and a Rayleigh wave height distribution with significant wave height H,. The effective
height is thus
Heff = [ H 4p(H)dH] (A.20)
in which all wave heights are in deep water and p(H) is the Rayleigh distribution,
p(H) = e(H/Hm)2 (A.21)
p = rms
Eq. (A.20) can be solved numerically to yield
Heff = Krm,Hrms = K,H, (A.22)
where Krms = 1.04 and K, = 0.735. Thus the longterm effective wave height Heff at a
particular location is
1
Heff= (KH,)4 (A.23)
n=1 I
A somewhat more appropriate but more cumbersome value of Heff is
N :I (K.,H,,)2.4 C 24,1
Heff2 N= 1  (A.24)
N n=1 c
c 1.2
and the effective value of to be used in Eq. (A.18) is the denominator of Eq. (A.24)
raised to the 2.4 power. The recommended values of effective deep water wave height around
the state of Florida are plotted in Figure 23.
APPENDIX B
PROGRAM LISTING
AND
SAMPLE INPUT AND OUTPUT
Program:
Input File:
Output File:
DNRBS.FOR
DNRBS.INP
DNRBS.OUT
(Note: Input and Output Files Presented for Numerical Example 2)
PROGRAM LISTING: DNRBS.FOR
C
C THIS PROGRAM DEVELOPED FOR DIVISION OF BEACHES AND SHORES,
C DEPARTMENT OF NATURAL RESOURCES FOR USE IN PREDICTING *
C THIRTY YEAR EROSION PROJECTIONS **
C
C *********************************************************************
C
DIMENSION YO(500),YN(500),X(500),Q(500),HB(500),ALP(500),
1 XER(40),EROSB(40),SUMA(50),VTOTA(50),YEARA(50),
2 ITNOUR(10),ISEG(10),IS(10,10),IE(10,10),DY(10,10),
3 WORD(20),YEAR(10),DV(10,10),NSEG(10),PCT(50),DYO(500)
4 ,QBACK(500),YSTRUC(10),ISTRUC(10)
OPEN(UNIT=6,FILE='DNRBS2.OUT',STATUS='NEW')
OPEN(UNIT=5,FILE='DNRBS2.INP',STATUS='OLD')
OPEN(UNIT=7,FILE='DNRBS2.DAT',STATUS='NEW')
55 FORMAT('***** IT = 1, I=1, EROSION RATE = ',E12.2)
120 FORMAT(6(I4,F8.2))
121 FORMAT(/,5X,'NTIME = ',16,' HB = ',F8.2,' ALP = 'F8.3,' SUM =
1 F8.2,' STDEV = ',F8.2,/)
122 FORMAT(//)
123 FORMAT(5F8.2,416)
124 FORMAT(8F8.2)
125 FORMAT(4(E8.2,F8.2))
126 FORMAT(20A4)
127 FORMAT(20A4,/)
160 FORMAT(816)
162 FORMAT(F8.2,3I6,2F8.2)
164 FORMAT(816)
165 FORMAT(/)
166 FORMAT(I6,3F8.2)
167 FORMAT(' INITIAL SHORELINE (INCL. NOURISHMENT) POSITION',/)
168 FORMAT(I6,F8.1,2E12.4,F8.2)
170 FORMAT(' HO =',F6.2,' FT., T =',F6.2,' SEC., ALPO = ',F6.2,' DEG.
1, BTAO = ',F6.2,' DEG., '
2 ,5X,' XMU =',F8.2,' DEG., DX = ',F8.2,' FT., DT = ',F8.2,' SEC.')
172 FORMAT(' HSTR = ',F8.2,' FT., B = ',F8.2,' FT., XK = ',F8.2,
1' VFACT = ,F8.2,14X,'QBKREF = ',F8.2,' FT.**3/SEC.')
173 FORMAT(' IREF = ',6,', IMAX = ',16,', NTIMES = ',18,
1 ', NS = ',16)
444 FORMAT(20X,'TIME = ',18,' YEARS')
446 FORMAT(' NYEARS = ',18,' DYSITE = ',F8.2)
447 FORMAT(' BACKGROUND EROSION TRANSPORT RATES',/)
448 FORMAT(5(I6,F8.3))
449 FORMAT(216,8F8.3)
GRAV=32.2
NER=7
SG=2.65
POR=0.35
PI=3.14159
PI02=PI/2.0
ITNM=1
XKAP=0.78
QBACK(1)=0.0 73
LCUR=0
READ(f .126)1 WORD(IT) .=1.20)
WRITE(6,127)(WORD(I),I=1,20)
WRITE(7,126)(WORD(I),I=1,15)
READ(5,124)HO,T,ALPO,BTAO,XMU,DX,DT
READ(5,123)HSTR,B,XK,VFACT,QBKREF,IREF,IMAX,NTIMES,NS
IF(NS.GT.O)READ(5,448)(ISTRUC(I),YSTRUC(I),I=1,NS)
WRITE(7,170)HO,T,ALPO,BTAO,XMU,DX,DT
WRITE(7,172)HSTR,B,XK,VFACT,QBKREF
WRITE(6,170)HO,T,ALPO,BTAO,XMU,DX,DT
WRITE(6,172)HSTR,B,XK,VFACT,QBKREF
WRITE(6,173)IREF,IMAX,NTIMES,NS
WRITE(6,165)
IF(NS.GT.0) WRITE(6,448)(ISTRUC(I),YSTRUC(I),I=1,NS)
ALPO=ALPO*PI/180.0
BTAO=BTAO*PI/180.0
XMU=XMU*PI/180.0
READ(5,124)(XER(I),EROSB(I),I=1,NER)
WRITE(6,165)
WRITE(6,125)(XER(I),EROSB(I),I=1,NER)
WRITE(*,125)(XER(I),EROSB(I),I=1,NER)
READ(5,160)NNOUS,NNOUE
WRITE(*,160)NNOUS,NNOUE
DO 60 I=NNOUS,NNOUE
READ(5,166)I,DYO(I)
60 DYO(I)=DYO(I)*VFACT
TOTH=HSTR+B
IMM1=IMAX1
IMP1=IMAX+1
DO 30 I=1,IMP1
X(I)=(I1)*DX
YN(I)=0.0
30 YO(I)=0.0
C**** FOLLOWING IS BACKGROUND EROSION AND ASSOCIATED TRANSPORT
DO 240 I=1,IMAX
CALL INTERP(EROSB,ERC,NER,X,XER,I,DT,QBACK,TOTH,DX,IREF)
240 CONTINUE
DQ=QBACK(IREF)QBKREF
DO 241 I=1,IMP1
241 QBACK(I)=QBACK(I)DQ
CALL WVNUM(HSTR,T,CC)
CO=GRAV*T/(2.0*PI)
CGO=CO/2.0
ALPSTR=BTAOASIN(CC/CO*SIN(BTAOALPO))
C WRITE(6,124)HSTR,T,CC,CO,CGO,ALPO,BTAO,ALPSTR
CALP=COS(ALPOALPSTR)
SALP=SIN(ALPOALPSTR)
WRITE(6,165)
WRITE(6,447)
WRITE(6,448)(I,QBACK(I),I=1,IMP1)
WRITE(6,165)
WRITE(6,160)NNOUS,NNOUE
WRITE(6,167)
C ***** FOLLOWING IS TIME LOOP
DO 300 NT=1,NTIMES
IF(MOD(NT,10).EQ.0) WRITE(*,*) NT,NTIMES
BB=XK*HO**2.4*CGO**1.2*GRAV**0.4*COS(BTAOALPO)**1.2/
1 (8.0*(SG1)*(1.0POR)*CC*XKAP**0.4)
SUM=0.0
SUM2=0.0
NFLAG=0
IF(NFLAG.EQ.1) GO TO 302
IF(NT.EQ.1.OR.NT.EQ.0) CALL NOUR(NT,ITNM,YO,IMAX,ITNOUR,
1 NSEG,IS,IE,DY,VTOT,IT,DV,X,NNOUS,NNOUE,DYO,DX,TOTH)
C YO(1)=0.0
C YO(IMAX)=0.0
C*****FOLLOWING IS TRANSPORT LOOP
BTA=XMUATAN2((YO(I)YO(I1)),(X(I)X(I1)))PI02
COSC=COS(BTAALPO)
SINC=SIN(BTAALPO)
Q(I)=BB*SIN(BTAALPSTR)
QB=QBACK(I)
QSAVE=Q(I)
CALL STR(NS,YSTRUC,I,YO,Q,IMAX,DX,ALPC,XMU,QB,BB,PI02,
1 ISTRUC,ALPSTR)
IF(NT.EQ.100.AND.I.EQ.116)WRITE(6,449)NT,I,Q(I),QB,
1 YSTRUC(1,Y(1)O(I,YO(I),QBACK(I),QSAVE
Q(I)=Q(I)+QB
100 CONTINUE
YN(1)=YO(1)
YN(IMAX)=YO(IMAX)
Q(1)=QBACK(1)+Q(2)QBACK(2)
Q(IMP1)=QBACK(IMP1)+Q(IMAX)QBACK(IMAX)
C******FOLLOWING IS FOR CONTINUITY EQUATION
DO 200 I=1,IMAX
IF(I.GT.1)GO TO 266
DX=X(2)X(1)
GO TO 268
266 DX=(X(I+1)X(I1))/2.0
268 CONTINUE
AA=YO(I)
YN(I)=YO(I)DT/(DX*TOTH)*(Q(I+I)Q(I))
YO(I)=YN(I)
IF(I.NE.1.OR.NT.NE.10)GO TO 200
WRITE(7,449)I,NT,AA,YN(I),DT,DX,TOTH,Q(I+1),Q(I)
200 CONTINUE
C WRITE(6,120)(I,YN(I),I=1,IMP1)
C WRITE(6,120)(I,Q(I),I=1,IMP1)
IF(MOD(NT,365).NE.O) GO TO 300
C IF(MOD(NT,3650).NE.0) GO TO 301
NYEARS=NT/365
NZC=NYEARS
IF(NZC.NE.1.AND.NZC.NE.5.AND.NZC.NE.10.AND.NZC.NE.30)GO TO 301
WRITE(6,444)NYEARS
WRITE(6,120)(I,YN(I),I=1,IMAX)
301 CALL PERCT(YN,DX,SUM,PCT,VTOT,LCUR,LCURM,SUMA,VTOTA,TOTH,X
1 ,NNOUS,NNOUE)
YEARA(LCUR)=1990.0+(NT1)*DT/31536000.0
300 CONTINUE
WRITE(7,168)(L,YEARA(L),SUMA(L),VTOTA(L),PCT(L),L=1,LCURM)
DYSITE=0.5*(YN(26)+YN(27))62.06NYEARS*2.31
C NZC=NYEARS
C IF(NZC.NE.1.OR.NZC.NE.5.OR.NZC.NE.10.OR.NZC.NE.30)GO TO 302
C WRITE(6,446)NYEARS,DYSITE
WRITE(7,120)(I,YN(I),I=1,IMP1)
C WRITE(6,120)(I,Q(I),I=1,IMP1)
302 CONTINUE
CLOSE(UNIT=5)
CLOSE(UNIT=6)
CLOSE(UNIT=7)
STOP
END
C
C r*******X'**********I1*
C
SUBROUTINE INTERP(EROSB,ERC,NER,X,XER,I,DT,QBACK,TOTH,DXB,IREF)
DIMENSION EROSB(40),XER(40),X(400),QBACK(400)
100 FORMAT(216,6F10.3)
101 FORMAT(6E12.4) 75
XC=X(I)
CON=DT/31536000.0
DO 10 IER=2,NER
*iT / Tm TI r n vr m rn \ \ Iin mrr 1 A____.n II ^n /m 7 nIr nNI
DX=XER(IER)XER(IER1)
DXX=XCXER(IER1)
AA=DXX/DX
BB=1.0AA
ERC=CON*(BB*EROSB(IER1)+AA*EROSB(IER))
QBACK(1+1)=QBACK(I)DXB*TOTH*ERC/DT
IF(I.NE.2)GO TO 6
C WRITE(6,100)I,IER,ERC,DT,TOTH,DX,AA,BB
C WRITE(6,101)QBACK(I),QBACK(I1),QBACK(I+1),CON,DXB
6 GO TO 20
10 CONTINUE
20 RETURN
END
C
C *********
C
c
SUBROUTINE NOUR(NT,ITNM,YN,IMAX,ITNOUR,NSEG,
1 IS,IE,DY,VTOT,ITC,DV,X,NNOUS,NNOUE,DYO,DX,TOTH)
DIMENSION YN(500),ITNOUR(10),NSEG(10),DY(10,10),
1 IS(10,10),IE(10,10),DV(10,10),YNT(500),
2 X(500),DYO(50)
24 FORMAT(' OUTPUT FROM SR NOUR ',16,' ISC = ',16,' IEC = ',16)
26 FORMAT(' REACHED SR NOUR',216,F8.2)
28 FORMAT(' NOUR EVENT = ',16,' YEAR = ',F8.2,
1 VOL ADDED = ',F8.3,' MILL YDS**3',/)
30 FORMAT(2(I6,F10.0,F8.2))
32 FORMAT(' TOTAL VOLUME ADDED = ',F12.1 ,' CUBIC YARDS',/)
VTOTT=0.0
FACT=1.0
C IF(NT.NE.1)FACT=0.5
DO 6 I=NNOUS,NNOUE
6 YN(I)=YN(I)+DYO(I)*FACT
DO 12 I=NNOUS,NNOUE
12 VTOTT=VTOTT+(X(I+1)X(I1))/2.0*YN(I)
VTOT=VTOTT
C WRITE(6,32)VTOT
C WRITE(7,32)VTOT
WRITE(6,30)(I,X(I),YN(I),I=1,IMAX)
RETURN
END
C
C ************* THIS SUBROUTINE CALCULATES PERCENTAGES OF
C TOTAL VOLUME REMAINING
SUBROUTINE PERCT(YN,DX,SUM,PCT,VTOT,LCUR,LCURM,SUMA,VTOTA,TOTH,X
1 ,NNOUS,NNOUE)
DIMENSION YN(400),PCT(50),SUMA(50),VTOTA(50),X(200)
24 FORMAT(5X,'LCUR = ',16,' PCT(LCUR) = ',F8.2)
SUM=0.0
DO 20 I=NNOUS,NNOUE
20 SUM=SUM+(X(I+1)X(I1))/2.0*YN(I)
LCUR=LCUR+1
LCURM=LCUR
SUMA(LCUR)=SUM
VTOTA(LCUR)=VTOT
PCT(LCUR)=SUM/VTOT
WRITE(6,24)LCUR,PCT(LCUR)
WRITE(*,24)LCUR,PCT(LCUR)
RETURN
END
C
C*********THIS SUBROUTINE CHECKS PFR Amn ACCOUNTS FOR THE TRANSPORT
C AROUND STRUCTURES 76
C
SUBROUTINE STR(NS,YSTRUC,I,YO,Q,IMAX,DX,ALPC,XMU,QB,BB,PIO2,
1 ISTRUC,ALPSTR)
,____ h T n X T I t \ I rI i \
18 FORMAT(316,6F8.2)
C WRITE(*,18)NS,I,I,YSTRUC(1)
DO 20 IS=1,NS
IC=IS
20 IF(I.EQ.ISTRUC(IS))GO TO 40
GO TO 80
40 DYP=YO(I)YSTRUC(IC)
DYM=YO(I1)YSTRUC(IC)
C WRITE(6,18)I,ISTRUC(IC),IC,DYP,DYM
DXC=DX/2.0
IF(DYP.GE.0.0.AND.DYM.GE.0.0)GO TO 80
IF(DYM.LT.0.0.AND.QB.GT.0.0)QB=0.0
IF(DYP.LT.0.0.AND.QB.LT.0.0)QB=0.0
IF(DYM.GE.0.0.OR.DYP.GE.O.O)GO TO 42
Q(I)=0.0
GO TO 80
42 IF(DYM.LT.0.0)GO TO 44
C TO HERE IF DYM.GT.0.0.AND DYP.LT.0.0
BTA=XMUATAN2(DYM,DXC)PI02
GO TO 46
C TO HERE IF DYP.GT.0.0.AND.DYM.LT.0.0
44 BTA=XMUATAN2(DYP,DXC)PI02
46 Q(I)=BB*SIN(BTAALPSTR)
80 RETURN
END
C
C ****** THIS SUBROUTINE CALCULATES WAVE LENGTH AND CELERITY
C
SUBROUTINE WVNUM(HSTR,T,CC)
20 FORMAT(I6,8F8.3)
G=32.17
EPS=0.001
TWOPI=6.283185
SIG=TWOPI/T
XK=TWOPI/(T*SQRT(G*HSTR))
DO 100 IT=1,20
ARG=XK*HSTR
EK=(G*XK*TANH(ARG))SIG**2
SECHA=1.0/COSH(ARG)
EKPR=G*(ARG*(SECHA**2)+TANH(ARG))
XKNEW=XKEK/EKPR
IF(ABS(XKNEWXK).LT.ABS(EPS*XKNEW)) GO TO 120
XK=XKNEW
100 CONTINUE
120 XK=XKNEW
XL=TWOPI/XK
CC=XL/T
RETURN
END
INPUT FILE: DNRBS.INP
(Example No. 2)
EXAMPLE
2.00
17.0
0.0
90000.
80 10
80 11
81 11
82 11
83 11
84 11
85 11
86 11:
87 11
88 11:
89 11
90 11:
91 11
92 11;
93 11
94 11:
95 11
96 11;
97 11:
98 11i
99 11
100 11:
NO. 2 UNIF. BACK. EROS. NO STRUC. 2 MILE PROJ.
6.0 90.0 90.0 180.0 500.0 86400.0
6.0 0.77 1.0 0.0 1 180 10950
2.0 90000. 2.0 49500. 2.0 60000.
3.0 100000. 3.0 140000. 2.0
0
2.0
2.0
2.0
2.0
2.0
2.0
2.0
2.0
2.0
2.0
2.0
2.0
2.0
2.0
2.0
2.0
2.0
2.0
2.0
2.0
2.0
0
3.0
OUTPUT FILE: DNRBS.OUT
(Example No. 2)
EXAMPLE NO. 2 UNIF. BACK. EROS. NO STRUC. 2 MILE PROJ.
HO = 2.00 FT., T = 6.00 SEC., ALPO = 90.00 DEG., BTAO = 90.00 DEG.,
XMU = 180.00 DEG., DX = 500.00 FT., DT = 86400.00 SEC.
HSTR = 17.00 FT., B = 6.00 FT., XK = .77 VFACT = 1.00
QBKREF = .00 FT.**3/SEC.
IREF = 1, IMAX = 180, NTIMES = 10950, NS = 0
2.00 .90E+05
3.00 .10E+06
2.00 .50E+05
3.00 .14E+06
2.00 .60E+05
2.00
BACKGROUND EROSION TRANSPORT RATES
1
6
11
16
21
26
31
36
41
46
51
56
61
66
71
76
81
86
91
96
101
106
111
116
121
126
131
136
141
146
.151
156
161
166
171
176
181
.000
.004
.007
.011
.015
.018
.022
.026
.029
.033
.036
.040
.044
.047
.051
.055
.058
.062
.066
.069
.073
.077
.080
.084
.088
.091
.095
.098
.102
.106
.109
.113
.117
.120
.124
.128
.131
2
7
12
17
22
27
32
37
42
47
52
57
62
67
72
77
82
87
92
97
102
107
112
117
122
127
132
137
142
147
152
157
162
167
172
177
.001
.004
.008
.012
.015
.019
.023
.026
.030
.034
.037
.041
.044
.048
.052
.055
.059
.063
.066
.070
.074
.077
.081
.085
.088
.092
.096
.099
.103
.106
.110
.114
.117
.121
.125
.128
3
8
13
18
23
28
33
38
43
48
53
58
63
68
73
78
83
88
93
98
103
108
113
118
123
128
133
138
143
148
153
158
163
168
173
178
An 1 nn
.00E+00
.90E+05
3.00
.001
.005
.009
.012
.016
.020
.023
.027
.031
.034
.038
.042
.045
.049
.053
.056
.060
.063
.067
.071
.074
.078
.082
.085
.089
.093
.096
.100
.104
.107
.111
.115
.118
.122
.125
.129
4
9
14
19
24
29
34
39
44
49
54
59
64
69
74
79
84
89
94
99
104
109
114
119
124
129
134
139
144
149
154
159
164
169
174
179
.002
.006
.009
.013
.017
.020
.024
.028
.031
.035
.039
.042
.046
.050
.053
.057
.061
.064
.068
.071
.075
.079
.082
.086
.090
.093
.097
.101
.104
.108
.112
.115
.119
.123
.126
.130
5
10
15
20
25
30
35
40
45
50
55
60
65
70
75
80
85
90
95
100
105
110
115
120
125
130
135
140
145
150
155
160
165
170
175
180
.003
.007
.010
.014
.018
.021
.025
.028
.032
.036
.039
.043
.047
.050
.054
.058
.061
.065
.069
.072
.076
.079
.083
.087
.090
.094
.098
.101
.105
.109
.112
.116
.120
.123
.127
.131
.INL J. .Lj.iJ ..r>LT.iIjj.LII.
1
3
5
7
9
11
13
15
17
19
21
23
25
27
29
31
33
35
37
39
41
43
45
47
49
51
53
55
57
59
61
63
65
67
69
71
73
75
77
79
81
83
85
87
89
91
93
95
97
99
101
103
105
107
.109
111
113
115
117
119
121
123
125
127
0.
1000.
2000.
3000.
4000.
5000.
6000.
7000.
8000.
9000.
10000.
11000.
12000.
13000.
14000.
15000.
16000.
17000.
18000.
19000.
20000.
21000.
22000.
23000.
24000.
25000.
26000.
27000.
28000.
29000.
30000.
31000.
32000.
33000.
34000.
35000.
36000.
37000.
38000.
39000.
40000.
41000.
42000.
43000.
44000.
45000.
46000.
47000.
48000.
49000.
50000.
51000.
52000.
53000.
54000.
55000.
56000.
57000.
58000.
59000.
60000.
61000.
62000.
63000.
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00,
.00
.00
.00
112.00
112.00
112.00
112.00
112.00
112.00
112.00
112.00
112.00
112.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
2
4
6
8
10
12
14
16
18
20
22
24
26
28
30
32
34
36
38
40
42
44
46
48
50
52
54
56
58
60
62
64
66
68
70
72
74
76
78
80
82
84
86
88
90
92
94
96
98
100
102
104
106
108
110
112
114
116
118
120
122
124
126
128
500.
1500.
2500.
3500.
4500.
5500.
6500.
7500.
8500.
9500.
10500.
11500.
12500.
13500.
14500.
15500.
16500.
17500.
18500.
19500.
20500.
21500.
22500.
23500.
24500.
25500.
26500.
27500.
28500.
29500.
30500.
31500.
32500.
33500.
34500.
35500.
36500.
37500.
38500.
39500.
40500.
41500.
42500.
43500.
44500.
45500.
46500.
47500.
48500.
49500.
50500.
51500.
52500.
53500.
54500.
55500.
56500.
57500.
DO.
80 )0.
)0.
)0.
62500.
63500.
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
112.00
112.00
112.00
112.00
112.00
112.00
112.00
112.00
112.00
112.00
112.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
\irlru rvurirui;riurl~l
L VUIiIV11
2.00 2
2.00 8
2.00 14
2.00 20
2.00 26
2.00 32
2.00 38
2.00 44
2.00 50
2.00 56
1.96 62
.87 68
10.62 74
49.83 80
92.71 86
103.87 92
81.19 98
33.87 104
4.34 110
1.60 116
1.99 122
2.00 128
2.00 134
2.00 140
2.00 146
2.00 152
2.00 158
2.00 164
2.00 170
2.00 176
LCUR =
LCUR =
LCUR =
LCUR =
131
133
135
137
139
141
143
145
147
149
151
153
155
157
159
161
163
165
167
169
171
173
175
177
179
100
TIME =
2.00 3
2.00 9
2.00 15
2.00 21
2.00 27
2.00 33
2.00 39
2.00 45
2.00 51
2.00 57
1.93 63
.18 69
15.06 75
58.16 81
96.98 87
102.52 93
74.10 99
26.75 105
2.31 111
1.77 117
2.00 123
2.00 129
2.00 135
2.00 141
2.00 147
2.00 153
2.00 159
2.00 165
2.00 171
2.00 177
SPCT(LCUR)
PCT(LCUR)
PCT(LCUR)
PCT(LCUR)
TIME =
1 10.00 2 10.00 3
7 10.00 8 10.00 9
13 10.00 14 10.00 15
65000.
66000.
67000.
68000.
69000.
70000.
71000.
72000.
73000.
74000.
75000.
76000.
77000.
78000.
79000.
80000.
81000.
82000.
83000.
84000.
85000.
86000.
87000.
88000.
89000.
116
2.00 4
2.00 10
2.00 16
2.00 22
2.00 28
2.00 34
2.00 40
2.00 46
2.00 52
2.00 58
1.87 64
.84 70
20.45 76
66.34 82
100.24 88
100.24 94
66.34 100
20.45 106
.84 112
1.87 118
2.00 124
2.00 130
2.00 136
2.00 142
2.00 148
2.00 154
2.00 160
2.00 166
2.00 172
2.00 178
= .78
= 81 .68
= .60
= .53
5 YEARS
.uu
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.LU
132
134
136
138
140
142
144
146
148
150
152
154
156
158
160
162
164
166
168
170
172
174
176
178
180
084
'U z .1 ,J \J.
65500.
66500.
67500.
68500.
69500.
70500.
71500.
72500.
73500.
74500.
75500.
76500.
77500.
78500.
79500.
80500.
81500.
82500.
83500.
84500.
85500.
86500.
87500.
88500.
89500.
.000 .5
1 YEARS
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
42 .542
2.00 5
2.00 11
2.00 17
2.00 23
2.00 29
2.00 35
2.00 41
2.00 47
2.00 53
2.00 59
1.77 65
2.31 71
26.75 77
74.10 83
102.52 89
96.98 95
58.16 101
15.06 107
.18 113
1.93 119
2.00 125
2.00 131
2.00 137
2.00 143
2.00 149
2.00 155
2.00 161
2.00 167
2.00 173
2.00 179
2.00 156
2.00 162
2.00 168
2.00 174
2.00 180
10.00 4 10.00 5 10.00 6 10.00
10.00 10 10.00 11 10.00 12 10.00
10.00 16 10.00 17 10.00 18 10.00
.000
1
7
13
19
25
31
37
43
49
55
61
67
73
79
85
91
97
103
109
115
121
127
133
139
145
151
157
163
169
175
2.00
2.00
2.00
2.00
2.00
2.00
2.00
2.00
2.00
1.98
1.31
7.07
41.64
87.43
104.31
87.43
41.64
7.07
1.31
1.98
2.00
2.00
2.00
2.00
2.00
2.00
2.00
2.00
2.00
2.00
'" """'''' ~"

.084
2.00
2.00
2.00
2.00
2.00
2.00
2.00
2.00
2.00
1.99
1.60
4.34
33.87
81.19
103.87
92.71
49.83
10.62
.87
1.96
2.00
2.00
2.00
2.00
2.00
000
6
12
18
24
30
36
42
48
54
60
66
72
78
84
90
96
102
108
114
120
126
132
138
144
150
25
31
37
43
49
55
61
67
73
79
85
91
97
103
109
115
121
127
133
139
145
151
157
163
169
175
1
7
13
19
25
31
37
43
49
55
61
67
73
79
85
91
97
103
109
115
121
127
133
13.9
145
151
157
163
169
175
158
164
170
176
2
8
14
20
26
32
38
44
50
56
62
68
74
80
86
92
98
104
110
116
122
128
134
140
146
152
158
164
170
176
10.00
10.00
10.00
10.00
LCUR =
LCUR =
LCUR =
LCUR =
LCUR =
20.00
20.00
19.99
19.98
19.92
19.76
19.31
18.25
16.05
12.04
5.64
3.15
13.40
23.14
29.93
31.79
28.14
20.10
9.92
.01
8.06
13.62
16.96
18.70
19.51
19.83
19.95
19.99
20.00
20.00
LCUR =
LCUR =
LCUR =
3
9
15
21
27
33
39
45
51
57
63
69
75
81
87
93
99
105
111
117
123
129
135
141
147
153
159
165
171
177
10 PCT(LCUR)
11 PCT(LCUR)
12 PCT(LCUR)
10 YEAR!
20.00 4
20.00 10
19.99 16
19.97 22
19.89 28
19.65 34
19.05 40
17.68 46
14.95 52
10.19 58
2.96 64
6.48 70
16.82 76
25.85 82
31.16 88
31.16 94
25.85 100
16.82 106
6.48 112
2.95 118
10.19 124
14.95 130
17.68 136
19.05 142
19.65 148
19.89 154
19.97 1C'
19.9 82
20.0
20.0
= .26
S.23
.19
10.00 26
10.00 32
9.98 38
9.86 44
9.37 50
7.68 56
3.12 62
6.41 68
21.71 74
40.01 80
54.71 86
59.00 92
50.69 98
33.,93 104
16.05 110
2.57 116
5.09 122
8.46 128
9.61 134
9.92 140
9.99 146
10.00 152
10.00 27
10.00 33
9.97 39
9.82 45
9.21 51
7.17 57
1.93 63
8.58 69
24.70 75
42.92 81
56.29 87
58.45 93
48.29 99
30.84 105
13.41 111
.91 117
5.89 123
8.76 129
9.70 135
9.94 141
9.99 147
10.00 153
10.00 159
10.00 165
10.00 171
10.00 177
PCT(LCUR)
PCT(LCUR)
PCT(LCUR)
PCT(LCUR)
PCT(LCUR)
LU u
10.00
9.99
9.96
9.77
9.00
6.58
.59
10.91
27.75
45.69
57.54
57.54
45.69
27.75
10.91
.59
6.58
9.00
9.77
9.96
9.99
10.00
10.00
10.00
10.00
10.00
28
34
40
46
52
58
64
70
76
82
88
94
100
106
112
118
124
130
136
142
148
154
160
166
172
178
.47
.42
.38
.33
.30
10.00 29
9.99 35
9.94 41
9.70 47
8.76 53
5.89 59
.91 65
13.41 71
30.84 77
48.29 83
58.45 89
56.29 95
42.92 101
24.70 107
8.58 113
1.93 119
7.17 125
9.21 131
9.82 137
9.97 143
10.00 149
10.00 155
10.00 161
10.00 167
10.00 173
10.00 179
20.00 5
20.00 11
20.00 11
19.99 17
19.96 23
19.86 29
19.59 35
18.89 41
17.34 47
14.32 53
9.16 59
1.51 65
8.19 71
18.48 77
27.06 83
31.55 89
30.62 95
24.55 101
15.12 107
4.80 113
4.33 119
11.15 125
15.53 131
17.98 137
19.19 143
19.71 149
19.91 155
19.97 161
19.99 167
20.00 173
20.00 179
10.00
9.99
9.92
9.61
8.46
5.09
2.57
16.05
33.93
50.69
59.00
54.71
40.01
21.71
30
36
42
48
54
60
66
72
78
84
90
96
102
108
'''' "'' "' "' '~"
  
6.41 114
3.12 120
7.68 126
9.37 132
9.86 138
9.98 144
10.00 150
10.00 156
10.00 162
10.00 168
10.00 174
10.00 180
J V W V
10.00
9.98
9.90
9.50
8.10
4.17
4.40
18.82
37.00
52.84
59.18
52.84
37.00
18.82
4.40
4.17
8.10
9.50
9.90
9.98
10.00
10.00
10.00
10.00
10.00
10.00
20.00
20.00
19.98
19.94
19.80
19.42
18.49
16.53
12.86
6.88
1.54
11.66
21.66
29.10
31.87
29.10
21.66
11.66
1.54
6.88
12.86
16.53
18.49
19.42
19.80
19.94
19.98
20.00
20.00
20.00
20.00
20.00
19.99
19.95
19.83
19.51
18.70
16.96
13.62
8.06
.01
9.92
20.10
28.14
31.79
29.93
23.14
13.40
3.15
5.64
12.04
16.05
18.25
19.31
19.76
19.92
19.98
19.99
20.00
20.00
TIME
20.00
20.00
19.99
19.97
19.91
19.71
19.19
17.98
15.53
11.15
4.33
4.80
15.12
24.55
30.62
31.55
27.06
18.48
8.19
1.51
9.16
14.31
17.34
18.89
19.59
19.86
19.96
19.99
20.00
20.00
6
12
18
24
30
36
42
48
54
60
66
72
78
84
90
96
102
108
114
120
126
132
138
144
150
156
162
168
174
180
LCUR =
LCUR =
LCUR =
LCUR =
LCUR =
LCUR =
LCUR =
LCUR =
LCUR =
LCUR =
LCUR =
LCUR =
LCUR =
LCUR =
LCUR =
LCUR =
LCUR =
59.99
59.52
58.93
58.13
57.01
55.45
53.40
50.80
47.68
44.15
40.40
36.71
33.41
30.83
29.24
28.83
29.64
31.59
34.45
37.92
41.66
45.36
48.77
51.72
54.14
56.02
57.42
58.41
59.12
59.63
LCUR =
r lCT LCUR I
SPCT(LCUR)
i PCT(LCUR)
SPCT(LCUR)
SPCT(LCUR)
I PCT(LCUR)
SPCT(LCUR)
SPCT(LCUR)
SPCT(LCUR)
SPCT(LCUR)
I PCT(LCUR)
SPCT(LCUR)
SPCT(LCUR)
SPCT(LCUR)
SPCT(LCUR)
SPCT(LCUR)
SPCT(LCUR)
TIME =
59.92 3
59.43 9
58.82 15
57.97 21
56.78 27
55.15 33
53.00 39
50.31 45
47.11 51
43.53 57
39.77 63
36.13 69
32.92 75
30.49 81
29.08 87
28.88 93
29.89 99
32.01 105
34.99 111
38.53 117
42.28 123
45.95 129
49.30 135
52.16 141
54.49 147
56.28 153
57.61 159
58.55 165
59.21 171
59.71 177
59.84
59.34
58.69
57.80
56.54
54.83
52.59
49.81
46.54
42.91
39.15
35.55
32.46
30.17
28.97
28.97
30.17
32.45
35.55
39.15
42.91
46.54
49.81
52.59
54.83
56.54
57.79
58.68
59.31
59.78
30 PCT(LCUR)
4
10
16
22
28
34
40
46
52
58
64
70
76
82
88
94
100
106
112
118
124
130
136
142
148
154
160
166
172
178
.26
= .13
= .10
= .08
= .05
= .02
= .00
S .03
S .05
S .08
S .10
S .13
S .15
S .17
S .20
S .22
S .24
30 YEARS
1
7
13
19
25
31
37
43
49
55
61
67
73
79
85
91
97
103
109
115
121
127
133
139
145
151
157
163
169
175
2
8
14
20
26
32
38
44
50
56
62
68
74
80
86
92
98
104
110
116
122
128
134
140
146
152
158
164
170
176
59.76
59.24
58.56
57.62
56.29
54.49
52.17
49.30
45.95
42.28
38.53
34.99
32.01
29.89
28.88
29.08
30.49
32.92
36.13
39.77
43.53
47.11
50.31
53.00
55.15
56.77
57.96
58.80
59.39
59.85
5
11
17
23
29
35
41
47
53
59
65
71
77
83
89
95
101
107
113
119
125
131
137
143
149
155
161
167
173
179
59.68
59.14
58.43
57.42
56.02
54.14
51.72
48.77
45.36
41.66
37.92
34.45
31.59
29.64
28.83
29.24
30.83
33.41
36.71
40.40
44.15
47.68
50.80
53.40
55.45
57.00
58.12
58.91
59.48
59.92
6
12
18
24
30
36
42
48
54
60
66
72
78
84
90
96
102
108
114
120
126
132
138
144
150
156
162
168
174
180
59.60
59.04
58.28
57.22
55.75
53.78
51.27
48.23
44.76
41.03
37.31
33.92
31.19
29.42
28.81
29.42
31.19
33.92
37.31
41.03
44.76
48.23
51.27
53.78
55.74
57.21
58.27
59.02
59.56
59.99
APPENDIX C
NUMERICAL EXAMPLE 1

