• TABLE OF CONTENTS
HIDE
 Front Cover
 Title Page
 Table of Contents
 List of Figures
 Introduction
 Methodology
 Step-by-step 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...






Group Title: UFL/COEL (University of Florida. Coastal and Oceanographic Engineering Laboratory) ; 89/026
Title: Development of methodology for thirty-year shoreline projections in the vicinity of beach nourishment projects
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Permanent Link: http://ufdc.ufl.edu/UF00076124/00001
 Material Information
Title: Development of methodology for thirty-year 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 )
non-fiction   ( 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.
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Bibliographic ID: UF00076124
Volume ID: VID00001
Source Institution: University of Florida
Holding Location: University of Florida
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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
    Step-by-step 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
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    Appendix G. Numerical example 5
        Page 124
        Page 125
        Page 126
        Page 127
        Page 128
        Page 129
        Page 130
        Page 131
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    Appendix H. Numerical example 6
        Page 134
        Page 135
        Page 136
        Page 137
        Page 138
        Page 139
        Page 140
        Page 141
        Page 142
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    Appendix I. Numerical example 7
        Page 144
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Full Text











Development of Methodology for Thirty-Year 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/COEL-89/026


DEVELOPMENT OF METHODOLOGY FOR
THIRTY-YEAR 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

STEP-BY-STEP 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 Non-Dimensional Shoreline Advancement Ayo/W. with A' and V'.
Results Shown for h./B = 2.0. .......................... 18
12 Variation of Non-Dimensional 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 Pelnard-Considere. 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
C-1 Numerical Example 1, Ayo = 112 ft, Nourishment Length = 2 miles, Zero Back-
ground Erosion ................... .. ........ ..... 86
C-2 Numerical Example 2, Shoreline Position Variation with Time at Locations In-
dicated and Shown in Figure C-1. . . . .... ........ 87
D-1 Numerical Example 2, Ayo = 112 ft, Nourishment Length = 2 miles, Uniform
Background Erosion = 2 ft/yr. .......................... 96
D-2 Numerical Example 2, Shoreline Position Variation with Time at Locations In-
dicated and Shown in Figure D-1. ........................ 97
E-1 Numerical Example 3, Ayo = 112 ft, Nourishment Length = 2 miles, Variable
Background Erosion ............................... 106
E-2 Numerical Example 2, Shoreline Position Variation with Time at Locations In-
dicated and Shown in Figure E-1 .......................... 107









F-1 Numerical Example 4, Ayo = 112 ft, Nourishment Length = 1,000 ft, No Back-
ground Erosion ................... ................116
F-2 Numerical Example 2, Shoreline Position Variation with Time at Locations In-
dicated and Shown in Figure F-............. ... ............. 117
G-1 Numerical Example 5, 112 ft Long Structure at North End of Project, Nourish-
ment Length = 2 miles, No Background Erosion. . . . ... 126
G-2 Numerical Example 2, Shoreline Position Variation with Time at Locations In-
dicated and Shown in Figure G-1. . . . ... ...... 127
H-1 Numerical Example 6, 112 ft Long Structure at South End of Project, Nourish-
ment Length = 2 miles, Uniform Background Erosion. . . ... 136
H-2 Numerical Example 2, Shoreline Position Variation with Time at Locations In-
dicated and Shown in Figure H-1. ........................ 137
I-1 Numerical Example 7, 112 ft Long Structure at South End of Project, Nourish-
ment Length = 2 miles, Variable Background Erosion. . . ... 146
I-2 Numerical Example 2, Shoreline Position Variation with Time at Locations In-
dicated and Shown in Figure I-1.......................... 147









DEVELOPMENT OF METHODOLOGY FOR

THIRTY-YEAR 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

16B-33.024(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

THIRTY-YEAR 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

16B-33.024(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 nourishment-adjacent areas.











92.4m
7"r 7A B = 1.5m


45.3m
-^ 1h*


b) Non-intersecting Profiles
AN= AF= 0.1m1 /3


,5.9m


c) Non-Intersecting Profiles"
AN= 0.1m11/3AF = 0.09m1/3


d) Limiting Case of Nourishment Advancement, 1/3
Non-Intersecting 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) Non-Intersecting 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 Non-Dmensona' = 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 Non-Intersecting
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 Non-dimensional Shoreline Advancement Ayo/W., with
A' and -'. Results shown for h, /B = 4.0.









It is seen that for each non- dimensional volume, the non-dimensional 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 non-dimensional 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 so-called heat conduction or diffusion equation which is well-known

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) =- e-2 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 non-dimensional 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-
C-L--I0.0
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 Pelnard-Considere. Longshore
Sediment Transport Rate Used in Example =180,000 cubic
yards per year. Littoral Barrier Length = 160 ft.


Figure 19.















NON-DIMENSIONAL 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.

STEP-BY-STEP DISCUSSION OF METHODOLOGY


In this section the limitations and the step-by-step 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.

STEP-BY-STEP DISCUSSION OF METHODOLOGY


In this section the limitations and the step-by-step 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 step-by-step 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 non-dimensional 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 step-by-step 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 non-dimensional 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 non-dimensional 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 step-by-step 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 step-by-step 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 k-A* .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 4-rI 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+ C4-Ayrmh 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 Pa-Itrs of(tIJ.s+Ome-S,
J ErMos ;n a-s)


mrs
vfur--4


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 step-by-step 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.
'--l-r^


.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 case-by-case 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 10-7 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).




















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LIBERy WAKULLA 4 BAKER (
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.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--- 1------l-: 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; Non-Uniform Background Erosion

Graphical Example 4: Downdrift of a Littoral Barrier, Non-Uniform 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 ICA--L EXtmYPILe i
(z4ero A-Pc4lJ-qroC't4'1 Ey-osi ")

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)^ ioh-117 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 31-t1 0 3/.1
.5-2 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
CUnr-form %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.)
X-U L I= Dl0,f-7 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)
-._tQ5-2>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- ^YIqo-H^ 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,0-o4(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
5z-80 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. 82-1-IIa, 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 DEL-SG-15-78, University of Delaware,

Newark, DE.


Dean, R.G. (1987) "Additional Sediment Input to the Nearshore Region", Shore and

Beach, Vol. 55, Nos. 3-4, p. 76-81.


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)(1-p) 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.^C-12g04 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* = sin-1 C[ sin(Po ao)] (A.10)
rio s, = sin-J 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 sin-1 [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 time-varying 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 long-term 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=IMAX-1
IMP1=IMAX+1
DO 30 I=1,IMP1
X(I)=(I-1)*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=BTAO-ASIN(CC/CO*SIN(BTAO-ALPO))
C WRITE(6,124)HSTR,T,CC,CO,CGO,ALPO,BTAO,ALPSTR
CALP=COS(ALPO-ALPSTR)
SALP=SIN(ALPO-ALPSTR)
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(BTAO-ALPO)**1.2/
1 (8.0*(SG-1)*(1.0-POR)*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=XMU-ATAN2((YO(I)-YO(I-1)),(X(I)-X(I-1)))-PI02
COSC=COS(BTA-ALPO)
SINC=SIN(BTA-ALPO)
Q(I)=BB*SIN(BTA-ALPSTR)
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(I-1))/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+(NT-1)*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))-62.06-NYEARS*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(IER-1)
DXX=XC-XER(IER-1)
AA=DXX/DX
BB=1.0-AA
ERC=-CON*(BB*EROSB(IER-1)+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(I-1),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(I-1))/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(I-1))/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(I-1)-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=XMU-ATAN2(-DYM,DXC)-PI02
GO TO 46
C TO HERE IF DYP.GT.0.0.AND.DYM.LT.0.0
44 BTA=XMU-ATAN2(DYP,DXC)-PI02
46 Q(I)=BB*SIN(BTA-ALPSTR)
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=XK-EK/EKPR
IF(ABS(XKNEW-XK).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




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