 Report documentation page 
 Title Page 
 Table of Contents 
 List of Figures 
 Abstract 
 Introduction 
 Modified equilibrium beach... 
 Application of equilibrium beach... 
 Summary and conclusions 
 Acknowledgements 

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Title: 
Equilibrium beach profiles characteristics and applications 

Series Title: 
UFLCOEL 

Physical Description: 
v, 70 leaves : ill. ; 28 cm. 

Language: 
English 

Creator: 
Dean, Robert G ( Robert George ), 1930 United States  National Oceanic and Atmospheric Administration National Sea Grant College Program (U.S.) University of Florida  Coastal and Oceanographic Engineering Dept 

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

Place of Publication: 
Gainesville Fla 

Publication Date: 
1990 
Subjects 

Subject: 
Coast changes  Mathematics ( lcsh ) Beach erosion  Mathematics ( lcsh ) Shorelines  Mathematics ( lcsh ) Coastal and Oceanographic Engineering thesis M.S Coastal and Oceanographic Engineering  Dissertations, Academic  UF 

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government publication (state, provincial, terriorial, dependent) ( marcgt ) bibliography ( marcgt ) technical report ( marcgt ) nonfiction ( marcgt ) 
Notes 

Bibliography: 
Includes bibliographical references (p. 6670). 

Funding: 
Sponsored by the Sea Grant College Program, National Oceanic and Atmospheric Administration. 

Statement of Responsibility: 
by Robert G. Dean. 

General Note: 
"January 15, 1990." 

General Note: 
Final. 
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Bibliographic ID: 
UF00076132 

Volume ID: 
VID00001 

Source Institution: 
University of Florida 

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University of Florida 

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All rights reserved, Board of Trustees of the University of Florida 

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Table of Contents 
Report documentation page
Unnumbered ( 1 )
Title Page
Title Page
Table of Contents
Table of Contents
List of Figures
List of Figures 1
List of Figures 2
List of Figures 3
Abstract
Page 1
Introduction
Page 2
Page 3
Page 4
Page 5
Page 6
Page 7
Page 8
Modified equilibrium beach profile
Page 9
Page 10
Page 11
Page 8
Application of equilibrium beach profiles of the form: h=Ay2/3
Page 12
Page 13
Page 14
Page 15
Page 16
Page 17
Page 18
Page 19
Page 20
Page 21
Page 22
Page 23
Page 24
Page 25
Page 26
Page 27
Page 28
Page 29
Page 30
Page 31
Page 32
Page 33
Page 34
Page 35
Page 36
Page 37
Page 38
Page 39
Page 40
Page 41
Page 42
Page 43
Page 44
Page 45
Page 46
Page 47
Page 48
Page 49
Page 50
Page 51
Page 52
Page 53
Page 54
Page 55
Page 56
Page 57
Page 58
Page 59
Page 60
Page 61
Page 62
Page 63
Summary and conclusions
Page 64
Page 65
Acknowledgements
Page 65
Page 66
Page 67
Page 68
Page 69
Page 70

Full Text 
___REPORT DOCUMENTATION PAGE
1. Report No. 2. 3. Recipient'# Accession No.
4. Title and Subtitle 5. Report Date
EQUILIBRIUM BEACH PROFILES: January 15, 1990
CHARACTERISTICS AND APPLICATIONS 6.
7. Author(s) 8. Performing Organization aRport mo.
Robert G. Dean UFL/COEL90/001
9. Performing Organization iNae and Address 10. Project/Task/Work Unit no.
Coastal and Oceanographic Engineering Department
University of Florida 11. Ctrt or rant No.
336 Weil Hall R/CS22
Gainesville, FL 32611 13. Typ of report
12. Sponsoring Organization Name and Address
Sea Grant Program Final
National Oceanic and Atmospheric Administration
14.
15. Suppleentary Notes
16. Abstract
An understanding of equilibrium beach profiles can be useful in a number of types of coastal engineering projects. Empirical
correlations between a scale parameter and the sediment size or fall velocity allow computation of equilibrium beach
profiles. The most often used form is h(y) = Ay2/3 in which h is the water depth at a distance y from the shoreline
and A is the sedimentdependent scale parameter. Expressions for shoreline position change are presented for arbitrary
water levels and wave heights. Application of equilibrium beach profile concepts to profile changes seaward of a seawall
include effects of sea level change and arbitrary wave heights. For fixed wave heights and increasing water level, the
additional depth adjacent to the seawall first increases, then decreases to zero for a wave height just breaking at the
seawall. Shoreline recession and implications due to increased sea level and wave heights are examined. It is shown, for
the equilibrium profile form examined, that the effect of wave setup on recession is small compared to expected storm
tides during storms. Profile evolution from a uniform slope is shown to result in five different profile types, depending
on initial slope, sediment characteristics, berm height and depth of active sediment redistribution. The reduction in
required sand volumes through perching of a nourished beach by an offshore sill is examined for arbitrary sediment and
sill combinations. When beaches are nourished with a sediment of arbitrary but uniform size, it is found that three types
of profiles can result: (1) submerged profiles in which the placed sediment is of smaller diameter than the native and
all of the sediment equilibrates underwater with no widening of the dry beach, (2) nonintersecting profiles in which the
seaward portion of the placed material lies above the original profile at that location, and (3) intersecting profiles with
the placed sand coarser than the native and resulting in the placed profile intersecting with the original profile. Equations
and graphs are presented portraying the additional dry beach width for differing volumes of sand of varying sizes relative
to the native. The offshore volumetric redistribution of material due to sea level rise as a function of water depth is of
interest in interpreting the cause of shoreline recession. If only offshore transport occurs and the surveys extend over
the active profile, the net volumetric change is zero. It is shown that the maximum volume change due to crossshore
sediment redistribution is only a fraction of the product of the active vertical profile dimension and shoreline recession.
The paper presents several other applications of equilibrium beach profiles to problems of coastal engineering interest.
17. Oritgnator's Key Words 18. Availability Stament
Beach erosion
Equilibrium beach profiles
Nourishment
Sea level rise
Seawalls
1*9. U. S. Security Classif. of the Report 20. U. S. Security Classl. of This Page 21. No. of Pafes 22. Price
Unclassified Unclassified 1 75
EQUILIBRIUM BEACH PROFILES:
CHARACTERISTICS AND APPLICATIONS
Robert G. Dean
January 15, 1990
Sponsored by:
Sea Grant College Program
National Oceanic and Atmospheric Administration
Coastal and Oceanographic Engineering Department
University of Florida
Gainesville, FL 32611
I
TABLE OF CONTENTS
ABSTRACT 1
INTRODUCTION 2
Modified Equilibrium Beach Profile 8
Applications of Equilibrium Beach Profiles of the Form: h = Ay2/3 12
Effect of Sediment Size on Beach Profile ................. .... 12
Beach Response to Altered Water Level and Waves .............. 12
Equilibrium Profile and Recession Including the Effect of Wave SetUp 18
Profile Adjustment Adjacent to a Seawall Due to Altered Water Level and
W aves . . . . . . . . .. .. 22
Response from Initial Uniform Slope ................. .... .. 29
Type 1 Profile .............................. 32
Type 2 and Type 4 Profiles .................. ...... 34
Type 3 and Type 5 Profiles ........................ 35
Limits of Profiles Types ........................ 35
No Terrace Present ........................... 36
Terrace Present .............................. 36
Perched Beach ................... .................. 36
Profile Response to Beach Nourishment . . . ..... 45
Intersecting, Non Intersecting and Submerged Profiles . ... 45
Variation in Sediment Size Across the Surf Zone . . . ... 49
Continuous Arbitrary Distribution of Sand Sizes Across the Surf Zone 49
Piecewise Uniform Sand Size Across Surf Zone . . ... 53
Comparison with Empirical Orthogonal Functions . . ... 53
Effects of Sea Level Rise on Beach Nourishment Quantities . ... 55
Case I Nourishment Quantities for the Case of No Onshore Sediment
Transport ... ... ... ... ..... .. ... .. ... ..... 55
Case II Nourishment Quantities for the Case of Onshore Sediment
Transport ... ... ... ... .. .. .. ... ... ... ..... 56
CrossProfile Volumetric Redistribution Due to Sea Level Rise ...... .. 59
Trailing Beach Profile Signature Due to Sea Level Rise . . ... 64
SUMMARY AND CONCLUSIONS 64
ACKNOWLEDGEMENTS 65
REFERENCES 66
LIST OF FIGURES
1 Beach Profile Factor, A, vs Sediment Diameter, D, in Relationship h = Ay2/3
(modified from Moore, 1982) ................... ......... 5
2 Profile P4 from Zenkovich (1967). A Boulder Coast in Eastern Kamchatka.
Sand Diameter: 150 mm 300 mm. Least Squares Value of A = 0.82 m1/3
(from M oore, 1982). ................... ............. 6
3 Profile P10 from Zenkovich (1967). Near the End of a Spit in Western Black
Sea. Whole and Broken Shells. A = 0.25 m1/3 (from Moore, 1982) . 6
4 Profile from Zenkovich (1967). Eastern Kamchatka. Mean Sand Diameter:
0.25 mm. Least Squares Value of A = 0.07 m1/3 (from Moore, 1982). . 7
5 Variation of Sediment Scale Parameter, A, With Sediment Size and Fall
Velocity (Dean, 1987a). ................... ......... 9
6 Comparison of Calibrated Profile Response Model With Large Wave Tank
Data by Saville (1957), From Kriebel (1986). . . . ... 10
7 Comparison of Calibrated Profile Response Model With Field Profile Affected
By Hurricane Eloise as Reported by Chiu (1977). From Kriebel (1986). 11
8 Comparison of Equilibrium Beach Profile With and Without Gravitational
Effects Included. A = 0.1 m1/3 Corresponding to a Sand Size of 0.2 mm. 13
9 Definition Sketch for Profile Response Due to Sea Level Rise. . ... 14
10 Equilibrium Beach Profiles for Sand Sizes of 0.3 mm and 0.6 mm A(D = 0.3
mm) = 0.12 m1/3, A(D = 0.6 mm) = 0.20 m/3. . . . ... 15
11 Isolines of Dimensionless Shoreline Change, Ay', vs Dimensionless Storm
Breaking Depth, h./B, and Dimensionless Storm Tide, S'. . . 17
12 Profile Geometry and Notation For Shoreline Advancement Due to Lowering
of W ater Level ................... .............. 19
13 NonDimensional Wave SetUp and Equilibrium Beach Profile. ...... .. 21
14 Beach Recession Due to Waves and Increased Water Level, Including the
Effect of Wave SetUp ...................... .......... 21
15 Definition Sketch. Profile Erosion Due to Sea Level Increase and Influence
of Seawall .. .. . . . . . . . . 24
16 Isolines of Dimensionless Seawall Toe Scour, Ah'~ vs Dimensionless Storm
Tide, S' and Dimensionless Breaking Depth h. . . . ... 26
17 NonDimensional Change in Water Depth at Wall, Ah',, as a Function of
NonDimensional Storm Surge. Example for h' = 6.0. . . .... 27
18 Definition Sketch, Profile Response Adjacent to a Seawall for Case of Lowered
Sea Level . . . . . . . . . 28
19 NonDimensional Shallowing Adjacent to a Seawall Due to Lowering of Wa
ter Level. Variation With NonDimensional Breaking Depth, h., and Non
Dimensional Water Level, S'. ........................... 30
20 Illustration of Five Equilibrium Profile Types Commencing From an Initially
Uniform Slope. ................... .................. 31
21 Regimes of Equilibrium Profile Types Commencing From An Initially Planar
Profile Showing Five Types of Equilibrium Profiles and NonDimensional
Profile Advancement, Ay'.............................. 33
22 Example of Type 1 (recession) Profile Response From An Initially Uniform
Slope and Associated Volumetric Transport. Note Only Positive (seaward)
Transport. ....................................... 37
23 Example of Type 2 (Recession) Profile Response From An Initially Uniform
Slope and Associated Volumetric Transport. Note Both Positive (Seaward)
and Negative (landward) Transport. . . . . .. 38
24 Example of Type 3 (Recession) Profile Response From An Initially Uniform
Slope and Associated Volumetric Transport. Note Only Negative (Landward)
Transport .. . . .. .. . . . . . 39
25 Example of Type 4 (Advancement) Profile Response From An Initially Uni
form Slope and Associated Volumetric Transport. Note Both Positive (Sea
ward) and Negative (Landward) Transport. . . . . ... 40
26 Example of Type 5 (Advancement) Profile Response From An Initially Uni
form Slope and Associated Volumetric Transport. Note Only Negative (Land
ward) Transport ................... .............. 41
27 Perched Beach, Demonstration of Nourishment Volumes Saved. ...... .. 42
28 Three Generic Types of Nourished Profiles. . . . ... 43
29 Effect of Nourishment Material Scale Parameter, Ap, on Width of Resulting
Dry Beach. Four Examples of Decreasing AF With Same Added Volume Per
Unit Beach Length. ............ ................... 46
30 Effect of Increasing Volume of Sand Added on Resulting Beach Profile. Ap
= 0.1m1/3, AN = 0.2 m1/3,h = 6 m, B = 1.5m................ 47
31 (1) Volumetric Requirement for Finite Shoreline Advancement (Eq. 54);
(2) Volumetric Criterion for Intersecting Profiles (Eq. 53). Variation with
AF/AN. Results Presented for h./B = 4.0. . . . .... 50
32 Variation of NonDimensional Shoreline Advancement Ay/W*, With A' and
V. Results Shown for h,/B = 2.0. ............ ........... 51
33 Variation of NonDimensional Shoreline Advancement Ay/W,, With A' and
V. Results Shown for h,/B = 4.0. ........................ 52
34 Comparison of Beach Profile Elevation Changes By Equilibrium Profile Con
cepts With Results From Field Measurements. Example in Panel b) for h,/B
= 4.0 and S/B = 0.5. ................... ............ 54
35 Possible Mechanism of Sedimentary Equilibrium (After Dean, 1987b). 58
36 Definition Sketch and NonDimensional Volume Redistributed as a Function
of NonDimensional Depth. Due to Sea Level Rise, S. Case of B/h. = 0.25,
S/B = 0.5 ................... ................ 60
EQUILIBRIUM BEACH PROFILES:
CHARACTERISTICS AND APPLICATIONS
R. G. Dean
Abstract
An understanding of equilibrium beach profiles can be useful in a number of types of
coastal engineering projects. Empirical correlations between a scale parameter and the
sediment size or fall velocity allow computation of equilibrium beach profiles. The most
often used form is h(y) = Ay2/3 in which h is the water depth at a distance y from the
shoreline and A is the sedimentdependent scale parameter. Expressions for shoreline
position change are presented for arbitrary water levels and wave heights. Application of
equilibrium beach profile concepts to profile changes seaward of a seawall include effects
of sea level change and arbitrary wave heights. For fixed wave heights and increasing
water level, the additional depth adjacent to the seawall first increases, then decreases to
zero for a wave height just breaking at the seawall. Shoreline recession and implications
due to increased sea level and wave heights are examined. It is shown, for the equilibrium
profile form examined, that the effect of wave setup on recession is small compared to
expected storm tides during storms. Profile evolution from a uniform slope is shown to
result in five different profile types, depending on initial slope, sediment characteristics,
berm height and depth of active sediment redistribution. The reduction in required
sand volumes through perching of a nourished beach by an offshore sill is examined for
arbitrary sediment and sill combinations. When beaches are nourished with a sediment
of arbitrary but uniform size, it is found that three types of profiles can result: (1)
submerged profiles in which the placed sediment is of smaller diameter than the native
and all of the sediment equilibrates underwater with no widening of the dry beach,
(2) nonintersecting profiles in which the seaward portion of the placed material lies
above the original profile at that location, and (3) intersecting profiles with the placed
sand coarser than the native and resulting in the placed profile intersecting with the
original profile. Equations and graphs are presented portraying the additional dry beach
width for differing volumes of sand of varying sizes relative to the native. The offshore
volumetric redistribution of material due to sea level rise as a function of water depth
is of interest in interpreting the cause of shoreline recession. If only offshore transport
occurs and the surveys extend over the active profile, the net volumetric change is zero.
It is shown that the maximum volume change due to crossshore sediment redistribution
is only a fraction of the product of the active vertical profile dimension and shoreline
recession. The paper presents several other applications of equilibrium beach profiles
to problems of coastal engineering interest.
INTRODUCTION
A quantitative understanding of the characteristics of equilibrium beach profiles is central
to rational design of many coastal engineering projects and to the interpretation of nearshore
processes. Several features of equilibrium beach profiles are wellknown: (1) they tend to
be concave upwards, (2) smaller and larger sand diameters are associated with milder and
steeper slopes, respectively, (3) the beach face is approximately planar, and (4) steep waves
result in milder slopes and a tendency for bar formation.
In a broad sense, it is obvious that sand particles are acted upon by a complex sys
tem of constructive and destructive generic "forces" with the constructive forces acting to
displace the sediment particle landward and vice versa. Constructive forces include land
ward directed bottom shear stresses due to the nonlinear character of shallow water waves,
landward directed "streaming" velocities in the bottom boundary layer (Bagnold, 1946;
Phillips, 1966), the phasing associated with intermittent suspended sediment motion, etc.
The most obvious destructive force is that of gravity coupled with the destabilizing effects of
turbulence induced by wavebreaking; others include the effect of seaward directed bottom
undertow currents and forces due to wave setup within the surf zone (e.g. Svendsen (1984),
Stive and Wind (1986)). Indeed, the above represents only a partial listing of the complex
force system acting on sediment particles and serves to illustrate the difficulty of a rational
physicsbased prediction of equilibrium beach profiles.
Several approaches have been pursued in an attempt to characterize equilibrium beach
profiles. Keulegan and Krumbein (1919) investigated the characteristics of a mild bottom
slope such that the waves never break but rather are continually dissipated by energy losses
due to bottom friction. Bruun (1954) analyzed beach profiles from the Danish North Sea
coast and Mission Bay, CA and found that they followed the simple relationship
h(y) = Ay2/3 (1)
in which h is the water depth at a seaward distance, y, and A is a scale parameter which
depends primarily on sediment characteristics. Eagleson, Glenne and Dracup (1963) de
veloped a complex characterization of the wave and gravity forces acting on a particle
located outside the zone of "appreciable breaker influence" and developed expressions for
the seaward limit of motion and for the beach slope for which a sand particle would be in
equilibrium.
Swart (1974) carried out a series of wave tank tests and developed empirical expressions
relating profile geometry and transport characteristics to the wave and sediment conditions.
The active beach profile was considered as four zones and empirical expressions were de
veloped for each zone. Vellinga (1983) investigated dune erosion using wave tank tests and
developed the following "erosion profile" which included the effect of deep water significant
wave height, Ho, and sediment fall velocity, w,
(7.6\.) h 7.6 18 w 0.56 0.5
0.47 68 y+ 18 2.0 (2)
Ho., Ho, .0268
in which the values of all variables are in the metric system. It can be shown that Eq. (2)
is in reasonably good agreement with Eq. (1). Sunamura and Horikawa (1974) examined
and characterized beach profiles for two sizes of sediments, and ranges of wave heights and
periods and initial slopes of planar beaches. Three beach profile types were established
by laboratory experiments including one erosional and two accretional types. Suh and
Dalrymple (1988) applied concepts of equilibrium beach profiles to address the same problem
as Sunamura and Horikawa and identified one erosional profile type and one accretional
type. Comparison of laboratory data demonstrated good agreement with their criteria and
predictions of profile changes.
Numerous investigations have been carried out to develop appropriate scale modeling
criteria including Dalrymple and Thompson (1976), Noda (1972), Hughes (1983) and Van
Hijum (1975). Hayden, et al. (1975) apparently were the first to apply the concept of
empirical orthogonal functions (EOF) to extract the principal modes of change from a set
of beach profile data. Numerous later investigations have used this approach to investigate
the character of the dominant modes of profile change, e.g. Winant et al. (1975), Weishar
and Wood (1983), Aubrey et al. (1977), and Aubrey (1979). The EOF is a purely descriptive
method and does not address the causes or processes of profile change.
Hayden, et al. (1975) assembled a data set comprising 504 beach profiles along the
Atlantic and Gulf coasts of the United States. Dean (1977) analyzed these profiles and used
a least squares procedure to fit an equation of the form
h = Ay" (3)
to the data and found a central value of n = 2/3 as Bruun had earlier. It was shown that
Eq. (3) with n = 2/3 is consistent with uniform wave energy dissipation per unit volume,
D.. It can be shown that P, and A are related by
A 24 D. (D) /3(4)
A 5 pg3/22 (4)
in which p is the water mass density, g is gravity, D is sediment particle diameter and C
is a constant relating wave height to water depth within the surf zone. The interpretation
of Eq. (3) was that a particle of given size is characterized by an associated stability and
that the wave breaking process results in the transformation of organized wave motion
into chaotic turbulence fluctuations; these fluctuations are destructive forces and, if too
great, cause mobilization of the sediment particle with resulting offshore displacement and
a milder beach slope, which reduces the wave energy dissipation per unit volume eventually
resulting in an equilibrium profile. Later Moore (1982) collected and analyzed a number
of published beach profiles and developed the relationship between A and D as shown in
Figure 1. As expected, the larger the sediment size, the greater the A parameter and the
steeper the beach slope. Figures 2, 3 and 4 present several profiles employed by Moore in
establishing the relationship presented in Figure 1. The profile in Figure 2 is of particular
interest as the sediment particle size ranges from 15 cm to 30 cm, approximately the size
of bowling balls! Dean (1987a) has shown that when the relationship presented in Figure 1
is transformed to A(w) rather that A(D), where w is the fall velocity, the relationship is
1.0
wI Suggested Empirical 
1H Relationship 
w
< From Hughes'
(L Where a Range of Sand Si;
SWas Given
_Z From Swart's
IW Laboratory Results
L 0.01
0.01 0.1 1.0 10.0 100.0
SEDIMENT SIZE, D (mm)
Figure 1. Beach Profile Factor, A, vs. Sediment Diameter, D, In Relationship
h = Ay23 (Modified from Moore, 1982).
DISTANCE OFFSHORE (m)
Figure 2. Profile P4 From Zenkovich (1967). A Boulder Coast in Eastern
Kamchatka. Sand Diameter: 150 mm 300 mm. Least Squares
Value of A = 0.82 m113(from Moore, 1982).
DISTANCE OFFSHORE (m)
0.00 60.00 120
0.00 I I
E  Least Squares Fit
Actual Profile
I
LU
S 3.00
UI
6.00
Figure 3. Profile P10 From Zenkovich (1967). Near the End of a Spit In
Western Black Sea. Whole and Broken Shells. A = 0.25 m/3
(from Moore, 1982).
DISTANCE OFFSHORE (m)
0.00
0.00 1N
1.00
2.00
Figure 4. Profile From Zenkovich (1967). Eastern Kamchatka. Mean Sand
Diameter: 0.25 mm. Least Squares Value of A = 0.07 m'l3(from
Moore, 1982).
surprisingly linear (on a log log plot) as presented in Figure 5.
Kriebel (1982), Kriebel and Dean (1984, 1985) and Kriebel (1986) have considered
profiles out of equilibrium by hypothesizing that the offshore transport is proportional to
the difference between the actual and equilibrium wave energy dissipation per unit volume,
i.e.
Q = K(D PD) (5)
Eq. (5) and a sand conservation relationship have been incorporated into a numerical
model of sediment transport with generally good confirmation between laboratory profiles
(Figure 6) and field results (Figure 7). Larson (1988) and Larson and Kraus (1989) have
considered the active region of sediment transport in four zones and have developed em
pirical counterparts to Eq. (5) for each zone, thus allowing solution of the transient beach
profile problem including a capability for generating longshore bars.
Modified Equilibrium Beach Profile
An unrealistic property of the form of the equilibrium beach profile represented by Eq. (1)
is the predicted infinite slope at the shoreline. Large slopes induce correspondingly large
gravity forces which are not represented in Eq. (1). A slight modification to include gravi
tational effects is
D, ah 1
+ (ECG) = D (6)
m ay hay
Gravity Effect Turbulence Effect
in which the two terms on the left hand side represent destabilizing forces due to gravity
and turbulent fluctuations due to wave energy dissipation, m is the beach face slope, and
as before D. represents the stability characteristics of the sediment particle but now the
interpretation of ,D is broadened beyond equilibrium energy dissipation per unit volume to
include gravity as an additional destabilizing force. Eq. (6) can be integrated to:
y = A3/2h (7)
SEDIMENT FALL VELOCITY, w (cm/s)
1.0 10.0
0.0
1.0
0.10
0.01
0.0
100.0
100.0
SEDIMENT SIZE, D (mm)
Figure 5. Variation of Sediment Scale Parameter, A, With Sediment Size and
Fall Velocity. (Dean 1987a)
1I
10.0
Suggested Empirical
Relationship A vs. D
(Moore)
From Hughes'
Field Results 0.44
From Individual Field 0
Profiles where a Range
of Sand Sizes was Given _
Based on Transforming
A vs D Curve using
Fall Velocity Relationship
/ From Swart's
Laboratory Results
1
10.00
0.00
z
Predicted
Sa40hat 40 hrs :
J 5.00 Observed
Si at40hrs
10.00
15.00 I I I I I II
0.00 40.00 80.00 120.00 160.00 200.00 240.00 280.00 320.00
DISTANCE (ft)
Figure 6. Comparison of Calibrated Profile Response Model With Large Wave
Tank Data by Saville (1957), From Kriebel (1986).
150
100
.. I.... I...... ...I .*.....*...... ...
S. i.... ....;.... .... .... .. ....I... .... .. .. .... I ... i.... I ..i... I...I. .... I ..i .... .... .. I I .... I.... I.... ..i I I... ..* I .. ...
L i i I i I 1 i .
i T !" !"! T" T!.... ....!.." P i l ..:". e T i I '!"!
...... ......... ...... Prestorm Pro .e: :....
.... .. ... observe ... ii ... ....
Predicted ......
II** *
... .... ... .. .... .. ... .... .... I....... ,....... .. .. ... .. .. ... re ic t. .... ... ......... .. .... .... .....I .. ... ...... ........ ...
:.. : : : i : :: i:: 1:: :,, : 1::: ., : i: i;: i :: i: :: :: i... : :: .. : .
. .j. ....:.: I: ;i..l..l .....I.... .j...v...e d I...j..... ......... ..........
50 100
150
200
MONUMENT (ft)
Figure 7. Comparison of Calibrated Profile Response Model With Field Profile Affected
By Hurricane Eloise as Reported By Chlu (1977). From Kriebel (1986).
10
250
200
250
surprisingly linear (on a log log plot) as presented in Figure 5.
Kriebel (1982), Kriebel and Dean (1984, 1985) and Kriebel (1986) have considered
profiles out of equilibrium by hypothesizing that the offshore transport is proportional to
the difference between the actual and equilibrium wave energy dissipation per unit volume,
i.e.
Q = K(D PD) (5)
Eq. (5) and a sand conservation relationship have been incorporated into a numerical
model of sediment transport with generally good confirmation between laboratory profiles
(Figure 6) and field results (Figure 7). Larson (1988) and Larson and Kraus (1989) have
considered the active region of sediment transport in four zones and have developed em
pirical counterparts to Eq. (5) for each zone, thus allowing solution of the transient beach
profile problem including a capability for generating longshore bars.
Modified Equilibrium Beach Profile
An unrealistic property of the form of the equilibrium beach profile represented by Eq. (1)
is the predicted infinite slope at the shoreline. Large slopes induce correspondingly large
gravity forces which are not represented in Eq. (1). A slight modification to include gravi
tational effects is
D, ah 1
+ (ECG) = D (6)
m ay hay
Gravity Effect Turbulence Effect
in which the two terms on the left hand side represent destabilizing forces due to gravity
and turbulent fluctuations due to wave energy dissipation, m is the beach face slope, and
as before D. represents the stability characteristics of the sediment particle but now the
interpretation of ,D is broadened beyond equilibrium energy dissipation per unit volume to
include gravity as an additional destabilizing force. Eq. (6) can be integrated to:
y = A3/2h (7)
where, as before, A is related to D, by Eq. (4). In shallow water, the first term in Eq. (7)
dominates, simplifying to
h = my (8)
i.e. the beach face is of uniform slope, m, consistent with measurements in nature. In
deeper water, the second term in Eq. (7) dominates with the following simplification
h = Ay2/s (9)
as presented earlier.
Figure 8 presents a comparison of Eq. (7), which includes the planar portion near the
water line and Eq. (1) which has an infinite slope at the water line. A form similar to Eq.
(7) was adopted by Larson (1988) and Larson and Kraus (1989).
Applications of Equilibrium Beach Profiles of the Form:
h = Ay2/3
The utility of Eq. (1) will be illustrated by several examples. A definition sketch of the
system of interest is presented as Figure 9. In results presented here, it will be assumed
that within the surf zone the wave height is proportional to the local water depth with the
proportionality factor, i, i.e. H = ich(. w 0.78). In particular, the breaking wave height,
Hb, and breaking depth, h,, are related by Hb = Kh,.
Effect of Sediment Size on Beach Profile
Figure 10 presents two examples of the effect of sediment size on beach profile. The
scale parameter A is determined for various sediment sizes from Figure 1 and the profiles
computed from Eq. (1).
Beach Response to Altered Water Level and Waves
The effects of elevated and lowered water levels will be treated separately.
An elevated water level, S, with wave and sediment conditions such that the profile
is reconfigured out to a depth, h*, is assumed to result in equilibration with the final state
DISTANCE OFFSHORE (m)
100
200
*I I I I I

SGravitational Effects Included
Only Wave Dissipation Included
~~ ~ . .
300
Figure 8. Comparison of Equilibrium Beach Profiles With and Without Gravitational
Effects Included. A = 0.1 m1/3 Corresponding to a Sand Size of 0.2 mm.
1
3
Figure 9. Definition Sketch For Profile Response Due to Sea Level Rise.
B~  
FjS
q~17.
DISTANCE OFFSHORE (m)
100
200
Figure 10. Equilibrium Beach Profiles for Sand Sizes of 0.3 mm and 0.6 mm
A(D = 0.3 mm) = 0.12 ml'/A(D = 0.6 mm) = 0.20 m1/3.
being the same profile form as before, but relative to the elevated water level. This situation
could pertain to a storm tide of long duration or to sea level rise.
Referring to Figure 9, the sand volume eroded, VE, is equal to the volume deposited,
VD
VE =VD (10)
When Eqs. (1) and (10) are combined the following implicit equation for the shoreline
change, Ay, is obtained
3 h,W, Ay\ 5/3 =3 hiWf /S S
5B W 5 B B
^ Ay 5. (1" s}; (W (11)
in which W* is the seaward limit of the active profile (* = (H)3/2) and for this case of
shoreline recession Ay < 0. Eq. (11) can be expressed in nondimensional form as
Ay' [ (+ Ay')/] S' =0 (12)
in which the nondimensional variables are
Ayy
W*
BAy =
B' =
h*
S
Eq. (12) is plotted in Figure 11. For small values of Ay', Eq. (12) can be approximated by
W
Ay = S (14)
(h, + B)
first proposed by Bruun (1962) and now referred to as the "Bruun Rule".
For the case of lowered water levels, there will be an excess of sand in the active system
and a resulting advancement of the shoreline. The profile equilibration depth, h,, will occur
at a distance, W2, from the original shoreline. The equilibrium depth is considered to extend
1.0
,.. o.8  ^ 7T7 
cn / /
0.8 
0 0.6 . ..... ....
I
S 0.4 "" ,,
z
. 0.2 I
o .2 0.0
5 0.0 : II I I,, I I I I II
0.1 0.2 0.5 1.0 2.0 5.0 10.0
DIMENSIONLESS STORM BREAKING DEPTH, =
,BB'
Figure 11. Isolines of Dimensionless Shoreline Change, Ay', vs.
Dimensionless Storm Breaking Depth, h,/B, and Dimen
sionless Storm Tide, S'.
17
; I
as a horizontal terrace from the distance W2 noted above to a landward location consistent
with the equilibrium profile and the shoreline advancement, Ay. For this case, shown in
Figure 12, the nondimensional shoreline advancement can be shown to be
S2 (1 S'B')6/2 1 (15)
5 (B' SB' + 1)
where it is emphasized that for this case, S' < 0. Additionally,
W= = (1 S'B')/2 (16)
and the landward location of the terrace, W1, is
W' = W l1 + Ay' (17)
W.
The other nondimensional parameters are as defined by Eqs. (13).
Equilibrium Profile and Recession Including the Effect of Wave SetUp
The equilibrium beach profile, h = Ay2/3, interpreted as resulting from uniform wave
energy dissipation per unit water volume does not include the effect of wave setup. As a
precursor to developing recession predictions due to increased water levels and wave setup,
it is useful to first establish the equilibrium beach profile, including the effect of wave setup,
q].
We start with the wellknown solution for wave setup across the surf zone (Bowen et
al., 1968)
q(y) = b + J[hb h(y)] (18)
in which hb is the breaking depth (hb = h. qb S), qlb is the setup (actually negative) at
breaking, and
3m'/8
J 2/ (19)
1 + 3i2/8
1 kH2 (20)
8 sinh 2kh.
I I JIIIM ,UI. T, Y
H L 
SL S<
hi *1 h
\ Original h
Profile
Horizontal
Terrace
Figure 12. Profile Geometry and Notation For Shoreline Advancement Due to Lowering
of Water Level.
where k is the wave number. Since i is positive over most of the surf zone, it is reasonable
that it contributes much like a tide in causing recession, especially for the larger breaking
wave heights. The equilibrium profile based on uniform wave energy dissipation per unit
volume commences from
((h)) (ECG)= D (21)
(h + S +q 4) aye
in which ye is directed landward, E is the wave energy density, and CG is the group velocity.
Assuming shallow water,
ECG = g'(h+S+ q g)/ [(h+S+ 4)
If the algebra is carried through, the nottoosurprising result is obtained
h + S + f = Ay2/3 (22)
where the yorigin is now the location where h + S + 4 = 0. Figure 13 presents a plot of
the nondimensional wave setup and the associated equilibrium beach profile.
In the following development, the effects of a uniform storm tide, S, and the setup
which varies across the surf zone (Eq. (22)) will be considered as presented in Figure 14.
Following procedures similar to those used for determining recession with a water level
which is uniform across the surf zone, volumes are equated as
f 0 W .+Ay &2W.+Ay /W.+Ay
[BSq(y)]dy+ A(yAy)23dy = Ay 2/3+f [S+(y)]dy (23)
y Ay J0 0
which after considerable algebra yields
3 5/3 1 (3/5 J)
Ay' + [1+ Ay'] I S 176 (24)
5B' B' (1J) (
in which Eqs. (13) have been used for nondimensionalization and 14 = 7b/B.
The question of the relative roles of breaking waves and storm surges can be addressed
by simplifying Eq. (24) for the case of relatively small nondimensional beach recessions,
Ay'l << 1), which upon adoption of n = 0.78 and expressing in dimensional form yields
Ay 0.068( + (
1+ (25)
W. 1 + 1.28H
B
Figure 13. NonDimensional Wave SetUp and Equilibrium Beach Profile.
U"
Ay
.4
Figure 14. Beach Recession Due to Waves and Increased Water level,
Including the Effect of Wave SetUp.
It is necessary to exercise care in interpreting Eq. (25), as the surf zone width, W. includes
the effect of the breaking wave height,
W. = (26)
As shown by Eq. (25), the dimensionless beach change (Ay/W,) is much more strongly
related to storm surge than wave height with the storm surge being approximately 16
times as effective in causing the dimensionless beach recession. However, during storms the
breaking wave height may be two to three times as great as the storm tide and the larger
breaking waves may persist much longer than the peak storm tides. The reason that the
storm tide plays a much greater role than that due to breaking wave setup is evident from
Figure 13 where it is seen that the wave setup (actually the setdown) acts to reduce the
mean water level over a substantial portion of the surf zone.
For the case in which storm surge is not important and the ratio of breaking wave height
to berm height is large, Eq. (25) can be simplified to represent only the effects of waves
and waveinduced setup
= 0.053
W.
or
( /b 3/2 (27)
Ay = 0.053 ( ) (27)
Thus for a doubled wave height, the recession induced by wave set up increases by a factor
of 2.8.
Profile Adjustment Adjacent to a Seawall Due to Altered Water Level
and Waves
We first consider the case of profile lowering adjacent to a seawall due to an elevated
water level.
It is wellknown that during storms a scour trough will occur adjacent to a sea wall.
For purposes here it is appropriate to consider this scour as a profile lowering due to two
components: (1) the localized and probably dominant effect due to the interaction of the
seawall, waves, and tides, and (2) the effect due to sediment transport offshore to form a
profile in equilibrium with the elevated water level. Applying equilibrium profile concepts,
it is possible to calculate only the second component.
The system of interest is presented in Figure 15. The profile is considered to be in
equilibrium with virtual origin yl = 0. For a water level elevated by an amount, S, the
equilibrium profile will now be different and will have a virtual origin at y2 = 0. We denote
the distances from these virtual origins to the wall as yw, and yw, for the original and
elevated water levels, respectively. As in previous cases, the approach is to establish the
origin, yw2, (now virtual) such that the sand volumes seaward of the seawall and associated
with the equilibrium profiles are equal before and after the increase in water level. In the
following, all depths (h values) are referenced to the original water level except h. which as
described previously is a reference depth related to the breaking wave height.
Equating volumes as before
/W.Yw,+Ywi rW
f .Y2+ hl(y)dyl = W h2(y2)dy2
"W1 fYw2
which can be integrated using hi(y) = Ay2/3, and h2(y) + S = Ay/3 and simplified to yield
h [h + S'] 5 h2 +S' 1 3/2 5/3 5/2
m 1 "
[S' (h',2+ S,)32 =0 (28)
in which the primes represent nondimensional quantities defined as
S hw,
h2 = hw
hW
hw1
S' = (29)
hw,
Yw2
Ey
m&
// / / // ///////
Virtual Origin, K,
Increased
Sea Level '
Virtual Origin,
Original
Sea Level
//
/
/r
W*
Seawall
hw
< Ahw h2 h*
h,
.
_ Increased Sea Level
 Y2
,,Original Sea Level
Figure 15. Definition Sketch. Profile Erosion Due to Sea Level Increase and Influence of Seawall.
Eq. (28) is implicit in h'y2 and must be solved by iteration. Defining the change in depth
at the wall, Ahw, as
Ahw = hw, hw1
and in nondimensional form
A Ahw
h, = 
hw,
The quantity AhM, is now a function of the following two nondimensional variables:
S' and h'. The relationship Ah',(h',S') is presented in Figure 16 where it is seen that
for a fixed h' and increasing S', the non dimensional scour, Ah'y, first increases and then
decreases to zero. Figure 17 presents a specific example for h, = 6. The interpretation
of this form is that as S' increases, the profile is no longer in equilibrium and sand is
transported seaward to develop the equilibrium profile and the water depth adjacent to the
seawall increases. However, as sea level rises further, with the same total breaking depth,
the active surf zone width decreases, such that less sand must be transported seaward to
satisfy the equilibrium profile. With increasing storm tide, the surf zone width approaches
zero at the limit
S + hw, = h,
or
S' = h' 1
which corresponds to the upper line in Figure 16. It is emphasized that the increased depth
at the seawall predicted here does not include the scour interaction effect of the seawall and
waves.
We next consider the case of lowered water level (S < 0) adjacent to a seawall as shown
in Figure 18. In this case sediment will move landward due to the disequilibrium caused
by the lowered water level. The notation is the same as in the previous case. Equating
volumes eroded and deposited is expressed as
/WAW, +Y hi(yi)dyi = h2 (2)dy2 + h*(WA W*) (30)
vW1 f1W2
En 10 h .=
LF 8.0 hw
1.0.8  \ K 
S 6.80 2
_____ ___ \ 5.0
e 0.64 .0 
0.4 0.8
z 1.5
0 0.4 0.1
0.4
0.2 I I I I I
1 2 4 6 810 20 40 6080100
DIMENSIONLESS STORM BREAKING DEPTH, h',
Figure 16. Isolines of Dimensionless Seawall Toe Scour, Ah'w vs Dimensionless
Storm Tide, S' and Dimensionless Breaking Depth, h,.
ZLU
00
z<
Sw
ZO
OZ 0
.5
.0
Figure 17.
1 2 3 4 5
NONDIMENSIONAL STORM SURGE, S'
NonDimensional Change in Water Depth at Wall Ah'w as a Function
of NonDimensional Storm Surge. Example for h' = 6.0.
Original Water Level
Virtual Origins 
Hori
Figure 18. Definition Sketch, Profile Response Adjacent to a Seawall
for Case of Lowered Sea Level.
Terrace
which can be integrated and simplified to yield
(h' + S')3/2 [ (h + S') h 2(h S')5/2 + h S' + h = 0 (31)
The nondimensional distance WI( WA/W*) is given by
WA 3/2 t t (32)
A= (1~ T1) + yW2 yWI (32)
The nondimensional depth change (decrease) at the wall is
Ah', = h' 1 (33)
which can be established by solving Eq. (31). Figure 19 presents a plot of Ahty (h', S').
In Eq. (31), the term h'y + S' represents the nondimensional total depth at the
seawall on the equilibrium profile for the lowered water level. A limiting case for the above
formulation is for this water level to be zero, i.e. h'~ + S' = 0, yielding
2(h S)5/2 + (h' S' + h' /= 0 (34)
which is plotted as the upper dashed line in Figure 19.
Response from Initial Uniform Slope
For simplicity, many wave tank tests commence with an initially planar beach slope,
mi. It is of interest to examine the relationship of the equilibrium and initial profiles. As
presented in Figure 20, there are five types of equilibrium profiles that can form depending
on the initial slope, mi, and the sediment and wave characteristics. Wave tank tests by
Sunamura and Horikawa (1974) identified profile Types 1, 2 and 5. Shoreline responses
for Types 1 and 3 have been investigated by Suh and Dalrymple (1988) using equilibrium
profile concepts and good agreement with wave tank data was demonstrated.
Referring to Figure 20, for the Type 1 profile the initial slope is much steeper than
that for the equilibrium profile and only seaward sediment transport occurs. An additional
characteristic is that a scarp is formed at the shoreline and no berm is deposited. The
wave tank profile presented in Figure 6 is an example of a Type 1 profile. Type 2 profile,
Limiting Case
1%% h'2+ S' = 0
0.4 = A h'w
0.3
0.2
0.1
S I i I I IlI I I I I i 1111
5 10 20
50 100
NONDIMENSIONAL BREAKING DEPTH, h' = h,/hw
Figure 19. NonDimensional Shallowing Adjacent to a Seawall Due
to Lowering of Water Level. Variation With NonDimensional
Breaking Depth, h and NonDimensional Water Level, S'.
1.01
0.5
0.2
0.1
0.05
0.02 F
' '"~
Type 1 Profile
(Recession, No Terrace)
Type 2 Profile
(Recession, No Terrace)
Ay< 0
Profile
Ay.* 0
Type 4 Profile
(Advancement, No Terrace)
 *Initial Profile
h,*
Ay> 0
Type 5 Profile
Figure 20. Illustration of Five Equilibrium Profile Types Commencing
From an Initially Uniform Slope.
Initial
Shoreline
also a case of shoreline recession, occurs for a somewhat milder relative slope (initial to
equilibrium), sediment transport occurs in both the onshore and offshore directions and a
berm is formed at the shoreline. With still milder relative slopes, sediment transport occurs
only shoreward resulting in a Type 3 profile characterized by shoreline recession with all of
the sediment transported deposited as a berm feature. A terrace or "bench" (here assumed
horizontal) is formed at the seaward end of the equilibrium profile. This type profile is
probably the least likely to occur due to the unrealistically high berm elevation required.
Type 4 profile is one of shoreline advancement, occurring for still milder initial slopes and is
characterized by sediment transport in both the landward and seaward directions. Finally,
Type 5 profile is one of shoreline advancement with only landward sediment transport
and leaving a horizontal terrace or bench at the seaward end of the equilibrium profile.
The following paragraphs quantify the profile characteristics and shoreline changes for each
of these five types. As described previously, shoreline recession and advancement will be
denoted by negative and positive Ay, respectively. It will be shown that the nondimensional
shoreline change, Ay'(= Ay/W,) is a function of the non dimensional depth of limiting
profile change, h',( h,/W.mi) and nondimensional berm height, B'(= B/W.mi). The
developments associated with these profile types will not be presented in detail. Methods
are similar to those applied earlier in this report for example for the case of shoreline
recession due to an elevated water level. Figure 21 presents Ay'(h',, B') and the associated
regions of occurrence for the five profiles types.
Type 1 Profile
The nondimensional advancement, Ay', can be expressed in terms of the nondimensional
depth of limiting motion, h', as
Ay' = 3h 1 (35)
5 2
2.0
.4'0 / ,,S0.2
.. ,, ,
1 5,y ./ /3A.
1. 5 0 .8 0.1 /
Sy1.5 / / / T //
S/ /
. / y/ / +0.1 0 /
ri/ +0.3
S10 / / .5 0.4
, / +o.6 /
/ /+0.7
/ / / pe /
U 0+5 0e
Sype / / / / //+0.
 /
o ^ T ype 4 / / / / 1.0
0 1.0 2.0 3.0
h,
NONDIMENSIONAL SEAWARD DEPTH OF ACTIVE MOTION, h',= m,
Figure 21. Regimes Of Equilibrium Profile Types Commencing From An initially
Planar Profile Showing Five Types Of Equilibrium Profiles And Non
Dimensional Profile Advancement, Ay'.
where for this profile type, Ay' < 0. Because no berm is formed, the berm characteristics
do not appear in this expression and B is simply a reference quantity when plotted in
Figure 21.
It can be shown that the nondimensional volume, V, transported seaward past any
location, y', is
= 1 [ 12 3 h2r 3 ,
VBW 2B' ( 2] + AY')5/3 Ay' < y' < 1 + Ay' (36)
in which
y' (37)
For this type profile, Suh and Dalrymple (1988) have denoted h,/W. as the equilibrium
slope, me, and have shown a correlation between the ratio of initial to equilibrium slopes
and the resulting profile changes.
Type 2 and Type 4 Profiles
For these profile types, the nondimensional change, Ay', depends on h', and non di
mensional berm height, B',
Ay'= (B' + 1)+ 2B' + h, (38)
For Type 2 and Type 4 profiles, the values of Ay' will be negative (recession) and positive
(advancement), respectively.
The equations for nondimensional volumes transported seaward past any location, y',
for Type 2 and Type 4 profiles are
y = ([B'2 y2] (y'+ B') B' < y' < Ay' (39a)
(Ay'B') 2B'(y'A Ay'< y'< 1+ Ay (39b)
S= (Ay' + B') (/y1 B')+ 3 y Ay')5/s Ay' < y' < 1 + Ay' (39b)
Type 3 and Type 5 Profiles
Both of these equilibrium profile types include a horizontal terrace at their seaward
limits. The expressions for Ay' are identical:
1 h B2 4 h'
Ay' = (40)
2 h'. + B'
where again Ay' is negative and positive for profile Types 3 and 5, respectively.
The equations for nondimensional volumes transported seaward past any location, y',
for Type 3 and Type 5 profiles are
V = 1(B y'2)(y'+ B') ,B' < y' < Ay' (41a)
2B'
v = '( B ) + 3 '(y Ay)5/3 (Ay'+ B') Ay' < y' < 1+ Ay' (41b)
2Bh' h B' 12 h'
4"V= Ay' (+ yz l+ Ay < y' < h' (41c)
5 B' 2 B' 2 2 B' B
Limits of Profiles Types
The criterion for berm formation is
B' + 3h' 1 < 0, No Berm Formed (42)
5 2 > 0, Berm Formed
where here B' is interpreted as the nondimensional berm height that would be formed if
the berm height exceeds the scarp height cut by the shoreline recession into the uniform
slope profile. For berm formation, Ay > B/mi.
The limit for terrace formation is
62 < 0 No Terrace
(B' + h')2 2B' Terrace (43)
5 {> 0, Terrace
The limit for no shoreline change, Ay' 0, depends on whether or not a terrace is
present.
No Terrace Present
 f= 0, No Shoreline Change
B'2 h, + 1 > 0, Shoreline Recession (44)
< 0, Shoreline Advancement
Terrace Present
= 0, No Shoreline Change
B'2 h + h, > 0, Shoreline Recession (45)
< 0, Shoreline Advancement
Figures 2226 present examples of profile response and associated volumes transported
for profile Types 15.
Perched Beach
The offshore extent of a perched beach is terminated by a shore parallel structure which
prevents the sand from moving seaward. In conjunction with a beach nourishment project, it
is possible in principle to obtain a much wider beach for the same volume of added sediment.
Denoting the "native" and "fill" sediment scale parameters as AN and Ap respectively and
referring to Figure 27 for terminology, the required volume, V, for the case of the sand just
even with the top of the submerged breakwater is
V = B[( h, )"'3 h2 )3/2
3 r h, \s/2 h ( _2 5/2
+ AN ) A([A (46)
5 \AN AF
and the added beach width, Ay, is
&Y ( h, 3/2 2 3/2 (47)
AN Ap
Referring to Figure 27, it is possible to calculate the reduction in required volume
through a perched beach design. The results can be developed for intersecting and non
intersecting type profiles (c.f. Figure 28a and b). For simplicity, only the results for non
0.3.
/0.2
0.2
h 0.4
h* 0.6
BWi.. ~
y
W*
0.6
a) Volumetric Transport Past Any Location
b) Profile Response
Figure 22. Example of Type 1 (Recession) Profile Response From An
Initially Uniform Slope and Associated Volumetric Transport.
Note Only Positive (Seaward) Transport.
F
I \ I
i I


0.2  0.2 0.4 0.6 0.8 1.0
a) Volumetric Transport Past Any Location
Equilibrated Profile
b) Profile Response
Figure 23. Example of Type 2 (Recession) Profile Response From An Initially Uniform
Slope and Associated Volumetric Transport. Note Both Positive (Seaward)
Transport and Negative (Landward) Transport.
BW
BW*
0.2
SI I I I I I
y
.0.8 1.
a) Volumetric Transport Past Any Location
.6 0.8 1.0 Y
 W*
Inital Profile
Equilibrated Profile
< b) Profile Response
Figure 24. Example of Type 3 (Recession) Profile Response From An
Initially Uniform Slope and Associated Volumetric Transport.
Note Only Negative (Landward) Transport.
BV
BW*
0.8
0.6  0.4
0.2
0.1
\ 0.2
I ,
S.
w*
S  I'
0.4 0.6
1.0 w*
a) Volumetric Transport Past Any Location
0.8 1.0 Y
II I
:ial Profile
Equilibrated Profile
b) Profile Response
Figure 25.
Example of Type 4 (Advancement) Profile Response From An
Initially Uniform Slope and Associated Volumetric Transport.
Note Both Positive (Seaward) and Negative (Landward) Transport.
BW
Bw,
0.1 
0.2 
,i N
0.2
 ""
0.2 I
y
ill
V
BW*
I
^^0.2
%% 0.1 
0.3
0.4
0.5 1
y
1.6 W*
._ ... a)Volumetric Transport Past Any Location
1.6. Y
I aI I I I I I
SInitial Profile
Equilibrated Profile
Figure 26.
Example of Type 5 (Advancement) Profile Response From An
Initially Uniform Slope and Associated Volumetric Transport.
Note Only Negative (Landward) Transport.
I I
0.6
0.6
0.
BW 012
S
0.6
0.8
1.0
I I I
 ~I1
I
1.2,,714
w,
m
Yi
<AyI y2
Perched Beach
h2
Toe Structure
Figure 27. Perched Beach, Demonstration of Nourishment Volumes Saved.
m~
A:yj
.. .F
B
w,
<Ay
Added Sand :
b) NonIntersecting Profile
Ay
1 L
Virtual Origin of
Nourished Profile
Added Sand
c) Submerged Profile AF
Figure 28. Three Generic Types of Nourished Profiles.
intersecting profiles (without the sill) will be presented here. The fractional reduction in
volume, AV /V, is given by
AV Ay' + [(h)5/2 (A )32 )
A 5B, [ (h )/2]5/3 N (h i)/2
A V = 1" I I A 2 ( 4 8 )
in which VI = the volume that would be required to advance the shoreline seaward by an
amount, Ay, without the sill and
y' = y2/w.
h' = hi/h,
h' = h2/h.
Ay' = Ay/W*
B1 B
h,
As an example of the application of Eq. (48), consider the following parameters
AN = AF = 0.15 m/3
h, = 6m
hi= 4m
h2 = 3m
B = 2.0 m
Ay = 48.3 m (From Eq. (47))
The width of the surf zone without the sill, W., is
W, = )3/2 = 253.0m
AN/
and
/(2 3/2
1Y2 3/2 = 89.4m
\AF/
The fractional reduction in volume is
AV
V 0.342
i.e., there is a 34% reduction in sand volumes with the perched beach design.
Profile Response to Beach Nourishment
Intersecting, NonIntersecting and Submerged Profiles Beach nourishment is con
sidered with sediment of arbitrary but uniform diameter. As indicated in Figure 28,
nourished beach profiles can be designated as "intersecting", "nonintersecting" and "sub
merged" profiles. A necessary but insufficient requirement for profiles to intersect is that
the placed material be coarser than the native. Similarly, nonintersecting or submerged
profiles will always occur if the placed sediment is the same size as or finer than the na
tive. However, nonintersecting profiles can occur if the placed sediment is coarser than
the native. For "submerged" profiles to occur, the placed material must be finer than the
native. Figure 29 illustrates the effect of placing the same volume of four different sized
sands. In Figure 29a, sand coarser than the native is used, intersecting profiles result and a
relatively wide beach Ay is obtained. In Figure 29b, the same volume of sand of the same
size as the native is used, nonintersecting profiles result and the dry beach width gained is
less. More of the same volume is required to fill out the milder sloped underwater profile.
In Figure 29c, the placed sand is finer than the native and much of the sand is utilized in
satisfying the milder sloped underwater profile requirements. In a limiting case, shown in
Figure 29d, no dry beach is yielded with all the sand being used to satisfy the underwater
requirements. Figures 30a through 30d illustrate the effects of nourishing with greater and
greater quantities of a sand which is considerably finer than the native. Figure 30d is the
case of formation of an incipient dry beach, i.e. the same as in Figure 29d. With increasing
volumes, the landward intersection of the native and placed profiles occurs closer to shore
and the seaward limit of the placed profile moves seaward.
92.4m
7 '_ ~B = 1.5m
a) Intersecting Profiles, 
AN= 0.1mi/AF = 0.14m1/3
45.3m
b) Nonintersecting Profiles
AN= AF= 0.1m1/3
c) NonIntersecting Profiles'
AN= O.lm 3,AF = 0.09m1/3
d) Limiting Case of Nourishment Advancement, 1 3
NonIntersecting Profiles, AN= 0.1ml/3,AF = 0.085m1
I I i I! I
100
200
300
400
500
h,= 6m
600
OFFSHORE DISTANCE (m)
Figure 29. Effect of Nourishment Material Scale Parameter, A F,on Width of
Resulting Dry Beach. Four Examples of Decreasing A F With Same
Added Volume Per Unit Beach Length.
I
S115.9m
OFFSHORE
200
DISTANCE (m)
300 400
b) Added Volume = 490 m3 Im
Added Volume = 1660 m3/m
Case of Incipient Dry Beach
Figure 30. Effect of Increasing Volume of Sand Added on Resulting
Beach Profile. A F= 0.1 ml/3,AN= 0.2 m1/3,h, = 6.0 m, B = 1.5 m.
100
,+4
I 0
0
O
" 10
.J
500
I 1 1 I
We can quantify the results presented in Figures 28, 29 and 30 by utilizing equilibrium
profile concepts. It is necessary to distinguish the three cases noted in Figure 28. The first is
with intersecting profiles such as indicated in Fig. 28a and requires Ap > AN. For this case,
the volume placed per unit shoreline length, V1 associated with a shoreline advancement,
Ay, is presented in nondimensional form as
J/ j = A + A)5[/ ( 3/212/3 (49)
in which V1 ( V /BW,) is the nondimensional volume, B is the berm height, W. is a refer
ence offshore distance associated with the breaking depth, h., on the original (unnourished)
profile, i.e.
W= ( 32 (50)
(hN
and the breaking depth, h, and breaking wave height, Hb are related by
h, = Hble/
with ni(s 0.78), the spilling breaking wave proportionality factor.
For nonintersecting but emergent profiles (Figure 28b), the corresponding volume V2
in nondimensional form is
V, = Ay' + 3/2 5/3 3/2 (51)
It can be shown that the critical value of (Ay') for intersection/nonintersection of
profiles is given by
A' + (AN )3/2 1 < 0, Intersecting Profiles
y' + F 1 (52)
Ap > 0, NonIntersecting Profiles
The critical volume associated with intersecting/non intersecting profiles is
'^ = + 3 ) 1_(A) (53)
and applies only for (AF/AN) > 1. Also of interest, the critical volume of sand that will
just yield a finite shoreline displacement for nonintersecting profiles (AF/AN < 1), is
( )2= AN5 32 i 1 (54)
Figure 31 presents these two critical volumes versus the scale parameter ratio AF/AN for
the special case h,/B = 4.0, i.e. B' = 0.25.
The results from Eqs. (49), (51) and (52) are presented in graphical form in Figures 32
and 33 for cases of (h./B) = 2 and 4. Plotted is the nondimensional shoreline advancement
(Ay) versus the ratio of fill to native sediment scale parameters, AF/AN, for various isolines
of dimensionless fill volume V' (= W ) per unit length of beach. It is interesting that the
shoreline advancement increases only slightly for AF/AN > 1.2; for smaller values the
additional shoreline width, Ay, decreases rapidly. For AF/AN values slightly smaller than
plotted, there is no shoreline advancement, i.e. as in Figure 29d.
Referring to Figure 28c for submerged profiles, it can be shown that
A 1 AN \3/2
=i (1 (55)
yi AF
where Ay < 0 and the nondimensional volume of added sediment can be expressed as
S(3 (AN //2(
5B' A 32 23 A
Variation in Sediment Size Across the Surf Zone
All cases presented earlier have considered the sand size to be of uniform size across the
surf zone. In most cases, there is some sorting with the sand grading to finer sizes in the
seaward direction. With the relation of A(D), and thus PD(D), known (c.f. Eq. (4)) it is
possible to calculate equilibrium profiles for cases of a continuum of sand sizes across the
surf zone and a distribution composed of piecewise uniform diameter segments.
Continuous Arbitrary Distribution of Sand Sizes Across the Surf Zone The
differential equation for an equilibrium beach profile is given by
=2 A3/2(D) (57)
00
Fn >
z
0
oe
Figure 31.
1 2
AF/AN
3
LL
z
200
1
o
z O
0
z
(1) Volumetric Requirement for Finite Shoreline Advancement
(Eq. 54); (2) Volumetric Criterion for Intersecting Profiles
(Eq. 53). Variation with AF/AN. Results Presented for h,/B = 4.0.
10.
1.0
1 . .
B %/ V' = 0.5
0.10 V' = 0.1
V' = 0.05
0.01' = 0.01
y V' = 0.005
:; V = V/BW, = 0.002
hA .. .. ..
0.001 Definition Sketch 
0 1.0 2.0 2.8
A'= AF/AN
Figure 32. Variation of NonDimensional Shoreline Advancement Ay /W,, With
A' and : Results Shown for h.B = 2.0.
1.0
W Asymptotes "IA M
for Ay = 0
0'= 0.02
^ ^npt^^E ;F^
0.01
0.001 AV'= VIBB = 0.001
Definition Sketch
I _
0.0 001 00
AY f AF V
I B N__ I W*
0 1.0 2.0 2.8
A' = A/AN
Figure 33. Variation of NonDimensional Shoreline Advancement Ay I/W, With
A' and V: Results Shown for hi/B = 4.0.
from which Eq. (1) is obtained readily. Integrating across the surf zone yields the equilib
rium profile
h3/2(y) = j AS/2(D)dy (58)
Piecewise Uniform Sand Sizes Across the Surf Zone Denoting the sand size as,
D,, over the segment y, < y < yn+l, the water depth in this region is obtained from a
slight variation of Eq. (58)
h3/2(y) = h3/2(yn) + A /(Dn)[y yn] (59)
which applies for yn < y < yn+i.
Comparison with Empirical Orthogonal Functions
It is instructive to compare results of profiles obtained from a simple application of the
equilibrium beach profile with those developed by various researchers (e.g. Winant, Inman
and Nordstrom (1975) and Weishar and Wood (1983)) in their application of Empirical
Orthogonal Function (EOF) methods to time series of natural beach profiles. This method
has been described by Hayden, et al. (1975). For purposes here, we note that the first EOF
is analogous to the equilibrium beach profile and the second EOF is termed the "bermbar"
function.
We will consider the change in profile elevation resulting from a single elevated water
level and wave and sediment conditions that would mobilize sediment out to a depth h..
Consideration of Figure 9 and utilizing Eq. (12), the first EOF is the average profile, and
the second EOF can be shown to be approximately
Ah = Ah' B'(1 S') + (/' Ay')2/3 Ay < y' < 0
h. (y' Ay')2/3 y2/ S'B 0 < y' < 1 + Ay (60)
and where the primed (nondimensional) quantities are as defined by Eqs. (13). Figure 34
presents a comparison between Eq. (60) and the second EOF as determined by Winant et
al. (1975) based on field measurements at Torrey Pines, CA. The similarities between the
EOF obtained by these investigators and those developed by equilibrium profile synthesis
are quite evident.
SEAWARD DISTANCE, y(m)
a) Most Significant Eigenfunction of Profile Change
(From Winant, Inman and Nordstrom, 1975)
Y
b) Nondimensional Elevation Changes Based
On Equilibrium Beach Profiles
Figure 34.
Comparison of Beach Profile Elevation Changes by Equilibrium
Profile Concepts With Results From Field Measurements. Example
In Panel b) for hJB = 4.0 and S/B = 0.5.
o c
> C
i
J
w~
z
uJ*
wo
0
z
Effects of Sea Level Rise on Beach Nourishment Quantities
Recently developed future sea level scenarios (Hoffman, et al., 1983) have been developed
based on assumed fossil fuel consumption and other relevant factors and have led to concern
over the viability of the beach nourishment option for erosion control. First, in the interest
of objectivity, it must be stated that the most extreme of the scenarios published by the
Environmental Protection Agency (EPA) amounting to sea level increases exceeding 11 ft
by the year 2100 are extremely unlikely. While it is clear that worldwide sea level has
been rising over the past century and that the rate is likely to increase, the future rate
is very poorly known. Moreover, probably at least 20 to 40 years will be required before
our confidence level of future sea level rise rates will improve substantially. Within this
period, it will be necessary to assess the viability of beach restoration on a projectby
project basis in recognition of possible future sea level scenarios. Presented below is a basis
for estimating nourishment needs for the scenarios in which there is no landward sediment
transport across the continental shelf and there is a moreorless welldefined seaward limit
of sediment motion; in the second case the possibility of onshore sediment transport will be
discussed.
Case I Nourishment Quantities for the Case of No Onshore Sediment Transport
Bruun's Rule (1962) is based on the consideration that there is a welldefined depth limit,
h,, of sediment transport. With this assumption, the only response possible to sea level
rise is seaward sediment transport. Considering the total shoreline change Ay, to be the
superposition of recession due to sea level rise Ays and the advancement due to beach
nourishment, AyN,
Ay = Ays + AlN (61)
and, from Bruun's Rule (Eq. 14)
W,
s = Sh, (62)
in which S is the sea level rise, W, is the distance from the shoreline to the depth, h,,
associated with the seaward limit of sediment motion and B is the berm height. Assuming
that compatible sand is used for nourishment (i.e. Ap = AN)
AY N = (63)
h, + B
and V is the beach nourishment volume per unit length of beach. Therefore
1
Ay = [V SW.] (64)
(h. + B)
The above equation can be expressed in rates by,
dy 1 [dV dS 1
1 B) W. (65)
dt (h + B) dtW dt
where S now represents the rate of sea level rise and H is the rate at which nourishment
material is provided. It is seen from Eq. (65) that in order to maintain the shoreline stable
due to the effect of sea level rise the nourishment rate is related to the rate of sea level
rise dS by
dv dS
t= W. t (66)
Of course, this equation applies only for crossshore mechanisms and therefore does not
recognize any other causes of background erosion or longshore transport losses from the
project area. It is seen that W, behaves as an amplifier of material required. Therefore, it
is instructive to examine the nature of W, and it will be useful for this purpose to consider
the equilibrium profile given by Eq. (1),
W* Hb )3/2(67)
i.e. W, increases with breaking wave height and with decreasing A (or sediment size).
Case II Nourishment Quantities for the Case of Onshore Sediment Transport
Evidence is accumulating that in some locations there is a substantial amount of onshore
sediment transport across the continental shelf. Dean (1987b) has noted the consequences
of the assumption of a "depth of limiting motion" in allowing only offshore transport as a
response to sea level rise and proposed instead that if this assumption is relaxed, onshore
transport can occur leading to a significantly different profile response to sea level rise.
Consider that there is a range of sediment sizes in the active profile with the hypothesis
that a sediment particle of given hydraulic characteristics is in equilibrium under certain
wave conditions and at a particular water depth. Thus, if sea level rises our reference
particle will seek equilibrium which requires landward rather than seaward transport as
required by the Bruun Rule. Figure 35 summarizes some of the elements of this hypothesis.
Turning now to nourishment requirements in the presence of onshore sediment transport,
the conservation of crossshore sediment yields
= + sources sinks (68)
ay at
in which h is the water depth referenced to a fixed vertical datum and the sources could
include natural contributions such as hydrogenous or biogenous components, and suspended
deposition or human related contributions, i.e. beach nourishment. Sinks could include
removal of sediment through suspension processes. Eq. (68) can be integrated seaward
from a landward limit of no transport to any location, y
(y) o (sources sinks)dy = Y dy (69)
If only natural processes are involved and there are no gradients of longshore sediment
transport, the terms on the left hand side of Eq. (69) represent the net rate of increase of
sediment deficit as a function of offshore distance, y. For y values greater than the normal
width, W., of the zone of active motion, the left hand side can be considered as representing
the "ambient" deficit rate due to crossshore sediment transport resulting from longterm
disequilibrium of the profile and source and sink terms.
In attempting to apply Eq. (69) to the prediction of profile change and/or nourishment
needs under a scenario of increased sea level rise, it is reasonable to assume that over the
next several decades the ambient deficit rate (or surplus) of sediment within the active zone
will remain constant. However, an increased rate of sea level rise will cause an augmented
demand which can be quantified as W, [(s) ()o] in which (!) is the reference
POSSIBLE MECHANISM OF SEDIMENTARY EQUILIBRIUM
Increased Sea Level
SO al Sea Level___
StS & Orialnlal Sea Level
Sediment
Particle
Subjected to a Given Statistical Wave Climate, A Sediment
Particle of a Particular Diameter is in Statistical Equilibrium
When in a Given Water Depth
Thus When Sea Level Increases, Particle Moves Landward
Figure 35. Possible Mechanism of Sedimentary Equilibrium (After Dean, 1987b).
sea level change rate during which time the ambient demand rate is established. Thus the
active zone sediment deficit rate will be
New Deficit Rate = [ dy + W, ( ] (70)
[f ol at 10 dt dt dt
in which t represents the nourishment rate and the subscript "0" on the bracket represents
the reference period before increased sea level rise. In order to decrease the deficit rate to
zero, the required nourishment rate is
dV I* \fdS\ fdS\
dV [W ahd]W [(dS) (dS)] (71)
dt at dt dt
These models may assist in evaluating the vulnerability of various shoreline systems to
increased rates of sea level rise.
CrossProfile Volumetric Redistribution Due to Sea Level Rise
Eq. (12) was developed earlier to describe the shoreline change, Ay, due to a sea level
rise, S. Associated with this recession is a crossshore transfer of sediment from the upper
to the lower portions of the profile. However there will be no net change of volume across
the entire active profile. This statement would not hold, of course, if a portion of the eroded
profile were peat or lagoonal muds that would be transported by suspension well beyond
the normal limits of the active profile.
It is instructive to consider the change in volume that would be measured due to surveys
extending out to a depth, h,, which is less than the active profile depth, h,. Referring to
Figure 36b, the volumetric change per unit length due to profile equilibration as a result of
sea level rise, S, must be considered for four regions.
_h.
VA = Aydh B < h, <S
B
VB = VA (S) Ay+ (h 3 dh S < h, < 0
JS L A
0.050 . 10 S
0 .%, 1 BW,*
0.5
1.0
a) NonDimensional Volume b) Definition Sketch.
Redistributed vs. NonDimensional
Depth.
Figure 36. Definition Sketch and NonDimensional Volume Redistributed as a Function
Of NonDimensional Depth. Due to Sea Level Rise, S. Case of B/h, = 0.25,
S/B = 0.5.
VC = VB(O) + Ay+ ()3/2 ( )3/2 dh O O
yD = c(h)+ Ay+ d h, S
A.h,S [_s A;l
(72)
The depth, he, at which the profiles cross will represent a maximum volume, Vma, and
is given by
h, = h, Ay +h' 3/2 2/3 (7)
h =h S (73)
and must be solved by iteration. The maximum volume, Vma, which occurs at he is
ma= Ay (B + h ) _S W* S 3/(74)
The height A can be shown to be
A = h I + 2 1+ S (75)
The above integration results can be cast into non dimensional form with the following
nondimensional parameters
Ay A = y
M h
h'e = h
h*
A
h*
St S
B
Bi= B
S/ B (76)
WB
The nondimensional volumes are
A = Ay'(l+ B'< h<S'B' (77)
S= Ay'(1+ (h' S 'B') SB < hB < 0 (78)
v = A (l+2+ [(h)/2 (, B)5/2 < h' < S'B' (79)
B'] 5B'E 5/ 5B
4 = Ay'(1+ 5 \Bh)5/2' 5 1 <.^<1I'B'+A'
(80)
It can be shown by substituting the value of A' from Eq. (75) and Ay from Eq. (12)
into Eq. (80), that V', (h' = 1 S'B' + A') = 0 as would be expected.
To illustrate the volume changes that would be determined by surveying to various
depths, h., consider the following example,
B' = B/h = 0.25
S' = S/B=0.5
For this case, the following results are obtained
Ay' A 0.10285
W,.
h', = he = 0.23947
h,
A
A' = =0.05520
h,
Jmax = VmaZ /(WB) = 0.11795
The variation of the crossshore volumetric transport versus depth below the elevated
water level, S, is shown in Figure 36a for this example. It is seen as expected that the total
volumetric change at h = h, + A is zero.
It is of interest to compare the maximum volumetric change, Vmaz, with the volume
deficit associated with a sea level rise, S. The latter quantity is simply WS. Thus this
ratio, r, is
V ,,Vmax W* "Bmax
WS S'
and for our example
(0.11795)
r (0.11795) = 0.2359
0.5
Thus, the maximum volumetric survey error that could occur due to the surveys not ex
tending to a sufficient depth is approximately 24% of the volume deficit associated with the
sea level rise.
It is also possible to develop approximate expressions describing the volumetric redis
tribution. First, we approximate the non dimensional recession (Eq. (12)) by
S S'
(h, + B) + ( )
which for our example yields Ay' = 0.10 vs the complete equation result of 0.10285. The
approximate expression for h' is
4 2Ayr2
h = (81)
9S'B'
which for our case yields ht = 0.30089 versus the complete equation result of 0.23947.
The value of A' obtained from the approximate value of Ay' is A' = 0.05717 vs 0.05520
obtained from the complete equation. Finally, the maximum nondimensional volume, Vmaz,
is 0.092036 versus 0.093497 obtained from the full equation. In general, it is seen that for
this example the approximations to the full equations are quite reasonable.
Trailing Beach Profile Signature Due to Sea Level Rise
If, as discussed by Bruun (1962), and as implied by Eq. (14), the beach profile moves
landward and upward in response to sea level rise, it is possible to infer a simple trailing
beach profile signature, which can then be compared with measured cross continental shelf
profiles. The processes are complex due primarily to the landward boundary of the profile
and the implicit assumption in Eq. (14) that the sand is transported only offshore. These
potential shortcomings aside, Eq. (14) predicts that for each unit of vertical rise, the
landward retreat of every element on the equilibrium beach profile will be, W./(h. + B),
a value usually considered to be in the range of 50 to 100. Thus the trailing profile slope
should be the inverse of this ratio. A comprehensive investigation of profiles may provide
insight into conditions under which Eq. (14) is most valid and/or of the ratio W,/(h. + B).
SUMMARY AND CONCLUSIONS
Equilibrium beach profile concepts provide a useful basis for application to a number
of coastal engineering projects. In addition to addressing conditions at equilibrium, these
concepts establish a foundation for considering the response of profiles out of equilibrium.
Based on analysis of numerous profiles representing laboratory and field scales, a reason
able approximate and useful form of a monotonic beach profile appears to be h(y) = Ay2/3
in which h is the water depth at a distance, y, offshore and A is a scale parameter depending
on sediment characteristics. A representation of the sediment scale parameter is presented
and provides a rational basis for assessing the relative merits of utilizing beach sand of
different sizes.
Methods are presented for quantifying the shoreline response due to elevated water levels
and wave heights on natural and seawalled shorelines. Additionally, results are presented for
calculating nourishment quantities for sand of uniform but arbitrary diameter. Depending
on volumes and sizes of sediment added, three types of profiles can occur: intersecting,
nonintersecting and submerged. The advantages of using coarser sand are quantified and
equations are presented expressing the volume of a particular sand size required to yield
a desired additional beach width. Many laboratory studies of beach profiles commence
with a planar slope which could be much steeper or coarser than the overall equilibrium
slope consistent with the sand size in the experiments. Applying equilibrium beach profile
concepts, it is shown that five profile types relative to the initial profile can occur. Three
of these types are erosional and two are accretional. Three of these profile types have
been identified in laboratory studies. Conditions under which profile type will occur are
quantified and all results including shoreline change are incorporated into a single graphical
representation.
The volumetric redistribution of sediment across the profile due to sea level rise is
examined in detail and compared with the total sediment "demand" as a result of the sea
level rise. An application is the possible error if the survey does not extend over the full
depth of effective motion. It is shown that the maximum error is only a fraction of the
sediment "demand".
The effects of sea level rise on nourishment needs are evaluated for cases with and without
onshore sediment transport across the continental shelf. It is shown that the sediment
volumes required to maintain a shoreline position vary directly with wave height and sea
level rise rates and inversely with profile slopes.
It is hoped that the results presented herein will provide guidance for coastal engineering
projects and serve as a framework for interpretation of project performance and behavior
of natural beach systems.
ACKNOWLEDGEMENTS
Support provided by the Sea Grant College Program of the National Oceanic and Atmo
spheric Administration under Project R/C S22 is hereby gratefully acknowledged. This
support has enabled the author and students to pursue a range of problems concerned with
rational usage of the shoreline. Ms. Cynthia Vey and Lillean Pieter provided their usual
equations are presented expressing the volume of a particular sand size required to yield
a desired additional beach width. Many laboratory studies of beach profiles commence
with a planar slope which could be much steeper or coarser than the overall equilibrium
slope consistent with the sand size in the experiments. Applying equilibrium beach profile
concepts, it is shown that five profile types relative to the initial profile can occur. Three
of these types are erosional and two are accretional. Three of these profile types have
been identified in laboratory studies. Conditions under which profile type will occur are
quantified and all results including shoreline change are incorporated into a single graphical
representation.
The volumetric redistribution of sediment across the profile due to sea level rise is
examined in detail and compared with the total sediment "demand" as a result of the sea
level rise. An application is the possible error if the survey does not extend over the full
depth of effective motion. It is shown that the maximum error is only a fraction of the
sediment "demand".
The effects of sea level rise on nourishment needs are evaluated for cases with and without
onshore sediment transport across the continental shelf. It is shown that the sediment
volumes required to maintain a shoreline position vary directly with wave height and sea
level rise rates and inversely with profile slopes.
It is hoped that the results presented herein will provide guidance for coastal engineering
projects and serve as a framework for interpretation of project performance and behavior
of natural beach systems.
ACKNOWLEDGEMENTS
Support provided by the Sea Grant College Program of the National Oceanic and Atmo
spheric Administration under Project R/C S22 is hereby gratefully acknowledged. This
support has enabled the author and students to pursue a range of problems concerned with
rational usage of the shoreline. Ms. Cynthia Vey and Lillean Pieter provided their usual
flawless typing and illustrative services, respectively. I appreciate the many discussions and
collegial support provided by present and former students.
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