Front Cover
 Title Page
 Table of Contents
 List of Figures
 List of Tables
 Model test apparatus, procedure...
 Experimental results and discu...
 Volume change analysis
 Shoreline and hydrographic...
 Prototype application
 Concluding remarks
 Appendix A. Contour maps
 Appendix B. Results of empirical...

Group Title: UFL/COEL (University of Florida. Coastal and Oceanographic Engineering Laboratory) ; 89/015
Title: Effects of seawalls on the adjacent beach
Full Citation
Permanent Link: http://ufdc.ufl.edu/UF00076139/00001
 Material Information
Title: Effects of seawalls on the adjacent beach
Series Title: UFLCOEL
Physical Description: xii, 130 leaves : ill. ; 28 cm.
Language: English
Creator: Toue, Takao
University of Florida -- Coastal and Oceanographic Engineering Laboratory
Publication Date: 1989
Subject: Beach erosion   ( lcsh )
Sea-walls   ( lcsh )
Littoral drift   ( lcsh )
Coastal and Oceanographic Engineering thesis M.S
Coastal and Oceanographic Engineering -- Dissertations, Academic -- UF
Genre: non-fiction   ( marcgt )
Thesis: Thesis (M.S.)--University of Florida, 1989.
Bibliography: Includes bibliographical references.
Statement of Responsibility: by Takao Toue.
Funding: This publication is being made available as part of the report series written by the faculty, staff, and students of the Coastal and Oceanographic Program of the Department of Civil and Coastal Engineering.
 Record Information
Bibliographic ID: UF00076139
Volume ID: VID00001
Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
Resource Identifier: oclc - 20311918


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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
        List of Figures 4
    List of Tables
        List of Tables
        Abstract 1
        Abstract 2
        Page 1
        Page 2
        Page 3
        Page 4
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    Model test apparatus, procedure and condition
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    Experimental results and discussion
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    Volume change analysis
        Page 56
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    Shoreline and hydrographic changes
        Page 71
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    Prototype application
        Page 99
        Page 100
    Concluding remarks
        Page 101
        Page 102
        Page 103
    Appendix A. Contour maps
        Page 104
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    Appendix B. Results of empirical eigenfunction analysis
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Full Text




Takao Toue
Hsiang Wang

Sea Grant Project No. R/C-S-26
Grant No. NA86AA-D-SG068




Takao Toue
Hsiang Wang

Coastal and

Oceanographic Engineering Department
College of Engineering
University of Florida
Gainesville, Florida 32611

Sea Grant Project No. R/C-S-26
Grant No. NA86AA-D-SG068




LIST OF FIGURES ....................

LIST OF TABLES ....................

ABSTRACT ........................


1 INTRODUCTION ...................

1.1 Statement of the Problem ............

1.2 Effects of Seawalls -Literature Review .....

1.2.1 Model Test ................

1.2.2 Field Survey ...............

1.2.3 Numerical Simulation ..........

1.3 Possible Mechanisms of Seawall's Effects ...

1.4 Objectives and Procedure ............


2.1 Model Test Apparatus ..............

2.1.1 Wave Basin ................

2.1.2 The Beach and Seawall Model .....

2.1.3 Measurement Apparatus .........

2.2 Experimental Procedures ..... ......

2.3 Test Condition ..................


3.1 Evaluation of Model Test ...........



,. . .




. .

. . o

. . .


. .

. .

. . ii

3.1.1 Design of Model Beach ...................... 18

3.1.2 Assessment of Equilibrium Beach Profile . .... 21

3.1.3 Assessment of Normal and Storm Profiles Classification 21

3.1.4 Flow Regime and Mode of Sediment Transport . ... 27

3.2 M odeling Law ................... ........... 30

3.2.1 Modelling Requirement from Dean's Equilibrium Profile 32

3.2.2 Hughes' Modeling Law ...................... 33

3.2.3 Wang's Modeling Law Revised . . . .... 34

3.2.4 Noda's Modeling Law ...................... 40

3.2.5 Summary ............................. 42

3.3 Comparison to 2-D and 3-D Model Test Results . . .... 42

3.3.1 Beach Profile Comparison ... ....... .... .. 43

3.3.2 Offshore Breaking Bars ..................... 45

3.3.3 Reflection Bars .......................... 48

3.3.4 Scour .. .. ... . .. .. .. .. .. .. 50

4 VOLUME CHANGE ANLYSIS ....................... 56

4.1 Definition of Volumetric Changes . . . .. .. 56

4.2 Results and Discussion ......................... 59


5.1 Empirical Eigenfunction (EEF) Analysis . . . ... 71

5.1.1 Literature Review ....................... .. .. 71

5.1.2 Basic Concept on Empirical Eigenfunction Analysis . 78

5.1.3 Formulation and Procedure of EEF for Contour Lines .. 81

5.1.4 Results and Discussion ...................... 82

5.2 Shoreline Changes Based on One-Line Theory ........... ..92

5.3 Correlation Between Shoreline Changes and Volumetric Changes 96

6 PROTOTYPE APPLICATION ....................... 99


7 CONCLUDING REMARKS ......................... 101

7.1 Important Findings ................... ....... 101

7.2 Recommendation for Future Study . . . ... 103


A CONTOUR MAPS ................... ........... 104


BIBLIOGRAPHY .......................... ....... 126

BIOGRAPHICAL SKETCH ................... ....... 130


1.1 Erosion Mechanisms in the Presence of Seawall . . 7

2.1 Beach and Seawall System in the Experiment . . 9

2.2 Geometry of the Initial Profile in the Model Test . ... 10

2.3 General View of the Beach Profile Measurement System (after
Bodge, 1986) ............................. 12

2.4 Assembly of Beach Profile Measurement System . ... 13

2.5 Locations of the Wave and Current Measurement Points . 14

3.1 Correlation of Equilibrium Beach Profile Scale Parameter, A,
with Combined Sediment Wave Parameter, Hb/TW ...... ..20

3.2 Comparison of Equilibrium Beach Profile to the Final Beach
Profile in the Model Test ...................... 22

3.3 Comparison of Equilibrium Beach Profile to the Beach Profile
in the Model Test for each elapsed time (Case 1) . ... 23

3.4 Comparison of Equilibrium Beach Profile to the Beach Profile
in the Model Test for each elapsed time (Case 3) . ... 24

3.5 Comparison of Equilibrium Beach Profile to the Beach Profile
in the Model Test for recovery condition (Case 1) . . 25

3.6 Criterion of Normal and Storm Profile (after Kriebel at al., 1986) 26

3.7 Criterion of Normal and Storm Profile (after Sunamura and
Horikawa, 1974) ........................... 27

3.8 Criterion of Normal and Storm Profile (after Hattori and Kawa-
m ata, 1980) .............................. 28

3.9 Criterion of Normal and Storm Profile (after Wang, 1985) 28

3.10 Jonsson's Flow Regime ....................... 29

3.11 Classification of Sediment Transport Model (after Shibayama
and Horikawa 1980) ......................... 31

3.12 Reduction of Settling Velocity in the Oscillatory Flow (after
Hwang 1985) ............................. 39

3.13 Moore's Diagram for Scale Parameter A . . .. 41

3.14 Comparison of Initial Beach Profiles between 2-D and 3-D Model
Test . . . . . . . . 46

3.15 Comparison of Beach Profiles after 4 Hours Duration between
2-D and 3-D Model Test ....................... 47

3.16 Relation between HB and ht adopted by Horikawa(1988) .. 48

3.17 Breaking Wave Index adopted by Horikawa (1987) . ... 49

3.18 Relation ht and he .................. ....... 49

3.19 Definition of Scour Depth and Other Parameters . ... 51

3.20 Relation of SI and ho ........................ 53

3.21 Relation between St/Ho to ho x tan/Lo . . ... 54

4.1 Sketch of Coordinate System . . . .. .. 57

4.2 Definition of Three types Volumetric Changes . ... 58

4.3 Rate of Volumetric Change along a Profile, p . ... 61

4.4 Cumulative Rate of Volumetric Change Referenced to Down-
wave Boundary, i6r ......................... 62

4.5 Rate of Volumetric Change in a Local Control Area, i; . 63

4.6 Ratio of the Rate of Volumetric Changes with and without Sea-
w all . . . . . . . .. 67

4.7 Ratio of Volumetric Changes with and without Seawall for 4
Hours Duration ............................ 68

4.8 Sketch of 12 Sections Surrounding Seawall . . .... 69

4.9 Volumetric Changes in Sections for 4 hours Duration . 70

5.1 Contour Maps for Four Hours Elapsed Time . .... 72

5.2 Schematic of Seasonal Sand Volume Changes at Torry Pine Beach
California, based of a Dual Pivotal Points (after Aubrey 1979) 74

5.3 Spatial Eigenfunction in Birkemeier's Study . . ... 75

5.4 Definition of d" ........................... 83























Contour Map for Erosive condition t=0 hour .

Contour Map for

Contour Map for

Contour Map for

Contour Map for

Contour Map for

Contour Map for

Contour Map for

Contour Map for

Eigenfunction for

Eigenfunction for

Eigenfunction for

Eigenfunction for

......... 105

Erosive condition t=1 hour . . ... 106

Erosive condition t=2 hour . . ... 107

Erosive condition t=4 hour . . ... 108

Recovery condition t=1 hour . ... 109

Recovery condition t=2 hour . ... 110

Recovery condition t=4 hour . ... 111

Recovery condition t=8 hour . ... 112

Recovery condition t=12 hour . ... 113

Case 1 ...................... 115

Case 2 ...................... 116

Case 3 ...................... 117

Case 4 ...................... 118

Real Shoreline Changes, Net Shoreline Changes and Results of
EEF (Case 1) .............................

Real Shoreline Changes, Net Shoreline Changes and Results of
EEF (Case 2) .............................

Real Shoreline Changes, Net Shoreline Changes and Results of
EEF (Case 3) .............................

Real Shoreline Changes, Net Shoreline Changes and Results of
EEF (Case 4) .............................

Real Shoreline Changes, Net Shoreline Changes and Results of
EEF (Case 5) .............................

Real Shoreline Changes, Net Shoreline Changes and Results of
EEF (Case 6) .............................

Schematic Concept of One-Line Theory . . . .

Ratios of Ultimate Mean Shoreline Position with and without
Seaw all . . . . . . . .

Compariosn of Shoreline Change between Measured and Calcu-
lated . . . . . . . . .

B.5 Eigenfunction for Case 5 ...................... 119

B.6 Eigenfunction for Case 6 ................... .. .. 120

B.7 Eigenfunction for Case 1 (recovery) . ..... . .. 121

B.8 Eigenfunction for Case 2 (recovery) . .... .. 122

B.9 Eigenfunction for Case 3 (recovery) . . ..... 123

B.10 Eigenfunction for Case 4 (recovery) . ..... . .. 124

B.11 Eigenfunction for Case 6 (recovery) . ... .. 125


2.1 Physical Parameters of the Model . . . .... 11

2.2 Test Condition ................... ......... 16

3.1 2-D Model Test Condition (Barnett 1987) . . .... 43

3.2 Comparison of Scale Parameter "A" in 2-D and 3-D Model Test 55

4.1 On/Offshore and Longshore Transport Rate for Natural Beach 64

4.2 On/Offshore and Longshore Transport Rate for Seawall Backed
Beach .. . . . . .. . . 64

5.1 Contribution of Each EEF .............. ....... 85

5.2 Estimated Values of e, C1, C2, k and 6 . . .... 96

6.1 Example of Prototype Quantities Based upon Laboratory Re-
sults ................... ................100




This study was carried out to examine the effects of seawalls on the adjacent

beach by three dimensional model test. The results obtained from model test were

analyzed in terms of volumetric changes and shoreline and hydrographic change to

quantify the effects of seawalls.

The experiments were carried out in the wave basin of Coastal and Oceano-

graphic Engineering department, University of Florida. A model seawall was in-

stalled on the test beach (19mxl4m) which was initially molded into equilibrium

shapes. During the test, hydrographic surveys were conducted at regular time in-

tervals. The main variable in the experiment is the wave angle. Cases both with

and without seawall were tested.

Before examining the effects of seawalls, the problems inherent to model test

were examined. First, assessment of equilibrium beach profile concept, flow regime

and modes of sediment transport were examined. The experimental set up is found

to be reasonably representative of prototype phenomena, for the erosive conditions

but not for the recovery conditions. Second, using equilibrium beach profile as a

prototype template, several modeling laws were examined. Again, it was found

that modeling law for erosion is far more firmly established than for that of accre-

tion. From this analysis, it was also confirmed that the physical meanings of scale

parameter "A" of the equilibrium profile is the settling velocity scale parameter.

The effects of seawalls were examined in terms of volumetric changes and shore-

line changes. In the volumetric change analysis, three types of volumetric changes

were defined and examined. Especially, the volume change in a control area sur-

rounding seawall showed that the erosion rate in front of and adjacent the seawall

was larger than that without seawall for oblique incident waves, but is smaller for

normal incident waves. Although the rate of erosion was larger with seawall, the

analysis also showed that the influence of seawall was localized.

In the shoreline and hydrographic change analysis, first of all, the dominant

modes of shoreline movement were examined using empirical eigenfunction. The

dominant mode of eigenfunction of shoreline was represented by the simple retreat

mode and the rhythmic feature mode. These dominant modes did not differ between

cases with and without seawall. Moreover, the temporal eigenfunction without

seawall is very similar to that with seawall. The most obvious effect of seawall which

appeared in the first spatial eigenfunction mode was the groin effect at the downdrift

side. The hydrographic change analysis revealed that the first spatial eigenfunction

was much more irregular and the contribution of the first eigenfunction was smaller

than that of the shoreline.

Based on the empirical solution of one-line theory, alongshore diffusivity was

calculated. The calculated value was compared with the existing formula and was

found to be reasonable for natural beach.



1.1 Statement of the Problem

Beach erosion is found along many portions of the coast of the world. The

causes of the erosion could be sea level rise, reduction in sediment supply, interrup-

tion of the littoral drift by structures. There are several conventional engineering

solutions to combat such erosion. Those are (1) coastal structures such as groins,

seawalls, breakwaters and coastal dikes, and (2) non-structural solutions, such as

beach nourishments. Among them, seawalls might be the most efficient and direct
method to protect the up-land property provided that they are designed adequately.

Recently, the adverse effects of seawalls on their fronting and adjacent beaches

have gained great attention and raised criticism about the use of seawalls in the

coastal area. The most often alleged effects are (1) offshore profile slope steepening,

(2) intensified local scour, (3) transport of sand to a substantial distance offshore,

(4) adverse down drift erosion and (5) delay post-storm recovery (Dean, 1986).

Although numerous example can be found from articles in newspaper or popular

magazine reporting the adverse effects of seawalls, reliable and scientific based doc-

ument is actually scarce. Moreover, the conclusion derived from the few technical

reports on the adverse effects of seawalls remain controversial. Considering the

merits of seawall as the reliable structure to protect upland erosion,abandoning or

prohibiting seawalls altogether as means of coastal protection without firmly estab-

lishing their effects might be irrational. Therefore, there is a need to examine the

effects of seawalls carefully and to quantify them if possible. Also, to examine the

causes and effects of seawalls might lead to more rational design in the future.

1.2 Effects of Seawalls -Literature Review

A summary of an evaluation on the coastal armoring effects was given by Dean

(1986). Subsequently, Kraus (1987) also presented a general review. Although

Kraus titled his paper as the review of effects of seawall, the subject was extended

to revetments, breakwaters and dikes. In the present study, the terminology of

seawall is defined as the vertical structure with backfills and is normally located

onshore from the shoreline. Based on this definition, the literatures concerning

effects of seawalls are reviewed.

1.2.1 Model Test

There were many two dimensional model tests related to seawall, but most

studies paid attention to the local scour and the stability of seawall. Hattori and

Kawamata's (1977) and Barnett's (1987) two dimensional wave tank studies were

the few dealing with beach evolution in front of seawalls. Hattori and Kawamata

concluded that except in the immediate vicinity of seawall, the erosional and recov-

ery process of beaches with or without the presence of seawall was similar. Barnett

(1987) also carried out two dimensional model test, and concluded that the ma-

jor transport process was not significantly influenced by the presence of seawall.

Furthermore, he examined the volume change of beach profile, and found that the

volume of sand retained upland of the structure which would be eroded under iden-

tical wave condition without seawall was found to be greater than the additional

volume eroded at the toe of the structures The ratio of sand volume saved versus

additional volume eroded was found to be approximately 2 to 1.

Three dimensional experiment for seawall effects was carried out by McDougal,

Sturtevant and Kommer (1987) in a relatively small basin (7m x 7m). Only flanking

effects were examined. Combining with field survey results, they established an

empirical relation between flanking and the length of seawall as

s = 0.101 x L,



where s is the flanking erosion, and L, is the length of the seawall.

Equation (1.1) does not include the forcing factor such as the wave height, period

and so on. Besides, the data, upon which the equation was based, was considerably


1.2.2 Field Survey

Field observations of beach changes under the influence of seawall were con-

ducted by only a few. Based on aerial-photograph, Birkemeier (1980) estimated the

bluff and shore erosion along a stretch of beach in the southeastern section of Lake

Michigan from 1970 to 1974. To compare erosion rate of both protected and unpro-

tected shorelines, he selected five reaches along the beach, each approximately 1.6

km long and each with different characteristics; one of which contained a section of

seawall 579 meter long. Erosion due to flanking at both downdrift and updrift of the

seawall were evident. These erosions appeared to be localized as all the five reaches

sustained approximately the same rate of shoreline and bluff recession. Since his

study is based on interpretation of aerial-photos the beach changes in front of a

seawall cannot be evaluated.

Berigan's (1985) study was based on beach surveys from the water line to the

seawall front. The Traval Seawall is located in San Francisco. It was built to protect

a 220m long badly eroding beach front. The toe of structure is above the highest

astronomical tide. By analyzing beach profiles taken over a seven year period and

relation of beach changes to wave energy, Berigan found that: low wave energy

has little effect; intermediate wave energy built up the beach and high wave energy

removed sand in front of seawall. He also concluded that while beach rebuilding

is a slow process, the existence of the seawall accelerated the rebuilding process

considerably compared with a beach with no seawall, which is quite contrary to

common belief.

Kana and Svetichny (1982) monitored the beach profile change in South Car-

olina coast where the beach nourishment projects were carried out. They compared

the sculptured beach to the seawall backed beach, and concluded the seawall backed

beach results the more significant erosion than the sculptured beach but flawed.

Kriebel, Dally and Dean (1986) also monitored the beach profile at Clearwater

in Florida after hurricane Elena. They examined the recovery process at down-drift

side of seawall, in front of seawall and the up-drift side of seawall. After comparing

the profiles, they made the general remark that the toe scouring at the base of a

seawall associated with a storm event was clearly evident but also suggested that

the presence of seawall did not considerably alter the beach recovery. The same

amount of sand that would have supplied to beach through upland was probably

deprived from the beach in the form toe scouring.

Walton and Sensabaugh (1985) investigated the flanking effects in Florida coast,

and suggested the suitable return wall length.

Dette and Gartner (1987) presented the time history of the island of Sylt/North

Sea and also by examining the changes in the beach topography which were mea-

sured in 1869, 1953 and 1967; the long term resonance of the coastal structures on

the foreshore bed features were estimated. They obtained the following conclusion:

(1) The characteristic feature of the foreshore consisting of longshore bars and

troughs has not been "destroyed" due to the presence of coastal structures.

(2) In the vicinity of the coastal structures, it is obvious that the trough has deep-

ened with time and also seems to be shifted landward.

(3) With respect to a final judgement about the interaction of structure and fore-

shore topography probably the considered time space is not yet sufficient.

1.2.3 Numerical Simulation

The models developed by Ozasa and Brampton (1980) and Hanson and Kraus

(1980) have been applied on seawall-backed beaches. The authors made the sim-

plified assumption in their model that the beach profile simply moves seaward or

shoreward in parallel to itself without changing form. Hanson and Kraus (1986)

in their shoreline evolution model incorporated a seawall as a solid boundary to
compute plan form changes. Local effects such as sea bottom scouring and flanking

were not modeled. As Kraus (1987) observed, the simulation does not follow the

real sediment transport.

Kraus and Larson (1988) and Larson (1988) incorporated in their beach profile

model a two dimensional longshore bar configuration and allowed the bar to move

and to change shape and volume based on wave properties. A seawall is permitted

in the model as a solid boundary. Again, similar to Hanson and Kraus's approach,

local scour and wave reflection effects are not modelled.
1.3 Possible Mechanisms of Seawall's Effects

The effects of seawall are not well understood, but several possible mechanisms

can be deduced from our general knowledge in coastal engineering. These are illus-

trated in Fig. 1.1 and are also described below:

(1) flanking effects: Flanking due to wave refraction and diffraction is expected to

occur on the corners of the seawall to cause local erosion.

(2) cross wave effects: During storm surge period, the water depth in front of sea-

wall is likely to be larger than that along the beach and wave reflection will occur

as shown in Fig. 1.1 b. Consequently, the longshore current together with more

reflected wave energy trapped in the trough will remove sand in front of seawall and

transport them to down drift location.

(3) groin effects: If the shoreline retreats due to littoral drift, the seawall will even-
tually protrude seaward and act like a groin. Although this groin effect does not

remove sand from the system it increases downdrift erosion pressure.

(4) sand supply cut off: Seawall prevents sand from being added to the littoral

system, which again adds erosional stress downdrift and could result in lower bar

profile in front of seawall during the storm surge period.

1.4 Objectives and Procedure

The main objective of the present study was to attempt to quantify the three di-

mensional effects of seawall on beach changes described in previous section. In order

to gain a fundamental understanding it was decided to conduct three dimensional

model tests in the laboratory environment. In analyzing the model test results, the

following problems have been addressed.

(1) Effects of seawall on the volumetric changes in the system

(2) Effects of seawall on the shoreline and hydrography

Furthermore, the inherent problems associated with model testing such as scale

effects, modeling law and so on are also examined to insure the adequacy of the

model test results.

This report consists of seven chapters. After introduction, model test proce-

dure is described in chapter 2. In chapter 3, the problem inherently related to the

model test is examined carefully including modeling laws. The effects of seawall on

volumetric changes and shoreline changes are examined in chapter 4 and 5, respec-

tively. In analyzing the shoreline changes, both empirical eigenfunction analysis and

a one-line concept are employed. Then, in chapter 6, based on the modeling law

established in the present study, model test results are translated into prototype.

Finally, in chapter 7, conclusions and recommendations for future studies are given.

ErodinK Shoreline
with Flanking

b) Cross Wove Effect


Storm Profile-
without Seawoll


'... Normal Profile

/Storm Profile
with Seawall

d) Sand Supply Cut Off (Shaded Area
indicates Sand denied to the System)

Figure 1.1: Erosion Mechanisms in the Presence of Seawall


a) Flanking





g r


2.1 Model Test Apparatus

2.1.1 Wave Basin

The model test is carried out in the wave basin of the Coastal and Oceanographic

Engineering Department, University of Florida. The dimension of the basin is

approximately 28 m x 28 m and 1 meter deep. It is equipped with a snake-type

wave maker which consists of 88 paddles of 24 cm width each. By adjusting the

phase of each individual paddle, waves of various oblique angles can be generated.

The wave maker is capable of generating waves of wave heights ranging from 1 cm

to 12 cm and wave periods from 0.89 to 1.89 sec.

2.1.2 The Beach and Seawall Model

The beach and seawall system used in the experiments is shown in Fig.2.1 The

beach is composed of 125 tons of well sorted fine quartz sand. The total length of

the beach is about 19 m and the distance from the back shore to the toe of the

beach is 14 m. The orientation of the shoreline is 10 degree to the wave maker. The

back shore of the beach is supported by a block wall of 81 cm high.

Both sides of the beach are constrained by wooden templates cut into a design

beach profile shape. This design allows wave induced long-shore current to circulate

unimpeded through the backside of the beach.

The beach profile is shaped in accordance with the concept of equilibrium beach

profile (Dean, 1977). Based upon a fall velocity of Ws = 1.7 cm/sec, which corre-

sponds to the median size of the beach sand used in the model, the geometry of the


Figure 2.1: Beach and Seawall System in the Experiment


-0.2 h(Y)= 0.089Y


Figure 2.2: Geometry of the Initial Profile in the Model Test

profile is given by

-0.152y -2.50m < y < -1.22m
-0.135y -1.22m < y < -0.61m (2.1)
-0.130y -0.61m < y < 0.095m
-0.089y2/3 0.095m < y

Figure 2.2 shows the geometry of the profile according the above equation. The

details concerning the model beach design and the physical parameters involved can

be found in Bodge (1986). Table 2.1 summarizes the physical parameters related

to model design.

The model seawall is made of sheet metal. It is installed parallel to the beach

near its center. The length of the seawall is 3.0 m and with 1.0 m return walls on

each side to prevent flanking. The height of the seawall is 45 cm which is sufficient

to prevent wave over toppings. The toe of the seawall is located at 45 cm above the

basin bottom which corresponds to the mean water level of storm conditions in the

present test configurations. Details of test conditions will be discussed later.

Table 2.1: Physical Parameters of the Model

Median Sediment Size Dso = 0.16 mm
Sorting Coefficient So = 1.27
Median Fall Velocity W, = 1.7 cm/sec
Shoreline Length 19 m
Back-shore to Toe Length 14 m
Time Scale Nt w 6
Vertical Length Scale N,v 9
Horizontal Length Scale Nh = 18
Distortion (N, : Nh) 2 : 1
Design Water depth 45 cm and 35 cm
Design Submerged Profile h(x) = A x2/3
Design A Parameter A = 0.089 m1/3

2.1.3 Measurement Apparatus

There are three basic quantities to be measured: the beach profile and hydro-

graphic changes, the input wave conditions and the near shore current.

The measurement of beach profile and nearshore contour was carried out by

means of an automated profile measurement system designed by the Coastal and

Oceanographic Engineering Laboratory (COEL). The assembly, as shown in Fig.

2.3, consists of four major components: the supporting platform, the wheeled truss,

the profiler carriage and the profiler.

The supporting platform is a rectangular steel frame spans the entire area to

be surveyed. The wheeled truss, as the name implies, is a steel truss structure

with guided wheels.The truss is placed on the pair of shore-parallel rails of the

supporting platform and is free to traverse from one end of the beach to the other.

The cross-shore movement is facilitated by the motorized profiler carriage mounted

on the truss. The profiler, which is the heart of the system, is a pivoting arm

mounted on the carriage. At the lower end of the arm, a freely rotating PVC wheel

is attached. As the carriage moves forward, the profiling arm swings behind and

below the carriage following the beach face as the bottom wheel rolls along the


Figure 2.3: General View of the Beach Profile Measurement System ( after Bodge,

Figure 2.4: Assembly of Beach Profile Measurement System

bed. This assembly is shown in Fig. 2.4. The horizontal position is determined

by the voltage change across a potentiometer on the carriage as it is at different

locations on the truss. The vertical position is determined by the voltage change

across a potentiometer owing to the change of angle of the arm. This angle is then

converted into vertical distance. All the signals were collected and stored in the

COEL computer system and also recorded on strip charts.

Waves were measured by two capacitance-type wave gauges. One is located

in the offshore region between the wavemaker and the toe of the beach and the

other one is mounted on the carriage. The offshore gauge monitors the incident

wave conditions and the carriage gauge provides the local wave conditions, usually

in the breaking zone. Near shore currents were also measured using an electric-

magnetic current meter. Video and still photographs were also employed to estimate

wave direction and current patterns. Rhodamine dye was used for flow pattern

Y *



4 -- -

3 6 9 12 15 X(m)
current& wav wave UrEn

Figure 2.5: Locations of the Wave and Current Measurement Points


The locations where wave and current measurements were performed are given

in Fig. 2.5 together with beach profile measurement lines.

2.2 Experimental Procedures

The experiments were carried out following the steps outlined here:

Step 1 The initial beach profile was molded in the equilibrium shape according to

Eq.2.1 for every test case. First, wooden templates cut into the initial profile

were inserted into and across the beach at a spacing of about 3.6 m. The beach

between the templates was then molded in shape by a wooden bar along the

templates. After completion of one section, the templates were removed and

reinserted into the next section and so on until the complete beach was molded

into equilibrium profile.

Step 2 The basin was filled to the required water depth and the wave generat-

ing machine was adjusted to produce the test wave condition including wave

height, wave period and wave incident angle.

Step 3 The initial beach profiles were surveyed after the beach attended certain

degree of saturation. A total of 21 profiles were surveyed at equal spacing of

75 cm. The grid system was as shown in Fig. 2.5 with the origin at the left

lower corner, the x-axis parallel to shore and y-axis perpendicular to shore.

Step 4 The beach was then subject to the test wave conditions for a designated

duration. Each test usually began with waves of erosional nature for about 4

hrs or until the beach reached a state while the erosional process became very

slow. The wave condition was then changed to affect beach recovery. The

recovery process was much slower and usually took 12 hrs to reach a state of

Squasi-equilibrium. During this test period, profile measurements were carried

out at regular intervals. In the erosional phase, surveys were conducted at 0,

1, 2, and 4 hrs whereas in the recovery phase, the intervals were 1, 2, 3, 8 and

12 hrs.

2.3 Test Condition

A total of six tests were conducted. Table 2.2 summarizes the test conditions.

In all tests, the wave height was set at 11cm and the wave period at 1.74 sec during

the erosion phase; the wave height was changed to 3.0 cm while keeping the period

the same at 1.74 sec during the recovery phase. The corresponding water depths

were 45 cm for erosional phase and 35 cm for recovery phase. One of the main

parameters was the wave incident angle which was changed from 0 degree to 5

degrees and, finally, to 10 degrees. Of the six tests, three were with seawall and

three without so that the effects of seawall can be examined in reference to cases

without seawall.

Table 2.2: Test Condition

Case wave wave water wave seawall elapsed
height period depth angle time
(cm) (sec) (cm) (o) (hour)
Casel 11.0 1.74 45.0 0 no 4.0
Casel[R] 3.0 1.74 35.0 0 no 12.0
Case2 11.0 1.74 45.0 0 yes 4.0
Case2[R] 3.0 1.74 35.0 0 yes 12.0
Case3 11.0 1.74 45.0 5 no 4.0
Case3[R] 3.0 1.74 35.0 5 no 12.0
Case4 11.0 1.74 45.0 5 yes 4.0
Case4[R] 3.0 1.74 35.0 10 yes 12.0
Case5 11.0 1.74 45.0 10 no 12.0
Case6 11.0 1.74 45.0 10 yes 5.0
Case6[R] 3.0 1.74 35.0 10 yes 12.0


3.1 Evaluation of Model Test

Laboratory experiments serve two basic purposes:

1. To discover and understand the fundamental underlying principle of a physical

phenomenon or process such that knowledge gained from the experiments can be

applied to prototype project.

2. To faithfully portray the prototype condition so that the laboratory results can

be extrapolated to prototype application.

To achieve the first purpose, laboratory experiments are often designed in such

a way that the pertinent factors affecting the process are isolated. Experiments

are then carried out to establish the relationship between these factors and their

corresponding effects. The physical rules derived from the experiments are then

applied to the actual engineering works. The difficulty with this approach lies in

the fact that we must identify the most pertinent factors which is not always an

easy task for a complicated system as the coastal beach. Furthermore, to study

the effect of an isolated factor gives no assurance or information on the interactions

among the factors.

The second approach does not make an explicit attempt to isolate the factors.

Rather, its attempts to duplicate the prototype process as faithfully as possible

so that the experimental results can be directly applied to the prototype. The

difficulty with this approach is the degree of success of faithfully reproducing the

natural system at a reduced scale.

The approach of the present experiments lies in between the above two in that

the experiment is designed to portray the prototype condition as closely as possi-

ble. However, owing to the inherent difficulty that will be discussed later complete

similarity can not be achieved between the model and the prototype. We must

supplement our approach by attempting to identify the most pertinent factors in

the process and to establish their roles in the whole process. Again owing to the

complexity of the phenomenon these factors and their interactions can not always

be easily sorted out. Therefore, before the results are presented, it is instructive

to first examine here the relevance of the experiment to the prototype process it

attempts to replicate.

3.1.1 Design of Model Beach

The experiment is not intended to model a specific prototype condition. How-

ever, the model beach must fulfill two basic requirements: it represents initially an

equilibrium condition and it realistically portrays a plausible prototype condition.

To fulfill these requirements, the original design by Bodge (1987) was closely fol-

lowed. Bodge's laboratory experiments were designed to model a stretch of sandy

beach along Atlantic-coast near Duck North Carolina where he conducted field tests

on short-term impoundment tests of longshore sediment transport. Hughes' (1983)

moveable-bed modeling law was utilized to scale the model beach based upon typi-

cal profiles measured in the field. In Hughes' modelling law, there are two criteria

that involve four scale parameters:

Tn = /2 (3.1)

A = (3.2)

where A, 6, Tn, and W., are the ratios of prototype to model of the horizontal

length, the vertical length, the time scale and the sediment particle fall velocity,

respectively. Bodge established W, = 1.47 and selected Tn = 6. Thus, based

upon the above criteria, he arrived at A = 9 and 6 = 18, and therefore a

2:1 horizontal/vertical distortion. The details of the model design parameters are

given in Table 2.2. The time scale was so chosen that the corresponding vertical
and horizontal length scales, when applied to the field profiles, produced a model

geometry which closely resembled an equilibrium profile for the median grain size.

Therefore, the two basic requirements stated above are fulfilled. Specifically, the

submerged portion of the beach profile follows:

h(y) = Ay2/3 (3.3)

where h(y) is the water depth at y, y is the distance from the shoreline and A is

the scale parameter.

The scale parameter "A" is shown to be a function of grain size diameter (Moore,

1982), or more appropriately, a function of fall velocity. Based upon Moore's dia-
gram, A = 0.078 m for the model sand. The best fit of the field profiles obtained by

Bodge yielded a value of A = 0.13 m. This, when scaled to the model, would give

a value of A = 0.098 m. Bodge compromised the selection by choosing A to be the
average of the two values, or A = 0.089 m. He also observed that the profile was

fairly stable for all experiment waves, of which wave heights were 0.03 m to 0.1 m,

and wave periods were 0.85 to 0.19 sec. In the present model design, there appears

to be no reason to change the value A as suggested by Bodge; the value of A =

0.089 m is used. Dean (1986) re-plotted the value of A as a function of fall velocity

as shown in Fig.3.1. Thus, for the test range of Hb/TW from 1.93 to 7.23, the value

of "A" is no longer a constant but varies from 0.15 to 0.06. Consequently, there is

no tangible advantage to select one value over the other as long as the A value falls

in the mid range of the extremes.

,Normal Profile I Storm Profile

.v I I 1 I

0.5 -
Hb= Breaking Wave
T =Wave Period
Sw =Sediment Fall






I I 11

(No Bar) I


Bar Present

1 I I 1 I I 1 I.

- Recommended

From Hughes'
Field Results

From Swart's --N
Laboratory Results

I I I I iii

I I I I I i l l

(Hb / wT)


-;-ri -

Figure 3.1: Correlation of Equilibrium Beach Profile Scale Parameter, A, with
Combined Sediment Wave Parameter, Hb/TW


- ---- -(No Bar) II

' '""' ~L `'----~--~II


3.1.2 Assessment of Equilibrium Beach Profile

To assess the applicability of equilibrium beach profile at this laboratory scale,

profiles measured at the center of the test section (x = 7.5 m) without seawall were


For the erosive wave condition, the fall velocity parameter assumes the value:

Hb 0.13
= 4.39 (3.4)
TW 1.74 x 0.017 = 4)

According to Fig. 3.1, the value of "A" is 0.09. Under this condition, it is found


a. The over all submerged profiles at the final stage of the tests match well with

the equilibrium profile for all wave angles tested 0 5 and 100. Fig. 3.2 shows

the comparisons of the experimental results and the theoretical profile.

b. Moreover, the equilibrium profile also represents the model beach profiles

well during the transient stages. Figs.3.3 and 3.4 shows the comparisons for elapsed

time of 1 hr., 2 hrs., and 4 hrs. respectively.

For the recovery wave condition, the fall velocity parameter assumes the value

of 1.01 and the corresponding "A" value is 0.24. As can be seen in Fig.3.5 the

comparison is poor.

3.1.3 Assessment of Normal and Storm Profiles Classification

The classification of normal and storm beach profiles is a tool used by many in-

vestigators to determine whether a beach is undergoing or will undergo an erosional

or accretional process. The topic has been quite extensively researched since the

first attempt by Johnson (1952). A literature review is given recently by Kriebel

et al. (1986). Four criteria purposed by four different authors-Dean (1973, 1977),

Sunamura (1974), Hattori and Kawamata (1980) and Wang (1985)-are tested here.

Common to all the four criteria are two parameters, the wave steepness, Ho/L,,

for the wave condition and the sediment fall velocity, W, or sediment particle size,



w 0








Equilibrium Beach Profile

- 0.



Equilibrium Beach Profile

( RA=0.09 m'

I 1s

0 O









t = 1 hour

n t
0 (q









t = 4 hour

Equilibrium Beach Profile

; 0,09 m"'' )




/.0 2.0 3.0 4.0 5.0 6.0

Equilibrium Beach Profile

t 2 hu( A = 0.09 m" )
t = 2 hour

I I I -i I I I --- 1- I
1.0 2.0 3.0- 4.0 5.0 6.0

Equilibrium Beach Profile


X=9.0 m

X=8.25 m

X=7.50 m

X=6.75 m

X=5.25 m

X=4.50 m

h =Ay

Figure 3.5: Comparison of Equilibrium Beach Profile to the Beach Profile in the
Model Test for recovery condition (Case 1)

Iwagaki and Noda,1963 o
-- Saville, 1957 A A
Rector, 1954 a



oooo, __I fC.I1 ', U3

0.001 0.010 0.100
Ir W

Figure 3.6: Criterion of Normal and Storm Profile (after Kriebel at al., 1986)

D, for sediment characteristics. With the exception of Dean, the other three authors

all included an additional parameter of beach slope, tanf, but for different reasons.

Sunamura included tan# as the effect of initial condition. Wang and Hattori and

Kawamata, on the other hand, utilize this parameter to account for wave breaking

forms in the surf zone. Dean (as modified by Kriebel et al.) and Sunamura have

different values for laboratory and field data whereas Hattori and Wang have one

condition for all the data (Wang, however, used the same data set as Dean's).

In the present comparison, the profiles at x = 7.5 m(at the center of test beach)

are used. The results are shown in Figs.3.6 to 3.9. All the four criteria past the test

for erosional condition as the data point falls well within the zone of erosive profile.

For the accretive profile, Hattori and Kawamata's, and Sunamura's criterion passed

the test but both Dean's and Wang's failed.

Figure 3.7: Criterion of Normal and Storm Profile (after Sunamura and Horikawa,

3.1.4 Flow Regime and Mode of Sediment Transport

It was remarked by Wang (1985) that practically all the proposed beach classi-

fication criteria were based upon the premise that in the surf zone the suspended

sediment transport mode dominates as signified by the presence of the fall velocity

parameter. The equilibrium beach profile is also based upon the same concept.

This, while well may be the situation under prototype condition, may or may not

be true at the laboratory scale. In fact, the results presented above seem to sug-

gest otherwise. To examine the flow regime of the experiment, Jonsson's(1966) flow

regime chart is used. His diagram as shown in Fig.3.10 consists of three regimes and

three transition zones. The position is determined by two parameters; a roughness

parameter defined as



ws / gT

Erosive Profile ..

--- ,r-1/1

A/ Pr
4 1/
,,/ | Accretive Profile
.LE. Lucaion
mbhaOT. 0 o MaiLCAS (JAAN)
/ A TO IT .. A A KlluA CoaSt (JPaM)
SoOu ao KAu Iko (U.S.A)
OQuA 0a HlDuA CoAST J*A) I
SSUnwAmAsAU *-* SurmTHA. SAc. (INOIA)I
, ,1 1 ( 1 ,



3.8: Criterion of Normal and Storm Profile (after Hattori and Kawamata,

r' (W/T)

Figure 3.9: Criterion of Normal and Storm Profile (after Wang, 1985)





10' Laminar __ _Z
Sio *
0/ *

10 Transition
Si/ Rough turbulent


10 2 5 10'2 5 10' 2 5 10' 2 5 10' 2 5 10'


Figure 3.10: Jonsson's Flow Regime

and a Reynolds number given by

R = am (3.6)

where ub is the amplitude of the fluid particle velocity near the bed,am is the am-

plitude of the fluid particle displacement, k, is the roughness length and v is the

kinematic viscosity. Assuming k, = D N 1.5D and using linear wave theory, the

flow status at wave breaking point was decided and plotted in Fig.3.10. Therefore,

for erosional tests, the flow is in the transition zone between rough turbulent to

smooth turbulent and, for accretional tests, the flow is in the transition zone be-

tween smooth turbulent to laminar. Since in all likelihood, the field flow condition

would be in rough turbulent regime, the results in the recovery tests might not have

practical value.

Now we turn our attention to the mode of sediment transport. Shibayama and

Horikawa (1980) proposed to use two parameters to classify mode of transport.

These parameters are relative fall velocity and the Shields parameter; they are

defined as follows:

Ub/W (3.7)

Tm = (3.8)
where ,m, is the Shields parameter ,f, is the Jonsson's friction coefficient,

s is the sediment specific gravity in the fluid and g is the gravity acceleration.

Again, using linear wave theory, we obtain:

Xm = 0.828 for the storm condition


1m = 0.315 for the recovery condition.

The positions of these conditions are plotted in Fig.3.11. It is shown that for the

storm condition the mode of sediment transport is in the transition zone of bed

load and suspended load whereas for the recovery condition the mode is bed load.

This result further reinforces the observation that the recovery tests should be

viewed with caution to extract any qualitative or quantitative conclusions for field

applications. The laboratory results for the storm wave condition, on the other

hand, are more realistic and their application to prototype is possible provided

proper scaling laws can be established.

3.2 Modeling Law

Modelling law for beach evolution process has been studied by many investi-

gators, notably, Noda (1972), Kamphuis (1975), Hughes (1983), Dean (1983) and

Wang (1985). Yet, a completely correct modelling law is still not available. The

fundamental difficulty is that the basic mechanism of coastal sediment transport is

not well understood. In addition, practical considerations severely limit the choice

of material that can be used in the laboratory. Consequently, the scaling of sed-

iment size, specific weight and viscosity are all very limited. Finally, there lacks

0 No movement
I Bed load (BL)
2 Bed load-Suspended load intermediate iBSI) Transition 4
3 Suspended load (SL) 4
4 Sheet flow (SF) 4
0 Field data
S Honkawa et al. (1982) 3 3

.1- 33 SF(Sheet flow)

S8- 2 2 12 3
:a 0 12 2 ', SL(Suspended load)
2 22 -.2 2
4- 2 22
I 2
2 -i 2
o 'BL-
No movement' BL BSI
i i i i Ii i r II ___
0.02 0.04 0.06 0.08 0.1 0.2 0.4 0.6 0.81.0
Shields parameter 7,.

Figure 3.11: Classification of Sediment Transport Model (after Shibayama and
Horikawa 1980)

quality prototype data for verification purposes.

In the present study, the initial profile of the model is approximately scaled in

accordance with Bodge's field profile (1987) with the assumption that equilibrium

is attained at both laboratory and field conditions. There is, however, no field infor-

mation on beach evolution to compare with the laboratory results. An alternative

approach is taken here. Since Dean's (1977) equilibrium profile is derived from an

assemblage of field data, it can be treated as a prototype template of some sort. If

the beach response is slow, the equilibrium profile should at least match with the

final profile. If the beach response is fast, the equilibrium profile should also repre-

sent well the transient profiles (the entire profiles as well as the origin could move

as a function of time). Based upon this approach, a number of proposed modelling

laws are examined here.

3.2.1 Modelling Requirement from Dean's Equilibrium Profile

Let Pp and P, denote, respectively, the prototype and model values of a physical

quantity ,P, and let P, = Pp/Pm be the scale ratio. We further define A and 6 as

the horizontal length scale and vertical length scale of the geometry, respectively.

Then, if the equilibrium profile applies equally well to the prototype and the

model, we should have the following scale relationship for the factor "A":

An = h = 23 (3.9)

In addition, A should be a function of fall velocity, following the empirical rela-

tionship as given by Fig.3.1 or,

An = (3.10)

Therefore, we have two constraints among six physical properties two geometrical

properties ,A and 6 two flow properties,H and T, one sediment property, W, and

one scale coefficient, A. If H is the same as the geometrical vertical scale, the

actual variables reduced to five. The dynamic behavior in a gravitational flow field

is usually simulated by preserving the Froude number between prototype and model.

This scaling law yields for the time variable:

T 1/2 (3.11)

Therefore, we arrived at a system with two degrees of freedom provided the wave

period can be treated as time scale in the Froude similitude.

In the present experiments, if we select Wn = 1.47 and A = 18 to be consistent

with Bodge's values, the following scale ratios are then obtained from the above

three equations:

An = 1.29
6 = 8.88 (3.12)
T, = 6.04

These are very close to the actual values used by Bodge and the initial condition

in the present experiments. Therefore, the initial beach profile conforms with the

empirical modeling law as it should.

For the erosive wave conditions tested, the non-dimensional fall velocity is equal

to 4.39 and the corresponding Am value is 0.09. In the previous Section, it was

shown that the equilibrium profile represents well the model beach profiles for the

final stage as well as for the intermediate stages. This implies that the underlying

mechanism of sediment transport is properly modeled under the erosive wave condi-

tion. The fact that intermediate stages are also well represented by the equilibrium

profile seems to suggest that the morphological time scale is of the same order as

the wave time scale, a phenomenon which is often observed in the field under storm

wave conditions.

For the recovery wave conditions, on the other hand, the fall velocity parameter

becomes 1.01 and the corresponding value of Am should be 0.24. Neither the shape

of the equilibrium profile (controlled by the 2/3 power law) nor the scale of it

(controlled by the factor A) appears to compare well with the experiments. A value
of Am = 0.1 would be the best choice under this circumstance. Again, as observed in

the previous Section, under recovery wave condition, the flow regime is most likely

turbulent at field scale but is in a transition regime between laminar and smooth

turbulent in the laboratory. Thus, the sediment transport mechanism appears to

be not properly modeled in this case.

3.2.2 Hughes' Modeling Law

The Hughes' modeling law which the present model design is partially based

upon is examined here. The criteria as specified in Eqs.(3.1) and (3.2) are repeated


T. = 1- (3.1)

A = (3.2)
Equation (3.1) is essentially a statement of Froude similarity between prototype

and model. Equation (3.2) is the preservation of non-dimensional fall velocity pa-

rameter, used by Dean in his empirical formula to determine "A". This can be seen

by letting wave height H follow the vertical scale and wave period T follow the time

scale. Eliminating A from Eqs. (3.1) and (3.2) we have

( )P ( ) (3.13)
This is a more restrictive condition than the Dean's empirical relation given by Eq.

(3.10) which only requires An to obey the empirical functional relationship. The

Hughes' law also has two degrees of freedom and is equivalent to Dean's empirical


3.2.3 Wang's Modeling Law Revised

Based on the argument that the beach elevation changes must be constrained

by the sediment conservation law, Wang (1985) proposed a different set of Modeling

Law. A revised version of the Wang's law is discussed here.

The sediment conservation equation states that

+ K = 0 (3.14)
at ay
where z is the change of bottom elevation, q. is the sediment transport rate in the

direction of x, and K is the coefficient related to the porosity and is usually taken

as unity.

A set of non-dimensional variables are introduced here.
t z z- x
r=-; z= = -; x = (3.15)
tb -b qb Xb
Here the reference values are selected at breaking point signified by the subscript

b. Other reference values, of course, can be used as long as they are consistent

and represent the characteristics of the phenomenon. Now, substituting these non-

dimensional variables into Eq. (3.14), we arrived at the non-dimensional transport

equation in surf zone
9Z _= (t q (3.16)
8t h- b as t
To maintain similitude between model and prototype requires

qbtb qbti
(hO-b)m- ) (3.17)
hbxa hbxb

q 1 (3.18)
where h, and x, are the geometrical scales of 6 and A, respectively, t, is a morpho-

logical time scale and q, is the sediment transport scale.

Since the mechanisms of bed load transport and suspended load transport are

quite different, Wang argued that the Modeling criteria are also different depending

upon which mode of transport dominant. Here only the suspended load dominant

case is discussed.

Wang proposed that the suspended load transport rate can be expressed by

depth averaged properties:

q, = hVC (3.19)

Here h=depth; V= mean transport velocity and C = mean sediment concentration.

Following Hattori and Kawamata's approach, it is assumed that the suspended sed-

iment concentration is directly proportional to the stirring power due to turbulence,

Pt, and inversely proportional to the settling power due to gravity, Pr. The stirring

power per unit volume can be expressed as

Pt = pgu' (3.20)

where u' is the turbulent velocity The resisting power is

Pr = (Ps p)gW


with W being the settling velocity. Therefore, we have

C oc (3.22)

A = (- 1) (3.23)

Based on field measurement of kinetic energy and momentum flux in surf zone,
Thornton (1978) suggested that the ratio of turbulent velocity intensity and the

wave-induced velocity intensity is a function of surf zone parameter, i.e.,

= f/() (3.24)
where a' and a are the standard deviations of the turbulent and wave-induced
velocities, respectively, and e = tan/3//Ho/Lo. Thus, substituting Eq. (3.24) into
Eq. (3.22) and absorbing the proportionally constant into f(e), we arrived at

= f(w) (3.25)
For steep waves u can be assumed to be proportional to H/T the above equation
can, therefore, be expressed as
C= f(7) (3.26)

Physically, this equation states that the suspended sediment concentration in surf
zone is proportional to a function of surf zone parameter (or the breaker type),
and is inversely proportional to the specific gravity and the relative fall velocity.
Substituting Eq. (3.26) into Eq. (3.19), results in

S=hVf () (3.27)

Substituting Eq. (3.27) into Eq. (3.18) gives

V fA( )tn
n/ = 1 (3.28)

A number of assumptions are made here

a. The wave height is proportional to water depth, thus, can be treated as a ver-
tical scale, or, H, = 6.

b. The morphological scale is the same as the wave period scale, or, t, = T,. This
is justified in that during erosional process the rate of erosion is proportional
to the number of waves propagated in the duration.

c. V oc 61/2. This is based upon the results by Wang, et. al. (1982). That the net
transport velocity is found to be

V o c2'/gh (3.29)

where x is the wave height to water depth ratio.

d. ep = ,m. It is a more restrictive but sufficient condition to guarantee fp() =

fm(C). In fact, if ~p = em, we have f,,() = 1 and this parameter can be
eliminated from Eq. (3.28). However, for ep = em requires

(tan/3/H-/L-)p = (tanf3/VH,/Lo)m (3.30)

Since L, = gT2 and (tan),n = 6/A, Eq. (3.30) is equivalent to

Tn= 1/ (3.31)

which is the Froude similarity between prototype and model.

Now substituting the above four condition into Eq. (3.28), we arrive at the
following modeling criterion

6 = (,Y1W,)2/3A2/3


If we define A, as

A, = a('YW)2/3 (3.33)

with a a proportional constant, Eq. (3.32) can be written as

6 = (A,)A,2/3 (3.34)

This equation is almost the same as Dean's equilibrium profile equation, the dif-
ference being the functional dependency of the scale parameter, A. Based upon
empirical evidence, Dean and Moore suggested A to be a function of sediment par-
ticle size, D, or non-dimensional fall velocity, Hb/TW. Based upon sediment mass
conservation equation, the correct form of A is suggested here to be (see Eq. (3.28))

A, = (Y )2/3 (3.35)

Wang paid special attention to the scaling of settling velocity in a wave field as
the magnitude of the velocity is affected by the nature of the oscillatory flow field
(Fig.3.12). Hwang (1985) suggested that
W Vt V
W= f.( R, ) (3.36)

Wo is the terminal velocity in calm water,
Vf and V, are the amplitude of fluid and particle oscillations, respectively and
R is the particle Reynolds number.
Proper modeling requires

W, = (Wo)n[f,( R )] (3.37)

This is difficult to fulfill. At present, W is treated as equal to Wo. Thus,

W. = (Wo), (3.38)

It is further observed here that for small Reynolds number, WD/v < 1.0,

W oc D2


Fall Velocity (w)

HO'S Dotg(1964)
Reo pS
S2700 776
v 800 7.90
a 230 2.65
o 28 7.76
o 1.1 7.76

Vf /wo

W A? v D vs
= f ( ._.O__ ; o( ) IR = .Wo- Vs
Wo Wo Vf

Figure 3.12: Reduction of Settling Velocity in the Oscillatory Flow (after Hwang

for higher Reynolds number but less than 103,

W, oc D (3.40)

and finally for high Reynolds number,

Wo oc D1/2 (3.41)

The corresponding slope of A, in these ranges are D1'ss, D0.67 and D'.33, respec-

tively. They are consistent with the data compiled by Moore (Fig.3.13) in which

the slope of curve A decreases with increasing D.

For erosive wave condition, this model law is applied to the test results. For

horizontal scale A = 18 and W, = (Wo), = 1.47, the vertical scale should be

6 = (1.47)2/3 x 182/3 = 8.88 (3.42)

which is very close to the value of 9 used in model design. The corresponding

An = (W,)2/3 = 1.29. This is very close to the ratio determined from the empirical

diagram of Moore's, which gives a ratio of 1.31.

3.2.4 Noda's Modeling Law

Noda gave a set of empirical modeling law as follows:

Dn. (.)1.8 = (6)0.55 (3.43)

A = () 1.32 ()-0.386 (3.44)

This set of equation has two degrees of freedom. Eliminating from the above

equations, we have

6 = D~0.17X08S (3.45)

The shape factor 0.83 is in the same magnitude as in the previous model laws. The

scale factor, on the other hand, has a quite different functional dependency on D








0.1 1.0 10.0 100.0


given by Moore (1982). If we further let D, = 1.21, the value used in the present

experiments, Eq. (3.43) yields

6 = 1.41 (3.46)

This is a very restrictive criterion and is quite different from the value used here.

3.2.5 Summary

For the verification of the modeling laws, we have assumed that the equilibrium

beach profile can be treated as a prototype template for the model results to be

compared to. Four different modeling laws are examined-Hughes, Wang's, Noda's

and an empirical law derived from Dean's equilibrium profile concept.

The experimental results show that the model tests were carried out under

different flow regime, thus, required different scaling laws for the recovery conditions

and for the erosive conditions. Of the four modeling laws, only Wang's can address

the problem of different modeling laws for different sediment transport modes. For

comparison purposes, only results from the erosive wave conditions are tested.

Noda's model is definitely inadequate. Hughes' model, Dean's equilibrium con-

cept and Wang's model are consistent, although they are derived from significantly

different argument. All of them give reasonable results. Finally, Wang's model

provides a rational explanation of the scale parameter "A" as well as its functional

dependency and how it should be scaled without resorting to empirical graphs.

It also addresses bed load dominated case and transient profile modeling that, in

principle, can not be addressed by the equilibrium concept.

3.3 Comparison to 2-D and 3-D Model Test Results

Barnett (1987) conducted two dimensional (2-D) model tests of a vertical seawall

on beach in a wave tank in the same laboratory here in the Department of Coastal

and Oceanographic Engineering, University of Florida. His results are compared

with the results of the present study which are from three dimensional (3-D) con-

ducted in a wave basin.

3.3.1 Beach Profile Comparison

The 2-D and 3-D experiments were conducted under rather close but not iden-

tical conditions. To facilitate comparison, certain adjustment has to be made first.

The test conditions of 2-D experiment are given in Table 3.1. By comparing with

the test conditions of the 3-D model given in Table 2.2, a selected sets of data with

similar test conditions-Cases 1A and 9C are used for the natural beach without

seawall and Cases 3A and 12C are used for beaches with seawall.

Table 3.1: 2-D Model Test Condition (Barnett 1987)
Wave Condition
Wave Height(cm) 11.75 8.80 4.00
Wave Periods(s) 1.81 1.30 1.81
Sediment Condition
Diameter(mm) dso = 0.15
Settling Velocity(cm/s) W = 0.017
Water Depth(cm) 46.0 and 56.0
Seawall Location shoreline and 0.3m from the shoreline
Initial Profile A = 0.075m1/3

Case 1A and 3A are based upon an equilibrium beach profile with scale factor "A"

equal to 0.075. Cases 9C and 12C have the same initial profile as Case 1A but

with the water level raised by 10cm to represent storm surge condition. Therefore,

strictly speaking, the initial condition of Case 9C and 12C is not an equilibrium

profile. If it were to be treated as an equilibrium profile, the "A" value has to be

adjusted according to

Ac = AA(y/yA)(XA/C) 2/3 (3.47)

where subscripts A and C denote Case 1A or 3A and Case 9C or 12C, respectively.

If AA = 0.075, yA = 46cm, yc = 56cm and the x scales are the same then an

equivalent A is obtained as equal to 0.091. Table 3.2 summarized the condition of

these cases.

As can be seen from Table 3.2, the initial profiles of the cases to be compared

are usually not the same because of the slightly different test conditions. Therefore,

one can not be assured that the subsequent comparison of profile evolution are
meaningful. The first step is to attempt to establish the profile relationship between

2-D and 3-D test so that meaningful comparisons can be made. This is handled by

treating the 2-D experiment as a distorted model of 3-D case, or vice versa.

Based on the modeling requirement from Dean's equilibrium beach profile, we


Y2 = A h32/2 (3.48)


y 3 = A/2 (3.49)

where subscript 2 and 3 refer to 2-D and 3-D cases respectively.

Dividing Eq.(3.48) by Eq.(3.49) gives

2_ = (A2s /h2 )S/ (3.50)
Y3 A3 h3

This is the basic equation used for profile adjustment. All the scale adjustment were

made by using 3-D case as reference. Since the vertical scale (the water depth) and

the "A" values are known, the most convenient adjustment is the horizontal scale,


A = Y( h2)3/2 (3.51)

Figure 3.14 compares the initial conditions of 3-D and 2-D cases with and with-

out profile adjustment. Figure 3.15 compares profile 4 hrs after the test for cases

with and without seawall. For the cases without seawall, the agreement is very good

within the surf zone. The 2-D tests generally produced bottom undulation due to

wave reflections confined in the channel. For cases with seawall, the comparison is

less satisfying. The most significant difference between them is the bar formation.

Prominent multiple reflection bars appeared in all the 2-D tests, whereas in 3-D

test, no reflection bar were formed in case 2 and 6 and only small ones appeared in

case 4.

3.3.2 Offshore Breaking Bars

There are a few studies which examine the relation between the scale of the

breaking bar such as the bar height or width and the other property such as wave

height or sediment. Keulegan (1948) found in a laboratory experiment that the

ratio of the water depth of the bar trough, ht, to the depth of the bar crest, he, has

an average value of 1.69, i.e.,

ht = 1.69he (3.52)

Sunamura (1985) mentioned that the relation given by Eq. (3.52) was supported by

Kajima's large model test, and also suggested the simple relation between ht and

the breaking wave height HB as

ht = HB (3.53)

Combining Eq. (3.52) and Eq. (3.53), we have

he = 0.59HB = 0.59ht (3.54)

It is not always easy to examine the relationship between he and ht, because

the breaking point bar is not always stable; the breaking bar formed rapidly then

disappeared, ht sometimes is also unrecognizable. Selecting the maximum bar height

in the test, the relation given by Eq. (3.53) are examined by both test results as

shown in Fig3.16. Since the breaking wave height in 2-D model test is not available,

the breaking wave height were estimated by using the breaking wave index shown

in Fig.3.17.

The data are scattered but the relation given by Eq. (3.53) agreed with model

test results reasonably well. The difference between 2-D and 3-D, however, is not




........... 2-0 TEST S


"*"i --------------~~~~~~~---------------------------- -- ,,,
^\ -----------------1

-No.0 ---------..---------------- ....

-tfl. OO ------- --------- .__________________________" ~ : ::: > ^ ^ ___

-1.00 *3.00 -2.00 *I.00
0 1.00

2.00 3.00 4.00 5.00 8.00

7.00 8.O0 9.00




Figure 3.14: Comparison of Initial Beach Profiles between 2-D and 3-D Model Test




........... 2-0 TEST 9
--- 2-0 TEST I

---3-0 CASE I




-4.00 -3.00 -2.00 -1.00 0.00

1.00 2.00 3.00 4.00 S.00 6.00

7.00 8.00 9.00



----..........- 2-0 ES It

120.00 ----- -0 TEST 2

--- -0 CASE 2




. Ol.80

-4.00 -3.00 -2.00 -1.00 0.00

1.00 2.00 3.00 4.00 S.00

6.00 7.00

.0 .00 1.00

Figure 3.15: Comparison of Beach Profiles after 4 Hours Duration between 2-D and

3-D Model Test



Prototype exp.
S i Shimizu et a. (1985)
inner bar
./* athor
E*0 .
:: 1 o*o 3 D


E 0.1.- /
S v v
Small-scale exp.
S / Watanabe et al. (1980-)
v Yokotsuka (1985)
0.01 ,,
0.01 0.1 1 10
Breaker height, HB, m

Figure 3.16: Relation between HB and ht adopted by Horikawa(1988)

found. The relation between he and ht is also examined in Fig. 3.18. The data are

scattered again, but the dashed line in Fig. 3.18 calculated by using least square

sense is given by

h, = 0.6ht (3.55)

This coefficient is very close to the value proposed by Keulegan. The scatter of

the data points does not permit us to establish the difference between 2-D and 3-D


3.3.3 Reflection Bars

The reflection bars and the offshore bars (or breaking bars) are formed by

different mechanisms. The offshore bar is formed by breaking waves while the

reflection bars result from the standing waves due to reflection. The location of

latter usually coincide with nodes and antinodes of the standing wave system.

Xie (1981) found that there are two types of reflection bar, that is, for fine

sediment the bar forms at antinode, but for coarse sediment the bar forms at the

node. Irie et al. (1984) also came to the same conclusion and they noted that the

criterion of these two formations is a function of Ursell number and the ratio of

.s .. Beach slope
1 --/20 Goda
f 1/30 (1970a)
*So '----,_ ^1/50 or less
S-- Suna- mura(1983)

C1/10 5
0.5 1/20 Ostendorf and Madsen
4 05 \1/30 (1979)

nr "1/50 \

I 0.001 0.01 0.1 1.0
Relative water depth at breaking h./L,

Figure 3.17: Breaking Wave Index adopted by Horikawa (1987)

10 20

ht (cm)-

Figure 3.18: Relation ht and he


the fluid velocity at the bottom to sediment settling velocity. Both studies are two

dimensional. Irie et al. (1984) and Silvester (1987) also conducted three dimensional

tests on breakwaters and seawalls. Their results also showed similar reflection bar

pattern. Hsu and Silvester (1989) demonstrated that with angled waves bar and

trough occurrence also dependant upon the obliquity. These are determined in

fractions of the distance of island crests, formed at wave interactions, termed the

crest length. Katsui and Toue (1988) examined the bathymetric change around a

large offshore structure by 3-D model test and found that well defined reflection

bars formed in front of the structures. It should be noted here that all the studies

cited above were carried out at non-breaking waves.

In the present study, reflection bars in 2-D test are prominent even within

breaking zone. In 3-D tests, the reflection bar is far more difficult to maintain; it

appears and then disappears. This is probably because the energy flux reflected

from seawall tends to propagate along the wave crest (a diffraction phenomenon)

and the fact that the wave induced current is less organized to sustain stable bar


3.3.4 Scour

The scour depth and the other quantity are defined in Fig.3.19. Two scour

depths, SI and S,, are defined. SI is the depth from the elevation of the nearest bar

crest to the elevation of the scoured bed at the toe of the seawall, and Sg is the

depth from the initial bed to the bar crest. The reason for defining two scour depths

is that S1 and S, might be caused by the different mechanisms. As can be seen in

Fig.3.15, the beach is eroded from the entire region in front of seawall in Test 2A,

while the erosion occurred only in the vicinity of the seawall in Test 12C and Case

2. In other word, SI is caused by strong vortices due to interaction of wave and

structure, while S, is caused by the sediment depletion to form offshore bar. As to

the parameter s which is relevant to the scouring, (1) incident wave height,Ho, (2)

Figure 3.19: Definition of Scour Depth and Other Parameters

wave period, T, (3) initial water depth at the seawall, h, and (4) distance from the

seawall to the nearest bar, yfb are considered. After several attempts to examine

their relation, the clear relation between SI and h, was found in Fig.3.20. The

following observation based on physical reasoning are made:

(1) The scour depth might be directly related to the wave height.

(2) For the case that the elevation of the toe is above the still water level, if the

elevation of the toe is so high that the wave can not reach the seawall, the scouring

does not occur. Thus, some variables which define the relation of ho to the wave

run up height must be included, and the wave run up is related to the beach slope

and the wave steepness.

(3) The waves which reach the seawall must be influenced by the breaking wave


Consequently, based on the relation obtained in Fig.3.20 and the physical reasoning

above, the scour depth, S, is given as a function of Ho, Lo, h0 and the surf similarity

parameter based on the tanfb (Figure 3.21).


SI 10.0


0.0 10.0


Figure 3.20: Relation of SL and h,


i- I i

0.0 1.0 2.0 3.0 4.0 5.0

o Tan Pb

Figure 3.21: Relation between SI/Ho to ho x tanib/Lo


0.0 L

Table 3.2: Comparison of Scale Parameter "A" in 2-D and 3-D Model Test
3-D model test Cases 1A and 3A Cases 9C and 12C
water depth 45cm 46cm 56cm
wave height 11.75cm 11.0cm 11.0cm
wave period 1.73sec. 1.7sec 1.7sec
scale parameter 0.09m1/3 0.075m1/3 0.089m1/3


The effects of seawall on beach changes are evaluated using two major indices-
the effects on volumetric changes of beach material and the effects on shoreline and
contour-line changes in foreshore and backshore region. The effects on volumetric
changes is addressed here first.

4.1 Definition of Volumetric Changes

The basic equation used to compute the sediment volumetric changes is the
sediment mass conservation equation given by

az aq.q aq
= -( + + ) (4.1)
9t x By
Referring to Fig.4.1, the terms in Eq. (4.1) are defines as
x=shore-parallel axis, y=shore-perpendicular axis; z=profile elevation;
The origin is set at the undisturbed still water level at the down wave boundary
of the test region 7.5 m from the center of the seawall. Three different types of
volumetric changes are computed; their definitions are shown in Fig.4.2.
(1) Rate of Volumetric Changes along a Profile, ip(x, t)
The volume change along a profile can be obtained simply by integrating Eq.
(4.1) along a profile from y = 0 to y = yo

p(x,) = o iz

= -{q(yo) q (0) + xdy} (4.2)

where y, is the offshore boundary. Since q,(0) = 0, the above equation becomes

iP(x,t) = -qy(yo) f dy (4.3)
Jo ax

Xo X N

Figure 4.1: Sketch of Coordinate System

The units of Vp(x, t) are volume per unit length per unit time.

(2) Cumulative Rate of Volumetric Changes Referenced to Down-Wave Boundary,

If Eq. (4.3) is integrated along shore from x = 0 to a = zo, we have

Vc(xo,t) = / ( qUyo)dx {q,(xo) q(x0)}dy (4.4)
0 0y

This represents the cumulative rate of volumetric changes in a control area bounded

along shore from x = 0 to x = xo

(3) Rate of Volumetric Changes in a Local Control Area Centered around the Sea-

wall, 6 (t)

In this case, the integration of Eq. (4.3) is carried out from -W/2 to W/2,

where W/2 is the long shore distance measured from the centerline of the seawall

location, i.e.,
= ze+W/2
i(t) z-w/ i, p-(x,t)dx





xo x


r X

Figure 4.2: Definition of Three types Volumetric Changes

= Q.(x2 + W/2,t) Qx( W/2,t) + 2Iqy (o)dx (4.5)


Q, is the longshore transport rate, xz is the location of center of seawall and

-W/2 and W/2 are the variable down-drift and the up-drift boundary of the local

control area respectively.

4.2 Results and Discussion

Based on Eqs. (4.3), (4.4) and (4.5), p and vi are calculated. The integra-

tion are carried out by the following discrete formulas

( =_ (zk z,) (4.6)
(po)e = E(pAp):Ax (4.7)
) = ( ( "(4.8)


i,j and k denote y, x and t respectively,

Ay = 0.05m, Ax = 0.75m At = 3600s, 7200s, and 14400s,

I is the integer so that IAy = 7.0m

J is the integer so that JAx = 15.5m

ji and j2 are the integer so that jlAx = x W/2 and j2Ax = xz +W/2 respectively.

The results are presented in Figs. 4.3, 4.4 and 4.5. In the case of ii and Oe, the

abscissa is the longshore distance measured from origin. In the case of i), the

abscissa is the width of the local control areas non-dimensionalized with respect to

the seawall length, or W/L, where W is the width of the control area and L, is

length of the seawall. The ordinate of all three cases represents the rate of volume

change per hour. All the computations are carried out to a total of four hours

duration even though some of the experimental runs were carried out to a much

longer duration. Since the topographic changes were measured at 1 hour, 2 hours

and 4 hours after the test began, the computed points denoted as t=4 hours in the

graphs actually represent the average values from t=2 hours to t=4 hours.

The rate of volumetric change along individual profile is examined first. In

Fig. 4.3, the seawall is located at x = 6m to 9m, and the wave incident angle is

defined as shown. Therefore, the updrift boundary is at the right hand side for all

the cases shown. Under normally incident waves, the rate of erosion in the center

portion of the test region is almost uniform with or without the seawall. Also, in

this center region, the rate of volumetric erosion is seen to be smaller on profiles

fronting the seawall than that of natural beach. This is similar to the 2-D results

by Barnett. Towards the two sides of the test region, the volumetric rate of change

along individual profile becomes erratic, perhaps owing to boundary effects.

For the cases of oblique wave of 50 and 100 incident wave angles, Erosional rate

along individual profile becomes more irregular due to the three dimensional flow

pattern in the near shore zone. For natural beach, there is a definite appearance

of rhythmic feature. For beach with seawall, the rate of changes along individual

profiles are very irregular. However, if a line is drawn across the region to represent

the averaged rate, one clearly sees trend of increasing erosion forward the downdrift

side. And, the slope of the line also appears to increase with increasing wave angle,

which means the difference of the averaged rate of erosion between the updrift and

downdrift sides becomes larger as the incident wave angle becomes larger.

In the next case, the cumulative volumetric change with reference to a fixed

down-wave boundary is given in Fig.4.4. For the case of normal incident waves,

the slopes of the cumulative curves are almost uniform, an additional indication

of uniform erosional rate. For cases with oblique waves angles on natural beaches,

the cumulative curves reveal an undulation superimposed upon uniform erosion. If

mean straight lines are fitted on the curves, the effects on wave angle in rate of

. 000

. 0.000

- 0.000 oo I
0.000 o10.000


- 0.000


-11.000 .00I- II
0.000 10.000 ;




- "--- TOTAL

-1.000 '

Longshore distance (m)


U~ 000

I /w ve I-
-- -- /-- .



-WflMn ILL.Z.2~rL....L L

I.000 0



1 1.00 0



-1 -.o00oo
20.000 0.000

seawall /


Longshoe distance (m)

_--- TO/w ave ~

=^^^-"l ITOTL

__"__ _____ *----- a HOUR

- I I----& -I HOUR

M- -e-- -- HOUR

G ,- 2l HOUR

--- ---IHOUR
S^^S^Sa o "'--ou8

---- HOUR

-a- I HOUR

-I Ig(- N

-- I

0, '1 W ,S%

















4.000 Ii 0.000 10.000 I
O.000 0.000 10.000


0 0

II .





.000 0.1


distance from down-wave boundary. (m)


distance from down.wave boundary (m)

1 1 1 1 1 1 1 1





4- 1
- -----

-- --- ---- ---.--

------ --- i



-- ---TOTAL

--"-q HOUR

---- 2 HOUR


Z 'TWrr+-SbII z- I

---- TOTn

----- q HOUR

--6-- i HOUR





10.000 2


11 T. TI



1 0.000

-q.000 '
.000 0.000



. -1.000
20.000 0.I

- -*- TO TM.

----- -- -- -- -- -



00 10 20 000





-- ---- -- I---- ----_ ------_-_-_ ----- --I 8U












- -...- a OU

--- -0- HO




I non


---- TOTAi

---11 HOUR

----- a ROUR

------- -- -- B

i 1 1 1 1 ::J a ,-


II I ill I







Table 4.1: On/Offshore and Longshore Transport Rate for Natural Beach
wave angle (deg.) on/offshore transport rate gradient of longshore transport rate
m3/m/s x 10-6
0 8.18 0.
5 8.18 0.54
10 8.18 1.68

Table 4.2: On/Offshore and Longshore Transport Rate for Seawall Backed Beach
wave angle (deg.) on/offshore transport rate gradient of longshore transport rate
m3/m/s x 10-6
updrift downdrift updrift downdrift
0 9.75 9.58 0.0 0.0
5 9.75 9.58 -3.31 3.55
10 9.75 9.58 -4.49 5.1

erosion is clearly revealed. If we further assume that under normal wave incident

wave angle, the spatially averaged transport is in the on/offshore direction only,

the longshore and on/offshore component of transport rate can be separated by the

following equation

bT(a) = 'o + 6i(a) (4.9)

where OT(a) is rate of the gradient of the transport; V, is the on/offshore compo-

nent assumed here to be independent of wave angle, a, and Oz(a) is the longshore

component. Based upon this equation, the longshore and on/offshore component

for the three cases of the natural beach are given in Table 4.1. Now, for the cases

with seawall, the slope on the down drift and on the updrift are different. Com-

paring with the case of 00, the slope are steeper in the downdrift side but milder in

the updrift side. By using Eq. (4.9), the on/offshore component and the longshore

component are established separately for the downdrift and the updrift sections.

These results are also given in Table 4.2. The negative longshore components in the

updrift side are the consequence of groin effects.

We now turn the attention to examine the extent of seawall on the rate of volu-

metric change within a control region centered around the seawall. The results are

plotted in Fig.4.5 with various width of the control region. Using the same data

set, the ratio of volumetric change, with(6,) and without(6,,), is plotted against the

non-dimensional control width (W/L,)(Fig.4.6). If the ratio is larger than 1.0, the

rate of erosion is larger with seawall than without seawall and vice versa. From this

Figure, a number of observations can be made.

(a) For normal incident waves, the rate of erosion in the vicinity of seawall in-

cluding the fronting beach is generally smaller for the case of seawall than without.

This is particular apparent for beaches fronting the seawall (or when W/L, < 1)

(b) Under oblique wave condition, the rate of erosion is considerably faster for sea-

walled beach than natural beach in the early stage of wave attack, but this trend is

reversed as time progresses. In the final stage of reaching a new equilibrium con-

dition, this ratio is either less than 1 or approach 1 as the control width increases.

Thus, the effects of seawall in volumetric erosion appears to be localized.

Considering the final total volume change, the results are given in Fig.4.7. In

Fig.4.7a, the ratio of total volume change with(v,) and without (v,) seawall is plotted

for the updrift and downdrift region separately. In the updrift region, v,/v, is always
less than 1 irrespective the width of the control region and the wave angles. Thus,

in the case of normal incident wave, this value less than one because more sand is

retained by the seawall in the backshore than the additional material being eroded

in front of the seawall, when compared with the natural beach case. For cases with

oblique waves, on the other hand, sand is retained in the updrift due to groin effect;

they also makes this ratio less than unity. On the downdrift side, the situation is

different. As expected, under normal incident waves,v,/v, is less than onemuch the

same as the updrift side. Under oblique waves, v,/v,, is less than 1 when W/L, = 0.5,

or when the control region coincides with the seawall length. Apparently, even

under oblique wave, the material retained from the scouring trough in front of the

seawall. However, when W/L, becomes larger than 0.5, v,/v, also becomes larger

than 1. The presence of seawall now interrupt the normal longshore transport and

causes downdrift erosion to be greater than the natural beach condition. Finally, in

Fig.4.7b, the ratio of total volumetric changes including both updrift and downdrift

region are given. It can be seen that when W/L, < 1.25, v,/v, < 1.25. When

W/L, > 1, the ratio of v,/v, quickly becomes constant and approaches 1 as it

should when W/L, becomes large. Therefore, the effects of seawall appears to be

quite localized, certainly within 3 to 4 seawall length for the cases tested.

To examine the volume change in more detail, we divide the region surrounding

the seawall into 12 sections as shown in Fig.4.8. The seawall is located at the center

section C1; Sections L1, C1 and R1 are on the beach and the rest sections are in

offshore. The total volume changes after four hours duration within each sections

are then computed and the results expressed in percentages of change are presented

in Fig.4.9. From this Figure, it can be seen that for the case of zero incident wave

angle, flanking effects and toe scouring are evident. Material lost from flanking and

toe scouring are deposited offshore to form offshore bars. Since the material saved

on backshore is more than offset the increased material loss to the offshore owing

to the presence of the seawall the total volume loss in the control area is generally

less for the case with seawall than without seawall. However, as the control area

becomes large, this difference of losses becomes small. For the case of oblique waves,

the scouring hole in front of the seawall remains prominent. The decrease in updrift

erosion and increase in downdrift erosion, mainly in the beach sections (L1,R1) are

also evident.


2.00 ---





.--- --...... "-..........-- ----...---
-. -...........

--- --,-- "--- -.... ..... --F--...- -.- --D -

-- -g :---

- 0.0



0.00 I 1

- .. _____ L _____ U L _____ 0.......... U

ITIO r it oS CO2.0L RE T SE H (W

-- --- it NOU

-..----.-- r A IwU



---*- Ir HOU

SI--T-- I 0l0
-Q- ir I MM


Figure 4.6: Ratio of the Rate of Volumetric Changes with and without Seawall


- -- a HOUR





---. ........ .. -
..__ -* r^ ....... .... .- .... ....

_ i -
----**" .P ^ ^ -B "--S *- r --- --- ---

(a) Up-Drift and Down-Drift Separately

- -....-. .... E- .8 -



(b) Total Region

- --- 0 00tCRE

-. 0 DE0Rti


Figure 4.7: Ratio of Volumetric Changes with and without Seawall for 4 Hours








2.25 2.25 2.25

L4 C4 R4

L3 C3 R3

L2 C2 R2


unit: m



Figure 4.8: Sketch of 12 Sections Surrounding Seawall


2.0 + 0.2 1.U

1 -12.4 8.9 0.9

-11.0 6.0 -14.5

-13.1 -13.2 -16 6

(H) CRSE 1 00O

+ 1.5 1.2 -5.3

4.9 + 1.3 2.9

8.2 8.7 -18.0

6.1 -18.8 -28.7

(C) CRSE 3 5u

-2.4 + 2.0 -2.0

-11.3 9.5 8.1

-15.3 -11.9 6,4

-11.0 -10.8 -13.0

(E) CASE lO -



+6.5 +7.1 1.1

-13.9 4.7 + 8.5

-10.8 -19.0 -27.2

-18.2 SEAHALL -27.1

(B) CSE 00o

+ 0.5 +0.6 +3.8

-5.3 -0.5 -5.9

-15.8 -20.5 -27.8

-20.41 SEAWALL -20.4

(D CRSE 1 50

0.5 + 4.3 + 1.8

4.9 7.8 + 2.5

-24.1 -21.2 -10.5

-28.1 SEAWALL -8.5

(F) CRSE b 10o

Figure 4.9: Volumetric Changes in Sections for 4 hours Duration


As stated in Chapter 2, hydrographic surveys were carried out at regular time

intervals. In the erosional phase, surveys were conducted at 0, 1, 2, and 4 hours

whereas in the recovery phase, the intervals were 1, 2, 3, 8 and 12 hours. The

time-elapsed contour plots are illustrated in Fig. 5.1. The complete data sets are

presented in Appendix A. In this Chapter, the three dimensional shoreline and

hydrographic changes are examined by means of Empirical Eigenfunction (EEF)

analysis and One-Line Model (OLM) analysis.

5.1 Empirical Eigenfunction (EEF) Analysis

5.1.1 Literature Review

Since Winant et al. (1975) showed the usefulness of empirical eigenfunction

(EEF) analysis or empirical orthogonal function (EOF) analysis, many investiga-

tors have utilized EEF to examine beach profile changes. In applying EEF to the
bathymetric data, the solution is usually not unique, thus several EEFs for the

bathymetric changes can be defined and examined their physical meanings.
Winant et al. (1975) applied EEF to data set obtained from two years monthly

surveys at Torrey Pine Beach, California. The analysis separates the temporal and

spatial dependance of data, i.e.

h(x,t) = em(x)c,,(t) (5.1)


e,(x) is the spatial eigenfunction and cm(t) is the temporal eigenfunction.
They found that only first three eigen vectors explain the original data well. They

(a) CASE 1

(b) CASE 2

(d) CASE 4

.(e) ICASE 5






(o) CASE 3

(f) CASE 6

named the first, second and third spatial EEF as mean function, bar-berm function

and terrace function respectively. The physical meanings of these three function are

described below:

(1) mean function. Mean beach function is the arithmetic mean profile of the data.

If the beach is stable, it has a constant time dependance.

(2) bar-berm function. This is the seasonal eigenfunction, characterized by a strong

seasonal temporal dependance.

(3) terrace function. This function has a broad maximum near the position of the

low-tide terrace with complicated time dependance.

Aubrey (1979) applied EEF to beach profile changes. He found two pivotal

points where beach profile does not change seasonally. The pivotal points are defined

as the zero crossing points of the second spatial eigenfunction. Based on a dual

pivotal points, he calculated seasonal volume change or sediment movement at Torry

Pine Beach California as shown in Fig. 5.2

Hashimoto and Uda (1982) investigated the response of beach profiles to incident

waves at Ajigaura beach, Japan. Defining the original data set as the difference

from the mean value, they examined the relation between the eigenfunction and

wave characteristics. They found that the time derivatives of the first temporal

eigenfunction has clear relation to the wave angle given by

= -0.42 + 0.0670(t 12) (5.2)

where 0 is the wave angles and-denotes the average over five weeks. For the second

temporal function, they found the following relation to wave characteristics

d Q2
dt = 1.13 0.53H ,,,,,, (5.3)

where Hmean,maz is the weekly mean value of the daily maximum significant wave




____________________ "- MSL -

S_ -10

500 400 300 200 100

Figure 5.2: Schematic of Seasonal Sand Volume Changes at Torry Pine Beach Cal-
ifornia, based of a Dual Pivotal Points (after Aubrey 1979)

These two studies showed how to utilize EEF analysis to quantify or predict

beach profile changes.

Similar analysis to Winant et al. were carried out also by Birkemeier (1984),

Wright et al. (1984) and Kriebel et al. (1986). In Birkemeier's study, the original

data set is the same as that of Hashimoto and Uda. From the shape of the first eigen

function shown in Fig. 5.3, he concluded that the first eigenfunction describes the

changes of the profile from a single bar configuration to a double bar shape, and the

first and second eigenfunction appear to result from two unique sequence of storm

recovery activities, rather than being controlled seasonally. Moreover, although the

third eigenfunction accounted for 1/2 of the variance of the second eigenfunction,

its weightings (temporal vectors) have the most well-defined annual cycle.

Wright et al. conducted two types of eigenfunction analysis. The first type used

the same method as Winant's, and the second is called "floating datum" analysis. He

-- IST 37.6 X
..-- 2ND 27.2 '.
c 3\ -- 3RD 13.3 X
3 0.4 37 SPACE PTS.

N 0.0

: -0.2-

0 100 200 300 400 500 600 700

Figure 5.3: Spatial Eigenfunction in Birkemeier's Study

mentioned "floating datum" analysis best express profile changes and can be related

to beach profile. In the "floating datum" analysis, the variability is referenced to

the instantaneous positions of the shoreline and is independent of absolute degree

of accretion or erosion.

Kriebel et al used EEF to examine the beach recovery process at Clearwater,

Florida. They compared eigenfunctions of seawall-backed beach and natural beach,

and found the eigenfunction has no significant difference between them except in

the vicinity of seawall.

Barnett (1987) used EEF in his two dimensional laboratory experiment, and

examined EEF characteristics for the seawall-backed beach. The original data set

is defined as the difference from the initial beach profile. He found the first eigen-

function describes the bar trough features and the second describes the reflection


The method mentioned above is basically based on the same idea as Winant et

al. As the usefulness or powerfulness of EEF analysis to beach profile changes has

been demonstrated, different versions were developed.

Uda and Hashimoto (1982) and Dick and Dalrymple (1984) calculated the two

spatial eigenfunction in the direction of cross-shore and long-shore at a certain

time. Uda and Hashimoto examined the beach profile changes at Misawa fishery

port where 180 m long break water is constructed. They assumed the beach profile

data is represented by

h(x, y, to) = ek(y, t)c k(, to) (5.4)

ek(y,to) is the spatial eigenfunction in the direction of cross-shore.

Ck(x, to) is the spatial eigenfunction in the direction of long-shore.

They found all ek and cl are independent of time, and c2 and c3 are time dependant,

and have clear correlation to the shoreline changes. They described the character-

istics of the eigenfunction, as follows:

The first eigenfunction el corresponded to a mean profile. The second
eigenfunction e2 turned out to correspond to profile changes due to long-
shore sand transport, because its value was positive over a broad region
of the shore, and the second time function c2 corresponding to it, which
gave longshore change of the beach profile, was correlated with shoreline
position, y,. The third eigenfunction es corresponded to profile changes
due to the influence of the breakwater, because e3 took positive values
near the shoreline and negative value in the offshore zone, and function
c3 was correlated with the offshore distance y" of the four meter depth
line from the shore. (Uda and Hashimoto, 1982, p. 1414).

Then, combining one line theory, they developed the prediction model of beach

profile changes. The results obtained from their model explained the field data


Dick and Dalrymple calculated three types of EEF to examine the beach profile

changes at Berthany Beach, Delaware. The first type is the same as the Winant et

al. and called the "temporal analysis"; the second is the same as the Uda et al. and

named the "spatial analysis," and the third is very similar to the second, but the

original data sets are defined as the difference between two surveys, called "difference

spatial analysis." In their spatial analysis, the second spatial eigenfunction in the

direction of longshore was named the "rotation function" which explain the beach

rotation. They also examined the sensitivity of EEF. The seasonal movement of

beach can be determined by the "spatial analysis" from two surveys, one taken from

the summer profile and other from the winter, instead of using repeated surveys that

are necessary for the "temporal analysis."

Garrow (1984) applied EEF to examine the rhythmic feature of shore line and

contour lines in the field. First he examined the EEF calculated based on the hypo-

thetical data matrices representative of ideal beach topography. He found that if the

sinusoidal pattern remained stable but varied in amplitudes, the analysis produces

a sinusoidal mean and a single sinusoidal eigen vectors 180 degree out of phase with

the mean. From the analysis of the field data, three important morphologic compo-

nents are found, those are, overall accretion/erosion of the shoreline, configuration

of 800-850m wavelength beach rhythmic topography, and longshore profile or phase

of rhythmic features. But they could not be easily separated.

The same type of EEF analysis sometimes produced different results. For ex-

ample, the second eigenfunction obtained by Birkemeier explained the movement

of the single bar to the double bars or vise versa, while the function from Winant

et al. or Aubrey is defined as the bar-berm function. The spatial eigenfunction

in the direction of cross- shore calculated by Uda and Hashimoto is independent

to time, while the function calculated by Dick and Dalrymple is time dependant.

The main cause of these difference might be the differences of time scale, survey

intervals (sampling time) and local geographic characteristics.

Some comments on EEF by several examiner are summarized here.

Birkemeier: EEF has a number of significant limitations. First, the eigen vectors do

not necessarily have any physical significance. Secondly, the analysis assumes

that every survey is equally spaced in time and that all cases are equally

weighted, an assumption that is not usually the case with field data. Finally,

when analyzing data from a single profile line, the EEF analysis does not

separate cross-shore effects from long- shore effects

Dick and Dalrymple: The EEF method is an efficient way to describe beach profile

changes, however, it should be emphasized that it is a descriptive process and

therefore does not reveal any information regarding the governing process.

Garrow: This mathematical technique has two main goals. One goal is to deter-

mine a new set of variables which are independent each others. The second

goal is to define the fewest new variables possible that can completely describe

the original data.

EEF analysis, of course, is not used only in beach profile but other natural

phenomena such as wave spectrum parametric study (Vincent and Resio, 1977)

edge wave analysis (Kato 1984). In meteorology and geology, EEF analysis has

often been conducted to quantify stochastic data (Kutzbach 1967).

5.1.2 Basic Concept on Empirical Eigenfunction Analysis

The usefulness of EEF analysis is to reduce the number of data from original

data sets. In the present study, the data are the hydrographic survey, and since the

measurement is carried out four times at 21 x 140 points, the number of variables

would exceed 4 x 21 x 140 = 11760. If Uda and Hashimoto's representation (Eq.

5.4) is possible and the first three EEF can represent most of the original data, the

number of data needed would reduce to 4 x (3 x (21 +140)) = 1932. The EEF that

will be presented here, reduces the number of necessary data to 384. Furthermore,

In EEF analysis, since the two eigenfunction is assumed to be independent to each

other, the new set of variables can be examined independently. Hereinafter, the

basic concepts of EEF is described.

Suppose, for example, we have some survey data at M points, and N times

surveys are carried out. Thus, the number of data is MxN, this would compose M

by N matrix, say, F. F can be represented as

f1l f12 fiN
f21 f2N
F = (5.5)

fM1 fM2 fMN
In the analysis of bathymetric changes, the variables fnm would be the water

depth at a point of space and time. Defining a vector f, as

f = i (5.6)

then we have

F =[fif2... fN] (5.7)

One wants to determine which vector e has the highest resemblance to all the obser-

vation vectors f, simultaneously, where resemblance is measured with the squared

and normalized inner product between a vector f and e. This is the equivalent to


etRe (5.8)

subject to the condition

e'e = 1 (5.9)

where R is an M x M symmetric matrix whose element, rij is given by
rai = N-1 E finfi, (5.10)


R = N-(FFt) (5.11)

This problem yields to eigen value problem, i. e.

Re = eA (5.12)

In fact, one obtains not just one eigenvector, but series el, i = 1, 2....M associated

with M eigen values of R. It can be shown that e, are orthogonal and that A, are

real and positive. Thus, writing eq. (5.12) for all eigen vectors

RE = EL (5.13)

where E is an M by M orthogonal matrix whose column are eigen vectors, ei,....eiM.

Note that

EE = I (5.14)

and L is an M by M diagonal matrix whose ith diagonal element A, is the eigenvalue

associated with ei. Combining eqs. (5.11) and (5.12)

N-1FFtE = EL



= LN (5.15)


C = EtF (5.16)

where C is M by N matrix

it follows that

F = EC (5.17)

From eq. (5.17), it is clear that the observation vector f,, representing the nth

observation of the M variables, can be expressed as a linear combination of the M

eigenvectors. The cith element of C will be referred to as the coefficient associated

with the ith eigenvector for the nth observation. From Eq. (5.16), it can be shown

that ci, play the same role in Eq. (5.17) as the coefficients in, for example, a Fourier

series representation. The N observations refer to N different time and, therefore,

the elements in the ith row of C represent time variations of the coefficient associated

with the ith eigenvector. Different representations are possible, the basic concept is

the same as presented.

5.1.3 Formulation and Procedure of EEF for Contour Lines

In essence, EEF analysis is similar to harmonic analysis with the exception

that the functional form is not determined a priori but is part of the solution.

For a multivariate function such as the three dimensional contour line represented

by h(x, y, t) there are several possible combinations of eigenfunction representation

such as

h(x,y,t) = wmc,(t)em(x)f,(y) (5.18)

h(x,y,t) = WmCm(t)em(X, y) (5.19)

h(x, y, t) = E WmCm(, to)em(Yto) (5.20)

These combinations are, however, not independent of each other. The hope is to

have the right choice such that most the variance in the data set will be accounted

for in fewest terms. Unfortunately, at present, there is no criterion for making such

a choice and one has to rely on intuition and trial and error. After a number of pre-

liminary tests, it is decided to use the distance from a baseline to particular contours

as the dependent variable. Thus, we assume that this distance can be represented

by the linear sum of spatial eigenfunctions, S,(x), and temporal eigenfunctions,

Tt(t), of the following form
d"(x,t) = E tvS,(x)T (t) (5.21)

where w' is the weighting function, m is the mode and n denotes nth contour line.

But, if we apply EEF analysis based of Eq. (5.21), the influence of the initial profile

dominates the other modes. Since in the laboratory the initial profile is the same, it

is logical to eliminate this effect in the EEF analysis and to use the following form

D"(x,t) = dn(x,t) d(x, to) (5.22)
= E wnS(x)T (t) (5.23)
or, in the discrete form neglecting superscripts m for brevity

Dij = E wmSmTmi (5.24)
where i and j denote x and t respectively.

In applying the procedure outlined here to the present experiment data set, a num-

ber of problems are identified that require special attention:

(1) Owing to the existence of offshore shoals and bars, the dm(y, t) has, at times,

multi-values as illustrated in Fig. 5.4. Care must be exercised to select the depth

contour that is physically meaningful. (2) In EEF analysis, the contour lines are

assumed to be continuous. But, in the case of seawall, the contour lines near it are

clearly not continuous. In such case, the contour lines are divided into segmented

continuous lines and EEF analysis is performed to each line individually. (3) In

the recovery process, the final profile in the stormy wave condition is taken as the

initial profile and is eliminated in the EEF analysis.

5.1.4 Results and Discussion

EEf analysis is performed for eight contour lines for each test case, they are

-0.5, 0.0, 5.0, 7.5, 10.0, 12.5, 15.0, and 20.0 cm respectively. They are measured

with respect to the still water level in the erosional wave condition. Thus, 0.0

corresponds to shoreline in the erosional cases and 10.0 corresponds to shoreline in

the recovery cases. The results are presented in Figs. A.1 to A.9 in the appendix.

In these results, with the exception of Case 5, all the case have both erosional and


...... ... Y

Figure 5.4: Definition of d"

recovery phases. The recovery processes are identified with a "R" following the

caption. Case 5 has erosional phase only. Also, duration of test is 12 hours in Case

5, 5 hours in Case 6 and 4 hours for the rest. In the spatial eigenfunctions, the

ordinate represents the eigenvectors multiplied by the associated weighting value.

It should be noted here that if the spatial and temporal eigenfunction are of same

sign the contour line progresses whereas if they are of opposite sign the contour line


Figures 5.5 to 5.10 compare the real shoreline changes and shapes of eigenfunc-

tions for each tested. In the figures, the top graph gives the real shoreline changes

at different elapsed times as measured, the middle graph provides the net change of

measured profile at different time from the initial profile. The bottom graph plots

the first two spatial eigenvectors.

Table 5.1 summarized the contribution of each eigenvector (mode) to the total

variance. Based upon these results, a number of qualitative observations are made.

Shoreline Changes

Shoreline change is the primary signature of beach erosion and accretion. For

all the cases tested, the first eigenvector appears to account for more than 90%

of the variance. Therefore, there is no surprising that the real shoreline change is

similar to the first spatial eigenvector. For cases of erosional waves with normal

incident angle, ideally the shoreline would recede uniformly as there should be no

longshore component of sediment transport. By examining Fig. 5.5 for the case of

no seawall, the first spatial eigenvector is almost a parallel line with the exception

near the edges where the three dimensional effect comes into play. For the case with

seawall (Fig. 5.6), the first spatial eigenvector is also almost a parallel line. The

flanking effects, if any, is not visible. The temporal vectors are almost identical for

both cases.

Under oblique waves, for the case of no seawall, the first spatial eigenvector

exhibits a rhythmic feature in addition to a uniform recession (Fig 5.7). The first

temporal vector appears to be similar to the case of normal incident waves. With

the presence of seawall, the rhythmic feature on the updrift sides becomes more pro-

nounced. Again, the temporal vectors possess similar characteristics as the previous

cases. Now, as the wave angle increases (to 10 degrees), this rhythmic features di-

minishes in amplitude. The groin effect becomes more evident that results in severe

down- drift erosion immediately in the shadow of the seawall.

It is not clear at this moment whether this rhythmic feature is both induced and

controlled by the basin characteristics alone or it is induced by the basin charac-

teristics but eventually controlled by the nearshore topography. Since the rhythmic

feature is an often observed phenomenon, particularly immediately after storm at-

tack, there is a strong possibility that it is controlled by the nearshore topography.

Table 5.1: Contribution of Each EEF
case modes Water depth of Contour Lines(cm)
-5 0 5 7.5 10 12.5 15 20
Erosive Condition [D]:downdrift [U]:updrift
case 1 1 st 92.1 94.5 93.1 79.7 72.7 64.7 80.5 85.4
2 nd 5.9 4.7 5.6 18.7 18.7 23.2 17.7 10.9
case 2 1 st 95.4 96.3 90.4 89.2 88.2 87.5 83.1 70.9
[D] 2 nd 3.4 3.6 9.4 6.2 8.3 8.0 10.8 18.5
case 2 1 st 94.2 98.0 96.3 ** ** ** ** **
[U] 2 nd 4.3 1.4 2.1 ** ** *
case 3 1 st 95.8 97.8 95.8 83.7 86.5 85.7 86.3 76.9
2 nd 3.4 1.7 3.1 13.0 11.2 11.8 8.5 17.6
case 4 1 st 98.3 99.3 95.6 86.7 88.9 88.2 91.2 84.0
[D] 2 nd 1.5 0.5 3.5 10.5 7.1 8.3 5.8 11.8
case 4 1 st 97.3 98.4 96.7 ** ** ** ** **
[U] 2 nd 1.9 1.3 2.7 ** ** ** ** *
case 5 1 st 97.1 98.7 92.8 94.8 95.0 97.3 98.4 92.6
2 nd 7.0 9.0 7.1 5.2 5.0 2.7 1.6 7.4
case 6 1 st 98.6 99.7 96.3 92.8 92.0 93.7 91.2 92.8
[D] 2 nd 0.9 0.2 3.0 5.1 5.9 3.7 5.6 4.3
case 6 1 st 91.5 97.1 90.3 ** ** ** ** **
[U] 2 nd 3.9 2.3 6.2 ** **
Recovery Condition
case 1[R] 1 st ** ** ** 85.7 90.3 91.6 85.8 84.9
2 nd ** ** ** 9.9 7.3 6.2 7.2 9.1
case 2[R] 1 st ** ** ** 83.5 86.1 86.6 90.4 79.7
2 nd ** ** 11.9 10.3 10.5 7.3 9.7
case 3[R] 1 st ** ** ** 92.0 94.5 83.6 84.5 91.4
2 nd ** ** 6.1 4.4 12.8 10.1 5.7
case 4[R] 1 st ** ** ** 92.1 91.7 90.0 84.5 94.5
2 nd ** ** 5.7 6.2 8.4 8.4 4.8
case 6[R] 1 st ** ** ** 82.0 81.3 78.2 85.8 88.0
2 nd 11.1 12.4 18.3 9.1 7.0













....... ...... i HOUR
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0.7S 8.2s 15.75

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S....---- 2 HOUR
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.......... .... ..

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0 -2.001
0.5S .25 15.75



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0 -1." 1 =:::&=
C .25 25.75






n 3 ****~*** ~ A

. 8.25


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z 0.00

........ ...... .. HOUR
E------ HOUR

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