UFL/COEL 88/004
PERFORMANCE AND STABILITY OF LOW
CRESTED BREAKWATERS
BY
D.M. SHEPPARD
J.K. HEARN
FEBRUARY 1989
ACKNOWLEDGEMENTS
A portion of this work was supported by Mr. John P. Ahrens of the U.S.
Army Corps of Engineers Coastal Engineering Research Center, Wave Dynamics
Division, Wave Research Branch in Vicksburg, Mississippi.
TABLE OF CONTENTS
Page
LIST OF TABLES iii
LIST OF FIGURES iv
LIST OF SYMBOLS vi
CHAPTER 1  INTRODUCTION 1
1.1 BACKGROUND 1
1.2 BRIEF SUMMARY OF PREVIOUS WORK 2
1.3 PURPOSE AND ORGANIZATION OF THIS REPORT 3
CHAPTER 2  EXPERIMENTAL APPROACH 5
2.1 DESCRIPTION OF EXPERIMENTS 5
2.2 BRIEF SUMMARY OF AHRENS' ANALYSES AND RESULTS 11
CHAPTER 3  ANALYSES AND RESULTS 23
3.1 STRUCTURAL STABILITY 24
3.1.1 Volumetric Changes 27
3.1.2 Crest Height Changes 29
3.2 WAVE FIELD MODIFICAITON DUE TO THE BREAKWATER 30
3.2.1 Comparison of Transmission Gages 32
3.2.1(a) Wave height 32
3.2.1(b) Goda's spectral peakedness parameter 34
3.2.2 Energy Transmission 35
3.2.2(a) Variation of Kt within a test 35
3.2.2(b) Prediction of Kt 37
3.2.2(c) Comparison of data with predictive approaches
of other researchers 45
3.2.3 Energy Reflection 48
3.2.4 Changes in Wave Period and Spectral Peakedness 52
3.2.4(a) Change in T 52
3.2.4(b) Change in Ts and T 52
3.2.4(c) Change in Qp 57
CHAPTER 4  DESIGN AID PROGRAM 60
CHAPTER 5  SUMMARY AND RECOMMENDATIONS 72
REFERENCES 75
APPENDICES
A DESIGN AID PROGRAM
B EXPERIMENTAL DATA
C REPORT ABSTRACTS
LIST OF TABLES
Table Page
2.1 Summary of incident wave conditions for lowcrested
breakwater test. 8
2.2 Summary of structural and incident wave conditions for
type 1 (stability) tests 9
2.3 Summary of structural and incident wave conditions for
type 2 (previously damaged) tests 10
3.1 Summary of slopes and yintercepts for design curves in
Figure 3.3(b) 31
LIST OF FIGURES
Figure Page
2.1 Details of Experimental Setup (after Ahrens 1984). 7
2.2 Damage parameters as a function of the Hudson stability
number and the spectral stability number for subset 1. 12
2.3 Damage parameters as a function of the Hudson stability
number and the spectral stability number for subset 3. 13
2.4 Damage parameters as a function of the Hudson stability
number and the spectral stability number for subset 5. 14
2.5 Damage parameters as a function of the Hudson stability
number and the spectral stability number for subset 7. 15
2.6 Damage parameters as a function of the Hudson stability
number and the spectral stability number for subset 9. 16
2.7 Comparison of the measured and calibrated incident wave
heights. 19
2.8 General trend for transmission coefficient vs. relative
freeboard (after Ahrens 1984). 21
2.9 General relationship between energy reflection, trans
mission and dissipation as a function of relative free
board (after Ahrens 1984). 22
3.1 Relationship between initial structure height and total
crosssectional area. 26
3.2 Dimensionless damage as a function of the modified spectral
stability number. 28
3.3 Relationship between final relative height and spectral
stability number  (a) data from type 1 tests; (b) design
curves based on subsets 1, 3, and 5. 30
3.4 Comparison of transmitted wave heights measured at gages 4
and 5. 33
3.5 Comparison of spectral peakedness parameters (Q ) measured
at gages 4 and 5. 36
3.6 Change in transmission coefficient from the beginning to
the end of damage tests. 38
3.7 Transmission coefficient as a function of relative freeboard. 39
Figure Page
3.8 Transmission coefficient as a function of P for relative
freeboards greater than 1.0 41
3.9 Transmission coefficient as a function of relative free
board, R, for R < 1.0. Data are separated by both peak
incident wave period (a) and subset (b). 42
3.10 Design curves for the prediction of transmission coefficient
as a function of relative freeboard and P. 44
3.11 Definition sketch of idealized dmaaged structure. 46
3.12 Comparison of measured transmission coefficients with those
predicted by the approaches of Seeling (a, b) and Madsen and
White (c, d). Seelig (1980) accounts for transmission by
overtopping only. Madsen and White (1975) consider both
overtopping and transmission through the structure. 50
3.13 Transmission coefficients predicted by the approaches of
Seelig (a, b) and Madsen and White (c, d) as a function of
relative freeboard. Seelig (1980) accounts for transmission
by overtopping only. Madsen and White (1975) consider both
overtopping and transmission through the structure. 49
3.14 Design curves for the prediction of reflection coefficient
as a function of relative freeboard and relative depth. 51
3.15 Ratio of incident to transmitted peak period as a function
of relative freeboard. 53
3.16 Ratio of transmitted to incident significant wave period
as a function of relative freeboard. 54
3.17 Ratio of transmitted to incident significant wave period as
a function of relative freeboard. The limits used in the
design program (Chapter 4) to determine the upper and lower
bounds on the ratio are shown. 55
3.18 Ratio of transmitted to incident average wave period as a
function of relative freeboard. 56
3.19 Ratio of transmitted to incident spectral peakedness parameter
as a function of relative freeboard for each of the four wave
files. 58
3.20 Ratio of transmitted to incident spectral peakedness parameter
as a function of relative freeboard. The limits used in design
program (Chapter 4) to determine the upper and lower bounds on
the ratio as shown. 59
LIST OF SYMBOLS
a 0.5926 (Equations 310, 11, 49, 12, 18, 21, 22, 23, 25)
aj wave amplitude of frequency band in Equation 3.4
Ad area of original breakwater that displaced during test
At total crosssectional area of structure
B breakwater crest width
d water depth
Ad
D.D. dimensionless damage = 2
(D50)
w
D50 median stone diameter = w)
r
f wave frequency in Equation 3.4
F freeboard, h d, in Seelig's formula for Kt (Equation 312)
h structure crest height in Seelig's formula for Kt (Equation 312)
hf maximum damaged crest height
hf average final crest height
hi initial structure crest height
hf/d final relative crest height
Hc incident zeromoment wave height used in the calculation of Kt
Hs incident zeromoment wave height
Ht transmitted significant wave height calculated as average of two
transmission gages H3
K Hudson's dimensionless stability coefficient =
D ( 1) cot a
Kd energy dissipation coefficient D50 w
Kr energy reflection coefficient
Kt energy transmission coefficient
Lp wave length corresponding to T
h 1.5
i\l*J
M modified spectral stability number = N (s
H
N Hudson's stability number =
w
(H2 L )1/3
N spectral stability number = 
s w
A H
P 2w
D p
50
Q Goda's spectral peakedness parameter defined by Equations 33
and 34
h d
R relative freeboard =
H
S(f) value of energy density spectrum (Equation 33)
T average wave period of spectrum
T peak wave period of spectrum
Ts significant wave period of spectrum
w50 median stone mass
wr mass density of stone
ww mass density of water
Af frequency band width (Equation 34)
Ah change in crest height = hi hf
a slope of structure face
CHAPTER 1 INTRODUCTION
1.1 BACKGROUND
Traditional ideas about shore protection works embrace the philosophy
that damage to a structure is to be avoided for all but catastrophic
conditions. In the case of offshore breakwaters, this usually means
specifying the crest elevation such that little to no overtopping occurs,
since the volume of water overtopping the crest has been found to be an
important parameter in determining rear slope stability (Graveson et al.,
1980). This approach often results in costprohibitive shore protection,
because structure cost is integrally related to the volume of material
required for construction and maintenance. Thus, any reduction in crest
height results in a cost savings and an increase in project feasibility.
Recent field observations and subsequent laboratory studies indicate
that adequate shore protection can be achieved in some instances through
the use of lowcrested and/or "sacrificial" breakwaters. In 1976, com
bined wave and surge action due to Cyclone David caused severe damage to a
breakwater at Rosslyn Bay in Australia. Despite the fact that its crest
was battered to below mean water level, the breakwater continued to func
tion effectively for two and a half years until the structure was repaired
(Bremner et al. 1980). The unexpected success of this failed breakwater
prompted the concept of a "sacrificial" offshore structure which is used
to protect an inner breakwater or revetment and is designed to fail under
extreme wave conditions. Model tests on such a structure proposed for
Townsville Harbor, Australia were conducted, and it was shown that this
approach would save 40 percent over a conventional design (Bremner et al.
1980). Interestingly, these tests also suggested that the wave
transmission may not be very dependent upon the amount of structural
damage, because the increased energy transmission resulting from a lower
crest is balanced by the increased energy dissipation resulting from a
wider crest. Thus, the design parameters associated with these structures
are the prediction of damage levels and the subsequent performance of the
"failed" breakwater. Additional research is required in order to better
understand the influence of structural and wave parameters on these
criteria.
1.2 BRIEF SUMMARY OF PREVIOUS WORK
Ahrens (1984) investigated the stability and to some extent the per
formance of lowcrested breakwaters, with regard to certain structure
parameter and incident wave conditions. His data and findings are the
basis for this report and are discussed in greater detail in Chapter 2. A
brief summary of recent studies on the lowcrested design concept consti
tutes the remainder of this section. A more comprehensive list and
annotated bibliography of research on submerged and lowcrested structures
and related topics is included in Appendix A.
Foster and Haradasa (1977) conducted model tests on the original
Rosslyn Bay breakwater and on a proposed modification. Irregular incident
wave conditions were simulated using monochromatic waves with the same
height and period as the significant wave height and peak period of the
prototype spectrum. The tests closely reproduced the mode of damage seen
in the prototype structure, except that the initiation of damage occurred
earlier and the rate of damage after initiation was slower in the model
than in the prototype.
Foster and Khan (1984) studied overtopped structures in an attempt to
determine the variables most influencing their stability. They conclude
that the relationships between the parameters governing stability of an
overtopped structure are more complex than for a nonovertopped structure.
They recommended that rigorous physical model testing with the full range
of expected wave conditions and water depths be conducted prior to proto
type construction.
Seelig (1979) studied wave transmission by overtopping of regular and
irregular waves for subaerial and submerged smooth, impermeable, trape
zoidal structures. Seelig found that the dimensionless parameter, free
board divided by the incident significant wave height, is an important
factor governing energy transmission by overtopping. In a later investi
gation, tests were extended to include rubble mound and dolos armored
structures (Seelig 1980). An empirical method for determining wave trans
mission by overtopping that includes the effects of structure width and
wave runup, in addition to freeboard and wave height, was developed.
Allsop (1983) studied transmission, overtopping, and damage to low
crested, multilayered trapezoidal structures. He found that wave trans
mission, which was largely due to overtopping, is a function of wave
steepness. Although he did not find a similar period dependence in the
damage data, he notes that since stability is closely related to overtop
ping, it is possible that stability of overtopped structures is a function
of the wave period.
1.3 PURPOSE AND ORGANIZATION OF THIS REPORT
The purposes of this investigation were to provide additional
insights into the stability and wave transmission data from studies of
homogeneous, lowcrested breakwaters conducted at the Coastal Engineering
Research Center (Ahrens 1984), and to develop an interactive computer
program to assist in the design of these structures. In addition, an
extensive literature search was conducted. Pertinent papers and reports
are summarized in a series of abstracts which are presented in Appendix C
of this report.
Chapter 2 of this report outlines the experimental approach and
analytical techniques used by Ahrens (1984). Chapter 3 describes the
analyses used here and presents the results which are the basis for the
design program summarized in Chapter 4. Summary, conclusions and recom
mendations for future work are given in Chapter 5. Contained in the three
appendices are: the design aid program, the experimental data analyzed in
this report and the abstracts of related reports and technical papers.
CHAPTER 2 EXPERIMENTAL APPROACH
2.1 DESCRIPTION OF EXPERIMENTS
Laboratory experiments on the performance of lowcrested breakwaters
were conducted (by Ahrens) in the wave flume at the U.S. Army Corps of
Engineers Waterways Experiment Station in Vicksburg, Mississippi (Ahrens
1984). Structures were tested in a 61cm wide channel within a 1.2 m by
4.6 m by 42.7 m tank. Signals for the generation of irregular waves were
stored on magnetic tape and transferred to the wave paddle using a data
acquisition computer system (DAS). Four files with periods of peak energy
density ranging from 1.45 to 3.60 sec were used. A total of five wire
resistance wave gages recorded wave conditions in front of and behind the
structure. Records from three unequally spaced gages in front of the
structure were used to resolve the incident and reflected wave fields
using the method of Goda and Suzuki (1976). Two gages behind the struc
ture recorded the transmitted wave conditions. The DAS sampled the gages
sixteen times per second for 256 seconds.
A tenturn potentiometer in a voltage divider network was used to
regulate the signal amplitude to the wave blade. The signal amplitude is
related to the wave heights that are generated. An undamped signal pro
duced the depthlimited energy spectrum as described by Vincent (1981,
1982). The theoretical basis for this spectrum is taken from the work of
Phillips (1958) who proposed an expression for the upper bound on energy
density for deep water waves based on wave steepness. Phillips' limit is
proportional to f5 where f is wave frequency. Using Phillips' expression
as a starting point, Kitaigordoskii et al. (1975) derived an equation for
the depthcontrolled maximum energy density. The depthdependent limit on
energy density is proportional for f. Other characteristics of these
energy density is proportional for f .Other characteristics of these
spectra are a sharp drop in energy density at frequencies below the peak,
and, in the shallow water limit, wave heights that are proportional to the
square root of depth.
To ensure the most severe wave conditions possible at the structure,
waves were shoaled on a 1:15 slope from a water depth 25 cm greater than
the depth at the breakwater. Incident significant wave heights ranged from
one to eighteen centimeters. Details of the test setup are shown in
Figure 2.1. Incident wave conditions are summarized in Table 2.1.
Two types of tests were performed. The purpose of the first type was
to determine expected levels of damage under different wave conditions,
both mild and severe. The second type sought to evaluate the performance
of the damaged breakwaters under more typical, less severe wave attack.
Wave action for Type 1 tests lasted between 1.5 hr for File 1 spectra to
3.5 hr for File 4 spectra. Wave data were collected several times during
each run. Tests on the previously damaged structures lasted about
30 minutes and data were collected two to three times.
Structures were built with homogeneous stone. Two different stone
sizes were tested. Specific gravity and median mass were 2.63 and
17 grams, respectively, for the smaller stone, and 2.83 and 71 grams for
the larger stone. The undamaged structures were trapezoidal with front and
rear slopes of 1 on 1.5. The initial profile for a Type 2 test was the
same as the final profile of the proceeding test. Starting crest heights
ranged from 24.11 to 36.09 cm in a water depth of 25 cm and from 31.55 to
32.06 cm in a depth of 30 cm. Using test type, initial structure height,
stone size, and water depth as criteria, the 205 experiments were divided
into ten subsets. A summary of pertinent parameters for each subset is
given in Tables 2.2 and 2.3.
SCALE
0 1 2 3 4 5 Iml
o DENOTES WAVE GAGE LOCATION
WALL OF WAVE TANK
TO WAVE
GENERATOR
19 m
TRAINING
WALLS
GRAVEL WAVE ABSORBER GRAVEL WAVE ABSORBER BEACH
1 ON IS BEACH PONOING RELIEF CHANNEL WAVE
61 cm SCALING SLOPE oo o o o ABSORBER
REEF ONOING LIEF CHANNEL MATERIAL
BREAKWATER 
GRAVEL WAVE ABSORBER BEACH GRAVEL WAVE
AUXILIARY CHANNEL ABSORBER BEACH
6 GRAVEL WAVE ABSORBER BEACH
WALL OF WAVE TANK
PLAN VIEW
Figure 2.1 Details of Experimental Setup (after Ahrens 1984).
WAAA
r\rru
Table 2.1. Summary of incident wave conditions for lowcrested breakwater test.
Water Approx Range of Incident
File No. Depth (cm) Peak Period Wave Height (cm)
(Sec)
1 25 1.45 1.09 11.47
30 1.45 5.76 12.63
2 25 2.25 1.16 13.43
30 2.25 2.58 14.46
3 25 2.86 1.62 15.78
30 2.86 8.20 18.17
4 25 3.60 2.25 16.10
30 3.60 5.22 17.60
Table 2.2. Summary of structural and incident wave conditions for type 1 (Stability) Tests
Range of Range of
Subset No. of Median Stone Water CrossSect onal File Incident Initial Crest
No. Tests Diameter (cm) Depth (cm) Area (cm ) No. Wave Height (cm) Height (cm)
1 27 1.86 25 1170 1 2.87 11.45 24.11 25.39
2 2.91 13.43 24.41 25.48
3 3.89 15.78 24.44 25.73
4 5.46 16.10 24.14 25.12

3 29 1.86 25 1560 1 2.82 11.36 29.02 30.48
2 2.89 13.38 29.29 29.81
3 3.68 15.63 28.74 29.84
4 2.59 15.84 28.86 30.08

5 41 1.86 25 2190 1 2.75 11.35 34.38 35.57
2 4.03 13.02 34.41 36.06
3 1.81 15.61 34.93 36.09
4 2.56 15.99 34.59 36.03

7 38 2.93 25 1900 1 2.60 11.44 31.36 32.00
2 2.72 13.11 31.49 32.34
3 1.65 15.66 31.24 32.52
4 2.35 16.04 31.39 32.80
  
9 13 2.93 30 1900 1 5.76 12.63 31.55 31.82
2 5.80 14.46 31.58 31.67
3 8.20 18.17 31.61 32.06
4 5.22 17.60 31.61 32.13
 
Table 2.3. Summary of Structural and Incident Wave Conditions for Type 2 (Previously Damaged) Tests
Range of Range of
Subset No. of Median Stone Water CrossSectional File Incident Initial Crest
No. Tests Diameter (cm) Depth (cm) Area (cm ) No. Wave Height (cm) Height (cm)
2 3 1.86 25 1170 1 N/A N/A
2 5.87 5.95 15.88 19.99
4 12 1.86 25 1560 1 3.17 11.19 17.56 18.01
2 2.72 13.27 17.80 19.45
6 11 1.86 25 2190 1 2.84 11.47 19.54 19.81
2 2.49 12.88 19.78 19.96
8 26 2.92 25 1900 1 1.09 11.03 28.19 28.35
2 1.16 13.31 28.16 28.29
3 1.62 13.32 27.58 28.01
4 2.25 12.26 27.55 28.01
10 5 2.92 30 1900 1 N/A N/A
2 2.58 14.41 24.96 25.21
2.2 BRIEF SUMMARY OF AHRENS' ANALYSES AND RESULTS
Structural stability is defined by Ahrens (1984) in terms of both the
volumetric damage and the change in crest height. Volumetric damage is
described by a dimensionless damage parameter,
Ad A
D. D. = = d (21)
w50 2/3 ( 2
50) (Ds50
r
where
Ad = crosssectional area of the portion of the original break
water that was displaced;
w50 = median mass of the stone;
wr = mass density of the stone; and
D50 = the median stone effective diameter.
The change in crest height is represented by the final relative crest
height or ratio of final crest height to the water depth. Final crest
height is measured at the highest point on the structure.
Hudson's stability number defined as,
H
N s (2.2)
D 50( 1)
w
where
Hs = incident significant wave height; and
w, = mass density of water,
was initially used by Ahrens as a means of predicting the stability of the
structure. Plots of final relative crest height and dimensionless damage
as a function of Ns are presented in Figures 2.2 2.6(a,b) for the Type 1
tests. The expected trends are obvious; larger values of stability number
00 L
o o0 o
Jo o a h"
: oee Ronge of .
a eg d
0
0 00
0
1 0
.o00 2'00 u.00 6'.00 8'.00
HUDSON STABILITY NUMBER
(a)
0
Shz
Range of 
o 
J 0 0
'0 0
0
0
b.oo .0oo 8.00 12.00 16.00
SPECTRAL STABILITY NUMBER
(C)
0.
).ootb.oo
o
0
0
0..
I 1
20.00 o. oo
2'.00 .00 6.00 '.00
HUDSON STRBILITT NUMBER
(b)
4.00 8.00 12.00
SPECTRAL STABILITY
(d)
16.00 20.00
NUMBER
Figure 2.2 Damage parameters as a function of the Hudson stability
number and the spectral stability number for subset 1.
I00 ;
oo

CI.O
01a
00 0
08
0a
r
__ ? *' l u .1 .1 .. ,1.
W.oo 2'.o .00 o '.O 8.00 10
HUDSON STRBILITT NUMBER
(a)
D
m h
S Ronge of
++ +
S+ 44 
44
4.
V I _____________________4 .
.0oo0
0'.oo 8.00 1.oo 16.00
SPECTRRL STRBILITT NUMBER
(c)
0M
.oo0 .oo0
2b.00oob.o
2'.00oo .oo soo 0 800
HUDSON STRBILITT NUMBER
(b)
40oo 8.00 1t.00 16.00
SPECTRAL STRBILITT NUMBER
(d)
1b.00
0.0oo
Figure 2.3 Damage parameters as a function of the Hudson stability
number and the spectral stability number for subset 3.
13
h,
Range of '
*
+ +
+
+4
4*.
+
+
tt
"r
cv:
Uv4
tnU
LU
+
0 + +
z +
Lii+
'+
0
+
I
c
2
6* h7T
S** Ronge of
". o
cr
o *
0J
z
a:
.00 2000 4. 6'.00 8'.00 10
b.oo 2'.oo 0 'oo s.oo soo
HUDSON STABILITY NUMBER
(a)
0
V)
2
.oo 8'.00 1 0.o sb.oo0
SPECTRAL STABILITY NUMBER
(c)
0.
.0 .00
2.00 o .00 6.00 s8.00 1
HUDSON STABILITY NUMBER
(b)
.00 8'.oo 12.oo00 Ib. 2
SPECTRAL STABILITY NUMBER
(d)
Figure 2.4 Damage parameters as a function of the Hudson stability
number and the spectral stability number for subset 5.
6
e
*
6
*6
6
*
*
* 6
6
* 6
6
6 
* hi
Ronge of
a.. d
*
$3)
0 0
a: *
0
.o
ar !
c)
CJ
0
ZC9
i3
'
.oo00
01l
o__
*1
x "
xx
x.
Ronge of 
d
booo 2'.00oo lao soo e'o io
b.OO 2.00 '1.00 6.00 8.00 1i
HUDSON STABILITY NUMBER
(a)
Range of
xx
.oo 0 oo 8.00 .0o0 .oo0
SPECTRAL STABILITY NUMBER
(c)
0
0
1.00
b.00 2.00 '.00 6'.00 8.00 It
HUDSON STABILITY NUMBER
(b)
0
0
0 ______ ^ _  11 1. .
4.00 8.00 12.00 16.00
SPECTRAL STRBILITT NUMBER
(d)
20o. o.coo
Figure 2.5 Damage parameters as a function of the Hudson stability
number and the spectral stability number for subset 7.
tM
0
0
o
(0.
U,
_J
Zo
00
0.
en"
0
x
o _', x
).00
0.00
%C.oo
2
,
09 7
2
I
r
b.OO 2.00 4'.00 6.00 8.00 I
HUDSON STABILITY NUMBER
(o)
a
0,
b.0oo oo
Y
2.00 4.00 s.00 8.00
HUDSON STABILITY NUMBER
k'.oo .o00 1I.00 b.oo0
SPECTRAL STABILITY NUMBER
(c)
.00oo'.oo
4.:00 8.00 1.00
SPECTRAL STABILITY
(d)
t1.00 20.00
NUMBER
Figure 2.6
Damage parameters as a function of the Hudson stability
number and the spectral stability number for subset 9.
Y y hJ
Y Ronge of 
d
Y
 '  
y Range of
d
y
b.oo
y
I Y
Iy
00 = YYy
O
D ....
0 I
o
lead to greater damage and lower crest heights. The data, however,
exhibited considerable scatter. Scrutiny of the data with respect to peak
incident wave period suggested that some of the scatter might be elimi
nated by the inclusion of a term to account for wave period effects.
Graveson et al. (1980) present results of several different rubble mound
stability studies conducted at the Danish Hydraulic Institute (DHI). They
note that the dimensionless stability coefficient defined by
H3
K = s (23)
3 w
D50 (h 1) cot a
where w
w
a = slope angle,
is proportional to wave steepness, H/Lp where L is the wave length
corresponding to the peak period. Thus, Hudson's stability number was
modified to the following:
2 1/3
(H L )
N = sp (24)
s w
50 ( 1)
w
w
Using this parameter, Ahrens found that substantial reduction in scatter
could be achieved (Figures 2.2 to 2.6(c,d)). A general trend seen in all
the subsets is that the onset of damage occurs at about Ns* = 6. Expected
damage increases slowly as Ns* approaches 8, and increases rapidly for
Ns* > 8.
The manner in which wave energy is distributed can be described by
the equation
2 2 2
K + K + K = 1 (25)
t r d
where
Kt = transmission coefficient;
Kr = reflection coefficient; and
Kd = energy dissipation coefficient.
For these tests, Kr is given as the reflection coefficient measured during
the last period of wave sampling in a test; the Kt value is the average
transmission of all the sampling periods. Traditionally, Kt is defined by
the ratio of the measured transmitted wave height to the measured incident
significant wave height. This approach can lead to artificially low
values of Kt, however, since some energy is lost due to internal and
bottom friction between the wave gauges on the forward side of the break
water and the wave gages measuring transmission. In order to ascertain
the amount of energy transmission due to the breakwater only, the trans
mission coefficient was defined as
H
K (26)
t H=
c
where
Ht = the average value of significant wave height as measured at
the back gages, and
Hc = the average incident significant wave height at the loca
tions of the transmitted gages without the structure in
place.
This definition gives a more conservative estimate of Kt than the
traditional approach. Figure 2.7 illustrates the difference between the
incident and calibrated wave heights.
D.00 4 .00 8 .00 12.00 16.00 20.00
MEASURED INCIDENT WAVE HEIGHT
Figure 2.7 Comparison of the measured and calibrated incident wave
heights.
In agreement with other researchers (Seelig 1979), Ahrens notes that
the relative freeboard defined by
h d
R = (27)
s
where
hf = final crest height, and
d = water depth at structure site,
is the primary variable in the determination of Kt for overtopped and sub
merged structures, i.e., situations in which the dominant mode of trans
mission is overtopping. As R gets large, however, the dominant mode
shifts from overtopping to transmission through the structure. Ahrens
(1984) suggests that this transmission occurs at about R = 1.5. As the
mode of transmission changes, so do the variables affecting Kt. Wave
steepness, for example, becomes more important as R increases. Figure 2.8
shows the general trend exhibited by the data from these tests. The
dashed line indicates the region in which transmission through the
structure dominates. One should not interpret the dashed line to mean
that wave transmission increases as the freeboard increasesthis is
clearly contrary to intuitionbut rather that for a constant freeboard,
smaller wave heights give larger transmission coefficients.
Finally, Ahrens (1984) presents a schematic graph of the general
relationship between energy reflection, transmission and dissipation as a
function R (Figure 2.9). One notes that the energy of long waves is not
as easily dissipated as the energy in short waves. The difference is
particularly obvious for wave reflection for larger values of R.
I
p TRANSITION BETWEEN
I TRANSMISSION MODES
a
a
I I i 1 I I
2.0 1.0 0 1.0
hfd
RELATIVE FREEBOARD,
Hs
3.0
Figure 2.8
General trend for transmission coefficient vs. relative
freeboard (after Ahrens 1984).
1.0,
 0.8
0.6 k
0L
3
.0
100 u
~//////,/_ /. 1 SHORT
S / WAVES
90 10
LONG
WA VESREFLECTED
S. ENERGY 20
w I w
S70 30 >
,6 40 zr
z i 60 \ L
S60 ENERGY W
WH DISSIPATED O
> U) >
< 50 50 <
0 o
W 40 V 60 W
w n TRANSMITTED ww\
a. 30 ENERGY 70 _
< w
SHORT WAVES LONG
20 WA VES 80
10 90
0 iii100
4 3 2 1 0 1 2 3
hf d
RELATIVE FREEBOARD,
Hs
Figure 2.9 General relationship between energy reflection, transmission
and dissipation as a function of relative freeboard (after
Ahrens 1984).
CHAPTER 3 ANALYSES AND RESULTS
CERC provided the summarized damage and transmission data on 5 1/4
inch floppy disks in spreadsheet format for use with a personal computer.
In addition, some of the computer printouts from which the summaries were
compiled were supplied. Because it was faster, the spreadsheet files were
transferred to the Coastal and Oceanographic Engineering Department's VAX
750 computer.
The spreadsheet summary includes the following information for each
of the 205 tests.
Subset number (110);
Test type (stability or previously damaged);
File number and wave maker signal amplification;
Median stone mass, w50;
Stone mass density, wr;
Crosssectional area of breakwater, At;
Water depth, d;
Average incident significant wave height, Hs;
Average incident peak period, Tp;
Average transmitted significant wave height, Ht;
Average reflection coefficient, Kr;
Calibrated significant wave height, Hc;
As built structure height, hi;
Damaged structure height, hf;
Area of damage, Ad;
Peak, significant, and average incident and transmitted wave
periods,
Tp, Ts, and T, respectively, for the final sampling period of a
test;
Goda's spectral peakedness parameter, Q for the final sampling
period of a test;
Fraction of displaced stone found seaward of the structure;
Unsubmerged area of the damaged breakwater.
Discussion of the data analyses and results is in two parts. Section
3.1 deals with the mechanisms governing overall structure stability.
Section 3.2 reviews the parameters influencing spectral changes and energy
redistribution due to the breakwater.
3.1 STRUCTURAL STABILITY
Structural stability of overtopped breakwaters is complexly related
to many factors, including stone shape, density, and median mass; incident
wave height and period; ratio of structure height to water depth; storm
hydrograph; and currents. As discussed by Foster and Khan (1984), the
relationship between governing variables and stability is much more
complex for overtopped than for nonovertopped breakwaters, and careful
modeling of proposed structures is still the best way to obtain infor
mation about an individual structure's behavior. It is, however,
beneficial to be able to predict the general performance of the structure
in order to expedite testing.
Ahrens' data were examined exhaustively in an attempt to enhance
existing understanding of stability of lowcrested structures and to
develop a viable preliminary design procedure. Prior to a discussion of
analyses and results, it is enlightening to examine what is meant by
"stability" of lowcrested structures and how the expression of stability
is related to structure shape and size. As discussed in Chapter 1, the
design problem associated with stability of a lowcrested structure is not
necessarily prevention of damage, but rather prediction of damage for a
given set of incident wave conditions. For maintenance purposes, the
volume of material displaced is needed. For prediction of energy trans
mission, the reduction in crest height, and, to a lesser degree, final
structure width are important. It is obvious that structures of many
different shapes may have the same total volume of material (cross
sectional area), but entirely different relationships between that volume
and structural dimensions. For example, a rectangle with a given area has
an infinite number of height to width ratios, and a rectangle and
isosceles triangle with the same area and base dimensions have quite dif
ferent heights. Similarly the damage area associated with a given reduc
tion in crest height is a function of the initial structure size and
shape. Thus, the application of results that utilize quantities such as
damage area, total area, and structure height are necessarily restricted
to structures of like shape. The structures used in this study were
trapezoidal with front and back slopes of 1:1.5. The relationship between
the total area and initial structure height was linear (Figure 3.1) and is
described by the equation,
At = m hi + b (31)
where
m = 98.7769 cm2/cm; and
b = 1285.44 cm2
Converting to prototype units,
m = S (0.987769) m2/m; and
b = S2(0.128544) m2
2500
2000
N
<1500
E

LUJ
1000
I
500
500
I Range of Data
INITIAL HEIGHT (cm)
Figure 3.1
Relationship between initial
crosssectional area.
structure height and total
and
m = S (3.24067) ft2/ft; and
b = S2(1.38260) ft2
where
S = prototype to model length scale.
Based on Ahrens' observation that Ns* provides better definition of
stability than Ns, the relationship between damage and Ns was not con
sidered in the analyses. Instead, an extension and hopefully improvement
of the Ahrens (1984) work were sought.
3.1.1 Volumetric Changes
Examination of Figures 2.2 2.6(d) reveals that the rate at which
dimensionless damage increases as Ns* increases is dependent upon the
ratio of initial structure height to water depth, hi/d. Ahrens calls this
the "exposure parameter" (personal communication, 1986). This observation
is a reflection of the fact that the more of the structure that is exposed
to direct wave attack the greater the volume of displaced material will be
for the same Ns*. Often the structure stabilizes once the crest becomes
submerged and the water acts as a protective cushion (Bremner, et al.,
1980). Since dimensionless damage is a function of both Ns* and h /d, a
new relationship is proposed in which dimensionless damage is related to
the product of Ns* and (hi/d)n. It was found that the best correlation,
using a least squares curvefitting technique, is obtained for n = 1.5
(Figure 3.2). The curve shown is described by the equation,
D.D. = 19.4458 7.4546 m + 0.760505 m2 0.010478 m3, (32)
CD
CC
r:
CC
J
CI
0
LUJ
CO
0
Q
0 SUBSET 1
+ SUBSET 3
0 SUBSET 5
C X SUBSET 7
Y SUBSET 9
0
o ++
o
+
< ii~r8
1.00 8.00
MODIFIED
16.00
SPECTRRL
21L.00
STRBI
I
32.00 40.00
LITT NUMBER
(H Lp)
Ns* = (HL
D ( w 1)
5D wg
Figure 3.2
Dimensionless damage as a function of the modified spectral
stability number.
e~
where
M = Ns* (hi/d)1'5
and is valid within the limits 6 < M < 29.
3.1.2 Crest Height Changes
The typical final structure profile was irregular in that the crest
height varied along its length. For the purposes of this study, the final
crest height, hf, was specified by the highest surveyed point on the
crest; hf cannot, therefore, be geometrically related to Ad. For this
reason, an estimation of hf was made independent of Ad. Figures 2.2 
2.6(c) show that hf varies approximately linearly with N for Ns* > 6,
and constant hi/d. All data from Figures 2.2 2.6(c) are shown on
Figure 3.3(a). In a manner consistent with the observations in the last
section, the rate of decrease of hf/d increases as hi/d increases. The
data from subsets 7 and 9 appear to drop off more rapidly than is expected
based on results from subsets 1, 3, and 5. This does not suggest that the
structures built with larger stone suffered more damage. Rather, it may
be an artificial effect resulting from the difference in stone size, i.e.,
the removal of one large stone shows up as a greater decrease in height
than the removal of several smaller stones. Because the range of Ns*
tested and the number of tests where damage was measured were less for the
large stone structures, the average trend is not as welldefined. It was,
therefore, decided to use only data from subsets 1, 3, and 5 for this
analysis. Based on these data, and a least squares analysis, a set of
design curves is proposed as shown in Figure 3.3(b). Each line is for a
constant value of hi/d; interpolation is required for intermediate values
of hi/d. A summary of slopes and yintercepts for each line is given in
Table 3.1.
0SUBSET
^ +SUBSET
* SUBSET
aX X SUBSET "
^X^?04) xYSUBSET !
++ +
+ 
+ +
+ +f 4+
00 +++
Oe.
+ 0
0
I'.00 8'.00 1i.00
SPECTRAL STABILITY
(a)
1.oo0
NUMBER
Figure 3.3
Relationship between final relative height and spectral
stability number (a) data from type 1 tests; (b) design
curves based on subsets 1, 3, and 5.
L
F..40
'
LoI
o
1
Lu_
.4.
.00oo
0I
0
Table 3.1. Summary of slopes and yintercepts for design curves in Figure 3.3(b)
h.
Subset # Range of Slope 102 yintercept
1 0.9644 1.029 3.32322 1.21020
3 1.150 1.219 4.39221 1.44490
5 1.375 1.444 6.19724 1.80112
3.2. WAVE FIELD MODIFICATION DUE TO THE BREAKWATER
Several types of changes to the wave field due to the presence of the
breakwater were examined. Emphasis was placed on energy transmission, but
attention was also given to energy reflection, shifts in the peak, signi
ficant, and average wave periods, and changes in the spectral peakedness
parameter, as defined by Goda (1970). In addition, data from the two wave
gages in the lee of the structure, gages 4 and 5, were compared. The data
were also compared to the predictive methods of Seelig (1980) and Madsen
and White (1976).
Calculation of the transmitted Qp and the comparison of gages 4 and 5
are based on data from four runs from subset 3, all of subsets 5, 6, 8 and
9, and all but a few runs in subset 7.
3.2.1 Comparison of Transmission Gages
3.2.1(a) Wave height
Figure 3.4 shows the significant wave heights measured by gages 4 and
5 for each of the four data files. It is clear from these plots that the
magnitude of the discrepancy between the gages is in part a function of
the incident Tp. In general, the measured difference increases as T
increases and within each file as wave height increases. The differences
are probably caused by reflection from the absorbing material at the end
of the wave tank. As the wave period (wave length) increases, so do the
amount of reflection for a given wave height and the difference in wave
heights measured by gages 4 and 5. Little if any difference is seen in
the File 1 data. File 2 data show only small absolute differencesup to
about 3/4 cmbut these can translate into substantial percent
differences, particularly for the smaller waves. Usually, but not always,
the larger Ht was measured by gage 4. The File 3 and 4 data have about
2A50 S00H
WAVE HEIGHT
7.50 ab.oo
CM (GAGE 4)
'.so s 0'. 7'.50 tb.oo
HAVE HEIGHT IN CM (GAGE q)
(b)
Figure 3.4
Comparison of transmitted wave heights measured at gages 4
and 5.
N.
Wg
Ujo.
9
0"
cr
o
a
X:U
Ur.
Io
z
LDW
>. 3=)
0*/
3:
o
0d
File I
Tp =1.45sec
0/^
0o
File 2
Tp=2.25sec
A
A
A
b~ oo
TY
o
c
M 
'..ooD
the same maximum absolute difference (1 1/2 2 cm); Ht at gage 5 was always
greater than Ht at gage 4. Note also that the absolute difference for
File 3 increases gradually as wave height increases, but the discrepancy
in File 4 data grows rapidly to about Ht = 3 cm and more slowly
thereafter. The result is that the percent differences for File 4 are
greater than for File 3.
It is difficult to assess the error due to reflection that is intro
duced into the calculated value of Ht because the measured values of Ht
depend upon the location of the wave gages with respect to the partial
standing wave. Goda's resolution procedure should be used in order to
ensure that the most accurate transmitted wave height is obtained.
3.2.1(b) Goda's spectral peakedness parameter
Goda (1970) defines spectral peakedness as
2 f fS(f)2df
0
Q = 2 (33)
(I S(f)df)
0
where,
f = frequency; and
S(f) = value of the energy density spectrum.
In differential notation,
2 j fj a,_ (34)
Qp A ( aj2) 2
where
fj = frequency at the midpoint of the band; and
Af = spectral band width.
The higher the value of Qp, the more peaked the spectrum.
The Qp values for the wave spectra measured by gages 4 and 5 were
calculated and are presented in Figure 3.5. To maintain consistency for
comparison with the incident Qp, all Qp values were calculated using the
range of frequencies spanned by the incident spectrum. Except for File 1,
the trends observed are generally consistent with the observations of
Section 3.2.1(a). Figure 3.5(a) suggests that the spectral peak at gage 4
is greater than the peak at gage 5, but no corresponding difference in
wave height is observed (Figure 3.4(a)). Plots of the File 1 energy spec
tra show only small differences in the energy densities measured by the
two gages. Thus it seems that the Qp values are more sensitive than the
wave heights to small differences in energy density.
3.2.2 Energy Transmission
3.2.2(a) Variation of Kt within a test
As discussed in Section 2.2, the Kt value obtained by Ahrens for use
in subsequent analysis was calculated using,
1 m (Kt + K )
K = z (35)
t m n= 2
where
m = number of sampling periods;
Kt4 = transmission coefficient obtained using Ht from gage 4; and
Kt5 = the transmission coefficient obtained using Ht from gage 5.
Figure 3.5
Comparison of spectral peakedness parameters (Q ) measured
at gages 4 and 5.
This value gives the average Kt that can be expected for the given storm
event, but if Kt changes substantially from the beginning to the end of a
test, it may not be adequate for prediction of the maximum transmission.
Figure 3.6 presents typical examples of Kt vs. sampling period for
one test from each of the four files. The difference between the gages is
due to the difference in the measured Ht (Section 3.2.1(a). Due to the
reduction in freeboard, energy transmission increased as the tests
progressed. The increase is smaller, however, than would be expected if
the crest height had been reduced without the accompanying increase in
structure width. This observation is consistent with those of Bremner, et
al. (1980). The maximum absolute difference in Kt for the four files
ranged from about 0.06 to 0.17, but the percent change was as high as 46
percent. These changes are the same order of magnitude as the difference
between the values of Kt measured by gages 4 and 5. Without resolving the
discrepancy between the gages, it is inappropriate to do a detailed trans
mission analysis based on the "maximum" Kt. Even greater inaccuracies
could be introduced, because the existing errors may tend to cancel one
another. Figure 3.6 suggests, however, that the increase in Kt should be
addressed in future studies.
3.2.2(b) Prediction of Kt
The relative importance of the parameters governing energy trans
mission and, therefore, the method used to predict Kt change as the
dominant mode of transmission shifts from overtopping to flow through the
structure. Figure 3.7 presents the average Kt as a function of relative
freeboard, R. The relationship between Kt and R changes at about R = 1.0
because as R increases, transmission by overtopping approaches zero, and
the importance of freeboard in determining Kt is diminished. Ahrens
S2 P4 6R
SAMPLING PERIOD
Run 88
Tp= 2.84sec
_u .X
... :.......
...................... GAGE 4
 GAGE 5
SM 4 6
SAMPLING PERIOD
0
0
I
LU
4
4.
U_
U
LUo
CO
Z
o
ri
8
z
cc
8
8;
C
c
UJ
Lno
i
I
L00
CO
U,
(0
in
z
cc
cc
j i r I I I I
2 4 6
SAMPLING PERIOD
S.
Run 47
T =3.58 sec
x
/M
. ...N"
...................... GAGE 4
 GAGE 5
2 P 6
SAMPLING PERIOD
Figure 3.6 Change in transmission coefficient from the beginning to the
end of damage tests.
RUN 97
Tp 1.33 sec
...................... GAGE 4
..... .. GAGE 5
GAGE 5
.0
1 *
in
z
LU
8
IL
iL
O
U,
0
10
.1
o,
o~v
Run 101
Tp= 2.28 sec
...... .......... .... ."..
...................... GAGE 4
 GAGE 5
LO
t
z
LU
U')
4.
LL0
LU
0
1,
z
I
0
0
0
0 SUBSET 1
A SUBSET 2
+ SUBSET 3
X SUBSET 4
x SUBSET S
SUBSET 6
X SUBSET 7
x A Z SUBSET B
x x Y SUBSET 9
S X X SUBSET 10
A z
z
+ z x
+ 4 *
< 0 2
.k X
YI? :^ ^
ax ;xx
*^
*+ *a
*
I I
2.00 0.00
RELATIVE
2'.00 L.00
FREEBORRD
Figure 3.7.
Transmission coefficient as a function of
freeboard.
I
U
.L
LJO
LIJ C
C) C;
od
z
zo
*i
2O
OC
o
o
6.00
relative
;I
.oo0
(personal communication, 1986) found that Kt is a function of the
parameter, P, for R > 1.0, where
H A
P = (36)
L (D50)
p 50
(Figure 3.8). Note that, in this formulation, Kt is independent of free
board for R > 1.0. At values of R < 1.0, Kt depends upon both P and R,
i.e., transmission is by a combination of the two mechanisms. The
relative importance of P decreases as R gets smaller.
The parameter, P incorporates the influences of wave period, stone
size, structure area or width, and Hs all of which can be examined
separately. Figures 3.9(a) and 3.9(b) show the transmission data for
R < 1.0. From Figure 3.9(a), it is clear that the longer waves tend to
produce a higher Kt than the short waves, all else being equal. Unfor
tunately, there are no long wave data for R less than about 0.6. The
available data indicate, however, that the Kt for Files 3 and 4 and low R
values would be higher than for Files 1 and 2. Figure 3.9(b) shows that
the Kt for subsets 710 is consistently higher than the Kt for subset
16. The difference is probably due to the larger void spaces resulting
from the use of larger stones in the structures of subsets 710. Also, Kt
is generally smaller for subsets with larger crosssectional areas, all
else being equal, suggesting additional energy dissipation across the
crest and/or through the structure.
The approach used to predict Kt varies depending upon the value
of R. For R < 0.0, Kt is assumed to be a function of R only and is
predicted using an exponential curve of the form
+ SUBSET 3
SUBSET 5
X SUBSET 7
Z SUBSET 8
z
z
z
z
I
x
x5\
\X
^Y a
3t~~^X
0
C
LLo
LLJCO
Z
140
o
cno
C.)
o
0.
9J.
00
Ib.oo
15.00
2b.00
P= LP D50)
Figure 3.8
Transmission coefficient as a function
freeboards greater than 1.0.
of P for relative
5.00
25.00
S* 0 1
'x'A+ xX
0 0
2 A +
0 0A
o0 9o
0 ^x
o +
0 FILE I
A FILE 2
+ FILE 3
X FILE 4
x x
x
& x X 3+
* SUSSE *I 4.
A U
SUBSET
+ SUBSET
X SUBSET 1
2 SUBSET 
+ It
+ SUBS
X SUBSET
X SUBSEI 1
X SUISET IS
YZ S V BeSS : I
I I I
1.80 1.00 0.20 0.60
RELATIVE FREEBOARD
1.10
'.1
.80o 1.0o b.20o 0o6
RELATIVE FREEBORRD
(b)
Figure 3.9
Transmission
freeboard, R,
incident wave
coefficient as a function of relative
for R < 1.0. Data are separated by both peak
period (a) and subset (b).
LL*
C.
**
U
Oo
C()
CO
cr
zo
1*
0
..6
I
C,
 .60
0
Kt = All + A21 eR
where
All = 0.9 and
A21 = 0.358.
For 0.0 < R < 1.0
1.0
K =
t
A12 + A22 R
1.0
A12 = = 1.845 and
All + A21
A22 = (1 A12) + pa
= 0.845 + pa
(38)
(39)
(310)
where
a = 0.5926
For R > 1.0, Kt is a function of P only. The relationship shown in
Figure 3.8 is given by
1
K =
t 1 + pa
(311)
(Ahrens, personal communication, 1986). Design curves for different
values of P are shown in Figure 3.10. Details of this development are
discussed in Chapter 4.
where
(37)
x xg
+ x^
x
Iz
z
,
'R2
*
**. *
2.00 0.00 2. 00 4'.00
RELATIVE FREEBOARD
.1
6.00
L.00 2.00 0.00 2.00 4
RELATIVE FREEBORRD
S.00
Figure 3.10 Design curves for the prediction of transmission
coefficient as a function of relative freeboard and P.
1I
1._
I
u4
Lo
L.)
o
(ol
z
'4
C)
10
U) =!,
FE
0 SUBSEI I
SUBSEt S
+ SUBSET $
X SUBSET I
* SUBSET 5
+ SUBSET a
X SUBSET I
Y SU1SEIt
X SUBSET a
x SUsSE 1 1
2
* *4
0
0
F3
Zc0
4
LL
L)O
z
Cr)
zo
eLT
acc
P=50
P= 100
P= 15.0
6.0C
3.2.2(c) Comparison of data with predictive approaches of other
researchers
Seelig (1979, 1980) found that the transmission coefficient due to
overtopping for a structure fronted by a 1:15 slope is given by
Kt = C (1 F/U) (1 2C) F/U and (312)
C = 0.51 0.11 B/h
where
F = freeboard;
U = Wave runup;
B = crest width; and
h = structure height.
Madsen and White (1976) derived an analytical solution for transmission
through trapezoidal, permeable, multilayered structures. CERC program,
MADSEN, (presented in Seelig, 1980) calculates the total transmission com
bining both approaches using
Kt (total)2 = Kt (overtopping)2 + K (through)2. (314)
In order to compare these approaches with the data, the width of the
damaged structure had to be estimated. This was done by assuming that the
final structure shape is a trapezoid similar to the initial shape, e.g.,
parallel on top and bottom with side slopes of 1:1.5. The material
removed from the top is redistributed at the front and back sides. The
increased area at the sides is equal to the area of damage, Ad
(Figure 3.11). The total crosssectional area, At, is the same for both
profiles, so that At is expressed in terms of initial conditions as
45
Original Profile
Idealized Damage Profile
Figure 3.11. Definition sketch of idealized damaged structure.
I
hf
I
A = (B+4) h (315)
2
Damage area is given by
W +W
Ad = ( ) Ah (316)
2
where
Ah = the change in crest height (hi hf).
The values W1, W2, W3, W4, and B are given by
A
W1 h 1.5 hi, (317)
w2 = W1 + 3hi, (318)
W3 = W1 + 3Ah, (319)
W4 = B + 3f and (320)
A
B = 1.5 hf. (321)
hf
All variables in the above equations are known except hf, which represents
the average final crest height of the structure. The measured final
crest height is not used because it is the highest point on the crest and
would give a value for B that is inconsistent with the known values of At
and Ad. To solve for hf, equations 3 and 6 were combined to obtain
1
1.5(Ah)2 + W1Ah Ad = 0 (322)
which is a quadratic in Ah. The solution is the positive root of
1
W1 (W12 + 6 Ad)2
Ah = (323)
3
The average final crest height is thus, h = h Ah.
Comparison of the data with the predicted Kt by overtopping (Seelig,
1980) and with the predicted total Kt (Madsen and White, 1976) are shown
in Figures 3.12(a, b) and (c, d), respectively. Predicted Kt is plotted
against R in Figure 3.13. As expected, the predicted Kt due to
overtopping is less than the total measured Kt, with the discrepancy
increasing as Tp increases. Overprediction occurs at low relative
freeboards, and no transmission is accounted for when R > 1.0.
Consideration of transmission through the structure gives some
improvement, especially for cases of R > 1.0, but there is still
considerable scatter in the data. In particular, when the subsets are
considered individually the predicted values are unsatisfactory.
Predicted Kt has much less variation than the measured data show. A
possible explanation is that Seelig's formulation is valid for
0.88 < B/h < 3.2, but the range of B/h for these tests is 0.23 to 5.9.
For larger values of B/h, the relationship may overaccount for structure
width.
3.2.3 Energy Reflection
The reflection coefficient as a function of R is shown in Figure
3.14(a). There is a clear separation of the data into two groups based on
T with the longer waves of Files 2, 3, and 4 having a higher Kr than the
Figure 3.12.
0.40 0.60 U.BU I.u" o b. 00 0.20 0.40 0.60 0.80 1.00
MEASURED KT MEASURED KT
(c) (d)
Comparison of measured transmission coefficients with those pre
dicted by the approaches of Seelig (a, b) and Madsen and White
(c, d). Seelig (1980) accounts for transmission by overtopping
only. Madsen and White (1975) consider both overtopping and
transmission through the structure.
I
.00 0.00 2.00 4.00
RELATIVE FREEBOARD
(a)
.00o 0.00 2.00 4.00
RELATIVE FREEBOARD
(c)
Figure 3.13
6.00
0 FILE I
A FILE 2
+ FILE 3
SX FILE 'I
' c
0
S 4x
x
i* x
t^
?
'' ^ e
g~j
*> y
x 0 SUBSET I
A SUBSET 2
+ SUBSET 3
o f X SUBSET I
** SUBSET S
+o SUBSET B
B X SUBSET 7
e Z SUBSET B
Y SUBSET 9
1 X SUBSET 10
*+ z*
0)
C"
.H
0 4 4
1.00
 n.00 0.00 2.00 4'.00
RELATIVE FREEBOARD
(d)
6.00
6.00
Transmission coefficients predicted by the approaches of
Seelig (a, b) and Madsen and White (c, d) as a function of
relative freeboard. Seelig (1980) accounts for trans
mission by overtopping only. Madsen and White (1975)
consider both overtopping and transmission through the
structure.
 0.00 o oo 2.00 .00 oo
RELATIVE FREEBOARD
(b)
x 0 SUBSET I
A SUBSET 2
+ SUBSET 3
X SUBSET 4
x 0 SUBSET 5
+ SUBSET 6
+ X SUBSET 7
Z SUBSET 8
Y SUBSET 9
X SUBSET 10
zx
SO0 FILE I
A FILE 2
S ,+ FILE 3
o X FILE 4
)x
6
o
xx o
; ~~ ^ x< a
0
9.1
'l.00
 
m
I
I

r..
o O
0 0
0 FILE I
A FILE 2
+ FILE 3
O x FtILE 4
I U
U. x U0o d/L=0.0455
l.,C x LL",
LU XC +x +X X + 4 u
C o d/L= 00625
Sx A x A / d/L= 0.0800
cO* x / d/L=0.1000
O a d/L=01200
0 ^x\' eo o e 0
o A
o 0
o
00 '
o o
.00 .00 0'.00 2.00 4.00 6.00 l1.00 2.00 0'.00 2.00 4 '.00 6.01
RELATIVE FREEBOARD RELATIVE FREEBORRD
(a) (b)
Figure 3.14 Design curves for the prediction of reflection coefficient
as a function of relative freeboard and relative depth.
short waves in File 1. As R increases, Kr approaches a constant whose
value is a function of Tp only. Kr is predicted by fitting three lines
within the ranges of R < 1.0, 1.0 < R < 3.0, and R > 3.0 (Figure 3.14(b).
The lines were matched at R = 1.0 and R = 3.0 such that the overall good
ness of fit was as high as possible. The effects of Tp are included in
the linear coefficients which are a function of d/L. A detailed dis
cussion of the prediction of Kr is given in Chapter 4.
3.2.4 Changes in Wave Period and Spectral Peakedness
3.2.4(a) Change in T
As discussed earlier, Tp is the wave period in the spectrum that
contains the most energy. It is expected that energy will be lost at the
peak and redistributed to higher and lower frequencies, but that, in the
absence of breaking, the frequency at which the peak is located will not
change. This is generally the case for these data. The ratio of the
transmitted to incident Tp as a function of R is shown in Figure 3.15.
3.2.4(b) Change in T. and T
Ts is the average wave period of the onethird highest waves. T is
the average of all waves. Unlike Tp, Ts and T are expected to change
because they are directly related to the wave heights. As higher fre
quencies are introduced or filtered out, the change should be reflected in
Ts and T. The ratios of transmitted to incident Ts and T vs. R are shown
in Figures 3.16 and 3.18, respectively. The data in Figure 3.16(ad) are
plotted together in Figure 3.17. There is a distinct pattern to these
data that corresponds to the shift in transmission modes. For R less than
about 1.0, the ratio is less than one. This means that higher harmonics
are being introduced as waves pass over the structure. For R > 1.0, the
Itoo D.o o o .00 2.o00 4.00 6.
(a)
0
File I
Tp:1.45 sec
 .00oo .oo 0o'.oo00 2.00 I.oo 6.
RELATIVE FREEBOARD
(a)
s +
0
Tp = 2.86 sec
S.o 2.o 0.00ooo a'. oo00 '.oo
RELATIVE FREEBOARD
(C)
Cc)
Figure 3.15
C
a
I
c
0
C
C
a
c
it
a
U
a
00
N
3
C4
o
C
AA A AA
1.00 Do0 0'.00 2. 00 4.00 S.C
AI0 A
a
a
J
File 2
i Tp= 2.25 sec
.o00 2.00o o.oo 2. 00oo '.oo 6
RELATIVE FREEBOARD
(b)
o
0
0
0
CI
=CD
UJo
a"
File 4
a Tp=3.60 sec
0
1.00 2.00 0'00 22.00 11.00 .
RELATIVE FREEBOARD
(d)
Ratio of incident to transmitted peak period as a function
of relative freeboard.
*1
m O
E"
3
S 0
so
File I
0 Tp= 1.45 sec
.00.o .0 0 0'.00 2' o '00
RELATIVE FREEBOARD
0 10
r
zo
File 3
.00 Tp= 2.86 sec
.o1 0 .oo o'.00 2. 00 7L00 6.
RELATIVE FREEBOARD
(c)
o,
D
3Lo
RELATIVE FREEBOARD
(0)
Figure 3.16 Ratio of transmitted to incident
function of relative freeboard.
Cfr
a:
0
Oo
e 4
A&
LL.I A
zo
File 2
S= Tp=2.25sec
0
G .00 2.00 O.00 2.00 t00 6
RELATIVE FREEBOARD
(b)
0
0
X %
2 < X
x
File 4
STp3.60 sec
n . _nn u' nn a nn
RELATIVE FREEBOARD
(d)
significant wave period as a
w~uu
Figure 3.17
4.00 2.00 0.00 2.00 4.00 6.00
RELATIVE FREEBORRD
Ratio of transmitted to incident significant wave period as
a function of relative freeboard. The limits used in the
design program (Chapter 4) to determine the upper and lower
bounds on the ratio are shown.
File I
Tp= 1.45 sec
 00 2.00 I00
RELATIVE FREEBOARD
(a)
(a)
+ + +
+
File 3
STTp = 2.86 sec
S.00o 1.00 0.00 2.00 .o00 6.
RELATIVE FREEBOARD
(c)
Figure 3.18
00
IO
Fil 2
File 2
I
J x
File 2
o Tp32.25 sec
o00 2.00 0.00 2.00 i.o00 6.
RELATIVE FREEBOARD
(b)
0
D
o
File 4
 .0 I.0
00
RELATIV
E FREEBOARD
(d)
Ratio of transmitted to incident average wave period as a
function of relative freeboard.
ratio is greater than one, indicating that higher frequencies are being
filtered out. This is characteristic of waves passing through a struc
ture. The magnitude of the change is in part dependent on T Overall,
longer waves experience a greater change in Ts and T.
3.2.4(c) Change in Q
The Q ratios are plotted in Figure 3.19 and 3.20. Changes in the Qp
ratio are attributed to the same factors as the changes in T and T.
S
(0
0
o
0
o
0
(a)
.O O 20 'O '.0 *.0
RErIEFREOR
(a
0
S+
.t +
+ 4+
4.
0 + "
File 3
,Tp= 2.86 sec
.oo k.oo o'.oo 2o00 o oo0 6.
RELATIVE FREEBORRD
0
0
A AA
3.oo 00oo .00 2.00 '.00 6.
RELATIVE FREEBOARD
(b)
(b)
"M
SX XX
x
.0
File 4
o Tp=3.60sec
.00 D.oo 0'00 2.00 '.00 8.1
RELATIVE FREEBOARD
(d )
Figure 3.19 Ratio of transmitted to incident spectral peakedness parameter
as a function of relative freeboard for each of the four wave
files.
00
Figure 3.20
i4.00 2.00 o0.00 2 .00 u4.00 6.00
RELATIVE FREEBORRD
Ratio of transmitted to incident spectral peakedness
parameter as a function of relative freeboard. The limits
used in design program (Chapter 4) to determine the upper
and lower bounds on the ratio as shown.
CHAPTER 4 DESIGN AID PROGRAM
This chapter describes in detail the computer program, LCBDGN, which
is to be used as an aid in designing lowcrested breakwaters. Version 1.0
of this program is based on laboratory data only. Future versions will
incorporate field data as well.
The assumption is made that the designer knows the incident wave
conditions for which the breakwater is to be subjected and the desired
transmitted or reflected significant wave height for specified incident
wave conditions. Two sets of incident wave conditions must be specified
(referred to here as operational and extreme) as well as which of these
conditions are to be used as a basis for design. The program computes the
structure height needed in order product the desired results and the
height to which the structure must be constructed in order to achieve that
final height. The constructed height may or may not be the same as the
final height depending on the specified design conditions and structure
parameters.
Any two sets of incident wave conditions for which the designer would
like stability and performance information is acceptable as long as they
are within the range of the present data. As more and better data are
available the program can be upgraded and extended to include a wider
range of conditions. If sufficient statistical information is known about
the incident wave climate, the "operational" sea state may be taken as the
conditions (significant wave height, peak period, peakedness parameter)
that are not exceeded a high percentage (say 95%) of the time. "Extreme"
conditions refer to what is often called "design conditions" and are the
most severe conditions anticipated during the life of the structure.
LCBDGN computes the performance (i.e., transmitted significant wave
height, peak period, significant period range and peakedness parameter
range and reflected significant wave height) of the breakwater for both
sets of conditions before and after it has been subjected to the extreme
sea state.
Least squares curve fits to laboratory data have been made regarding
the stability and performance of lowcrested breakwaters. A description
of how these curves are used to compute structure heights and damage and
transmitted and reflected wave parameters is presented below.
Final Structure Height
The structure height required to produce the desired transmitted or
reflected significant wave height for a specified set of initial condi
tions is computed using lease squares curve fit equations of the data
shown in Figures 3.10, 3.3(b), and 3.14(b). Figures 3.10 and 3.3(b) are
used when transmitted wave conditions are specified and Figure 3.14(b)
when specific reflected wave conditions are desired.
First consider the case where transmitted waves are specified (refer
to Figure 3.10)
For R < 0.0, the transmission coefficient is given by
K = All + A21 eR (41)
All is fixed at 0.9 and A21 is determined using least squares curve fit
techniques
All = 0.9 (42)
A21 = (
K = 0.9
t
n R n R
S tiA
E e K All E e
i=1 t i=l
i
n 2R
iE e
0.358 e
 0.358 e
) = 0.358
(43)
(44)
Solving for hf in Equation 44 we get
h d 0.9 K
R = = and
s. 0.358
0.9 K
hf =d + H In ( t0 ) .
1 0.358
For 0.0 < R < 1.0, the transmission coefficient is given by
K 1.0
Kt = .
A12 + A22 R
In order to make the Kt vs. R curve continuous at R = 0.0,
expressed as
Al2 1.0
A12 l + A21
All + A21
1.845
(45)
(46)
(47)
A12 is
(48)
A22 is chosen so that the Kt vs. R curves will be continuous at
R = 1.0,
A22 = (1 A12) + pa
(49)
where, as defined in Chapter 3,
H A
s t
p 50
(410)
Thus
and
h
A = hi ( + b)
tan e
Substituting these expressions into the Kt equation results in
t
A12 + {1 A12 +
H h
H hi
( s
L ((D50)
p 50
hi can be expressed in terms of
Figure 3.3(b).
For N > 6.0
s
h
f *
= A13 A23 N
d s
h
(h +b))a} R
tan e
(412)
hf by using the curves from
(413)
where
h
A13 = A113 + A213()
d
h
A23 = A123 + A223 ()
h h h
= [A113 + A213()] + [A123 + A223( )]N
d d d
(411)
and
(414)
(415)
(416)
Solving for hi we have
hf
d[() A113 A123 N* ]
h = ds (417)
[A213 + A223 N ]
Substituting this expression into the Kt equation yields
h k h k h d
{k6 + [k5( k3 ( "k k + k )]a( H + k7) = 0 (418)
2 3 2 3 s
where
k. = A113 + A123 N
k2 A213 + A223 N
k3 tan e ,
k b ,
k Hi /Lp(D50)
k6 1.0 A12 (419)
k A12 
7 K A
t
A113 = 0.2338 ,
A213 = 1.436 ,
A123 = 0.03737 and
A223 = 0.06997 .
This transcendental equation can be solved using a NewtonRaphson
scheme, i.e.,
f(hf() )
hf(j+) =hf(j) f'[h )
where
f(hf)
h k h k hf d
{k6+ [k5 ( k ) k k )a H + k7) = 0
2 3 2 3 s
d f(hf)
f'(hf d h
hf k1 hf k1
{k6 + [k5( k k k + k4)]a}(1
2 3 2 3 s
hf k hf kk + a1
+ a[ki(2 k 2 kt+ k4)
S2 3 2 3
hf k1 1.0 k h k hf d
[k 2 T )+ fr k + k k k 4
5 k k k k k k k k 4
2 3 2 3 2 3 2 3 s
(420)
(421)
(422)
For R > 1.0
K 1.0
K =
t 1.0 + pa
where
H h
p = s [h.( e + b)]
S(D 2 50tan
p 50
(423)
(424)
Substituting this expression for P into the above Kt equation and solving
for hi results in
b tan 0
h = 
i
(b tan 2
(b tan O) +
where
K* E (1 
Kt
Next, equate hi in this expression to hi in Equation 4.17.
hf *
[ d A113 (A123) N]
d( d Ns)
(A213 + (A223) Ns)
b tan 6 1
2 2
(b tan +
(b tan 0) +
4L (D50)2 K* tan 6
(50
H
S
Solving for hf we get
L (D50)2 K* tan 8
H
s
(425)
(427)
h = A113 (A123) Ns
f s
+ [A213 + (A223)
2 *
b tan 6 1 2 4Lp(D50) K tan 0
N ][ +  (b tan 8) + H
s 2d 2d H 4
(428)
a = 0.5926
(429)
For the case where reflected
Figure 3.14(b) must be used.
For R < 1.0
significant
wave height is specified,
KR = A14 + (A24) R
A14 = A114 + A214
d
()
L
p
1.0
A24 = 
A124 + A224 ()
L
p
K A14
R = 4 and
A24
(K A14)
hf = d + Hs A24
f 5 A24
H {K [A114 +
s lr L
[A124 + A224
A214(d)]}
L
d
()
p
where
(430)
(431)
(432)
(433)
hf =d+
f
(434)
where
A114 = 0.5085 ,
A214 = 2.018 (435)
A124 = 1.019 and
A224 = 137.6 .
For 1.0 < R < 3.0
Kr = A15 + (A25) R (436)
d
A15 = All5 + A215() (437)
P
1.0
A25 = 1 (438)
A125 + A225()
P
H {Kr [A114 + A214()}
hf = d + 1.0 (439)
[A124 + A224( )]
P
where
A115 = 0.7195 ,
A215 = 3.400 (440)
A125 = 48.70 and
A225 = 268.1
For R > 3.0
hf(R = 3.0)
= A15 + A25(3.0) (441)
Initial Structure Height
Once the final structure crest height has been determined then the
initial or constructed crest height can be obtained from Figure 3.3(b)
(Eq. 417).
f *
() A113 (A123)N
hi = dS d
(A213 + (A223)N
s
Knowing the initial structure height, the "area of damage" can be
found from the least squares curve fit to the data in Figure 3.2.
Ad = (D50)2[A16 + A26M + A36M2 + A46M3] (442)
where
M N )15 (443)
s d
and
A16 = 19.45 ,
A26 = 7.455 (444)
A36 = 0.7605 and
A46 = 0.01048 .
With the initial and final structure heights established, the break
water performance can be computed for the circumstances and conditions of
interest. As stated earlier in this report, the performance parameters for
this version of the program include:
1. Transmitted significant wave height
2. Transmitted significant wave period range
3. Transmitted peakedness parameter range
4. Reflected significant wave height
These parameters are computed for the following circumstances:
1. Operational Incident Waves
a. Previous wave conditions not exceeding operational sea
state.
b. Previous wave conditions reaching but not exceeding
extreme sea state.
2. Extreme Incident Waves
a. Previous wave conditions reaching but not exceeding
extreme sea state.
Transmitted Significant Wave Period
The data for transmitted significant wave period is not sufficient to
allow prediction of specific values. Bounds on the range values for given
set of conditions can, however, be obtained from the data as shown in
Figure 55. The equations for these bounds are as follows:
T
ss 1.0 R < 3.5
= T11 + (T21)R 3.5 < R < 1.0 (445)
s T13 + (T23)R 1.0 < R
where
T11 = 0.6320 and
T21 = 0.106 (446)
Transmitted Peakedness Parameter
As with transmitted significant wave height, there is not enough data
to predict specific values of the peakedness parameter. Bounds on the
range of values were established by eye and equations fit to these bounds.
These equations are given below:
Pt 1.0 R < 3.5
= Q11 + (Q21)R 3.5 < R < 1.0 (447)
Pi 1.28 1.0 < R
where
Qll = 0.67 and
Q21 = 0.095 (448)
CHAPTER 5 SUMMARY AND RECOMMENDATIONS
Data from experiments on lowcrested breakwaters conducted at the
Coastal Research Engineering Center were analyzed with respect to
structural damage and modifications of the wave field by the breakwater.
The results were incorporated into an interactive breakwater design
program use with a personal computer. In addition, observations from
these analyses were used to develop recommendations for future studies.
Structural damage was assessed in terms of both crest height changes
and volumetric changes. The volumetric change, expressed as dimensionless
damage, was found to be related to a modified spectral stability number
defined by
h 1.5
M = N ()
=Ns
For values of 6 < M < 29 this relationship is given by a third order
polynomial (Equation 32). For M < 6, the damage is effectively zero.
Crest height changes are determined using a linear relationship between
hf i
 and N for a specified value of . Three design curves are based on
d s d
data from subsets 1, 3, and 5 having values of  of about 1.0, 1.2,
and 1.4, respectively, see Figure 3.3.
The modification of the wave field due to the breakwater is a complex
function of many variables including wave height, wave period, stone
characteristics, freeboard, and crosssectional area of the structure.
The relative importance of these paramterrs in the determination of Kt and
Kr depends upon the relative freeboard, R, which also serves to define the
transition from transmission by overtopping to transmission by flow
through the structure. Prediction of Kt is divided into three zones. For
R > 1.0, transmission is predicted using the parameter, P, which is a
function of wave steepness, crosssectional area of the structure, and
stone diameter, but is independent of R. The region 0 < R < 1 is a
transition zone in which Kt is a function of both P and R. At negative R,
Kt is primarily a function of R. The influences of stone and structure
parameters are less obvious in the reflection data. Below R = 3.0, Kr is
a function of R and incident wave period expressed as the relative depth,
d/L. At higher values of R, Kr depends of d/L only.
The shift in dominant mode of transmission also effects the ratio
between the transmitted and incident wave periods and spectral peakedness
parameters. When transmission is by overtopping (R < 1.0), higher
harmonics are introduced into the transmitted wave and the wave period and
spectral peakedness ratios are less than one. For higher relative
freeboards, high frequency (short period) waves are filtered out and the
ratio is greater than one.
Based on observations from these data and analyses, the following
recommendations for future studies are made:
1) Apply Goda's method for the resolution of reflected and incident
waves behind the structure as well in front of the structure;
2) Examine the increase in Kt as the damage tests progress in order
to determine the relationship between the average Kt and the
maximum Kt that can be expected during an extreme event;
3) Conduct experiments specifically designed to determine the rela
tionship between transmission at low relative freeeboards and
stone characteristics, structure size, and wave period; and
4) Conduct threedimensional tests so that a) (structure) end
effects can be examined and (b) nonorthogonal incident waves can
be tested.
A computer program has been written as part of the work reported
here. The program name is Low Crested Breakwater Design, LCBDGN, and its
purpose is to aid in the design of low crested breakwaters. The initial
version (Version 1.00) has certain limitations which need to be pointed
out at this point.
1. The data on which the design equations used in the program are
based are two dimensional wave tank data. There has been no
attempt to account for scale effects in going from the laboratory
to the prototype (field).
2. No attempt has been made to account for 3dimensional effects,
i.e., refraction and diffraction at the end(s) of the structure.
3. Incident reflected and transmitted waves are assumed to approach
and leave the structure at right angles to the structure (i.e.,
wave crests are assumed to be parallel to the structure).
In spite of the above qualifications, LCBDGN should be helpful to the
engineer designing a lowcrested permeable rubble mound breakwater. The
intent is to incorporate scale and other effects in later versions of the
program as field data (and its analysis) becomes available. In the mean
time, the program can be used as long as the results are used along with
"engineering judgment" and additional information regarding the factors
discussed above.
REFERENCES
Ahrens, J. P. (1983), Reef type breakwaters, Proc. of 19th Coastal
Engineering Conference, August, Houston, Texas, pp. 26482662.
Allsop, N. W. H. (1983), Lowcrest breakwaters, studies in random waves,
Proc. of Coastal Structures '83, March, Arlington, Virginia, pp. 94
107.
Bremner, W., D. N. Foster, C. A. Miller, and B. C. Wallace (1980), The
design concept of dual breakwaters and its application to Townsville,
Australia, Proc. of the 17th Coastal Engineering Conference, March 23
28, Sydney, Australia, Vol.2 pp. 18981908.
Foster, D. N. and D. Haradasa (1977), Rosslyn Bay Boat Harbour Breakwater
Model Studies, University of New South Wales Water Research
Laboratory, Technical Report No. 77/6, April, 12 pp. + figures and
tables.
Foster, D. N. and S. P. Khan (1984), Stability of overtopped rock armoured
breakwaters, University of New South Wales Water Research Laboratory,
Report No. 161, February, 38 pp. + figures.
Goda, Y. (1970), Numerical experiments on wave statistics with spectral
simulation, Report of the Port and Harbor Research Institute Ministry
of Transport, Japan, Vol. 9, No. 3, pp. 357.
Goda, Y. and Y. Suzuki (1976), Estimation of incident and reflected waves
in random wave experiments, Proc. of 15th Coastal Engineering
Conference, July, Honolulu, Hawaii, pp. 828845.
Graveson, H., 0. J. Jensen, and T. Sorenson (1980), Stability of rubble
mound breakwaters II, Danish Hydraulic Institute Technical University
of Denmark, 19 pp. (No report number).
Kitaigorodskii, S. A., V. P. Krasitskii, and M. M. Zaslavskii (1975),
Phillips' theory of equilibrium range in the spectra of windgenerated
gravity waves, Jour. of Physical Oceanography, Vol. 5, pp. 410420.
Madsen, 0. S. and S. M. White (1976), Reflection and transmission charac
teristics of porous rubblemounds breakwaters, U.S. Army Corps of
Engineers Coastal Engineering Research Center, Misc. Report No. 765,
March, 138 pp.
Phillips, 0. M. (1958), The Equilibrium range in the spectrum of wind
generated waves, Jour. of Fluid Mechanics, Vol. 4, pp. 426434.
Seelig, W. N. (1979), Effect of breakwaters on waves: laboratory tests of
wave transmission by overtopping, Proc. of Coastal Structures '79,
March, Alexandria, Virginia, pp. 941961.
Seelig, W. N. (1980), Twodimensional tests of wave transmission and
reflection characteristics of laboratory breakwaters, U.S. Army Corps
of Engineers Coastal Engineering Research Center, 187 pp.
Vincent, C. L. (1981), A method for estimating depthlimited wave energy,
U.S. Army Corps of Engineers Coastal Engineering Research Center,
Technical Aid No. 8116, November, 22 pp.
Vincent, C. L. (1982), Depthlimited significant wave height: A spectral
approach, U.S. Army Corps of Engineers Coastal Engineering Research
Center, Technical Report No. 823, 23 pp.
Appendix A
DOCUMENTATION FOR LCBDGN.FOR
"LCBDGN.FOR" is an interactive FORTRAN program the purpose
of which is to aid in the design of low crested breakwaters. Input
data can be entered 1) from the keyboard, 2) from a disk file or
3) by a user modified disk file (i.e. any number of the 18 input
quantities can be changed from within LCBDGN. An input data file
is created each time the program is run under the name LCBIN.DAT.
In order to make changes in the input data file you must
know the "sequence" number of the quantities you wish to change.
These numbers are given below:
SEQUENCE
NUMBER
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
QUANTITY
HSO
TSO
TPO
QPO
HSE
TSE
TPE
QPE
BS
WD
WIDTH
SS
WW
WR
W50
IDGNI
IDGN2
HSD
DESCRIPTION
Incident Operational Significant Wave Height (m)
Incident Operational Significant Wave Period (sec)
Incident Operational Peak Period (sec)
Incident Operational Peakedness Parameter
Incident Extreme Significant Wave Height (m)
Incident Extreme Significant Wave Period (sec)
Incident Extreme Peak Period (sec)
Incident Extreme Peakedness Parameter
Bottom Slope at Structure (del z)/(del x)
Water Depth at Structure (m)
Width of Structure Crest (Constructed) (m)
Structure Slope (forward and back) (del z)/del x)
Mass Density of Water (kg/m**3)
Mass Density of Stone (kg/m**3)
Mass of Mean Stone (kg)
Denotes Design Wave Condition
1 Operational Wave Conditions
2 Extreme Wave Conditions
Denotes Design for Transmitted or Reflected Wave
1 Transmitted
2 Reflected
Desired (Transmitted or Reflected) Significant
Wave Height (m)
CC
C PROGRAM LCBDGN (VERSION 1.00) 15 NOVEMBER 1986 C
C LOW CRESTED BREAKWATER DESIGN C
C C
C D.M. SHEPPARD ((904) 3921570) C
C C
C PURPOSE: C
C The purpose of this interactive FORTRAN program is to aid in C
C the design of Low Crested Breakwaters. The structure of the C
C program is such that new data and information regarding the C
C performance and stability of Low Crested Breakwaters may be C
C added with a minimum of effort. C
C C
C PROGRAM DESCRIPTION C
C The following assumptions are made: C
C 1. OPERATIONAL incident wave conditions are known C
C 2. EXTREME incident wave conditions are known C
C 3. The design is based on a desired transmitted or C
C reflected significant wave height for either C
C OPERATIONAL or EXTREME conditions. C
C C
C PROGRAM OUTPUT C
C 1. Constructed structure height C
C 2. Structure height after structure experiences OPERATIONAL C
C waves C
C 3. Structure height after structure experiences EXTREME C
C waves C
C 4. Transmitted and reflected wave parameters for OPERATIONAL C
C and EXTREME conditions C
C C
C C
C
DIMENSION DATA(18),ANDATA(18)
COMMON HSO,TPO,QPO,HSE,TPE,QPE,WIDTH,SS,WW,WR,W50,
1SSND,WD,WLO,WLE,D50,HCIOOT,HCIOOR,HCIEET,HCIEER
INTEGER ITEM(18)
C
OPEN(I1,FILE='LCBOUT.DAT',STATUS='UNKNOWN')
C
10 WRITE(*,20)
20 FORMAT(3X,'TYPE THE NUMBER IN FRONT OF THE DESIRED OPTION',
1/4X,'l) INPUT DATA FROM KEYBOARD '
2/4X,'2) INPUT DATA FROM DATAFILE LCBIN.DAT AS IS',
2/4X,'3) MODIFY DATA IN DATAFILE LCBIN.DAT',/)
READ(*,*)IOPT
IF(IOPT.EQ.1)GO TO 105
IF(IOPT.EQ.2)GO TO 30
IF(IOPT.EQ.3)GO TO 40
GO TO 10
C
30 OPEN(9,FILE='LCBIN.DAT',STATUS='OLD')
READ(9,*)HSO,TSO,TPO,QPO,HSE,TSE,TPE,QPE,BS,WD,WIDTH,
1SS,WW,WR,W50,IDGNI,IDGN2,HSD
GO TO 280
40 OPEN(9,FILE='LCBIN.DAT',STATUS='OLD')
WRITE(*,50)
50 FORMAT(3X,'REFER TO THE "INPUT DATA LIST" IN THE DOCUMENTATION',
1/1X,'FOR THIS PROGRAM !',//,3X,
2'HOW MANY QUANTITIES WOULD YOU LIKE TO CHANGE ?',/)
READ(*,*)INCNG
DO 70 I=I,INCNG
WRITE(*,60)I
60 FORMAT(3X,'TYPE THE NUMBER IN FRONT OF ITEM',12,IX,
1'TO BE CHANGED',/IX,'FOLLOWED BY THE NEW VALUE OF THE ITEM (SEPARA
2TED BY A COMMA)',/)
READ(*,*)ITEM(I),ANDATA(1)
70 CONTINUE
READ(9,*)(DATA(J),J=1,18)
CLOSE(9,STATUS='DELETE')
DO 90 K=1,INCNG
JK= ITEM(K)
DATA(JK)= ANDATA(K)
90 CONTINUE
OPEN(9,FILE='LCBIN.DAT',STATUS='NEW')
DO 100 L=1,18
WRITE(9,*)DATA(L)
100 CONTINUE
CLOSE (9)
GO TO 30
C
105 WRITE(*,110)
110 FORMAT(3X,'TYPE THE FOLLOWING "OPERATIONAL" INCIDENT WAVE',/IX,
1'INFORMATION SEPARATED BY COMMAS',/4X,
2'1. SIGNIFICANT WAVE HEIGHT (in meters)',/4X,
3'2. SIGNIFICANT WAVE PERIOD (in seconds)',/4X,
4'3. PEAK PERIOD (in seconds)',/4x,
5'4. Qp SPECTRAL PEAKEDNESS (Qp = 2.0 for PIERSONMOSKOWITZ)',
6//)
READ(*,*)HSO,TSO,TPO,QPO
WRITE(*,120)
120 FORMAT(3X,'TYPE THE FOLLOWING "EXTREME" INCIDENT WAVE',/IX,
INFORMATION SEPARATED BY COMMAS',/4X,
2'1. SIGNIFICANT WAVE HEIGHT (in meters)',/4X,
3'2. SIGNIFICANT WAVE PERIOD (in seconds)',/4X,
4'3. PEAK PERIOD (in seconds)',/4x,
5'4. Qp SPECTRAL PEAKEDNESS (Qp = 2.0 for PIERSONMOSKOWITZ)',
6//)
READ(*,*)HSE,TSE,TPE,QPE
WRITE(*,130)
130 FORMAT(3X,'TYPE BOTTOM SLOPE AT LOCATION OF STRUCTURE.',/IX,
1'( (DELTA Z)/(DELTA X) )'/)
READ(*,*)BS
IF(BS.GT.0.07)GO TO 135
WRITE(*,133)
133 FORMAT(/3X,'CAUTION! DATA USED IN THIS VERSION OF THIS PROGRAM',
1/1X,'IS BASED ON A BOTTOM SLOPE OF 1(VERT) ON 15(HOR) AT',/IX,
2'THE STRUCTURE')
135 WRITE(*,140)
140 FORMAT(3X,'TYPE WATER DEPTH (in meters) AT THE STRUCTURE SITE',
1/)
READ(*,*)WD
WRITE(*,150)
150 FORMAT(3X,'TYPE WIDTH OF STRUCTURE CREST (in meters)'/)
READ(*,*)WIDTH
WRITE(*,160)
160 FORMAT(3X,'TYPE STRUCTURE SLOPE (DELTA Z)/(DELTA X)',/)
READ(*,*)SS
IF(SS.LT.0.7.AND.SS.GT.O.6)GO TO 165
WRITE(*,163)
163 FORMAT(/3X,'CAUTION! DATA USED IN THIS VERSION OF THIS PROGRAM',
1/1X,'IS BASED ON A STRUCTURE SLOPE OF 1(VERT) ON 1.5(HOR)')
_ __~~ _____ ___ ___
165 WRITE(*,170)
170 FORMAT(/3X,'TYPE WATER DENSITY (in kg/m**3)',/)
READ(*,*)WW
WRITE(*,180)
180 FORMAT(/3X,'TYPE STONE DENSITY (in kg/m**3)',/)
READ(*,*)WR
WRITE(*,190)
190 FORMAT(/3X,'TYPE MEDIAN STONE WEIGHT, W50,(in kg)',/)
READ(*,*)W50
200 WRITE(*,210)
210 FORMAT(3X,'TYPE THE NUMBER IN FRONT OF THE CONDITIONS',
1/3X,'TO BE USED FOR DESIGN ',//4X,
2'1. OPERATIONAL CONDITIONS',/4X,
3'2. EXTREME CONDITIONS',/)
READ(*,*)IDGN1
IF(IDGNI.EQ.1.OR.IDGNI.EQ.2.)GO TO 220
GO TO 200
220 WRITE(*,230)
230 FORMAT(3X,'TYPE THE NUMBER IN FRONT OF THE CONDITIONS',
1/3X,'TO BE USED FOR DESIGN ',//4X,
2'1. TRANSMITTED SIGNIFICANT WAVE HEIGHT, HST,',/4X,
3'2. REFLECTED SIGNIFICANT WAVE HEIGHT, HSR,',/)
READ(*,*)IDGN2
IF(IDGN2.EQ.1.OR.IDGN2.EQ.2.)GO TO 240
GO TO 220
240 GO TO(250,260),IDGN2
250 WRITE(*,255)
255 FORMAT(/3X,'TYPE DESIRED VALUE OF TRANSMITTED',/3X,
1'SIGNIFICANT WAVE HEIGHT (in meters)',/)
READ(*,*)HSD
GO TO 270
260 WRITE(*,265)
265 FORMAT(/3X,'TYPE DESIRED VALUE OF REFLECTED',/3X,
I'SIGNIFICANT WAVE HEIGHT (in meters)',/)
READ(*,*)HSD
270 CONTINUE
C
OPEN(9,FILE='LCBIN.DAT',STATUS='UNKNOWN')
C
C WRITE INPUT DATA TO FILE
C
WRITE(9,*)HSTQTSO,TPO,QPO,HSE,TSE,TPE,QPE,BS,WD,WIDTH,
1SS,WW,WR,W50,IDGNI,IDGN2,HSD
C END INPUT
C C
C START COMPUTATION
C C
C
C COMPUTE DESIRED TRANSMISSION OR REFLECTION COEFFICIENT
C
280 IF(IDGN1.EQ.1)THEN
AK= HSD/HSO
ELSE
AK= HSD/HSE
C IF(AK.GE.0.75)THEN
C WRITE(*,281)AK
C 281 FORMAT(IX,'THE VALUE OF Kt (Kt = ',F8.3,') IS TOO LARGE FOR',
C 1 /IX,'THE RANGE OF VALIDITY OF THIS PROGRAM')
C GO TO 500
C ENDIF
___
ENDIF
C
C COMPUTE OPERATIONAL AND EXTREME WAVE LENGTHS, WLO AND WLE
C
CALL WAVLEN(WD,TPO,WLO)
CALL WAVLEN(WD,TPE,WLE)
050= (W50/WR)**.333333
SSND= D50*((WR/WW)1.)
SSNO= (((HSO**2)*WLO)**.3333333)/SSND
SSNE= (((HSE**2)*WLE)**.333333)/SSND
C
C COMPUTE STRUCTURE HEIGHT
C
CC
C IDGN1 = 1 FOR OPERATIONAL DESIGN
C = 2 FOR EXTREME DESIGN
C IDGN2 = 1 DESIGN BASED ON TRANSMITTED HS
C = 2 DESIGN BASED ON REFLECTED HS
C
C ICOND = 1 FOR OPERATIONAL CONDITIONS
C = 2 FOR EXTREME CONDITIONS
C
C IPREV = 1 FOR OPERATIONAL CONDITIONS NOT YET EXCEEDED
C = 2 FOR EXTREME CONDITIONS REACHED
CC
C
C COMPUTE STRUCTURE INITIAL AND FINAL HEIGHTS AND
C AREA OF DAMAGE FOR FOLLOWING SITUATIONS:
C
C TRANSMITTED
C
IFLAG= 0
IF(IDGN2.EQ.2)GO TO 310
C
IF(IDGNI.EQ.2)GO TO 290
C
C 1. DESIGN OPERATIONAL CONDITIONS
C TRANSMITTED HS
C CONDITIONS OPERATIONAL
C
IDGN1= 1
IDGN2= 1
ICOND= 1
CALL STRSTA(IDGN1,IDGN2,ICOND,AK,HSO,WLO,HI,HF,AD,RI,RF,IFLAG)
IF(IFLAG.GT.O)GO TO 500
HCIOOT= HI
HCFOOT= HF
DAMOOT= AD
RIOOT= RI
RFOOT= RF
C
C 2. DESIGN OPERATIONAL CONDITIONS
C TRANSMITTED HS
C CONDITIONS EXTREME
C
IDGNI= 1
IDGN2= 1
ICOND= 2
CALL STRSTA(IDGNI,IDGN2,ICOND,AK,HSE,WLE,HI,HF,AD,RI,RF,IFLAG)
IF(IFLAG.GT.O)GO TO 500
HCIOET= HI
HCFOET= HF
DAMOET= AD
RIOET= RI
RFOET= RF
GO TO 340
C
C 3. DESIGN EXTREME CONDITIONS
C TRANSMITTED HS
C CONDITIONS EXTREME
C
290 IDGNI= 2
IDGN2= 1
ICOND= 2
CALL STRSTA(IDGNI,IDGN2,ICOND,AK,HSE,WLE,HI,HF,AD,RI,RF,IFLAG)
IF(IFLAG.GT.0)GO TO 500
HCIEET= HI
HCFEET= HF
DAMEET= AD
RIEET= RI
RFEET= RF
C
C
C 4. DESIGN EXTREME CONDITIONS
C TRANSMITTED HS
C CONDITIONS OPERATIONAL
C
300 IDGNI= 2
IDGN2= 1
ICOND= 1
CALL STRSTA(IDGN1,IDGN2,ICOND,AK,HSO,WLO,HI,HF,AD,RI,RF,IFLAG)
IF(IFLAG.GT.O)GO TO 500
HCIEOT= HI
HCFEOT= HF
DAMEOT= AD
RIEOT= RI
RFEOT= RF
GO TO 340
C
C
C REFLECTED
C
C 1. DESIGN OPERATIONAL CONDITIONS
C REFLECTED HS
C CONDITIONS OPERATIONAL
C
310 IF(IDGNl.EQ.2)GO TO 320
IDGNI= 1
IDGN2= 2
ICOND= 1
CALL STRSTA(IDGN1,IDGN2,ICOND,AK,HSO,WLO,HI,HF,AD,RI,RF,IFLAG)
IF(IFLAG.GT.O)GO TO 500
HCIOOR= HI
HCFOOR= HF
DAMOOR= AD
RIOOR= RI
RFOOR= RF
C
C 2. DESIGN OPERATIONAL CONDITIONS
C REFLECTED HS
__ ___
C CONDITIONS EXTREME
C
IDGNI= 1
IDGN2= 2
ICOND= 2
CALL STRSTA(IDGNI,IDGN2,ICOND,AK,HSE,WLE,HI,HF,AD,RI,RF,IFLAG)
IF(IFLAG.GT.O)GO TO 500
HCIOER= HI
HCFOER= HF
DAMOER= AD
RIOER= RI
RFOER= RF
GO TO 340
C
C 3. DESIGN EXTREME CONDITIONS
C REFLECTED HS
C CONDITIONS EXTREME
C
320 IDGN1= 2
IDGN2= 2
ICOND= 2
CALL STRSTA(IDGN1,IDGN2,ICOND,AK,HSE,WLE,HI,HF,AD,RI,RF,IFLAG)
IF(IFLAG.GT.O)GO TO 500
HCIEER= HI
HCFEER= HF
DAMEER= AD
RIEER= RI
RFEER= RF
C
C 4. DESIGN EXTREME CONDITIONS
C REFLECTED HS
C CONDITIONS OPERATIONAL
C
330 IDGN1= 2
IDGN2= 2
ICOND= 1
CALL STRSTA(IDGNI,IDGN2,ICOND,AK,HSO,WLO,HI,HF,AD,RI,RF,IFLAG)
IF(IFLAG.GT.O)GO TO 500
HCIEOR= HI
HCFEOR= HF
DAMEOR= AD
RIEOR= RI
RFEOR= RF
C
CC
C COMPUTE THE PERFORMANCE OF THE BREAKWATER FOR
C THE FOLLOWING SITUATIONS:
C
340 IF(IDGN1.EQ.2)GO TO 350
C
C
C TRANSMITTED
C
C 1. DESIGN OPERATIONAL
C CONDITIONS OPERATIONAL
C PREVIOUS CONDITIONS OPERATIONAL NOT EXCEEDED
C
HSI= HSO
TSI= TSO
QPI= QPO
IF(IDGN2.EQ.1)THEN
HCP= HCFOOT
HCON= HCIOOT
ELSE
HCP= HCFOOR
HCON= HCIOOR
ENDIF
AT= HCON*((HCON/SS)+WIDTH)
ALP= WLO
CALL STRPER(HSI,TSI,QPI,HCP,AT,ALP,HST,TST,QPT,HSR)
IF(IDGN2.EQ.1)THEN
HTTOO=1 HST
TTTOOI= TST
QTTOO1= QPT
HTROO1= HSR
ELSE
HRTOO1= HST
TRTOO1= TST
QRTOO1= QPT
HRROO1= HSR
ENDIF
C
C 2. DESIGN OPERATIONAL
C CONDITIONS OPERATIONAL
C PREVIOUS CONDITIONS EXTREME CONDITIONS REACHED
C
HSI= HSO
TSI= TSO
QPI= QPO
IF(IDGN2.EQ.I)THEN
HCP= HCFOET
HCON= HCIOET
ELSE
HCP= HCFOER
HCON= HCIOER
ENDIF
AT= HCON*((HCON/SS)+WIDTH)
ALP= WLO
CALL STRPER(HSI,TSI,QPI,HCP,AT,ALP,HST,TST,QPT,HSR)
IF(IDGN2.EQ.1)THEN
HTTOO2= HST
TTTOO2= TST
QTTOO2= QPT
HTROO2= HSR
ELSE
HRTOO2= HST
TRTOO2= TST
QRTOO2= QPT
HRRO002= HSR
ENDIF
C
C 3. DESIGN OPERATIONAL
C CONDITIONS EXTREME
C PREVIOUS CONDITIONS EXTREME CONDITIONS REACHED
C
HSI= HSE
TSI= TSE
QPI= QPE
IF(IDGN2.EQ. )THEN
HCP= HCFOET
______ ____ __ __ ______ __ ___~___
HCON= HCIOET
ELSE
HCP= HCFOER
HCON= HCIOER
ENDIF
AT= HCON*((HCON/SS)+WIDTH)
ALP= WLE
CALL STRPER(HSI,TSI,QPI,HCP,AT,ALP,HST,TST,QPT,HSR)
IF(IDGN2.EQ.I)THEN
HTTOE= HST
TTTOE= TST
QTTOE= QPT
HTROE= HSR
ELSE
HRTOE= HST
TRTOE= TST
QRTOE= QPT
HRROE= HSR
ENDIF
GO TO 360
C
C 1. DESIGN EXTREME
C CONDITIONS OPERATIONAL
C PREVIOUS CONDITIONS OPERATIONAL NOT EXCEEDED
C
350 HSI= HSO
TSI= TSO
QPI= QPO
IF(IDGN2.EQ.1)THEN
HCP= HCIEOT
HCON= HCIEET
ELSE
HCP= HCIEOR
HCON= HCIEER
ENDIF
AT= HCON*((HCON/SS)+WIDTH)
ALP= WLO
CALL STRPER(HSI,TSI,QPI,HCP,AT,ALP,HST,TST,QPT,HSR)
IF(IDGN2.EQ.1)THEN
HTTEOI= HST
TTTEO1= TST
QTTEO1= QPT
HTREOI= HSR
ELSE
HRTEO1= HST
TRTEO1= TST
QRTEOI= QPT
HRREO1= HSR
ENDIF
C
C 2. DESIGN EXTREME
C CONDITIONS EXTREME
C PREVIOUS CONDITIONS EXTREME CONDITIONS REACHED
C
HSI= HSE
TSI= TSE
QPI= QPE
IF(IDGN2.EQ.1)THEN
HCP= HCFEET
HCON= HCIEET
ELSE
HCP= HCFEER
HCON= HCIEER
ENDIF
AT= HCON*((HCON/SS)+WIDTH)
ALP= WLE
CALL STRPER(HSI,TSI,QPI,HCP,AT,ALP,HST,TST,QPT,HSR)
IF(IDGN2.EQ.1)THEN
HTTEE= HST
TTTEE= TST
QTTEE= QPT
HTREE= HSR
ELSE
HRTEE= HST
TRTEE= TST
QRTEE= QPT
HRREE= HSR
ENDIF
C
C
C 3. DESIGN EXTREME
C CONDITIONS OPERATIONAL
C PREVIOUS CONDITIONS EXTREME CONDITIONS REACHED
C
HSI= HSO
TSI= TSO
QPI= QPO
IF(IDGN2.EQ. )THEN
HCP= HCFEET
HCON= HCIEET
ELSE
HCP= HCFEER
HCON= HCIEER
ENDIF
AT= HCON*((HCON/SS)+WIDTH)
ALP= WLO
CALL STRPER(HSI,TSI,QPI,HCP,AT,ALP,HST,TST,QPT,HSR)
IF(IDGN2.EQ.1)THEN
HTTEO2= HST
TTTEO2= TST
QTTEO2= QPT
HTRE02= HSR
ELSE
HRTEO2= HST
TRTEO2= TST
QRTEO2= QPT
HRREO2= HSR
ENDIF
C
CC
C OUTPUT
C
C INPUT DATA
C
360 WRITE(11,370)
370 FORMAT(IX,'
1 ',//10X,'DATA INPUT TO PROGRAM',/1X,'
2/)
WRITE(11,380)HSO,TSO,TPO,QPO,HSE,TSE,TPE,QPE,BS,WD,WIDTH,
1SS,WW,WR,W50
__ ___
380 FORMAT(4X,'OPERATIONAL INCIDENT WAVE CONDITIONS',/IX,
I'SIGNIFICANT WAVE HEIGHT (meters)  ',F7.2,/1X,
2'SIGNIFICANT WAVE PERIOD (seconds)  ',F6.2,/1X,
3'PEAK PERIOD (seconds)  ',F6.2,/1X,
4'SPECTRICAL PEAKEDNESS PARAMETER (Qp) 
5F6.2,//4X,
6'EXTREME INCIDENT WAVE CONDITIONS',/IX,
7'SIGNIFICANT WAVE HEIGHT (meters)  ',F7.2,/1X,
8'SIGNIFICANT WAVE PERIOD (seconds)  ',F6.2,/1X,
9'PEAK PERIOD (seconds)  ',F6.2,/1X,
I'SPECTRICAL PEAKEDNESS PARAMETER (Qp) 
2F6.2,//4X,'GENERAL PARAMETERS',/IX,
3'BOTTOM SLOPE AT STRUCTURE 
4F6.4,/1X,
5'WATER DEPTH AT STRUCTURE (meters)  ',F7.1,/1X,
6'INITIAL WIDTH OF STRUCTURE CREST (meters)  ',F6.1,/1X,
7'STRUCTURE SLOPE 
8F6.3,/1X,
9'MASS DENSITY OF WATER (kg/m**3)  ',F7.1,/1X,
1'MASS DENSITY OF STONE (kg/m**3)  ',F8.1,/1X,
2'MEDIAN STONE MASS (kg)  ',F8.1,/1X,
3' ",//)
C
C OUTPUT DATA
C
WRITE(11,390)
390 FORMAT(4X,'THE COMPUTATIONS MADE IN THIS PROGRAM ARE BASED',/2X,
1'PRIMARILY ON THE RESULTS OF LABORATORY TESTS WITH',/2X,
2'TRAPAZOIDAL STRUCTURES WITH FORWARD AND BACK SLOPES',/2X,
3'OF 1.5 (HORIZONTAL) ON 1 (VERTICAL). IN THIS VERSION ',/2X,
4'OF THE PROGRAM NO ATTEMPT HAS BEEN MADE TO ACCOUNT FOR',/2X,
5'SCALE EFFECTS.',//)
WRITE(11,400)
400 FORMAT(1X,'
1____ ',/1X,'
2 ',//2X,'STRUCTURE DESIGN BASED ON:')
IF(IDGNI.EQ.1)THEN
WRITE(11,410)
410 FORMAT(6X,'1) OPERATIONAL WAVE CONDITIONS')
ELSE
WRITE(11,420)
420 FORMAT(6X,'1) EXTREME WAVE CONDITIONS')
ENDIF
IF(IDGN2.EQ.1)THEN
WRITE(11,430)HSD
430 FORMAT(6X,'2) DESIRED TRANSMITTED SIGNIFICANT WAVE HEIGHT',/IOX,
1 'HST (meters) = ',F6.2)
ELSE
WRITE(11,440)HSD
440 FORMAT(6X,'2) DESIRED REFLECTED SIGNIFICANT WAVE HEIGHT',/1OX,
1 'HSR (meters) = ',F6.2)
ENDIF
C
450 IF(IDGN1.EQ.1.AND.IDGN2.EQ.1)THEN
WRITE(11,460)HSO,TSO,TPO,WLO,QPO,HCIOOT,RIOOT,HCFOOT,RFOOT,AT,
1SSNO,DAMOOT,HTTOO1,TSO,TTTOO1,TPO,QPO,QTTOO1,HTROOI,HCIOET,
2HCFOET,AT,DAMOET,HTT02,TSO,TTTOO2,TPO,QPO,QTTOO2,HTROO2,HSE,
3TSE,TPE,WLE,QPE,HCIOET,HCFOET,RFOET,AT,SSNE,DAMOET,HTTOE,TSE,
4TTTOE,TPE,QPE,QTTOE,HTROE
GO TO 500
_ __
ENDIF
IF(IDGNI.EQ.1.AND.IDGN2.EQ.2)THEN
WRITE(11,460)HS,TSTSO,TPO,WLO,QPO,HCIOOR,RIOOR,HCFOOR,RFOOR,AT,
1SSNO,DAMOOR,HRTOO1,TSO,TRTOO1,TPO,QPO,QRTOO1,HRROO1,HCIOER,
2HCFOER,AT,DAMOER,HRTOO2,TSO,TRT02,TPO,QPO,QRTOO2,HRROO2,HSE,
3TSE,TPE,WLE,QPE,HCIOER,HCFOER,RFOER,AT,SSNE,DAMOER,HRTOE,TSE,
4TRTOE,TPE,QPE,QRTOE,HRROE
GO TO 500
ENDIF
IF(IDGN1.EQ.2.AND.IDGN2.EQ.1)THEN
WRITE(11,460)HSO,TSO,TPO,WLO,QPO,HCIEET,RIEET,HCFEOT,RFEOT,AT,
1SSNO,DAMEOT,HTTEO1,TSO,TTTEO1,TPO,QPO,QTTEO1,HTREO1,HCIEET,
2HCFEET,AT,DAMEET,HTTEO2,TSO,TTTEO2,TPO,QPO,QTTEO2,HTREO2,HSE,
3TSE,TPE,WLE,QPE,HCIEET,HCFEET,RFEET,AT,SSNE,DAMEET,HTTEE,TSE,
4TTTEE,TPE,QPE,QTTEE,HTREE
GO TO 500
ENDIF
IF(IDGN1.EQ.2.AND. IDGN2.EQ.2)THEN
WRITE(11,460)HSOTSO,TSO,TPO,WLO,QPO,HCIEER,RIEER,HCFEOR,RFEOR,AT,
1SSNO,DAMEOR,HRTEO1,TSO,TRTEO1,TPO,QPO,QRTEO1,HRREOI,HCIEER,
2HCFEER,AT,DAMEER,HRTEO2,TSO,TRTEO2,TPO,QPO,QRTEO2,HRREO2,HSE,
3TSE,TPE,WLE,QPE,HCIEER,HCFEER,RFEER,AT,SSNE,DAMEER,HRTEE,TSE,
4TRTEE,TPE,QPE,QRTEE,HRREE
ENDIF
C
C OUTPUT FORMAT
C
460 FORMAT(/1OX,'FOR OPERATIONAL INCIDENT WAVES ',/2X,
I'SIGNIFICANT WAVE HEIGHT (meters)  ',F6.2,/2X,
2'SIGNIFICANT WAVE PERIOD (seconds)  ',F5.1,/2X,
3'PEAK PERIOD (seconds)  ',F5.1,/2X,
4'WAVE LENGTH (peak period) (meters) ',F6.1,/2X,
5'PEAKEDNESS PARAMETER, Qp,
6F5.1,///2X,
7'THE FOLLOWING STRUCTURE AND WAVE PARAMETERS ARE FOR',/2X,
8'A STRUCTURE THAT HAS ONLY EXPERIENCED WAVE CONDITIONS',/,2X,
9'NO GREATER THAN OPERATIONAL',//2X,
I'CONSTRUCTED STRUCTURE CREST HEIGHT (meters)  ',F5.1,/2X,
2'INITIAL DIMENSIONLESS FREEBOARD, R, ',F6.2,/2X,
3'STRUCTUE CREST HEIGHT AFTER OPERATIONAL WAVES(meters) ',F5.1,/2X,
4'DIMENSIONLESS FREEBOARD AFTER OPERATIONAL WAVES ,R, ',F6.2,/2X,
5'TOTAL STRUCTURE CROSSSECTIONAL AREA (meters**2)  ',F7.1,/2X,
6'OPERATIONAL SPECTRICAL STABILITY NO. ,Ns*, ',F6.2,/2X,
7'AREA OF DAMAGE (meters**2)  ',F7.1,//10X,
8'TRANSMITTED WAVES',/2X,
9'SIGNIFICANT WAVE HEIGHT (meters)  ',F6.2,/2X,
I'SIGNIFICANT WAVE PERIOD (seconds)  BETWEEN',F5.1,' AND',
2F5.1,/2X,
3'PEAK PERIOD (seconds)  ',F5.1,/2X,
4'PEAKEDNESS PARAMETER, Qp,  BETWEEN',F5.1,' AND',
5F5.1,//10X,
6'REFLECTED WAVES',/2X,
7'SIGNIFICANT WAVE HEIGHT (meters)  ',F6.2,/1X,
8' 
9///2X,
1'THE FOLLOWING STRUCTURE AND WAVE PARAMETERS ARE FAR',/2X,
2'A STRUCTURE THAT HAS EXPERIENCED EXTREME WAVE CONDITIONS',//2X,
3'CONSTRUCTED STRUCTURE CREST HEIGHT (meters)  ',F5.1,/2X,
4'STRUCTUE CREST HEIGHT AFTER EXTREME WAVES(meters)  ',F5.1,/2X,
5'TOTAL STRUCTURE CROSSSECTIONAL AREA (meters**2)  ',F7.1,/2X,
__ __ ___ ___ __ ___
6'AREA OF DAMAGE (meters**2)  ',F7.1,//10X,
7'TRANSMITTED WAVES',/2X,
8'SIGNIFICANT WAVE HEIGHT (meters)  ',F6.2,/2X,
9'SIGNIFICANT WAVE PERIOD (seconds)  BETWEEN',F5.1,' AND',
1F5.1,/2X,
2'PEAK PERIOD (seconds)  ',F5.1,/2X,
3'PEAKEDNESS PARAMETER, Qp,  BETWEEN',F5.1,' AND',
4F5.1,//1OX,
5'REFLECTED WAVES',/2X,
6'SIGNIFICANT WAVE HEIGHT (meters)  ',F6.2,//1X,
7' ,/ X
8,' '
9///,10X,
I'FOR EXTREME INCIDENT WAVES ',/2X,
2'SIGNIFICANT WAVE HEIGHT (meters)  ',F6.2,/2X,
3'SIGNIFICANT WAVE PERIOD (seconds)  ',F5.1,/2X,
4'PEAK PERIOD (seconds)  ',F5.1,/2X,
5'WAVE LENGTH (peak period) (meters) ',F6.1,/2X,
6'PEAKEDNESS PARAMETER, Qp, 
7F5.1,///2X,
8'CONSTRUCTED STRUCTURE CREST HEIGHT (meters)  ',F5.1,/2X,
9'STRUCTUE CREST HEIGHT AFTER EXTREME WAVES(meters)  ',F5.1,/2X,
1'DIMENSIONLESS FREEBOARD AFTER EXTREME WAVES,R, ',F6.2,/2X,
2'TOTAL STRUCTURE CROSSSECTIONAL AREA (meters**2)  ',F7.1,/2X,
3'EXTREME SPECTRICAL STABILITY NO. ,Ns*, ',F6.2,/2X,
4'AREA OF DAMAGE (meters**2)  ',F7.1,//1OX,
5'TRANSMITTED WAVES',/2X,
6'SIGNIFICANT WAVE HEIGHT (meters)  ',F6.2,/2X,
7'SIGNIFICANT WAVE PERIOD (seconds)  BETWEEN',F5.1,' AND',
8F5.1,/2X,
9'PEAK PERIOD (seconds)  ',F5.1,/2X,
1'PEAKEDNESS PARAMETER, Qp,  BETWEEN',F5.1,' AND',
2F5.1,//O1X,
3'REFLECTED WAVES',/2X,
4'SIGNIFICANT WAVE HEIGHT (meters)  ',F6.2,//1X,
5'///)
500 STOP
END
C END MAIN *
C**********************************************************************C

C
SUBROUTINE WAVLEN(WD,T,L)
C
C
C SUBROUTION WAVLEN C
C WAVE LENGTH C
C C
C PURPOSE: C
C The purpose of this subroutine is to compute the wave length, C
C L, (linear theory) of a wave in water depth, WD, and with a C
C period, T. C
C C
CC
C
REAL L,LN
PI=3.14159
ACC= .001
G= 9.800
10 RTEST= WD/(G*T**2)
IF(RTEST.LT..0025)THEN
L= T*SQRT(G*WD)
GO TO 20
ENDIF
IF(RTEST.GT..08)THEN
L= (G*T**2)/(2.*PI)
GO TO 20
ENDIF
L= (T*SQRT(G*WD)+ (G*T**2)/(2.*PI))/2.
20 A= G*T**2
B= A/(2.*PI)
C= A*WD
E= 2.*PI*WD
DEL= 1.0
30 IF(DEL.GE.ACC)THEN
CL= C/L**2
EL= E/L
SECH2= (1.0/(COSH(EL)))**2
LN= L((LB*TANH(EL))/(I.+CL*SECH2))
DEL= ABS((LNL)/L)
L= LN
GO TO 30
ENDIF
RETURN
END
C END SUBROUTINE WAVLEN *
C***********************************************************************
