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AN ABSTRACT OF THE THESIS OF
Claudio D. Fassardi for the degree of Master of Ocean
Engineering in Civil Engineering presented on May 14, 1993.
Title: Analysis of Wave Groups Effects on Rubble Mound
Breakwater Stability.
Abstract approved: Robert Turner Hudspeth
A series of large scale experiments were conducted at
the O.H. HinsdaleWave Research Laboratory (OHHWRL) located
on the campus of Oregon State University to evaluate the
influence of wave groups on the damage of rubble mound
breakwaters. The main goal of these experiments was to
resolve the controversy regarding the dependency of damage on
the characteristics of wave groups.
Realizations approximately 30 minutes long were
synthesized from truncated GodaJONSWAP spectra. Wave groups
were characterized by two parameters: 1) the peak enhancement
factor, 'y, of the GodaJONSWAP spectrum which correlates with
the length of runs; and 2) the envelope exceedance
coefficient, a, which correlates with the wave height
variability. Realizations were simulated with pairs of the
parameters {y, a} to test the influence of both short (y=l)
and long (y=lO) length of runs and high (a > 1) and low (a <
1) wave height variability. The envelope exceedance
Redacted for Privacy
coefficient a is computed from the wave height function H(t),
or from the envelope A(t), of the sea surface elevation (t).
The experiments were conducted on a rough quarrystone
breakwater divided in half longitudinally in order to test
simultaneously two armor layers with different armor rock
weights. The armor layers were tested with sequences of
realizations with increasing significant wave heights.
The experimental results indicated that damage on the
armor layers was not influenced by the length of runs
(spectral shape). However, it was observed that the wave
height variability affects the level of damage on the armor
layers.
The envelope exceedance coefficient a was correlated
with the groupiness factor GF = a[H2(t)]/8m0. Random wave
trains in wave channels may have exactly the same spectral
shape, but produce different levels of damage to the armor
layer. The groupiness parameters a or GF may be able to
resolve the variability of the mean damage for a given design
sea state.
Analysis of Wave Groups Effects
on Rubble Mound Breakwater Stability
by
Claudio D. Fassardi
A THESIS
submitted to
Oregon State University
in partial fulfillment of
the requirements for the
degree of
Master of Ocean Engineering
Completed May 14, 1993
Commencement June 1994
APPROVED:
Professor of Civil Engineering in charge of major
Head of /Department of Civ
Dean of Grad teSHoo
Engineer
Date thesis is presented: May 14, 1993
Typed by Claudio D. Fassardi
Redacted for Privacy
Redacted for Privacy
Redacted for Privacy
ACKNOWLEDGEMENTS
The financial support provided by the following agencies
is gratefully acknowledged: Dirección General de
Investigaciôn CientIfica y Técnica, under Grant PB880353;
the Office of Naval Research under the University Research
Initiative (ONRURI) Contract No. N0001486K0687; and the
Oregon Sea Grant and the National Oceanic and Atmospheric
Administration, Office of Sea Grant, Dept. of Commerce, under
grant NA85AADSG095 (project No. R/CE21) and appropriations
made by the Oregon State Legislature.
TABLE OF CONTENTS
Chapter Page
1. INTRODUCTION 1
2. ENVELOPE AND WAVE HEIGHT FUNCTIONS ........ 3
3. ANALYSES OF WAVE GROUPS ............. 12
4. TEST FACILITY AND MODEL DESCRIPTION ....... 17
5. DESCRIPTION OF THE EXPERIMENTS .......... 25
6. MEASURED DAMAGE ................. 30
7. ANALYSIS OF RESULTS ............... 38
8. SUMMARY AND CONCLUSIONS ............. 46
BIBLIOGRAPHY ..................... 48
Appendix A STATIONARITY ANALYSIS ........... 51
Appendix B  STATISTICAL ANALYSIS OF DAMAGE ...... 55
LIST OF FIGURES
Figure Page
1. Families of realizations, 1(t), with the same
envelope function, A1(t), and spectral density
function, S(f) ................ 7
2. Sea surface elevation spectra (unit variance),
S(f) and S(X), and envelope spectral densities
(unit variance), (f) and r(X), in time and
space domains respectively ........... 10
3. Classification of wave height time series by
Mase and Iwagaki (1986) ............ 13
4. Wave groups characteristics and parameters . . .16
5. Schematic of OHHWRL wave channel at OSU ....18
6. Schematic of the breakwater model cross section 21
7. Weight distribution of the armor layers ....23
8. Location of the instrumentation deployed for
the experiment ................. 24
9. Target values of the wave grouping parameters
selected for the experiment .......... 28
10. Realizations shifted by a constant phase iL' with
the same envelope function, A(t) ........ 29
11. Survey grid and armor layers with rock weights
WL=l28.S N and W=99.1 N ............ 31
12. Reference, P(L), and damage, P(L), profiles
corresponding to a realization with parameters
y=lO, a=1.027, 1=0, and H=0.73 m ....... 32
13. Measured, EV(7,L), and corrected, EV(7,L),
eroded volume functions, nonclosure value,
z(7), and damage level Ds(7) corresponding
to a realization with parameters y=1O,
a=l.027, =0, and H=0.73 m .......... 35
14. Comparison between measured damage
data and SPM(1984) damage function:
a & b) small rocks; c & d) large rocks;
a & c) long length of runs (y=lO);
b & d) short length of runs (y=l) ....... 42
LIST OF FIGURES (cont.)
Figure Page
15. Comparison between measured damage
data and SPM(1984) damage function:
a & b) small rocks; c & d) large rocks;
a & c) high wave height variability (a>l);
b & d) low wave height variability (a<l) ... . 43
16. Correlation between the groupiness factor GF and
energy exceedance coefficient a for: a) short
length of runs (y=l), and b) long length of runs
(y=1O) ..................... 45
LIST OF TABLES
Table Page
1. Wave groups characteristics and parameters 15
2. Summary of breakwater model characteristics 22
3. Parameters used in the truncated GodaJONSWAP
spectrum.................. 26
4. Measured sea state parameters of the
realizations tested ............. 36
5. Damage data D'(k) corresponding to the
realizations tested ............. 37
6. Sea states notation and corresponding wave
grouping parameters ............. 40
7. Number of reversed arrangements, PA, for
two realizations at wave gage locations 1,
2,and3 .................. 53
8. FTest results for damage of replicate
realizations ................ 59
9. FTest results for damage of phase shifted
realizations ................ 61
LIST OF SYMBOLS
Aa(e) = accreted (eroded) area,
A3 = amplitude of the jth wave component,
= area of active zone of damage of the 1th armor
rock weight,
A(x,t) = envelope function,
c1 = parameter, Eq. (30)
C= wave celerity,
C= mean wave celerity,
Cg = wave group velocity,
= mean group velocity,
CF'(k,L) = correction function for the jth armor rock weight,
= nominal diameter of the ith armor rock
weight,
df = frequency interval,
dX = inverse wave length interval,
D' = transformed dimensionless damage level, Eq. (37),
D'(k) = dimensionless damage level of the ith armor rock
weight and realization k,
D= damage vector,
DF = degrees of freedom,
E= energy flux,
E(x,t) = instantaneous energy flux,
E['] = expected value of ['],
EVM(c)(k,L) = dimensionless measured (corrected)
eroded volume function of the ith armor rock
weight,
f() = (peak) frequency,
LIST OF SYMBOLS (cont.)
f= frequency of the jth wave component,
max(mm) = maximum (minimum) cutoff frequency,
= mean (group) frequency,
= FTest statistic of the ith model,
g= acceleration due to gravity,
GF = groupiness factor,
h= water depth,
h1 = variable, Eq.[40],
H= monochromatic wave height,
H= design wave height of the ith armor rock weight,
H. = threshold wave height,
H(x,t) = wave height function,
H(k) significant wave height of the kth
realization,
H10 average wave height of the tenth highest
waves,
H= wave height ratios matrix,
= imaginary unit number,
J= total number of wave components,
K= number of subrecords,
KD = stability coefficient,
1a = horizontal location at which the acreeted area
starts,
1e = horizontal location at which the eroded area
ends,
L= longitudinal position at the surveying grid,
Inn =nth spectral moment,
LIST OF SYMBOLS (cont.)
N= total number of data points in the time
series,
N= stability number for 1th armor rock weight,
P(L) = nondamage reference profile function of the 1th
armor rock weight,
P,(L) = damage profile function corresponding to the
kth realization and jth armor rock weight,
RA = number of reverse arrangements,
RSS = residual sum of squares of the jth model,
SIWEH(t) = Smoothed Instantaneous Wave Energy History
function,
SWL = still water line,
S() = onesided variance spectral density function
of (x,t) in the (') domain,
SH2() = onesided variance spectral density function
of H2(x,t) in the () domain,
t= time,
T01 = mean wave period,
U(.) = Heaviside step function of (),
= mean weight of the ith armor rock,
x= horizontal coordinate,
z(t) = analytic function,
Z= regression analysis group identifier variable,
a= envelope exceedance coefficient,
a' = Eq. (23b),
= slope angle,
F(s) = (unit variance) envelope spectral density in the
(') domain,
LIST OF SYMBOLS (cont.)
= peak enhancement factor,
= relative mass density of submerged rock,
k) = EV(k,26) nonclosure value for the kth
realization,
= variable, Eq. (23c),
= discrete frequency interval,
= sampling time interval,
= horizontal space interval,
= inverse wave length interval,
= regression line parameter,
C= regression line parameter vector,
(x,t) = sea surface elevation,
(x,t) = Hubert Transform of 1(x,t),
x,t) = phase shifted realization,
OJ = random phase angle of the jth wave component,
8(t) = instantaneous phase function,
K= permeability,
= inverse wavelength of the 1th wave
component,
max(mm) = inverse wave length corresponding to
X(g) = mean (group) inverse wave length,
= parameter, Eq. (30),
v= level of significance,
Pw(r) = mass density of the water (rock),
a[5] = standard deviation of [],
a2[5] = variance of Es],
LIST OF SYMBOLS (cont.)
= parameter, Eq. (46),
= phase angle.
ANALYSIS OF WAVE GROUPS EFFECTS ON
RUBBLE MOUND BREAKWATER STABILITY
1. INTRODUCTION
The natural tendency of sea waves to form groups in a
random wave field has been recognized as a phenomena of a
significant influence in a variety of coastal processes and
ocean engineering problems. Medina and Hudspeth (1990)
reviewed most of these problems and interrelated in a unified
manner the most common methods of wave groups analyses.
Although the effects of wave groups on the damage of the
armor layer of rubble mound breakwaters have been considered
by many authors, a rational method for incorporating wave
groups into the design of rubble mound breakwaters has yet to
be developed.
Current design methods for armor layers follow the
design methodology proposed by the Shore Protection Manual
(SPM, 1984) which relates the design of the armor layer to a
single representative wave height which corresponds to the
design sea state. The SPM methodology does not incorporate
the effects of wave groupiness, wave period or duration of
the design sea state.
Bruun (1981) reviewed the design of mound breakwaters
and emphasized that the damage of these structures is
sensitive to wave groups and concluded that wave grouping
characteristics should be incorporated into their design.
2
Evidence of the importance of wave groups has also been
reported by Johnson et al. (1978) and Burchart (1979). Van
der Meer (1988) proposed 2new formulae, discriminating
between plunging and surging waves, which considered the
influence of the duration and wave periods of the design sea
state. However, he concluded that wave grouping
characteristics and spectral shape have only aminor
influence on damage. Recently, Medina and McDougal (1990)
reanalyzed the experimental data from Van der Meer (1988),
and found that rubble mound breakwaters were more stable
against sea states with long wave groups, which appears
contrary to intuition.
In order to evaluate the influence of wave groups on the
damage of rubble mound breakwaters, a series of large scale
experiments were conducted at the O.H. Hinsdale Wave Research
Laboratory (OHHWRL) located on the campus of Oregon State
University. The main goal of these experiments was to resolve
the controversy regarding the dependency of damage on wave
groups characteristics.
1
2. ENVELOPE AND WAVE HEIGHT FUNCTIONS
Rye (1982) reviewed the different wave groups parameters
and methodologies and concluded that wave groups measured
from field data compared quite well with those numerically
simulated with linear algorithms. The validity of the linear
model was also obtained by Goda (1983), Elgar et al.
(1984,1985) and Battjes and Vledder (1984) in nonshallow
waters.
In his classic treatise on random noise, Rice (1954)
developed an extensive theory that may also be applied to
linear surface gravity waves (see also Bracewell (1986),
Bendat and Piersol (1986), and Dugundji (1958)). The
envelope, A(t), of the sea surface elevation, (t), appears
to be an appropriate tool for analyzing wave groups. Medina
and Hudspeth (1987) and Hudspeth and Nedina (1988) used the
envelope, A(t), of the sea surface elevation, (t), to
analyze some of the most important aspects of wave groups in
a random sea state.
Assuming that the sea surface elevation at a given
location is a stationary, ergodic and Gaussian process having
a onesided variance spectral density, S(f), a realization
may be approximated by
(t) =YAcos(27rft+O) (1)
4
where J = total number of wave components in the realization,
A), f, and = the amplitude, the frequency and random phase
angle, respectively, of the jth wave component. The random
phase angle is uniformly distributed in the interval U[O,27r].
The amplitude of each wave component, A), and the onesided
variance spectral density, S(f), are related by [Tuah and
Hudspeth (1982)]
A2 = 2S(f)Lf (2)
where zf = frequency interval. The nth spectral moment is
defined by
=fS(f) di (3)
The Hilbert Transform f(t) of (t) is defined by
[Bendat and Piersol (1986)]
J
(t) = Asin(27Tit+O) (4a)
j=1
J(4b)
= EAcos(27ri3t+Oi)
j=1
An analytic function z(t) is a complexvalued function
defined by
z(t) = (t) +i(t) (5)
where I = [T. Equation (5) can be rewritten in polar form
as
5
z(t) =A(t)exp(i[e(t)+q]) (6)
where the envelope function, A(t), is defined by
A2(t) = p2(t) +2(t) (7)
and e(t)+ = instantaneous phase function is given by
9(t) + = ARCTAN (t) (8)
The wave height function, 11(t), is defined as
H(t) = 2A(t) (9)
A variety of methods have been introduced for analyzing
wave groups from one dimensional (lD) temporal records.
However, random waves in 2D (x,t) wave channels generate an
energy flux that varies both in space and time (x,t). Wave
group analyses in 2D wave channel experiments require a
precise description of the evolution in time and space of
this variable energy flux which is proportional to the local
squared wave height and group velocity. The wave celerity, C,
and wave group velocity, Cgi have been formally defined only
for monochromatic waves. However, useful approximations of
mean wave celerity, C, and mean group velocity, C, may be
obtained for random waves. The sea surface elevation in a 2D
wave channel may be approximated by
(x,t) =Acos[27r(ftX1K) +0.] (10)
where x and t = space and time parameters, and = the
inverse of the wave length corresponding to the jth wave
component computed from the dispersion relationship
= _X.tanh(27rX1h)
27r3 (11)
where h = water depth and g = acceleration due to gravity.
The Hubert Transform of ,(x,t) is defined as
(x,t) =EAsin[27T(ftX1x) 0] (12)
The envelope, A(x,t), and wave height, H(x,t), functions
in a 2D wave channel are given by
H(x,t) = 2A(x,t) = 2(?72(t) +2(t) )h/2 (13)
Families of phaseshifted realizations, j(x,t),
associated with a single envelope function, A(x,t), that have
the same energy flux may be expressed by
J
(x,t) = EAcos[27r(ftX1) +(Oi&)] (14a)
j=1
(14b)
= ,(x,t)cos +(x,t)sjn
where i1' = constant phase shift given to each wave component.
Figure 1 shows families of realizations that have the same
envelope function and onesided variance spectral density
functions.
t
£AMVWV11jvi
SS.
Ilvy VV V Ut7t
t
'
A.(t)
t)
(t)
S
S
S
A.(t)
t)
t
S 77(f)
Figure 1. Families of realizations, t1(t), with the same envelope function,
A(t), and spectral density function, S,1(f).
8
The onesided variance spectral density, S(X), in the
space domain is related to the onesided variance spectral
density, S,(f), in the time domain according to the
dispersion relationship, Eq.(ll), and
S(X) dX = S(f) df (15)
The envelope spectral density functions (unit variance)
and r(X), in time and space, respectively, are given by
[Hudspeth and Medina (1988)]
r'(f) = _JS(y+f)S(y)dy (16a)
r(X) = Js(z+X)s(z)dz (l6b)
Note that (f) and r(X) defined by Eqs. (16a,b) are not
related as the variance spectra of the sea surface
elevation are by Eqs. (15 and 11).
From aspectrum measured in a2D wave channel
experiment, estimates for the mean wave celerity, C, and
mean group velocity, Cgi may be computed from

;Cg= (17a,b)
x
where
)'fS(r)df
;
jS7(f)df jS(X)dX
Jfr(f)df JXF(X)dX
fg L;Xg=
fr(X)dX
(l8a,b)
(l9a,b)
where max(mm)= maximum (minimum) cutoff frequencies, >max(mmY
maximum (minimum) cutoff inverse wave lengths related to
max(mm) by Eq. (11), 1' = mean frequency, X = mean inverse wave
length, f= mean group frequency, and = mean group
inverse wave length. Figure 2 illustrates the (unit variance)
onesided variance spectral density functions S(f) and S(X),
and (unit variance) envelope spectral density functions F(f)
and r(X) for a GodaJONSWAP spectrum [Goda (1985)].
as
For monochromatic waves the energy flux, E, is defined
=Pw2g4 (20)
where p = mass density of water, and H = monochromatic wave
height.
A(lfm)
0.1 i U
max
0.125 II I0.2
Ii
II
II
ii II
it I
I1 min
I"mjn I
II
II
II
f
01. 0.6
max
f(Hz)
Figure 2. Sea surface elevation spectra (unit variance), S,jf) and S,,(X), and
envelope spectral densities (unit variance), F,(f) and F,1(X), in tune
and space domains respectively. H0
11
Equation (20) can be extended for random waves in 2D to
estimate the instantaneous energy flux, E(x,t), by
E(x,t) pgH2t) (21)
The variance spectra of H2(x,t), SH2(f) and SH2(X), in
time and space domains are given by [Medina and Hudspeth
(1987)]
SH,(f) 64 m02r(f) ;SH2(X) 64 m02r(X) (22a,b)
Damage to breakwater armor layers may be influenced by
the incident energy flux that is related to H2(x,t) by
Eq. (21). Based on this assumption, it would be reasonable
then to use H(x,t) to analyze the influence of wave groups on
the damage of breakwater armor layers.
12
3. AJALYSES OF WAVE GROUPS
The influence of wave groups on the damage of the armor
layer of rubble mound breakwaters do not appear to have been
treated in a consistent manner in previous studies. The
published results of these experiments do not always give a
precise indication of the wave grouping characteristics that
significantly affected damage. Johnson et al. (1978)
indicated that wave trains with long length of runs and high
groupiness factor, GF, are more damaging. However, Burchart
(1979) suggested that sea states with short length of runs
are the most damaging. Finally, Van der Neer (1988) could not
identify any significant differences between wave trains with
high GF, long length of runs, and narrow spectra and wave
trains with low GF, short length of runs, and broad spectra.
The apparent contradiction from these published
experimental results may possibly be resolved by the wave
grouping characterization model proposed by Mase and Iwagaki
(1986). They concluded that at least two parameters are
needed to characterize wave groups: 1) one to represent the
magnitude of the sequence of high waves or length of runs;
and 2) another to represent the magnitude of the variation of
the wave heights. Figure 3 shows four different wave height
time series classified by Mase and Iwagaki (1986).
Wave
Height
Variability H
High___________________t
S(f) 1f
Low
H
_t
Short
H
S(f)
H
Long Length of
Runs
Figure 3. Classification of wave height time series by Mase and Iwagaki (1986)
14
A run of waves heights is defined as a sequence of
waves, the heights of which exceed a predetermined value:
e.g., H, H10, etc. The length of a run is the number of waves
in that run. The length of runs may be characterized by a
spectral shape parameter. The relation between spectral shape
and length of runs has been shown by Goda (1970). The peak
enhancement factor, 'y, of the GodaJONSWAP spectrum [Goda
(1985)] was selected to characterize the length of runs. In
order to characterize the wave height variability, an
empirical envelope exceedance coefficient, a, is proposed.
This coefficient is a dimensionless measure of wave height
variability of a random sea state above a threshold level,
H.. The average of the tenth highest waves, H10, has been
selected as the threshold level H.. The envelope exceedance
coefficient is computed according to
where
E [ a'] (23a)
N
a'= (H)2 U(AH) (23b)
N1
H(nLt) 1(23c)
where E[] = the expected value of [], U[] = Heaviside step
function of ['], N = total number of data points in the time
15
series, t = sampling time interval, H(nt) = discrete wave
height function at x=O. The envelope exceedance coefficient,
a, may be used as a measure of wave height variability since
it depends on H2(x,t). In addition, since H2(x,t) is related
to E(x,t), a also represents a measure of the incident
energy flux which may have an influence on the damage of
armor layers.
Table 1and Figure 4show the wave grouping
characteristics proposed by Mase and Iwagaki (1986) and their
corresponding parameters.
Wave Groups Parameter
Characteristics
Length of Runs y
Wave Height a
Variability
Table 1. Wave groups characteristics and parameters.
The wave group characteristics listed in Table 1 may be
correlated with damage to the armor layer by comparing the
damage computed from each of the four realizations shown in
Fig. 3.
16
S(f)
LENGTH OF RUNS
WAVE HEIGHT VARIABILITY
AA(t) H(t)
(t)
Figure 4. Wave groups characteristics and parameters.
17
4. TEST FACILITY AND MODEL DESCRIPTION
A schematic of the wave channel at the O.H. Hinsdale
Wave Research Laboratory (OHHWRL) located at Oregon State
University is shown in Figure 5. The wave channel is 104 m
long, 3.7 m wide, and 4.6 m deep. The hinged wavemaker moves
in either periodic or random motion and is activated by
either an electronic function generator or a digital time
series synthesized on a digital computer through digital to
analog converters (DAC). The digital generation of periodic
or random waves uses an inverse Finite Fourier Transform
algorithm (FFT) described by Hudspeth and Borgmann (1979). A
112 KW, 24 Mpa oil pump controls the wavemaker through an
hydraulic servomechanism mounted at 3.05 m above the
wavemaker hinge.
The rubble mound breakwater was constructed at the end
of the wave channel with a crest high enough to prevent
overtopping. The rough quarrystone breakwater model was
divided in half longitudinally in order to test
simultaneously two different armor layers with mean rock
weights WL=l28.S N and W=99.1 N, respectively. The two armor
layers were tested with sequences of realizations with
consecutive increasing significant wave heights. The
significant wave heights, H(k) and H(k+l), corresponding to
realizations k and k+l respectively, were determined from the
armor rock weight ratio according to
IL__
5/9m I
I___
Wave Board
'w 5.4Bm k12.2Om
1.22
Section
76.22mW
Figure 5. Schematic of OHHWRL wave channel at OSU.
Beach
H
19
w11/3
H(kl1) L(24)
H(k)
According to the SPM (1984) the damage on the armor
layer is controlled by the stability number
where
H(k)
N(k) (25a)
x= [(i) 11 (25b)
L°' J
1/3
=
Pr (25c)
and, N(k) = stability number for the 1th armor rock weight
from the kth realization, H,(k) = significant wave height of
the kth realization, Pr and p = the mass density of rocks and
water respectively, d,=(Wj/pr)"3 is the nominal diameter of the
ith armor rock weight, and = mean weight of the ith armor
rock. According to Van der Meer (1988), the following factors
also influence the damage of the armor layer: slope,
permeability of the breakwater, number of waves, and
roughness and placement of the armor rocks. In the
experiments these secondary factors were held constant.
The stability number for the kth realization and armor
rock weight W is
H(k) is given by
20
H3 (k)
N(k) 1/3 (26)
L
tPrj
I
L(27)
H3(k) = H3(k+l)
Substitution of Eq. (27) into Eq. (26) yields to
w
H3(k+l) ki H3(k+l) = N(k+l) (28)
N(k)
ill3
1WL
IrPr
Pr
Therefore, since N(k) = N(k+l), the damage to the
armor layer with rock weight WL for realization k+l, should
be equal to the damage to the armor layer with rock weight W
for realization k. In this way, it was possible to duplicate
the amount of damage data collected during the experiment
having the same stability number.
The filter layer rock weight was W=2l.6 N and the rock
specific weight was Pr=274 kN/m3. Figure 6 shows a schematic
of the breakwater model cross section. Table 2 summarizes the
breakwater model characteristics where KD =stability
coefficient [SPM (1984)], j3 = slope angle, and H = design
wave height for the 1th armor rock weight computed according
to the Hudson formula [SPM (1984)].
GEOTEXTILE SAND CORE
Figure 6. Schematic of the breakwater model cross section.
Wj(LS)
WE
22
KD cotg/ WL [N] W[N] WF [N] p,[kN/m3] Hd''[m] Hds[m]
42128.5 99.1 21.6 27.4 0.599 0.544
Table 2. Summary of breakwater model characteristics.
Figure 7 shows the weight distribution for each armor
layer.
Figure 8 shows the location of the instrumentation used.
Cross section profiles of each side, sea surface elevations,
hydrodynamic pressures, video records of runup, and visual
observations of rock movements were recorded during the
experiments. The wave records obtained from the three
ultrasonic wave gauges located 10 m in front of the structure
were used to resolve the incident wave trains on the
structure. The incident and reflected wave trains were
resolved using the extension of Kimura (1985) to the method
of Goda and Suzuki (1976).
23
30
O/ 20
10
30
Q/ 20
10
W= 99.1N
= 19.6 N
ws
SMALL ROCKS
60 0100 120 140
WL 128.5 N
28.2 N
WL
LARGE ROCKS
(N)
100 120 140 160 180 200 (N)
Figure 7. Weight distribution of the ariiior layers.
VIDEO
/// \\\/// \\'///\\\
h3.O5m
WAVE
GAGES
123
,/, II,,
II.t2Om
I
II
k10m_*
/3
Pressure Transducers
Hydrophones
Figure 8. Location of the instrumentation deployed for the experiment.
25
5. DESCRIPTION OF THE EXPERIMENTS
Two fundamental wave grouping characteristics were
considered: 1) the length of runs, characterized by the
spectral shape; and 2) the wave height variability
characterized by the envelope exceedance coefficient, a. DSA
simulated random waves [Tuah and Hudspeth (1982)] were
synthesized from GodaJONSWAP spectra [Goda (1985)]. The peak
enhancement factor y in GodaJONSWAP spectrum was used to
characterize broad (y=l) and narrow (y=lO) spectra. The mean
period, given by
m
T01 m1 (29)
was a constant T01=3 sec for all realizations. The influence
of the wave height variability was analyzed by the envelope
exceedance coefficient, a. Two different phase spectra were
selected for each spectral shape ('y=l and y=lO) to simulate
realizations with relatively high and low a's. The low and
high values of a and their corresponding phase spectra were
selected from one hundred DSA random wave simulations from
two truncated GodaJONSWAP spectra (y=l and y=lO).
The incident wave time series were resolved from the
measured partially standing wave records using the method of
Kimura (1985). This method is able to resolve incident and
reflected wave time series only in afinite frequency
26
bandwidth. Accordingly, the DSA simulations were synthesized
from truncated GodaJONSWAP spectra [Goda (1985)] defined by
f4exp[_(_1)/2L21
S(f) =c1H2f!_exp[_1.25(y) ]y (30)
for mw max, where f = peak frequency, H =significant
wave height, and c1 is a dimensionless coefficient that
maintains the energy from the untruncated spectrum. Table 3
shows the values of the parameters used in Eq. (30).
yc1 [Hz] Y__I r(r.ç) (r.ç)
11.03746 0.27048 0.7 2.5 0.07L 0.09
10 1.01332 0.30578 0.7 2.5 0.07 0.09
Table 3. Parameters used in the truncated GodaJONSWAP
spectrum
The peak frequency, f, was selected so that the mean
period computed from Eq. (29) was a constant T01=3 sec for each
of the realizations.
Sequences of 7 realizations were simulated with the 4
pairs of parameters {'y, a}, shown in Fig. 7, and wave heights
in meters determined from
ki
H(k) o.43IWL1 ;k=1,2,..,7 (31)
where H8(k) =significant wave height in meters corresponding
to the kth realization. Each realization was approximately 30
27
minutes long and contained N = = 32768 values, at t =
0.06 sec. Figure 9shows the target values of the wave
grouping parameters and the notation used to identify the sea
states with those parameters: 1) El: long length of runs
(high y) and high wave height variability (a>l); 2) E2: short
length of runs (low y) and high wave height variability
(a>l); 3) E3: long length of runs (high 'y) and low wave
height variability (a<l); 4) E4: short length of runs and low
wave height variability (a<l).
Each of the seven realizations in a sequence was shifted
by a constant phase it'=0,21r/3,47T/3, and 271, according to
Eq. (14b) to determine the influence on damage from different
realizations with the same wave grouping characteristics.
Figure 10 shows different realizations having the same
envelope function, A(t), shifted by a constant phase .
At the time the experiment was conducted the OHHWRL did
not have direct digital control on the wavemaker that would
allow cancellation of waves reflected by the breakwater. The
rereflected waves added energy to the incident wave spectra.
In order to evaluate the stationarity of the wave generation
process under these conditions, reverse arrangements tests
[Bendat and Piersol (1986)] were performed. The results of
these tests are presented in Appendix A. These tests
demonstrated that the incident wave time series were
stationary.
(i'
3
2
(1,0.51) +
12
H
+
E1/
S,1( 1)
(10,2.52)
HE3
(10 0.23)
910
Figure 9. Target values of the wave grouping parameters selected
for the experiment. t')
A(t)
Figure 10. Realizations shifted by a constant phase /' with the same envelope
function, A(t) .
30
6. MEASURED DAMAGE
Cross section profiles on each side of the breakwater
were obtained after each run by measuring from the toe to the
crest the vertical distance below a reference level to the
armor layer in a 52x2 point grid using typical surveying
instruments. These vertical measurements were later
referenced to the bottom of the wave tank. Figure 11 shows
the survey grid.
The surveying procedure for each of the 16 sequences (4
pairs {y, a} x 4 constant phase shifts) of the 7 realizations
consisted of surveying each side of the breakwater as
follows: 1) prior to the first realization to obtain non
damage reference profiles; and 2) after each realization to
obtain damage profiles. Each side of the breakwater was
rebuilt after the seventh realization.
For each side of the breakwater the profiles measured
were used to compute average nondamage reference profiles,
P(L), and damage profiles, P,(L), from realization k at
stations L=1,2,..,26 for the th armor rock weight. Figure 12
illustrates typical nondamage reference and damage profiles
for the armor layer with armor rock weight W99.l N. The
accreted area starts at location 1a and the eroded area ends
at location l.
Wood Divider
Y.vs "1.
1
Wa v em a k e r
2
31
Figure 11. Survey grid and armor layers with rock weights
WL128.S N and W99.l N.
L.60
.00
3.20
E
E0
2.0
2.4
j
1.60
0.80
0.00
246810 12 14 14 18 20 22 24 26 L
tci te
Figure 12. Reference, P(L), and damage, P.(L), profiles corresponding to
a realization with parameters _y=1O, a=l.027,
and H=O.73 ni.
33
For each damage profile, dimensionless measured eroded
volume functions EV(k,L), for run k at station L, were
computed by
T(k,L) =7(k,L1) + _P(1)] _;L=2, . . ,26 (32)
l=L1 2(d,)2
where EV(k,l)=O; and zx=O.3O5 m horizontal space interval.
The kth eroded volume function is a measure of the normalized
averaged crosssectional area between the kth measured and
reference profiles for the 1th armor rock weight.
In most cases the eroded volume functions computed from
the measured profiles did not sum to zero at the crest of the
breakwater. When the nonclosure difference was positive,
EV(k,26)= z(k) > 0, it indicated that the measured accreted
area, Aai was greater than the measured eroded area, A,. When
EV(k,26)= (k) < 0, the opposite. A correction function,
CF'(]c,L), was applied by assuming: 1) a constant porosity; 2)
that the eroded and accreted crosssectional areas should be
equal after each run; and 3) the Sshaped crosssectional
profiles could be approximated by a sinusoid. Corrected
eroded volume functions, EV(k,L), were computed from
EVk(k,L), for the kth run at station L and th armor rock
weight, according to
34
EV(k,L) =EV(k,L) () CF(k,L) .>k1 (33)
a<AeJ
where CF'(k,L) is given by
27T(L1a)1 (34)
CF'(k,L) =____z.(k) [1_cos lela ]
4
when la L(la+le)/2 ;and
271(L1a)1 (k) (35)
CF'(k,L) = ____(k) [l+cos lela j2
4
when (la+le)/2 Lle
Figure 13 shows typical measured E'V(k,L) and corrected
EVc(k,L) eroded volume functions. The maximum value of
EV(k,L) defines the dimensionless damage level, D'(k), for
realization k for the ith armor rock weight. D'(k) is the
eroded (accreted) area normalized by (d,)2. Damage levels
defined in this way may be more reliable than ones obtained
by measuring only the eroded area since the measurement of
the accreted area provides additional information about the
rock distribution the armor layer. Table 4 summarizes the
measured sea state parameters of the 16 sequences of
realizations tested. has been computed according to
H105.091(1TL0)"2. Table 5summarizes the damage data
corresponding to the sea state parameters shown in Table 4.
DS (7)
FV
2 4 68%O U14 16 18 20 22 14 26 L
Figure 13. Measured, EV(7,L), and corrected, EV(7,L), eroded volume functions,
nonclosure value, L(7), and damage level Ds(7) corresponding to a
realization with parameters y=lO, a=l.027, '=O, and H=O.73 m.
01
i/,: 0 LI' 271/3 ):Ii7t/3 i/i: 27r
nv.(i) Run(k) IJ H,, H10 a's H0 a'1'H H19 a'
10.566 1.710 0.948 1.044 0.578 1.936 0.965 1.063 0.566 1.694 0.945 1.040 0.560 2.910 0.968 1.066
20.630 2.171 1.052 1.158 0.621 2.035 1.037 1.142 0.625 2.642 1.04) 1.149 0.632 1.998 1.055 1.162
30.688 2.079 1.149 1.265 0.678 1.653 1.13? 1.246 0.669 1.07) 1.117 1.230 0.678 1.606 1.132 .246
40.735 1.609 3.227 1.351 0.735 1.968 1.227 1.351 0.133 2.125 1.224 1.347 0.736 2.172 1.229 .353
50.708 1.630 1.316 1.449 0.806 2.1)2 1.346 1.462 0.801 2.008 1.337 1.412 0.797 1.001 1.3)1 .465
60.866 1.199 1.446 1.592 0.672 1.527 1.456 1.603 0.056 1.668 1.429 1.574 0.869 2.201 .451 .597
70.926 1.027 1.546 1.102 0.926 1.278 1.546 1.102 0.902 1.305 1.506 1.656 0.916 1.412 .533 .686
10.564 1.128 0.975 1.074 0.580 2.064 0.968 1.066 0.580 1.515 0.966 1.066 0.591 1.801 u.967 .086
20.6)1 .985 3.053 1.160 0.6)5 1.642 1.060 1.167 0.6)5 1.462 1.060 1.167 0.6)9 1.662 1.067 .175
30.685 2.069 1.144 1.259 0.697 2.166 1.164 1.281 0.681 1.31) 1.147 1.263 0.689 1.161 1.150 .263
240.744 1.716 1.742 1.368 0.756 1.681 1.262 1.390 0.7)8 1.714 1.232 1.351 0.750 1.554 1.252 1.319
50.004 1.758 1.342 1.478 0.804 1.1)6 1.342 1.478 0.196 1.309 1.329 1.463 0.825 1.221 1.371 1.517
60.870 1.750 .452 .599 0.866 1.68) 1.446 1.592 0.866 1.588 1.446 1.592 0.884 1.382 1.416 1.625
10.916 1.582 1.5)3 1.688 0.918 1.429 1.533 1.608 0.918 1.506 .5)3 1.688 0.918 1.506 1.5]) 1.688
10.504 0.515 0.975 .074 0.560 0.61) 0.968 1.066 0.505 0.516 ..,.971 1.075 0.564 0.558 0.975 1.074
20.63) 0.62) 1.057 .164 0.632 0.611 1.055 1.162 0.632 0.5)4 1.055 1.162 0.6)0 0.394 1.052 1.156
30.689 0.571 1.150 .761 0.687 0.1)0 1.141 1.263 0.665 0.519 1.144 1.259 0.669 0.441 1.150 1.261
340.747 0.512 1.241 1.373 0.749 0.570 1.250 1.371 0.145 0.461 1.244 1.369 0.746 0.449 1.245 1.311
50.809 0.645 1.351 1.487 0.811 0.646 1.354 1.491 0.615 0.581 1.361 1.498 0.814 0.448 1.159 .496
60.06) 0.11) 1.441 1.586 0.876 0.54) 1.462 1.610 0.816 0.606 1.466 1.614 0.885 0.513 .477 .627
10.951 0.700 1.598 1.759 0.950 0.424 1.5136 1.746 0.943 0.595 1.574 1.733 0.941 0.401 .581 .141
0.611 0.561 1.020 1.12) 0.617 0.455 1.0)0 1.1)4 0.615 0.6)2 1.021 1.13I 0.618 0.5)0 .0)2 .1)6
20.655 0.412 1.09) 1.204 0.667 0.433 1.114 1.226 0.664 0.459 1.109 1.22I 0.611 0.472 1.120 .73)
4
30.111 0.549 1.191 1.318 0.122 0.565 1.205
1.311 1.327 0.723 0.564 1.207 1.329 0.726 0.502 1.212 .335
40.785 0.411 1.311 1.44] 0.785 0.485 1.443 0.718
0.846 0.524 1.299 1.430 0.789 0.491 1.311 1.450
1.572
5
6
0.851
0.907 0.361
0.361 1.421 1.564 0.856
0.916 0.486
0.590 1.429 1,574 0.501 1.416 1.559 0.855 0.559 1.427
1.514 1.661 1.529 1.684 0.914 0.609 1.526 1.680 0.918 0.461 1.53] 1.688
10.959 0.365 1.601 1.763 0.974 0.522 1.626 1.790 0.967 0.426 1.614 1.718 0,969 0.392 1.618 1.181
Table 4. Measured sea state parameters of the realizations tested.
l/J27r/3 lJJ:47r/3 //:27f
Env.(i) Run(k) H, D(k) ';1S DS(k) 'i D1(k) <s DS(k) F1 D1(k) H, DS(k) D(k) HDS(k)
I0.95 2.1? 1.04 5.00 0.9? 2.29 1.06 1.66 0.95 0.00 1.04 0.00 0.9? 1.18 1.07 1.5)
21.05 2.35 1.16 2.90 .04 2.01 1.14 3.54 1.04 0.86 1.15 1.06 1.06
1.13 1.56 1.16 1.54
31.15 2.22 1.21 5.32 .13 2.26 1.25 1.82 1.12 0.84 1.23 4.43 3.30 1.25 4.41
41.23 3.41 1.35 1.52 .23 3.81 1.35 10.21 1.22 3.50 1.35 9.50 1.23 1.36 1.36 8.70
51.32 6.57 1.45 12.29 .35 5.03 .48 16.24 1.34 7.28 1.47 11.94 1.33 4.56 1.47 16.69
61.45 6.16 1.59 20.08 1.46 6.68 1.60 19.25 1.43 9.48 1.57 16.63 1.45 7.56 1.60 22.95
7.55 9.31 1.71 26.74 1.55 1L96 .70 32.92 1.51 16.27 1.66 26.95 1.53 9.61 1.69 34.12
10.90 0.72 1.01 2.44 .97 1.36 1.01 2.55 0.91 1.41 1.01 1.91 0.99 2.57 .09 4.42
21.05 1.58 1.16 4.64 1.06 2.02 1.11 4.20 1.06 0.56 1.17 3.11 1.07 3.04 1.18 6.97
31.14 1.59 1.26 6.16 .16 3.08 1.28 6.04 1.15 1.18 1.26 4.43 1.15 5.03 1.27 9.86
241.24 3.81 3.31 12.20 1.26 4.33 1.39 1.64 1.23 1.81 1.36 6.98 1.25 6.01 1.38 12.44
S1.34 5.43 1.48 18.18 1.34 1.01 1.48 1.33 5.52 1.46 11.57 1.30 8.26 1.52 19.02
61.45 6.82 1.60 20.11 1.45 9.19 1.59 1.45 6.07 1.59 18.60 1.48 8.03 1.63 21.95
71.53 9.68 1.69 36.15 1.53 13.24 1.69 1.53 11.65 1.69 31.30 1.53 11.03 1.69 37.55
10.98 2.67 1.01 2.62 0.91 1.10 1.0? 1.97 0.98 1.07 1.08 2.51 0.98 1.12 1.01 0.86
21.06 2.43 1.16 1.88 1.06 0.37 1.16 2.35 1.06 0.45 1.16 3.18 1.05 0.82 1.16 1.03
3
3
4
1.15
3.25 2.80 1.27 3.62 1.15 1.62 1.26 3.02 1.14 0.86 1.26 4.72 1.15
1.25 1.42 1.27 1.94
4.15 1.37 1.34 1.25 2.02 1.38 2.86 1.24 1.99 1.31 5.92 1.66 1.38 3.06
5.35 2.76 1.49 8.67 1.35 2.57 1.49 4.05 1.36 4.41 .50 6.06 1.36 4.42 1.50 1.55
6.44 6.57 1.59
1.76 13.23
22.18
.46 5.95 1.61 5.65 1.41 5.01 .61 8.66 1.48 6.82 1.63 12.45
11.82
7.60 14.50 1.59 7.69 1.75 14.64 1.51 6.25 1.73 11.78 1.58 10.36 1.74
1.02 1.01 1.12 1.63 1.03 0.89 1.13 4.26 1.03 0.44 1.13 .06 1.03 0.00 1.14 1.60
21.09 0.00
1.50 1.20
1.32
0.56
1.13
1.11
1.21
2.44 1.23 3.05 1.11 0.59 1.22 .16 1.12 0.33 1.23 2.33
3.20 1.63 1.33 2.60 1.21 0.61 1.33 .80 1.21 0.00 1.34 3.62
441.31 3.17 1.44 3.78 1.31 2.85 1.44 6.S3 1.30 1.81 1.43 .18 1.32 2.44 1.49 6.81
51.42 2.79 1.56 8.54 1.43 4.19 1.97 6.52 1.42 2.36 1.56 8.90 1.43 3.65 1.57 11.40
61,51 4.72 1.67 14.51 1.9) 4.66 1.68 10.64 1.53 5.42 1.60 18.17 1.93 3.68 1.69 12.70
71.60 9.31 1.76 21.12 1.63 5.82 1.79 29.79 1.62 6.19 1.78 35.49 1.62 3.12 1.78 32.98
Table 5. Damage data D'(k) corresponding to the realizations tested.
38
7. ANALYSIS OF RESULTS
Measurements of damage at different levels of wave
energy were recorded and compared to the damage data in the
SPM (1984). The percent damage in the SPM (1984) is defined
as the area of armor units displaced from the breakwater
active zone. Assuming that the elevation of the breakwater
crest is equal to the design wave height, the area of the
active zone of damage for the ith armor rock weight, A,t, for
a twolayer armor is given by
=4d, H /1ctg/3 (36)
where, His given by the Hudson formula [SPM (1984)] by
=d, (Kctgf)"3 (37)
According to Eq. (36) the area of the active zone of
damage for the armor layers tested in the experiment (KD=4,
cotg/3=2, =1.74), was approximately A' =3l(d,). The damage
data given in the SPM (1984) may then be reasonably
approximated by
1Hiol
'6[HJ (38)
where D =dimensionless damage.
39
It would also be possible to compare the experimental
data with the two damage formulae proposed by Van der Meer
(1988). However, the Van der Meer (1988) formulae were not
used for comparisons because they depend critically on the
value of the permeability parameter, K, which is too
difficult to estimate for design. According to the typical
values suggested by Van der Meer (1988), the permeability of
the two armor rock sections tested had estimated values of
0.1 < K < 0.4. Consequently, estimates of damage may vary as
much as 50% about the average value of K=0.25. Since the
selection of the parameter Kis too subjective to use for
design, the Van der Meer formulae have not been included in
these comparisons.
Damage observations from realizations shifted by t'=o and
=27r were assumed to be replicates that would demonstrate the
statistical variability of the experiments. Damage results
obtained from the phase shifted realizations having the same
length of runs and wave height variability (y and a)
demonstrated that no significant difference was observed for
different realizations with the same wave groups
characteristics. Damage data corresponding to the phase
shifted realizations were therefore considered as replicates.
A description of the analysis technique used and results are
presented in Appendix B.
Figures 14 and 15 compare the experimental results for
damage with Eq. (38) for similar and different values of a, y,
40
and for armor rock weights WL and W. A least square fit for
each of the four replicates (4 values of t,1')is shown for each
envelope. The sets of observations are identified by the name
of the envelopes, with the characteristics shown in Table 6.
Envelope Spectral Shape
y
Envelope
Exceedance, a
El 10 1.8
E2 11.6
E3 10 0.23
E4 10.51
Table 6. Sea states notation and corresponding wave grouping
parameters.
Figure 14 compares damage data, for both armor rock
weights WLl28.S N and W=99.l N, from realizations with both
high and low wave height variability for long length of runs
(narrow spectrum) (a and c) ; and short length of runs (broad
spectrum) (b and d).
Figure 14 shows that for both armor layers, the damage
corresponding to envelopes El and E3 (y=10; a>1 and a<l) were
similar to the damage corresponding to envelopes E2 and E4
(y=1; a>1 and a<1), respectively.
Figure 15 compares damage data, for both armor rock
weights WL=128.S N and W=99.1 N, from realizations with both
narrow and broad spectrum for a>1 (a and c); and a<l (b and
d).
Figure 15 shows that for both armor layers, the damage
41
corresponding to envelopes El and E2 (a>l; y=lO and y=l) were
consistently higher than the damage corresponding to
envelopes E3 and E4 (a<l; y=lO and 'y=l). This Implies that
envelopes with the same envelope exceedance coefficient a
produce approximately the same damage, regardless of their
spectral shape.
The experimental results shown in Figures 14 and 15
indicate that:
1) the spectral shape, which is a measure of the length of
runs, has no effect on the damage of the armor layers,
and
2) the wave height variability affects the level of damage.
The envelope exceedance coefficient, a, may be related
to a groupiness factor, GF, defined by [Medina and Hudspeth
(1987)]
GF a[H2(t)] (39)
8m0
where a[] = standard deviation of [').
0/E1('>i)
W = 99.1 N
30 SP M
00,"
E1 c,"
15 QJ 2" 'E3 ( <I)
.a
0
o0,
H1O/H
ago too .10 120 1,30 1.0) .00 .50 tOO .00 1.50
a)
D40
35 W = 128.5 N
30
25 QOE1
20
'5
to
o
SPMEl (>
E3(<)
H10/H
0
N
SPM
030 1l0 1.15 1i Iit) I .0 t.fO 1.t) 1.30 1.0 1.0
b)
040 '= 1
35 WL = 128.5 N
J
k_* E2
*o* E4
SPM
E2 (i>)
E4(Y <I)
H10/H
C) d)
Figure 14. Comparison between measured damage data and SPM(1984) damage function:
a & b) small rocks; c & d) large rocks; a & c) long length of runs (y=lO);
b & d) short length of runs (y=l).
a)
0
D
N
SPM
30 0;
"
:..T1:...
0.00 7.00 7.70 720 7.30 1,40 20 .80 1.70 IV 1.80
>1
WL = 128.5 N
S PM
E1()':lO)
E2()
H/H,
040 Wg.1N **SPM
30
eøE3 E4(J"l)
IIIf IH
b)
040
40 WL = 128.5 N 5PM
30
20 e'i.E3
20 *** E4 /
'5. E3t)'lO)
I0 E4(J':l)

H10/H4
c) d)
Figure 15. Comparison between measured damage data and SPM(1984) damage function: a & b)
small rocks; c & d) large rocks; a & c) high wave height variablity (a>l);
b & d) low wave height variablity (a<1).
44
If the GF was computed by the SIWEH(t), as proposed by
Funke and Mansard (1979), it would be biased and would depend
on the smoothing function used. The definition for GF given
by Eq. (39) was found to be highly correlated with the
envelope exceedance coefficient, a, computed from analyses of
1000 DSA random simulations for the two spectral shapes used
in the experiments (viz. y=l and 'y=lO). Figure 16 shows that
for both values of y, alinear relationship given by
GF(9+a)/lO correlates with the simulated data.
45
CF
1.20
0.90 + 0.6989
a
0.80 IIIIIIII I IIIIIII
0.0 0.5 1.0 1.5 2.0 2.5
a)
Li)
Figure 16. Correlation between the groupiness factor GF and
energy exceedance coefficient a for: a) short
length of runs (y=1), and b) long length of runs
(y=lO)
8. SUMMARY AND CONCLUSIONS
A series of large scale experiments were conducted at
the O.H. HinsdaleWave Research Laboratory (OHHWRL) located
on the campus of Oregon State University to evaluate the
influence of wave groups on the damage of rubble mound
breakwaters. The main goal of these experiments was to
resolve the controversy regarding the dependency of damage on
the characteristics of wave groups.
The experiments were conducted on a rough quarrystone
breakwater divided in half longitudinally in order to test
simultaneously two armor layers with different armor rock
weights. The armor layers were tested with sequences of
realizations with increasing significant wave heights. The
ratio between consecutive wave heights and armor rock weights
was selected so that the stability number of the small rock
armor layer for realization k was equal to the stability
number of the large rock armor layer for realization k+l.
Under these conditions it was possible to duplicate the
damage data for a given stability number.
Realizations approximately 30 minutes long were
synthesized from truncated GodaJONSWAP spectra. Wave groups
were characterized by two parameters: 1) the peak enhancement
factor, y, of the GodaJONSWAP spectrum which correlates with
the length of runs; and 2) the envelope exceedance
coefficient, a, which correlates with the wave height
47
variability. Realizations were simulated with pairs of the
parameters {y, a} to test the influence of both short ('y=l)
and long (y=lO) length of runs and high (a > 1) and low (a <
1) wave height variability. The envelope exceedance
coefficient a is computed from the wave height function H(t),
or from the envelope A(t), of the sea surface elevation,
(t). The armor layers were tested with 4 sequences of 7
realizations with increasing wave heights. Each of the seven
realizations in a sequence was shifted by a constant phase
1'=O,27r/3,47r/3, and 27r, to determine the influence of
different realizations with the same wave grouping
characteristics on the damage of the armor layer.
The experimental results indicated that:
1) different realizations with the same significant wave
height and envelope function produced the same damage on
the armor layers.
2) damage on the armor layers was not influenced by the
length of runs (spectral shape).
3) the wave height variability affects the level of damage
on the armor layers.
The envelope exceedance coefficient a was correlated
with the groupiness factor, GF [Eq. (39)]. Random wave trains
in wave channels may have exactly the same spectral shape,
but produce different levels of damage to the armor layer.
The groupiness parameters a or GF may be able to resolve the
variability of the mean damage for a given design sea state.
48
BIBLIOGRAPHY
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Bracewell, R.N. (1986): The Fourier Transform and Its
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Bruun, P. (Ed.), (1981): Port Engineering, (3rd Ed.), Gulf
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Shore Structures," Coastal Engineering, No. 2, pp. 189199.
Dugundji, J. (1958): "Envelopes and PreEnvelopes of Real
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Elgar, S., Guza, R.T., and Seymour, R.J. (1984): "Groups of
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Research Institute, Vol. 22, No. 1, pp. 341.
Goda, Y. (1985): Random Seas and Design of Maritime
Structures, University of Tokyo Press, p. 26.
49
Goda, 1. and Suzuki, Y. (1976): "Estimation of Incident and
Reflected Waves in Random Waves Experiments," Proceedings, 15
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"Effects of Wave Grouping on Breakwater Stability,"
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by Wave Height/Period Functions," Proceedings, 22nd IAHR
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Medina, J.R. and McDougal, W.G. (1990): "Deterministic and
Probabilistic Design of Breakwater Armor Layers,"
(Discussion), Journal of Waterway, Port, Coastal, and Ocean
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Linear Regression Models, (2nd Ed.), Irwin, p. 149.
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NY, pp. 123144.)
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50
Tuah, H. and Hudspeth, R.T. (1982): "Comparisons of Numerical
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pp. 6680.
APPENDICES
51
APPENDIX A. STATIONARITY ANALYSIS
The results of the wave data analyses may be
significantly influenced by the stationarity of the data.
Therefore, a stationarity test was performed on the wave data
collected.
Assuming that the nonstationarity characteristics of a
process may be revealed by time trends in the variance of the
data the stationarity of random time series can be tested as
follows [Bendat and Piersol (1986)]: a) divide the time
series into intervals of equal time length where the data in
each interval is considered to be independent; b) compute the
variance for each interval and align these sample values in
a time sequence; c) test the sequence of variances for the
presence of underlying trends or variations other than those
due to expected sampling variations.
The reverse arrangements test [Bendat and Piersol
(1986)] was the technique selected to detect monotonic time
trends in the measured wave data. It is hypothesized that the
sequence of sample variances represents independent sample
measurements of a stationary random variable with variance,
a2. If this hypothesis is true, the variations in the
sequence of sample values will be random and display no
trends. The number of reverse arrangements, RA, will be as
expected for a sequence of independent random observations of
the stationary random variable. If the number of reverse
arrangements is significantly different from this number, the
52
hypothesis of stationarity would be rejected. Otherwise, the
hypothesis would be accepted.
Following the procedure outlined by Bendat and Piersol
(1986) 6 wave time series were selected at random and tested
for stationarity. Each of the N = 32768 points long time
series were divided into K = 32 subrecords of N = 1024 points
each. The significant wave height of each subrecord was
computed and the number of reverse arrangements, RA, was
calculated.
From the set of sampled variances a21, I0 N' define
h'a>a (40)
lo otherwise
The number of reverse arrangements is defined as
where
RA = RA1 (41)
RA = (42)
j=i+1
Table 7 shows the number of reverse arrangements for the
time series recorded at wave gage locations 1, 2, and 3 for
2 realizations.
53
Hs(m) Hs(m)
Subrecord 123123
10.603 0.571 0.582 0.596 0.564 0.581
20.716 0.653 0.658 0.728 0.683 0.706
30.669 0.665 0.653 0.648 0.708 0.699
40.593 0.624 0.658 0.560 0.581 0.609
50.807 0.710 0.699 0.796 0.705 0.671
60.933 0.910 0.868 0.954 0.924 0.903
70.794 0.772 0.738 0.807 0.801 0.760
81.369 1.231 1.132 1.272 1.173 1.048
90.884 0.911 0.915 0.870 0.892 0.860
10 0.865 0.869 0.857 0.822 0.803 0.789
11 0.883 0.886 0.891 0.818 0.816 0.795
12 0.712 0.691 0.684 0.675 0.659 0.652
13 0.710 0.679 0.659 0.690 0.626 0.614
14 0.666 0.672 0.666 0.614 0.639 0.645
15 0.937 0.907 0.897 0.892 0.910 0.863
16 1.005 0.948 0.918 0.987 0.960 0.925
17 0.999 0.987 0.955 0.905 0.868 0.824
18 0.701 0.725 0.714 0.776 0.758 0.742
19 0.861 0.833 0.800 0.846 0.822 0.778
20 0.785 0.786 0.757 0.758 0.774 0,764
21 0.815 0.862 0.840 0.750 0.801 0.786
22 0.583 0.582 0.564 0.558 0.547 0.553
23 0.587 0.578 0.564 0.609 0.589 0.584
24 0.762 0.684 0.721 0.731 0.674 0.694
25 0.791 0.765 0.718 0.752 0.754 0.701
26 0.812 0.768 0.751 0.822 0.769 0.699
27 0.901 0.874 0.812 0.872 0.870 0.858
28 0.809 0.826 0.835 0.758 0.784 0.760
29 0.866 0.831 .0.824 0.810 0.767 0.759
30 0.654 0.636 0.632 0.651 0.645 0.632
31 0.570 0.567 0.552 0.575 0.577 0.549
32 0.694 0.663 0.673 0.715 0.682 0.661
RA I267 1261 253 I265 j268 278 I
Table 7. Number of reversed arrangements, RA, for two
realizations at wage gage locations 1,2, and 3.
54
The region of acceptance of the hypothesis is given by
RAKIP <RA<RAKP (43)
where K=number of subrecords, and i' =level of
significance. From the reverse arrangements distribution
table, and for K = 32 and ii = 0.05 the limiting values in
Eq.(34) are
RA(320975) = 191 RA(320025) = 312 (44)
Therefore, since the number of reverse arrangements
computed for each time series agrees with
191 < RA < 312 (45)
the hypothesis of stationary can be accepted with a 5% level
of significance.
55
APPENDIX B. STATISTICAL ANALYSIS OF DAMAGE
In order to test if different realizations with the same
wave groups characteristics produce the same damage on the
structure the following statistical analysis was performed.
First, the repeatability of the experiments was tested.
Damage data corresponding to realizations, with the same wave
groups characteristics {y, a} phase shifted by =O and i/i=27T
were compared. Second, it was tested if the 4 phaseshifted
realizations (=O, 27T/3, 4ir/3, and 2ir) with equal pairs {'y,
a} produce the same damage on the structure. Multiple linear
regression has been used to analyze the damage data.
In general, the damage data showed a power relation with
an increasing error variance. In order to apply the multiple
linear regression analysis a transformation on the damage
level variable D, of the form
= DX (37)
was needed to stabilize the variance of the error and to
obtain a linear relationship [Neter et al. (1989)]. BoxCox
transformations [Box and Cox (1964)] were applied to choose,
in each case, a transformation from the families of the power
transformations on D. The criterion for selecting the
parameter x with this approach consists in finding the value
of x that minimizes the residual sum of squares (RSS) for a
linear regression based on that transformation.
56
The proposed multiple linear regression analysis
consists of fitting regression lines to K groups of data with
three different models; model1: K nonparallel lines, model2:
K parallel lines, and model3: one line. For each model the
residual sum of squares (RSS) is computed. An FTest between
models is then performed to decide if one model is as good as
the other. Model(11) has been considered as the alternative
model. The FTest for models 1=2 and 3 is given by
(RSS1 RSS1.1) / (DF1 DF11) (47)
F1 RSS11 / DF11
where RSS = residual sum of squares, and DF = degrees of
freedom of the model. If the hypothesis provides as good a
model as does the alternative, then the F statistic will be
small when compared to the percentage points of the F(DFDF
,DF1, ii) distribution, where v = level of significance. If
the hypothesized models (2 and 3) are not rejected then it
can be concluded that differences between the damage data
collected are only due to random variability.
The models mentioned are conveniently formulated by the
following sets of equations:
a) Two groups of data (replicates)
Model1: 2 nonparallel lines.
=++ 2ZJ + e3ZH ;for j=l to 14 (48)
where D = damage level, c = regression
H!o/Hd for the corresponding damage, and
In this case, Z=1indicates the
otherwise. Since damage data corresp
replicate is uniquely identified by Z =
identified by this single variable.
57
line parameters, H =
identifies groups.
first replicate; 0
onding to the second
0 both groups can be
Model2: 2 parallel lines.
The same reasoning as before applies, but the model is
now constrained to compute a common slope for both groups,
D = + c1H3 + c2Z ;for j=l to 14 (49)
Model3: one line.
The simple regression equation is used,
=+ H ; for j=l to 14
b) Three groups of data (phase shifted realizations)
Model1: three nonparallel lines.
(50)
D = ++ 2Z1 + 3Z2 + 4Z1JHJ + c5Z2H ;(51)
for j=l to 28
Model2: three parallel lines
=++ 2Z1J + 3Z2J ;for j=l to 28 (52)
58
Model3: one line
D = + c1H ;for j=l to 28 (53)
Again, additional variables have been defined in order
to identify damage data corresponding to the phaseshifted
realizations,
Z1j 1if phase shift is t' = 0 and 27r; 0 otherwise
Z2j = 1 if phase shift is itt' = 2iT/3; 0 otherwise
All models explained above can conveniently be written
in matrix notation as,
(54)
where D = the damage vector, H = matrix that gives the
observed values of the predictors appended to a column of l's
as the leftmost column, and is the parameter vector. The
ith row of H and D corresponds to values of the ith case of the
data while the columns correspond to the different
predictors. The cases were ordered by increasing values of H1.
Table 8 shows the analysis of the replicates based on
the FTest.
W = 128.5 N W = 99.1N
E1 E2 E3 EEE3 IE
Model 1 flS51 0076 1170 0079 0920 0555 0025 0120 0115
Model 2 RSS 20094 1.330 0 094 0 920 I 070 0035 0.140 0.117
F2 2.07 1.37 1.90 0.00 9.28 400 1.66 0.20
F 11,10,0.025) 604 894 6.94 6.94 694 6.94 6,94
Model 11 RSS 0097 595(1 0140 1350 0043 0.330 0.158
FaDs
Table 8. FTest results for damage of replicate realizations.
It was concluded from the results shown in Table 8 that
the damage produced by two identical realizations was the
same. The influence on damage of different realizations with
the equal {y, a} was analyzed next. Damage data was grouped
according to the corresponding phase shift: group 1: t=O and
2ir, group 2: =27T/3, and group 3: =4ir/3. Table 9 shows the
analysis of the phase shifted realizations based on the F
Test.
It was concluded from the results shown in Table 9 that
the damage produced by the phase shifted realizations was the
same. Therefore it can be hypothesized that different
realizations which have the same wave groups characteristics
{y, a). will produce the same damage on the structure.
W = 128.5 N W = 99.1N
E3 E4 EEE3 EL
Model 1 ASS1 I 130 2025 0639 2638 3.320 0.010 0.150 0.304
Model 2 ASS 3.230 2.160 0.639 2.900 3.490 0,011 0240 0.350
r2 20.40 0.73 0.00 1.09 D36 1 22 8.60 106
F 1222,0.025) 4.38 4,38 4.38 43 438 .4.38
Model 3}

ASS) 2.750 0 060 3390 4.140
}
0022 0.413
=Falls
Table 9. FTest results for damage of phase shifted realizations.