SHORT COURSE
ON
COASTAL STRUCTURE DESIGN
LECTURE NOTES
NOVEMBER 2830, 1996
PREPARED BY
HSIANG WANG
COASTAL AND OCEANOGRAPHIC DEPARTMENT
UNIVERSITY OF FLORIDA
COURSE ORGANIZED
BY
DEPARTMENT OF CIVIL ENGINEERING,
HONG KONG UNIVERSITY
Short Course
on
Coastal Structure Design
Table of Contents
Section 1: Coastal Morphology and Coastal Sediment
Section 2: Environmental Factors in Coastal Region
Section 3: Shore Protection Structures
Section 4: Bank Protection and Earth Retention Structures
Section 5: Principles and Applications of Beach Nourishment
20 pages
42 pages
54 pages
44 pages
50 pages
Coastal Morphology and Coastal Sediment
1. The Coastal Area
From engineering point of view, the coastal area is the zone where waves, tides and wind
attack the land. Figure 1 shows a typical coastal cross section and the related terms describing the
system.
Coastal Area
, J)ffshore
Coast eacCn ana
Backshore
Dunel
or
Bluff
snore
Nearshore zone
=
%_I W M. 
Inshore zone
I Surf Zone
~I^Rrean r
JgL"maIr. x
L^ l __ Lais water.  
Figure 1 Coastal Area and Related Terms
Thus, one can see that the coastal area as defined here is within the wave breaking zone where
rapid transformation of momentum and energy of incoming waves takes place in a relatively narrow
strip of land. Strong material transport could also occur within this zone causing dynamic land form
changes.
There are a variety of coastal land forms and many kinds of classifications. For coastal
structure design the main interest is the material and the dynamic behavior of the coast at the
structural site. Based on material, coasts can be roughly classified as muddy, sandy and rocky.
Although coasts of other material do exist such as coral, mangrove, volcanic ashes, pebbles and
gravels, they are of far less magnitude and are usually treated as subcategories of the former as far
as their dynamic behaviors are concerned. Table 1 provides a rough guide line on the nature of coast
based on material size.
__ __
oresh
I n___~ _~~~ _1~__._
I
r
For rocky coast, its dynamic behavior is largely ignored in coastal structure design, the design
factors are mainly environmental forces and foundation treatment. For muddy and sandy coast, their
dynamic behavior and the resulting interactions with the man made structures must be taken into
consideration in addition to environmental forces and foundation.
An important difference between sandy and muddy coast is that former is differentiated by
grain size analysis by means of sieves whereas the latter requires consistency test using hydrometers.
Also, the engineering properties are largely established by different means. For instance, for mud and
clay, the commonly required engineering properties are shear strength, compressibility and so on
whereas for sandy material the main interests are size distribution and effective weight, among others.
Table 1 Beach Classification based on material size
Muddy Sandy Gravel Rocky
colloid clay silt fine median coarse Cobble Boulder
40200 1040 410
sieves sieve s sieves
0.3p1 5 74p 0.25mm 0.5mm 2.0mm 100mm 30 cm
2. Influence of Material on Nearshore profiles
Nearshore profile plays an important role in coastal engineering. A flat and shallow profile in
front of a structure could offer protection to the structure by dissipating significant among of incident
wave energy. On the other hand, extensive shallow water could be vulnerable to high storm surge.
It also influences wave breaking forms causing large or small amount of material transport as well as
varying degrees of structural scouring. Therefore, knowledge on nearshore profile and its interaction
with environmental factors and structures is of fundamental importance.
It is known that the foreshore slope of a sandy beach is largely influenced by the grain size
of the bottom material, beaches of fine sand have much flatter slope than beaches of coarse material.
Figure 2 shows the field data relating foreshore slope to the median sandgrain size.
A more rational approach to address the nearshore profile is the equilibrium concept. Under
a constant wave condition, beaches and nearshore profile will eventually reach an equilibrium state
when conceptually the destructive force balances the constructive forces. In nature this equilibrium
shape is certainly a dynamic quantity. Nevertheless the concept of equilibrium beach profile is a useful
engineering tool.
0.8
0.6
0.2 
1:10 1:20 1:30 1:40 1:50 1:60 1:70 1:80 1:90
Foreshore slope
Figure 2 Relationship between sand size and foreshore slope (SPM, 1984)
For an uninterrupted long stretch of beach, this equilibrium beach profile is found to be
mainly a function of bottom material. The functional form is taken as:
h =Ay (1)
where h = water depth measured from MWL; y = distance from shoreline toward the ocean; n and
A are two empirically determined constants. The value n is taken to be around 2/3 to 4/5 based on
large amount of field data taken around the world. A is known as the shape factor which governs the
flatness or steepness of the beach. A smaller A yields a fatter profile. Again based upon field evidence,
A was found to be largely a function of sediment grain size (or sediment fall velocity to be more
correct dynamically). The results from a number of laboratory and field profiles were assembled
representing a wide range of sediment size and are plotted in Figure 3. An example shows the
equilibrium profiles for sand sizes of 0.3 mm and 0.6 mm is given in Figure 4.
Although the equilibrium profile is derived for two dimensional uninterrupted shoreline it is
a useful tool to estimate the nature ofnearshore profile even for complicated contours. Its application
will be addressed later, particularly in beach nourishment design.
One practical problem is the selection of the reference mean grain size from the field sample.
Grain size could vary greatly over the beach face into the foreshore zone. The recommended
reference sample location is in between the MHHW and MLLW. Average over the length of the
project is also recommended.
SEDIMENT FALL VELOCITY, w (cm/s)
o.ol o.1 1_______ lo.__ loo.o________
0.01 0.1 1.0 10.0 100.0
FFrM ask Az0.067 w
Pure eh rile a F A
of SromDea, 1987)
0.10
^ BuredaonThnlermining
SA vsA D Cuve using
d FrOml ftrts
LabomoryResuA
0. 01 
0.0 0.1 1.0 10.0 100.0
SEDIENT SIZE, D (mm)
Figure 3 Beach Profile Scale Factor, A, vs. Sediment Diammeter, D, and Fall velocity, w.
(from Dean, 1987)
DISTANCE OFFSHORE (m)
E
1
uj
Figure 4 Equillibrium Beach Profiles for Sand Sizes of 0.3 mm and 0.6 mm
A(D = 0.3 mm) = 0.12, A(D = 0.6 mm)= 0.2.
3. Surf Zone Parameter and Fall Velocity Parameter
Surf zone parameter is a nondimensional term defined as
S= tanpl/ Lo, or =tan(2)
where tan p = mean foreshore beach slope; H = wave height and L = wave length. The subscripts 0
and b refer to deepwater and breaking conditions, respectively.
This parameter was first suggested by Iribarren for breakwater stability, therefore, it is also
known as the Iribarren number. The surf zone parameter is useful index to characterize the dynamic
interaction between waves and beaches or between waves and sloped mound structures. Dynamically,
it can be viewed as the ratio of the natural response period of the sloped face to the oscillatory
incident wave period. When the value of is small, the dynamic interaction is weak; as the value
increases the interaction becomes stronger and reaches a peak when the excitation period matches
with the natural period of the system; further increasing of E beyond this point again causes the
interaction to diminish. Table 2 shows a number of beachrelated phenomena that utilize the surf zone
parameter to characterize their properties. The most frequently used one is the characterization of
breaker types. The following domains were proposed based on Galvin's (1968) experimental data:
Spilling Breakers: o0 < 0.5 or E, < 0.4
Plunging Breakers: 0.5 < < 3.3 or 0.4 < < 2.0
Surging Breakers: 3.3 < or 2.0 < F
While surf zone parameter is useful for characterizing the interactions between waves and
sloped structure it does include sediment property. In the treatment of sediment transport
mechanics, one of the most prominent parameters is the Shields parameter which can be expressed
in a variety way. Some of the more familiar forms are:
0 'b 112
== U (3)
pg(S1)d g(S1)d
or,
e= U
(4)
[g(S1)dj (4)
where Tb = bottom shear stress; S = specific gravity of sediment; d= characteristic particle
diameter; u = characteristic flow velocity.
Table 2. Flow Characterization Utilizing Surf Zone Parameter, (
Surf Zone Parameter
Breaker type
Surface profile
Wave oscillation
Fluid particles movement
Energy dissipation
Ru/H
Rd/H
Reflection coefficient
Phase Difference
Front of the wave
Turbulence level
u/C (at the crest)
spilling
symmetric
progressive
oscillatory
progressive turb.
small
negative
zero
1.3
top of the crest
S1
plunging
asymmetric
translation
mass transfer
strongly turb.
increasing 1.0
increasing 0.5
increasing ~ 0.3
Decreasing 1.0 (resonance)
increasing vertical
all over
increasing to 2
collapsing surging
almost symmetric
almost standing
almost oscillatory
weak turbulent
1.5
1.0
0.8
0.3
decreasing angle
bottom of the profile
decreasing
reflection
asymmetric
standing
oscillatory
little dissipation
 1.0
 1.0
~ 1.0
turb. mainly at boundary
Ru runup, Rd rundown, T wave period, t uprushdownrush period, u horizontal fluid particle velocity, C phase speed of wave.
In surf zone sediment transport, a useful nondimensional parameter is known as the fall velocity
parameter which represents the ratio of sediment particle fall velocity, w, to water particle excursion
velocity in a wave field given by:
1w
V, (5)
gT
or simply as:
w
V, (6)
gT
4. Nearshore Sediment Transport
The subject of sediment transport in a combined current and wave field such as in the coastal
region is a difficult one. Our present knowledge on the subject is still rather rudimentary. A comprehensive
or even a reasonably abbreviated treatment on the topic is beyond the scope of this course. However,
certain fundamental understanding on this subject is essential in coastal structure design. One of the
questions often has to be answered first is whether the structure site is erosional or accretional and their
temporal and spatial scales. What is the littoral environment? How the site will respond under storm
conditions? What is the effect on the structure and vice versa?
From the sediment transport view point the coastal area can be artificially divided into two zones
as shown in Figure 5; the inshore zone from the wave breaking point landward to the end of the swash
and the offshore zone from wave breaking point seaward to the socalled depth of closure. The depth of
closure is the water depth where the wave or current induced fluid motion cease to mobilize the bottom
sediment. The inshore zone can be further divided into surf zone and swash zone. The time scales of
sediment transport are quite different in various zones with various modes. One task is to sort out these
scales for the region of interest then to reach for the appropriate tools to analyze the problem in hand.
The quantity of sediment transport is governed by two factors: the degree of sediment
mobilization and the magnitude and direction of transport agency. The former is due to the combined
effect of flow induced bottom shear stress and turbulence whereas the latter is simply the current field.
In the inshore zone, wave, undergone intense transformation, is the dominant influence for both
mobilization and transport. Therefore, it is logical to simply relate sediment transport directly to the wave
parameters. Furthermore, since the inshore zone is usually a narrow strip, it is convenient to separate the
transport into shore parallel (also known as longshore transport or littoral drift) and shore normal
component ( also known as crossshore transport). In the nearshore zone, the current field could be due
to the combined factors of waves, tides and winds and the bottom topography also plays a role in
influencing current patterns. Therefore, the formulation of sediment transport follows the conventional
approach of combining mobilization with transport similar to the river sediment transport. The problem,
however, becomes significantly more complicated due to the nonlinear combination of oscillatory and
Shoaling water
 
Surf zone
Nears hore I Inshore zone
Run up
Swash
Ox~ne~
TI
h, Closure depth
S
Figure 5 Schematics ofNearshore Sediment Transport Zones
unidirectional flow motions. Various simplifications are necessary. The description given above are shown
schematically in Fig. 5. The procedures of computing these transport quantities are briefly described here
with emphasis on inshore transport. The computation ofnearshore sediment transport often has to resort
to numerical modeling.
4.1 Crossshore Sediment Transport in Inshore Zone
One of the most common applications of crossshore sediment transport computation is to
determine the limit of beach erosion and profile response under storm conditions. Therefore, the temporal
scale coincides with thestorm period. The computation involves two steps: 1). to determine the condition
is erosional 2). to estimate the rate of erosion and the resulting profile evolution.
Beach State Classification
Over the years various criteria have been proposed classify the state of the beach as erosional or
accretional. A summary is given in Table 3.
Deepwater
Offshore
I Isoezo ne 
i
Vl
Table 3 Summary of Criteria Governing the State of Beach
Authors Parameters Criteria for Erosional State
Johnson, 1952 Ho/Lo Ho/Lo > 0.025 0.03
Saville, 1957 Ho/Lo varies
Iwagaki & Noda, 1963 Ho/Lo ; H/ds, graph
Nayak, 1970 Ho/Lo; Ho/ds; p,/p, graph
Dean, 1973 H/Lo,; xw / gT Ho/Lo> 1.7(w / gT)
Sunamura & Horikawa, 1974 H/Lo4;tan P; d, /Lo Ho/Lo > c (tan P)27 (d, /L,)0.67
Wang & Yang, 1985 He /(g T tan P); w / gT Hb /(g T tan P)>0.5(nw / gT)"
Hartori & Kawamata, 1980 H/L, tan P; tan P H]Lo tan P >xw / gT
Larson, 1988 Ho/L ; He / wT HoILo >0.0007 (He / wT)
Ho = deepwater wave height; L0= deepwater wave length; T = wave period; ds = mean grain size; p,
= sediment density; p,, = water density; tan P = mean beach slope; w = particle fall velocity.
Most of these criteria, particularly the later ones, were derived from similar data set. Figure 6
shows an example of the criterion as proposed by Larson (1988). Figure 7 shows the data plots by Wang
and Yang (1985).
Ho/Lo
1.0000
0.1000
0.0100
0.0010
0.0001
CE Erosion
0 CE Accretion
CRIEPI Erosion
0 CRIEPI Accretion
 Ho/Lo0.00070(Ho/wT)
. . AcicretFo==
0
C.*
 
U
 .
 _________Erosion HE~

Figure 6 Criterion to Distinguish Erosional (bar) and Accretional bermm) profiles (from Kraus, 1990)
10
I 
'
0.1 1.0 HOT 10.0
=
r'(W/gT)
Ho/9TZ
Figure 7 Classification ofNormal (accretional) and Storm erosionall) profiles Using Surf Zone
Parameter and Fall Velocity Parameter (wang, 1985).
Rate of Crossshore Transport and Profile Response
At present, crossshore sediment transport equation in the inshore zone suitable for engineering
application is limited to erosional case, i.e., net offshore sediment transport. No creditable transport
equation is currently available to address accretional case (net onshore transport). The transport equation
is largely empirical and many variations. Basically, they all follow a similar approach resulting in similar
forms of the following:
e ah e ah
q=K[DDq+" a for D > [Da1 (7)
and
e Sh
q=O for D<[D, (8)
where
q: offshore volumetric sediment transport rate per unit width in m3 /sec m.
K: is the transport rate coefficient on the order of 10" m'/N
D: wave energy dissipation per unit volume, D = (dF/dx)/h, F = Ec, is wave energy flux and
h is the water depth.
D : energy dissipation per unit volume corresponding to equilibrium profile. D, =
(5/24)(pgA)32 y2. Where A is the beach shape parameter discussed earlier; y is the breaking
depth index defined as the ratio of wave height to the water depth at breaking point, or,
S'= H~/hb.
e: is a sediment diffusion coefficient due to bottom slope effect taken on the order of 0.00006
m2/sec.
Assuming a straight shoreline the profile response can then be computed by the sediment
conservation equation:
3q ah
ax at (9)
There are many numerical schemes developed by different investigators such as the SBEACH
model by Kraus and Larson (1991), the modified version by Wu et.al. (1994) and others. Most of these
models yield reasonable and similar results for open sandy coast. They are largely untested for muddy
coast.
Longshore Sediment Transport (Littoral Drift) in Inshore Zone
The time scale of interest for longshore sediment transport is at least one order of magnitude larger
than the crossshore transport. Engineers are usually more interested in annual longshore sediment
transport rate rather than daily rate. The accepted formula was based on the assumption that the rate of
longshore sediment transport is proportional to the longshore wave energy flux. This leads to the
following well known formula (SPM, 1984):
K
Q, =KP =  (ECcosasina),
(P,P)g(1n)
(10)
= KECsin2a
where Qj: is the total volumetric sediment transport rate within the inshore zone; Pb is longshore energy
flux with in the breaking point; n is the porosity and a is breaker angle. K, is a dimensionless transport
coefficient. Since wave energy, E, is equal to pgiP /8, K, value depends on the reference wave height used.
in the computation. If wave energy is computed from significant wave height (H), K, is suggested to be
around 0.39. On the other hand ifRMS value is used (H.) the value of 4 is roughly in the order of
0.77. The values of other parameters in the above equation are given in Table 4. The coefficient K, is
dimensional and we have,
Q(m 3/yr) = 1290(m 3 s/Nyr)PNm/m s) (11)
Q/yd3/yr) = 7500(yd3 slbyr)Pftlb/fts)
using H, as basis for energy computation.
Table 4 Values of Parameters in Equation (10)
Terms Metric U.S. Customary
p. 2,650 kg/m3 5.14 slugs/ft
p (sea water) 1,025 kg/m3 1.99 slugs/ft3
p (fresh water) 1,000 kg/m3 1.98 slugs/ft3
n 0.4 0.4
g 9.8 m/s2 32.2 ft/s2
The value of coefficient as suggested is for straight is sandy coast. The actual value of K for a
specific shoreline is influenced by the material, foreshore geometry, manmade structures and other natural
changes, etc., and is, therefore, expected to vary from the suggested value.
Based upon the wave information and the longshore transport equation, longterm (or shortterm)
lateral drift environment can be estimated. Table 5 shows an example of estimated annual longshore
transport along the mid section of the east coast of Florida. This information is useful for establish
sediment budget for a project site and for assessing long term mutual effects of coastal structure and its
adjacent beaches.
4.3 Sediment Transport in Nearshore Zone
The approach to computing sediment transport in nearshore zone is different from that in the
inshore zone but similar to river sediment transport with the exception that the flow field is unsteady and
nonuniform owing to the presence of the waves. In addition, the problem is inherently two dimensional
and can not be simplified to one dimensional such as often applied to the case of river sediment transprot.
At present, we are still at the initial developing stage. The effort is mainly in the development of numerical
models; they can be either general purpose models or special purpose models. The basic approach is rather
similar consisting of coupling a flow model with a sediment transport model which in turn computes the
morphorlogical changes. Depending upon the intended time and spatial scales, various formulations were
were proposed. A few examples were given here based upon a model recently developed in the University
of Florida ( Kim, 1995). This model can be classified as a short to median term model in that it can
simulate near shore morphorlogical changes from storm events to few months even up to one year real
Table 5 Estimated Anual Longshore Sediment Transport Values Along Forida Atlantic coast
(in cubic yards per year for 20 years period from 1956 to 1975)
Month A 7a, 1 1of
Jan (South) 9,124 14,273 7.949 1.850
(North) 14,061 6,081 7.162 1.861
Feb (South) 7,352 11,406 7.772 1.788
(North) 4,486 6.929 7.291 1.829
Mar (South) 6,810 10,196 7.806 1.677
(North) 4,788 8,048 7.329 1.732
Apr (South) 968 1,402 6.105 1.377
(North) 1,132 1,298 6.043 1.946
May (South) 3,125 6.150 6.874 1.905
(North) 3,055 4,398 6.780 2.041
Jun (South) 1,462 3,150 5.919 1.875
(North) 2,077 4,102 6.132 1.937
S Jul (South) 964 1,247 5.597 1.997
(North) 1,384 2,357 6.075 1.913
Aug (South) 1,974 4,784 6.208 1.929
(North) 1,466 2,367 6.212 1.701
Sep (South) 3,870 7,428 7.057 1.626
(North) 2,642 5,201 6.704 1.811
Oct (South) 5,687 8,965 7.647 1.522
(North) 3,387 5,125 7.119 1.594
Nov (South) 7,609 10,948 7.859 1.714
(North) 3,992 5,197 7.322 1.734
Dec (South) 7,516 10,997 8.024 1.557
(North) 4,442 6,829 7.287 1.798
(a) Magnitude, p = mean value, a = standard deviation.
time. The structure of this morphodynamic model is shown as follows:
Like many others of similar kind, the flow model is depth integrated and timeaveraged over wave cycles.
Current and wave interactions are included. The sediment model is based on a common approach similar
to many others in that the total transport is the sum of bed and suspended load as follows:
Q = qb+q,
q, = A c(r,r) pg + Ab,,( ,,O)/pg (13)
q,= fcf cdz = Aurb) = Aturb,
h
In this scheme, all the transport components are linearly combined. This is a much simplified assumption.
There are, of course, many other possibilities to formulate the problems.
Profile Change(Surge=2m, H=2m)
150 200
Distance(m)
CrossShore Transport Rate
150 200
Distance(m)
350
350
Figure 8 (a) Model simulation of beach erosion under 16hr. storm with 2 m storm surge and 2
m wave height; (b) Crossshore sand transport rate distribution after 1 hour and 16 hours.
0
0
C
( 
I13
Depth Change(Present Model)
After 000 minutes
After 040 minutes
After 120 minutes
5 10
Longshore(m)
After 020 minutes
After 080 minutes
After 160 minutes
6 .3030
4
.10 10
.0 5 10 15
Longshore(m)
Figure 9 Numerical model simulation of topographic changes of a coastal inlet under storm wave attack;
the simulation is for a scaled movable bed model tests conducted at the University of Florida.
4
82
O
x
Figure 10 The test condition used in the experiment with test waves of 8 cm high and 1 sec and an oblique
angle from 5 to 10 degrees.
a) 6
5
13
22
0~
Depth Difference(Experiment)
. . .. . . . . .
...:: :::.. '...::: ..
Em
rI
2 4 6 8 10
Longshore(m)
12 14 16
Depth Difference(Present Model)
 accretion
6 8 1l
Longshore(m)
erosion
Figure 11 Comparison ofbathyemetric changes after 160 minutes test time (four tidal cycles).
'* . .. .. 3.:.:. 
''".'.'**** ,3. ....
.  
' ' .'*
.......... .
20C
E
10
10
5C
C
"n1n
'15C
10C
5C
5C
r
Depth Change(After 70days)
15
10
.......... ...... ... 
25
........ 4 .................... ... ....................................
0 200 400 600 800 1000
Longshore(m)
Depth Change(After 140days)
200
400
600
Longshore(m)
800
1000
Figure 12. Response of a single groin after 70 days and 140 days persistent wave attack from single wave
direction.
.15
10
...
~VV'
'0
Environmental Factors in Coastal Region
This Chapter deals with the methodologies of how to determine and describe the major
environmental factors that influence coastal structural design. The discussion is restricted to
physical factors only.
1 Coastal Wind and Wind Forces
Wind plays a number of important roles in coastal environment. It exerts a direct
loading on structures and incites structural motions. It also causes water level change,
enhances wave motion, produces nearshore current and induces long period oscillation; all in
turn affect the performance and the survivability of coastal structures. Therefore, assessment
and determination of wind condition are essential in the planning and design of coastal
structures.
1.1 Overwater Wind Field
In coastal engineering applications one mainly deals two basic types of wind field: the
persistent large scale winds and the intense but fast moving small scale winds. The
northeasters experienced along the east coast of the United States and monsoon winds in south
ease Asia are of the former type whereas hurricanes and typhoons are of the latter category.
Determination of design wind in the first category can best be accomplished by statistical
analysis such as various techniques of extreme value analysis. Design wind field of the second
category is more conveniently prescribed by numerical simulation such that the surface wind
field is entirely specified in terms of a few storm parameters. This is a typical approach of
treating severe storms of limited extent such as hurricanes and cyclones, in which little direct
data is available and the storm size is smaller than the typical resolution of standard pressure
analyses.
1.2 Design Wind by Extreme Value Analysis
The wind condition at any site varies widely from year to year, and it is not possible to
forecast any maxima 'from the annual frequency distributions. Nevertheless, if the annual
maxima are recorded for a number of years, then it is possible to forecast the maximum wind
speed likely to be attained in a given number of years by utilizing the statistics of extreme
values. A variety of statistical functions have been suggested by various investigators to
describe extreme values. Table 1.1 lists a number of them. It has not been demonstrated in a
rigorous manner one is superior than the other. Basically, the oceanic data can be collected and
presented in two ways as illustrated in Fig. 1.1.
One of them represents the maxima in a set time period such as the maximum annual wind
speed; the others are equal timespaced data such as threehourly wind or wave. As a general
guideline, if the data are presented in the former manner, they tend to follow lognormal or the
Gumbel distribution. On the other hand, the data in the latter format follow the Fr6chet,
Weibull, exponential or chisquared probability distributions. It is advisable, however, to try
various distributions to select the one with the best data fit.
Here we like to illustrate how to apply these statistics by using the Gumbel distribution
to describe and extrapolate annual extreme wind. Before we explain the Gumbel's distribution
function, let us first examine the meaning of return period.
TABLE 1.1. FORMULAS FOR EXTREMEVALUE STATISTICS
Type Distribution Function Remarks
Gumbel
Log
normal
Poisson
Fr6chet
Weibull*
Weibull
ChiSquare
F,(x) Exp[Exp(a(xu))] oo
1 .,1 I (fnSA} 2
F(x) = >O) d
x>0
v 3eV
F(x,) = . SI
x>0
x>O0
F(x) = Exp [(
x>0
F(x)= 1 Exp "
x > 0 
x>O 0
F(x) = 1 e')'
r O> 
Two parameters
a,u data related constants a>0
Two parameters
A,X data related constants
both positive
One parameter
v data related positive constant
Two parameters
1,y data related
positive
Two parameters
b,k data related
positive
constants both
constants both
Three parameters
Xo,x,,y data related constants
One parameter
p data related constant, positive
*Rayleigh distribution is a special case of Weibull distribution when k = 2.
For a certain probability distribution function F(x), there is a probability 1F(x) of a
value to be equaled or exceeded by x. Its reciprocal,
1 1
T(x) = F()
1 F(x) G(x)
(1.1)
X>,
tL
Figure 1.1 Illustration of Data Samplings
is called the return period. It is the number of observations such that, on the average, there is
one observation equaling or exceeding x. As an example, let us consider the outcome of an
unbiased dice. We all know that the probability of the outcome less than 5 for instance, is
1
4/6 or F(x) = 4/6. The return period of equal to or exceed 5 is T(x) = = 3. In other
14/6
words, in every three trials, one observation is expected to be larger than 4 and equal or
exceed 5.
Suppose that we have in hand n years annual extreme data wind which can be arranged
in a descending order:
U1 > U2> U, ... >UN
We can, based on existing evidence, immediately assign a return period for each wind speed,
such that the return period of U, is n, U2 is n/2, and so on. Evidently, the conclusion one can
draw upon a limited experimental evidence is restricted in two respects: the .basis of
experiment and the arbitrary nature of extending and extrapolation of information. Gumbel's
first asymptotic distribution method is a device to interpret experimental data in a rational
manner.
Gumbel's first asymptotic distribution function has two basic features: First, if the
initial distribution is of exponential type such as normal distribution, then the trend of
logarithmic increase of the extreme values is a straight line. Secondly, the asymptotic
distribution of the extreme value (largest or smallest) i.e., when the number of traits become
large, is a universal one and has the form
F(x) = e"1'
where
y = a(x u) (1.3)
is the reduced variate of largest value, and x is the actual largest values. The parameter u is
the mean value, and 1/a is a measure of dispersion; both of them have to be determined by
experiment. Equation (1.2) can be rewritten as
y = ln[ln F(x)] (1.4)
which states that if F(x) values are plotted on a doublelog scale (known as the Gumbel's
extreme probability scale), they should form a linear relationship with the reduced variate y.
Now let's apply the Gumbel's distribution function to the set of data listed in Table 1.2 which
is the annual maximum wind taken at Baltimore Airport, Maryland, arranged in ascending
order.
The probability of nonexceedance of the m* ranked wind can be calculated from
m
F(V,) (1.5)
N+1
where N is the total number of years of observations. From F(V,), y is calculated by
Eq.(1.2), and V is plotted against y in Fig. 1.2. The probability distribution can also be
plotted on the same diagram for quick reference. These points should, in theory, lie on a good
straight line. The various statistical tests are now to be performed in the following order:
1. The Expected Value Based upon data of the maxima speed, the sample mean and standard
deviation can be computed and we denote them by x, and S,, respectively. We also know
that the actual extreme value is related to the reduced variable by Eq. (1.3), i.e.,
1
x=y+u
a
Gumbel suggested that the parameters a and u are computed from
1 S (1.6)
a a(N)
and
u = x, y(N)l / a (1.7)
(1.2)
TABLE 1.2. ANNUAL EXTREME WIND STATISTICS AT BALTIMORE AIRPORT, MARYLAND
MAX. WIND SPEED (M.P.H.)
BALTIMORE WE AP MD
JAN FEB MAR APR MAY JUNE JUL AUG SEPT OCT NOV DEC MAX
1953 50 52 46 45 42 36 42 34 37 34 36 57 57
Dir. W NW W NW W SW NW NE NW NW NW W W
1954 60 54 52 70 40 35 38 50 43 73 57 42 73
Dir. W NW W W W NW N NW NW SE W NW SE
1955 41 43 60 53 65 42 35 54 31 56 52 45 65
Dir. NW NW SW SW SW NE NW NE NE NW NW NW SW
1956 52 68 52 43 39 40 36 25 38 42 50 37 68
Dir. NW W W NW NW W NW NW W N W W W
1957 60 40 37 56 42 52 43 31 29 65 40 40 65
Dir. NW NW NW W W NW E N W NE S SE NE
1958 63 52 34 49 33 52 36 47 37 40 38 49 63
Dir. NE NW NW NE SW SW S SW N NE NW W NE
1959 52 43 47 42 36 40 38 34 30 33 38 56 56
Dir. NW W W NW NW W W N SE SE NW W W
1960 34 49 48 37 37 33 26 40 33 36 34 34 49
Dir. W NW NW W W W NW NW W W NW NE NW
1961 37 49 38 43 65 41 28 26 42 30 37 50 65
Dir. NW NW W NW SW NW NW SW N NW NW NW SW
1962 34 32 38 39 26 22 57 22 27 31 32 34 57
Dir. SW NW NE W W SW NW N W NW NW NW NW
1963 40 33 45 56 37 35 23 30 26 28 41 38 56
Dir. NW NW NW NW W N NW NW NW NW W W NW
1964 45 44 57 36 33 41 22 27 29 30 32 37 57
Dir. W S NW NW W W SE NW W NW W NW NW
1965 34 52 34 41 33 40 30 41 36 35 41 32 57
Dir. W SW S NW NW SW W NW W SW W N SW
1966 47 44 45 34 37 40 30 37 26 35 30 37 47
Dir. W S NW S W SW W W W W S W W
1967 41 49 40 36 37 43 39 36 28 38 37 29 49
Dir. W W NW NW NW SW NW SE SE W W NW W
1968 35 47 45 42 37 39 42 42 20 32 35 43 47
Dir. NW NW W SW W NW NW N W NW W W NW
1969 35 42 35 33 29 32 49 29 21 34 42 42 49
Dir. W W NW SW W SE NW W N NW SW NW NW
1970 35 36 38 56 34 40 42 23 34 30 33 45 56
Dir. NW NW NW NW NW NW NW N W SW NW W NW
1971 54 36 54 38 33 34 33 23 18 24 42 40 54
Dir. NW W NW SW SW SW SW W W E NE NW NW
1972 38 38 34 33 26 37 26 30 31 32 36 38 38
Dir. SW NW NW W NW NW NW W NW NW W W NW
1973 31 35 46 35 36 28 33 26 37 24 40 34 46
Dir. NW SE W W W W SW SW N SW W N W
AVE. 41.76 44.90 41.90 43.67 37.95 38.19 35.62 33.67 31.09 37.24 39.19 40.90 55.90
I
RETURN PERIOD IN
YEARS
.1 .3 .5 1 .8 .9 '.95. .98 .99.995
PROBABILITY DISTRIBUTION
1 0 1 2 3 5
6 7
aDUCED VARIATE
Figure 1.2 Extreme Wind by Gambel Distribution (Baltimore, MD. USA)
44
aw
IA
a
CD
U,
where o(N) and y(N) are, respectively, the expected values of the dispersion and reduced
variate for N trails (while S, and x, are the actual outcomes from the experiment). These
expected values are shown graphically in Fig. 1.3. The asymptotic values of
and y() is 0.5772 (Euler's number).
Sample size. N
Figure 1.3 Gumbel Distribution Sample Size Correction Factors
Thus, based upon the actual sample mean and variance, the straight line fit can be
constructed with known sample size. This straight line allows extrapolation beyond existing
data.
2. Control Curves The control curves is a test of the theoretical fitness of the past
observations. It also projects the deviations of future events from the estimated expected
values.
It is very similar to the use of confidence intervals. There are two segments of the
control curves:
This first segment of the control curve is sample size related and is useful to judge the
dispersion of the samples and to establish the confidence that the assumed law of extreme
statistics is valid. For instance, if all the data points plot well within one standard error from
the expected mean, the assumed extreme function is judged to be good. On the other hand, if
the data scatters beyond three standard deviations, the validity of the assumed extreme function
is doubtful. For example, in a normal distribution with zero mean for 90% confidence, the
observation will be within the band between x = 1.645a to x = 1.645a. The real
observations within this range are allowable under .9 probability. In a similar way, two
control curves can be drawn in the upper and lower part of an estimated faired line of the
extreme value distribution function.
The first segment of the control curve consists of two portions. The lower portion is
valid for F,(x) in the range between 0.15 to 0.85. In this range, the asymptotic errors of order
statistics are obtained from the standard errors' under the condition of normal distribution.
This portion of the curve is located on either side of the x coordinate (vertical direction in Fig.
1.2) at the distance
= [y (y) / ]n (1.8)
where n = number of standard deviations and
y (y) = reduced standard error
= ( I) / e (1.9)
The upper portion of the first segment is for F(x) between 0.85 and N/(N+1). In this
region, the distribution of the order statistics will not converge toward normal distribution and
the equation stated above no longer applies. The control curve is continued by locating Ax on
either side of the faired curve in accordance with
Ax = a/, (n)
where f,(n) depends on the index of rank, i, and on the number of standard deviations of
dispersion, n. The values of f, (n) are given in Table 1.3 for different ranks. There i = 1 is'
the top ranked value (extreme value); i = 2 is the second ranked value, etc.
TABLE 1.3: VALUES OF f,(n)
n
i 1 2
1 1.140 3.07
2 0.754 1.78
3 0.589 1.35
4 0.538 1.17
The second segment of the control curves is for extrapolating beyond available data, or
for F(x) beyond N/(N+1). In this region, there is no theoretical basis to follow. The
current practice follows Gumbel's suggestion that parallel lines be drawn along the faired mean
line starting from the top extremes of the first segment. The control curves so constructed
represent 68.3% confidence of nonexceedance (or 31.7% confidence of exceedance for the
lower curve) for one standard deviation, and 95.5% confidence of nonexceedance (or 4.5%
confidence of exceedance for the lower curve) for two standard deviations.
1 Standard error is based on the distribution of sample estimate as opposed to standard
deviation which is based on the original population.
3. Return Period and Risk of Encounter With the probability distribution function obtained,
the associated return period of extreme wind can be computed by Eq. (1.1). These values
can also be plotted in the same probability graph such as on the upper abscissa. An
explicit expression of the return period for Gumbel can also be obtained as
1
T= (1.10)
I e"
For engineering application, the relationship between return period and extreme wind is an
important piece of information. The return period is defined as the average time interval
between the events being considered. The designers must clearly understand its meaning to
avoid misuse. For example, the probability that the event (X = xj will occur in one trial
is given by 1/n, and the probability the event will not occur is 11/n. Hence the
probability that the event will not occur for m trail is
P. (X < x) = (1 1/n)"
and the above quantity can be approximated by e" for large values of n and m. Thus, if
n=m, this quantity becomes e"' which is approximately equal to 0.37. In other words, the
chance that the event will not be encountered during the entire return period is
approximately equal to 0.37. Thus, the probability that the event will occur (at least once)
during one return period is 1 0.37 =0.63. Let n= the return period of an extreme
event, Td, and m= the design structural life, T., the risk of encounter, R, can be/
expressed as distribution:
R=lPr(x
or
T = T In(l R) (1.12)
provided both Td and T. are large. This is a Poisson distribution function. This risk factor can
also be incorporated inL the same probability graph shown in Figure 1.2. As an example, for a
design structure life of 10 years, at a risk of encounter of 30%, the corresponding design
return period is about 28 years. For the data shown, the design wind speed should be 76 mph.
1.3 Storm Parameterization
The basic governing equation for wind field is known as the gradient wind equation
given as follows:
vZ 1 dp
+ 2m sinAv (1.13)
r p On
for a coordinate system shown in Fig. 1.4. Where v = wind velocity along a streamline; r =
radius of streamline; sin A = latitude; w = earth rotation = 0.525 rad/hr (7.29 x 10
rad/sec); p = atmospheric pressure. The term 2co sin Xv which arises from earth rotation is
known as Coriolis acceleration. Therefore, this equation balances inertial force due to
centrifugal acceleration and Coriolis acceleration with the pressure gradient. Since the friction
term is missing this equation governs wind field at elevation where friction effect can be
neglected. This elevation is known as the gradient height and the wind velocity above this
elevation is known as gradient wind. Over water, the gradient elevation is around 500 m.
Streamline
Figure 1.4 Natural Coordinate System for Eq. (13)
In the case, 20o sin Xv > v2/r the resulting wind field as known as geostrophic wind,
the wind speed can be determined from Eq.(1.13) with the following approximate solution:
V I O 1 (* (1.14)
2pw sin A On 8pwr sin3 n)
The "+" sign on the RHS corresponds to the case around high pressure center and the "" sign
to that around low pressure center. The higher order terms are the corrections due to the
curvatures of the isobars. In the northern hemisphere the X value is positive. Therefore, the
wind speed is enhanced around high pressure center and reduced around low pressure center.
In the northern hemisphere, this wind field is counterclockwise around a low pressure center
known as cyclonic and is clockwise around a high pressure center known as anticyclonic. The
relationships are exactly opposite in the southern hemisphere.
In Eq.(1.13) if the v2/r is the dominant term, then the wind field is classified as
cyclostrophic wind. For this term to dominate r has to be small. Therefore, cyclostrophic wind
field is usually spatially considerably smaller than the geostrophic events but could be very
intense. Hurricanes and Typhoons are typical cyclostrophic events. Unlike the geostrophic
wind field, cyclostrophic wind field is much more organized, thus, can be prescribed by
idealized mathematical models.
The most common design events used by engineers is a design "hurricane", or a
"design storm". The terminology "maximum possible hurricane (MPH) and standard project
hurricane(SPH)" are widely used in the United States. The meaning of MPH is
selfexplanatory. The name SPH is, on the other hand, loosely defined as the most severe
hurricane (storm) that is considered representative to the region. Many engineers treat SPH as
synonymous to 100year hurricanes, or typhoons in the case of south east Asia. An idealized
hurricane field can be represented by a Rankine vortex as shown in Fig. 1.5. It consists of a
rotational core of radius R and an outer irrotational wind field.
pup
Figure 1.5 Pressure and Velocity Distribution of a Rankine Vortex
Within the rotational core, the wind speed increases with the radius Outside, R, the
wind speed gradually decreases. The pressure, on the other hand, is lowest at the center and
increases continuously to ambient pressure. Accordingly, the pressure and velocity field can
be expressed as:
2 1
p(r) = 2P
2 for r < R(Rotational) (1.15)
v = kr
and
p.v
p(r) + 2 p
for n > R(Rotational) (1.16)
k'
v=
r
where p(r) is the pressure at radius r; p. and p, are, respectively the central and ambient
pressures, v is velocity, p. is air density, and k,, k2 are constants of proportionality. This
Rankine Vortex model has a continuous pressure distribution along the raxis but the velocity
has a discontinuity at r = R.
A modified model which avoids the velocity discontinuity is to assume the pressure
field of an exponential form:
p(r) = po + (P. Po)e(R') (1.17)
In this case, the pressure gradient is
p= (P.P)e<" (1.18)
Substituting Eq.(1.18) into Eq.(1.13) gives
V+f R
+LfV, ( po) eR (1.19)
where V, is the gradient wind and f= 2o) sin X is the Coriolis parameter defined earlier. For a
pure cyclostrophic wind field, we have, by letting fV, =0,
I R R
vJ ( po)er" (1.20)
The maximum wind occurs at r=R, or,
.= (p. po)e' (1.21)
which is smaller than velocity from the Rankine model by almost 40% (1 e/). Thus, the
maximum cyclostrophic wind at gradient height is a function of pressure only (assuming p. is
approximately equal to constant).
The complete solution of Eq.(19) with the inclusion of fV, term can also be easily
obtained. For instance, at the maximum wind location, i.e., r=R, the solution is
V, = (p, po)e' +(fR/2) fR/2 (1.22)
Since both f and R are small values, neglecting the higher order term, we have
V (p p)e fR / 2 (1.23)
Pa
In northern hemisphere around a low pressure center, the positive sign should be selected.
Equation (1.23) is also often expressed in the following form:
V, = K (p p,) 05fR (1.24)
where K is dimensional coefficient and is a function of temperature among other physical
factors. In metric system, for V, in km/hr, R in km. and p in mb., the corresponding value
of K is 23.2 at 18*C.
Based upon the idealized model discussed above a synthesized hurricane wind field can
be developed. Four input parameters need to be provided:
1. Central Pressure (p.) or Central Pressure Index (CPI). The central pressure is the
minimum pressure at the center of a hurricane wind field, which usually governs the
hurricane intensity. It is a data based climatological variable.
2. Radius of Maximum Wind (R). The radius of maximum wind is an index of hurricane
areal size and is an important factor in the generation of waves and tides. Whenever
possible, the value of R should be determined by available data in the region.
3. Peripheral Pressure (p.). p, is the sea level pressure at the outer limits of the hurricane.
There is no statistics to guide the selection for design purpose. A high ambient pressure
usually produces smaller and more intense hurricane. A low ambient pressure, on the other
hand, reduces wind intensity but spreads the wind field wider. For lack of statistical
information the standard atmospheric pressure value of 1013 mb or 29.92 inches of
mercury is commonly used.
4. Forward velocity (P,). The forward velocity is the speed and direction of translation of
the hurricane center. The magnitude is mainly influenced by the latitude but the direction
appears to have no strong correlation with other major hurricane indices
With the input conditions specified, a synthesized wind field can be constructed to
provide the following information:
1. Maximum Wind Speed for Stationary Storm
The maximum gradient wind (VpJ is the actual wind speed at gradient height of a storm
with zero forward speed. It is the cyclostrophic wind with the correction of Coriolis
effect. Based upon Eq.(1.24), we have
v, = 23.2 (p, po)2 0.5Rf (1.25)
The maximum gradient wind, in theory, is a steady wind maintained at gradient height or
above. The actual wind measured by meteorological stations is, however, at a much lower
elevation well within the atmospheric boundary layer where the frictional effect cannot be
neglected. To correlate these two wind speeds, a frictional correction must be made on V,.
If we define V, as the maximum wind speed at standard instrument height (a height of 30
ft. or 10 m above sea level), an empirical correlation coefficient is obtained as
V,.= 0.868 V,
or
V = 0.868 (p po)aRf (1.26)
where a is a coefficient adjusted for frictional effect.
The term "maximum wind" is rather loosely defined here usually taken as the greatest
average speed over a five to ten minute interval. In many engineering applications, wind
speed sustaining over a longer period is more reasonable, further adjustment can be made.
A "maximum sustained wind", for instance, is referred to as the maximum
hourlyaveraged wind which is calculated as:
V, = 0.865 V, (1.27)
for a stationary hurricane at instrument height.
2. Moving Storm and Wind Field
The complete wind of a stationary storm can be constructed in accordance with Eq.(1.19).
For hurricane with a forward speed, P,, the wind field becomes skewed and the actual
wind speed should be the vectorial summation of the rotational component and the forward
speed component. For a counterclockwise wind field cyclonicc wind in northern
hemisphere) with a forward direction as illustrated in Fig. 1.6, the wind field is stronger
on the right side than the left side. The theoretical maximum velocity line should be about
90* clockwise from the forward direction. The actual angle where the maximum wind
occurs may be any value from 15* to 1650. A maximum velocity line (OM) is customarily
S;;s;nca c le n ars cr aClu3 cnx:mum wNinr. f
a. Horizontal Model of Hurricane Wind Dstribution.
Figure 1.6 Idealized Cyclostrophic Storm Wind Field
30
20
Tropopause
%;_ .<,l honic Eddv
"* '" ^Secondary Outflow
Wall of Eye
Primary Outflow
SSurface 1E
selected as the line 115" clockwise from the forward direction. The magnitude of velocity
along this line is adjusted by the following equation, be:
V. V, + 03 V, (1.28)
where V. is the theoretical rotational velocity at the instrument height. For instance, the
maximum sustained wind speed at the instrument height for a moving storm is
V, = 0.865 V + 05 V, (1.29)
The wind velocity around the radius R can then be constructed according to
v= V. (1cosO) (1.30)
2
where 8 is measured from the maximum velocity line. The complete wind field can also be
constructed in similar manner.
1.4 Wind Field Correction
Elevation Correction
Below the gradient height, the viscous and turbulence of the air motion causes a
frictional force which modified the wind field. The frictional force is generated as soon as the
motion starts; the force vector is always opposite to the velocity vector.
Thus, the wind vector rotates towards the low pressure center and the wind speed
reduces. As the frictional force increases with lowering altitude, these effects becomes more
pronounced. This change of wind vectors with the elevation forms a spiral known as the
Ekman spiral. Figure 1.7 illustrates this phenomenon.
Figure 1.7 Wind Spiral in the Friction Layer
Figure 1.7 Wind Spiral in the Friction Layer
The adjustment of the wind direction and magnitude is largely based upon empirical
relationships; among them the logarithmic law and the power law widely used.
Logarithmic Law: The logarithmic law, commonly known as the Prandtlvon Karman
universal velocitydistribution law, is given for rough (turbulent) flow as
U 1 z
U in
U, k zo
z zo
(1.31)
where U, = wind speed at elevation z above the surface
U, = friction velocity = Jp where r = shear stress at the surface and p is
the density of the fluid.
z = elevation above the surface.
4 = friction length.
k = 0.4, the von Karman constant.
The standard anemometer level is referred to as the 10m elevation above the free surface,
therefore,
U, 1, 10
In
U, k z,
Dividing Equation (1.31) by Equation (1.32), we have
U, lnz/ z
U, InlO/zo
This equation can be used to construct the wind profile for
the typical values of zo for various types of surfaces.
(1.32)
(1.33)
given U, and zo. Table 1.4 gives
TABLE 1.4 VALUES OF zo FOR VARIOUS SURFACES
Type of Surface z0 cm C (at 10m elv.)
Very smooth (mud flat, ice) 0.00010 0.01
Lawn grass up to 1.0 cm high 0.01 0.014
Downland thin grass up to 10 cm high 0.05 0.16
Thick grass up to 10 cm high 0.20 0.019
Thin grass up to 50 cm high 0.40 0.020
Thick grass up to 50 cm high 0.80 0.022
Water surface with waves 0.20.5 0.0190.021
In engineering practice, the shear stress is commonly related to the velocity by the empirical
relationship
S= CU' (1.34)
where C, is the friction coefficient. This friction coefficient is related to friction length by
substituting Eq.(1.28) into Eq.(1.31):
C, = K 2n (1.35)
Since zo is constant for a specific situation, it can be seen that C, depends upon the reference
level. The typical values of C, at standard instrument height are also listed in Table 1.4.
Power Law The power law for the vertical distribution of wind speed is
U z = z(1.36)
U, (TO)
The exponent m is a function of surface roughness and Reynolds number. For coastal region,
it varies from 0.3 for U, = 60 mph to 0.143 (1/7) for U, = 130 mph. The wind direction can
be adjusted according to the following empirical relationship:
0, = O + arcsin Bo / U, (1.37)
where 8, is the theoretical wind angle (direction of svector) and Bo is an empirical coefficient
in the order of 3.5.
Duration Conversion
As noted earlier wind is an unsteady phenomenon; its intensity depends heavily on the
averaging time. In general, the shorter the averaging time, the higher the peak speed for a
selected time duration. The differences among peak speeds for various averaging time indicates
the wind's gustiness. Since in weather reports, various averaging time intervals are used,
conversion must be made to uniform duration before data analysis. This conversion is
particularly important when data from more than one source are used. In offshore work, the
international practice is to use 10min. averaging wind, although fastestmile wind is also used
in the United States. For structural design, the time period selected for averaging should
correspond to the minimum response time of the structure, which is of the order of a few
seconds. For structural components, even shorter durations might be required. The gustiness
of wind depends upon a number of factors; among them, the thermal stability, topographical
peculiarity and the origin of storms. The duration conversion factor, or gust factor, varies for
varying cases. Figure 1.8 shows a graphical relation for windspeed ratios for various wind
to a
O F
5i' F 

EE IL
0
c8
Figure 1.8 Ratio of windspeed of any duration, U,, to the Ihour windspeed, U3,w.
durations. The center curve for a 10min. (600 sec.) average shows that the data are very close
to a straight line of the equation
UIomin
0.0561nt + 0.64 (1.38)
U,
where t is duration in seconds. If one wishes to determine U, from Uio.i, the following
equation is more convenient:
U, = (1.45 0.07 In t) Uomi (1.39)
For example, the maximum wind tabulated in the United States is based upon 5min.
average intervals. Such speed when converted to 30min. average a factor of 0.88 should be
applied. As another example, if the extreme value of Uom0 for the structure life is
determined to be 75 knots and the proposed structure has a response time of 6 sec., then the
design wind speed should be, according to Eq. (1.39).
U, = (1.45 0.07 In6) x 75 knots = 993 knots
Table 1.5 provides the conversion factors among the maximum Ulo.., maximum speed
(5min. average), extreme speed (fastest mile) and strongest gust. These values appear to be
borne out for the U.S. Continent but have not been critically tested for offshore regions.
TABLE 1.5 WIND SPEED CONVERSION FACTORS
Conversion Factors
Based on Uomi. Based on Usm.a
Max. UIo.. 1.0 0.95
Max. Speed (Us ) 1.05 1.0
Extreme Speed (fastest mile) 1.16 1.09
Extreme Gust (3 sec.) 1.37 1.31
Spatial Correlation
Most of the information on wind measurements has been based on data gathered over land
areas. This information has to be extrapolated to offshore sites where the structure is to be
built. Little information is presently available to deal with this spacial correlation. In general,
because the overwater wind meets far less obstructions, speeds higher than coastal stations
should be expected. Based upon hurricane data from Lake Okeechobee, Florida and limited
data comparisons in the Chesapeake and Delaware Bays, a spacial correction factor of the
order of 1.1 to 1.2 was purposed.
1.5 Turbulence and Wind Gustiness
Turbulence and wind gustiness are the fluctuating and unsteadiness features of the wind field.
The engineering expressions for these features are wind spectrum and gust factor.
The frequency and the spectral density function of wind turbulence are generally
expressed in dimensionless form. The following definitions are introduced,
Dimensionless frequency:
f = fz U,, (1.40)
Dimensionless spectral density:
S(f) = f S(f)/U (1.41)
where f = frequency in cps, z =height above sea level in meter, U,=mean wind speed at
height z in m/sec, S(f)=spectral density function in m2/sec, and U, =shear velocity in m/sec.
A number of empirical spectral formulas have been proposed in the past based on
measured wind spectra. The Davenport spectrum (Davenport, 1961) is the oldest and the most
well known. The formula is given as,
4x2
S(f)=) ( (1.42)
(1+ x2) 43
where x=1,200 f/Uo1. It was pointed out by Ochi (1988) that the Devenport spectrum and
many modified spectra by others tend to ignore the contribution in the low frequency
components (See Fig. 1.9). For coastal and offshore structures, the low frequency components
sometimes are important such as in harbor resonance and response of moored vessels. A
modified spectral function was proposed by Ochi and is shown in Fig. 1.10.
0.002 0.004
0.01 0.02 0.04 0.06 0.1 0.2
DIMENSIONLESS FREQUENCY f. fz/0z
Figure 1.9 Comparison of various wind spectral formulation
0.1 1
0001 0002 0004CI06001 002 004006 01
DIMENSIONLESS FREQUENCY f~=fz/O
0.2 04 06081.0
Figure 1.10 Ochi's wind spectral formula
I
O.
0
0.
( 0.
z
UI
i. i (./ o.o0 for f.< 0.003
420f0
4 S(f (1 f for 0.003; f.40.1
4 SO) (1 +_f.
/ 838 f,
3 + f03s .s for f. 0.1
2 d ~ (1 +f ?.35 11.5
8 Average of Measured Spectra _
6
2   "_   __
4
2
0.4 0.6 1.0
2 Coastal Waves
Waves are of primary concern in coastal engineering. They are the necessary input to
establish nearshore sediment transport, to assess coastal morphodynamics and to design coastal
structures, both in terms of structural integrity and their performance. The design wave
information requirement varies with objectives. For instance, to establish nearshore sediment
transport environment one needs long term directional wave information; for structural
integrity extreme design wave parameters are used. In this Section, general design wave
information requirement and method of analysis are presented.
2.1 Description of Ocean Waves
Ocean waves are irregular. Therefore, characterizing ocean waves often resort to
statistically based parameters. There are two ways to deduce wave parameters from measured
data; from the actual time series or from energy spectrum in frequency domain.
In the time series, the standard method of defining wave height and wave period of
each individual wave is the zero upcrossing analysis. The zero upcrossing point is defined
when the surface wave profile crosses the zero mean from the upward direction. An individual
wave height is defined by the vertical difference between the minimum and maximum levels
on two sides of upcrossing point. The corresponding period is defined by the interval of the
two crossing points. Figure 2.1 illustrates the definitions.
` T. ; T, I Ts
Figure 2.1 Definition of Waves Height and Wave Period of Irregular Ocean Waves
The commonly used wave parameters are as follows:
1. The significant wave (Hlr, T1n), which corresponds to the average of the heights
and periods of the onethird highest waves of a given record.
2. The mean wave (H., T,,), which corresponds to the mean wave height and period
of a given record.
3. The onetenth highest wave (Hi/1o, T1110), which corresponds to the average of the
highest onetenth waves in the record.
4. The root mean square wave (Hm,Tm ), which is defined as
H,., = H ]" (2.1)
The mean wave energy per unit surface area is,
E g F, H (2.2)
8N
LonguetHiggins showed that in deepwater the wave height distribution should be
expressible by the Rayleigh distribution. Therefore, the probability density function(p.d.f) for
wave height can be expressed as,
p(H)dH =: H e l /.1dH (2.3)
2 Hg
For this distribution function, all other wave height parameters can be related to the mean
value,
H,= .6 H.
Hmo= 2.03H. (2.4)
H,, H,
H. T=H.
There is no theoretical distribution function for wave period. An accepted assumption is
to assume deepwater wave length obeys Rayleigh distribution. Since in deepwater L=gTl the
following p.d.f is proposed for wave period in fully developed sea,
pCT)dT = 2. 7 erpo6r.7s/TIdT (2.5)
If one uses energy spectral based parameters, the root mean square wave height is
related to the variance of the spectrum by the following basic relationship,
HL. = 8 o (2.6)
where
S= I E()df (2.7)
is the variance of the time series and E(f) is wave energy spectrum. Other wave height
parameters can be in terms of wave energy spectrum using Rayleigh distribution assumption,
for instance, H,,=1.416 H.,=4 a and so on. As to wave period, the commonly used is the
peak energy period (T) which is the period corresponding to peak energy density in the wave
spectrum. An empirical relationship for significant wave period is also used occasionally,
T, = 3.86 H (2.8)
where T is in sec. and H is in meter.
There are a number of spectral functions proposed for deepwater and shallow water
applications. A few of them are given here.
(a) Pierson and Moskowitz (PM) Spectrum This spectrum is expressed in terms of
wind speed:
8.1Ox 10 8g g
E() = exp[0.74( )] (2.9)
294 f 21rUasjf
here U9,,. is the wind speed in m/sec at 19.5 m above mean sea level.
(b) Bretschneider Spectrum This spectrum is expressed in terms of wave parameters:
H g3 (
E(f) = 0.430( ) Yexp[0.675( j/] (2.10)
g7' T.f
4
where H, and T, are mean values of wave height and wave period, respectively.
(c) JONSWAP Spectrum This spectrum is modified version of PM spectrum based
on extensive wave observations carried out under the Joint North Sea Wave Project. The
spectrum is peak enhanced.
E( ag' 5 f [y (2.11)
E69 = (p[ ( r17'] !2 f. (2.11)
(2rx) f 4 f,
where
A =0.07 f f. (2.12)
and
A = 0.09 f > (2.13)
and f. is the peak frequency, a the phillips constant which is normally taken as 0.0080, and y
the ratio of the maximum JONSWAP spectral energy to that of the corresponding Pierson and
Moskowitz spectrum. As evident from the equation this spectrum contains five parameters.
The shapes of the three wave spectra are shown in Fig. 2.2.
2.2 Design Wave Estimation
Design Wave Based upon Measured Data
If measured wave data is available, then the statistical method of extreme value analysis
introduced in Chapter 1 can be applied for design wave estimation. The Gumbel distribution
function and the Weibull distribution function are two popular ones for extreme wave analysis.
The method of constructing Gumbel distribution curve has been described in Section 1. An
example of extreme wave statistics along Florida coast is shown in Fig.2.3 using Gumbel
distribution.
110
100
90 R 12.5m
70
60
50
40
30
20
10
0 0.02 0.04
0.06 0.08 .10 0.12 0.14 0.16 0.18
f= I/T(Hz)
Pn bls spectru~m.
! 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20
/= I/T(Hz)
PleonMoskowitz spectrum.
EJONSWAr
r E.t
E,*ai
/ I 
/.
Definition or parameters or JONSWAP spectrum (afier Houmb and
Ovwvik, 1977).
Figure 2.2 Shapes of Three Different Types of Wave Energy Spectra
RETURN PERIOD (TERR)
I1 2. s. ip 2p 9 ipo
STATION: MARINELANO
8 (OH,)IEXPCEXP ( ) 
C* C t.l d. o.M97
TYPE I LINEC(GwumLs mioM nc)
8 **CONTROL ANDO(99X C.I.) ,
* MONTHLY ORT (C.O.N.)
J1.
. 
0.0
l ov_
oo00 4.00
REDUCED
2.0oo .oo
VRRIRTE,
6. 00
.00
o.ool d.os b.i os o'.os o g'.s o'.e o:sl .' eso.s'b.
PROBRBILITT, 0 (Hs)
Figure 2.3 Example of Extreme Wave Statistical analysis Based on Measured Wave Data at
Marineland, Florida, USA
Cn
I
LL
cc
X
ll,
Similar approach can be employed for constructing other types of distribution
functions. As an example, if one uses the twoparameter Weibull distribution to plot extreme
significant waves, we have,
F(H) = 1e1 H (2.14)
or,
ln[ln{lF(H,)}]= (ln H,In H,) (2.15)
which reveals a linear relationship between H, and F(H) when plots on the appropriate log
scale known as the Weibull distribution scale. The values of H, and y are estimated from the
data using the following relationships,
H = H r( + 1)
2 (2.16)
2 1
S= H!(7+1)[ + (+ + )]
7 7
where H, and y'o are the sample mean and variance, respectively and r represents the Gamma
function.
Design Waves Based upon Design Wind Information
For locations where longterm wave data does not exist, design wave information has
to be established indirectly from wind information. The procedures are known as wind
forecasting and hindcasting. There are three basic types of techniques: empirical formulas,
wave energy spectrum method and hurricane wave description.
A. Empirical Wave Prediction Models
The representative empirical ocean wave prediction model is the SverdrupMunk
Bretschneider (SMB) method. The model relates wave parameters (Significant wave height and
wave period) to wind speed, fetch and duration by empirical formulas. The model is only
suitable for a stationary system of rather uniform wind field over the fetch in deepwater
condition. Because of its simplicity this method is still widely used.
The steps for applying this method are rather simple but certain subjectivity are
involved. The steps are outlined here.
(1) Delineate Wind Fetch. A fetch is defined subjectively as a region in which the
wind speed and direction are reasonably constant. As outlined in SPM, confidence in the
computed results begins to deteriorate when wind direction variations exceed 15" and
deteriorates significantly when direction deviations exceed 45. For practical wave predictions
the wind speed variations should be kept within 2.5 m/sec. (5 knots) from the mean.
(2) Estimate Duration. Wind speed from synoptic weather charts are prepared at 6hour
intervals. Linear interpretation is usually applied for determining the duration of constant wind
for the region of wind generation.
(3) Determine Fetch or DurationLimited. This is determined by computing the fetch
for the corresponding wind duration,
gF,= 0.015( / (2.17)
U A UA
where t = wind duration
F = fetch corresponds to duration t
UA = wind stress factor which is computed by,
UA = 0.71 U'" (U inm/s) (2.18)
and U is wind speed at instrument height adjusted with the corresponding wind duration and, if
necessary, thermo stability. In the absence of temperature information, a factor of 1.1 is
recommended to apply to U.
(4) Compute Significant Wave Parameters. The significant wave height and the
corresponding significant wave period are computed by the following pairs of equations,
gH,,3, (gFY
= 1.6x 10(o5 (2.19)
UA UA
and
=g/ 2.7x10 (gF )F (2.20)
UA UA
The F value used in the above equations should be the smaller one of the actual fetch
delineated from the weather map and F, computed from Eq.(2.16).
The formulas given above are for deepwater wave predictions. If the water depth
begins to affect the surface waves, the waves generated by the same given set of wind
condition would be of smaller wave height and shorter wave period. A number of shallow
water wave prediction models have been developed.
For the case of constant depth, a set of modified equations are given,
gF 12
gH gd 0.00565(2 )/
= 0.283tanh[0.530( f]tanh({ d } (2.21)
FA A 9 /14
U4 U tanh[0.530( /' ]
and
and
0. 0379( gF ),,
gT gd O U8A
= 7.54 tanh[O.833(u )/] tanh 0  (2.22)
U U tanh[0.833(j )j ]
Since the terms containing Tanh all approach unity as d approaches zero, these terms are
clearly shallow water corrections. For the case of sloped bottom, one can still utilize the above
set of equations to compute shallow water waves by finding an equivalent constant water
depth.
The computational procedure is rather straight forward. Design charts based on the
above listed equations have been prepared and can be found in many standard text books and
manuals. Computer programs are also available.
B. Spectral Wave Models
This type of model is based on following the development process of the wave energy
spectrum. The Pierson, Neumann and James (PNJ) method is the oldest one of this kind. Most
of the contemporary numerical wave prediction models are based on this approach. The
numerical models can accommodate spatial and time varying input wind field; can incorporate
various dissipation and energy transfer mechanisms such as wave breaking, bottom friction,
nonlinear wavewave interaction, etc. and can trace the wave field as well. Presentation of
this type of model is beyond the scope of this course.
C. Description of Hurricane Waves
A set of formulas have been developed to estimate wave conditions in deepwater at the
point of maximum wind due to hurricanes.
Rp 0.29aVp
Hj = 5.03 e4m[+ ]1 (2.23)
and
RAp 0.145aV,
T,= 8.6 e [1+ ] (2.24)
where Hi,3 is meters, T in seconds, R in kilometer, Vp in m/s., Ap in millimeters, U, is the
sustained wind at instrument height in m/s. The factor a is known as the resonance factor and
is depending on the forward speed of the hurricane and the suggested value for a slowly
moving hurricane is 1.0.
Once the significant wave height is determined for the point of maximum wind from
Eq. (2.22) it is possible to obtain the approximate deepwater significant wave height for other
area by constructing isolines using different Ap values. A nondimensional plot of significant
wave height isolines based on idealized hurricane wind model is shown in Fig.2.4
Figure 2.4 Isolines of Relative Significant Wave Height for Slow Moving Tropical
Storm.
31
The ranks of waves at a location during a hurricane passage can be established by
assuming a wave height distribution function, usually a Rayleigh distribution. For such
distribution function the ranks of waves are,
H.= 0.707 H,, In (2.25)
n
where N is the total number of waves passing the point during the hurricane and n is the rank
in the wave train. Therefore, the most probable maximum wave is for n=l and the second
largest wave is for n=2, etc. The total number of waves is determined by,
N= R (2.26)
v, T,
where N = number of waves, R = radius of maximum wind, Vp = forward speed and T, =
significant wave period.
2.3 Wave Kinematics and Dynamic Properties
With given design wave height and wave period (or wave length) the kinematics and
dynamic properties of wave induced flow field can be prescribed by the application of various
wave theories. Linear wave theory is by far the most common one for engineering
applications. A list of equations based on linear wave theory are provided here.
Relationship between Wave Period. Wave Length and Wave Phase Sueed
In water waves, wave period and wave length are not independent quantities. They are
related by the socalled dispersion relationship given by.
2 = gk tanhkh (2.27)
where o is defined as the wave angular frequency = 2 t/T with T being the wave period.
k is defined as the wave number = 2n/L with L being the wave length.
h is the water depth as shown in the definition sketch in Fig.2.5.
Direction of progressive
S wave propagation
Figure 2.5 Definition Sketch of Gravity Waves
The wave phase speed, C, also known as wave celerity, is given by,
C = = [/tanhWkh (2.28)
T k
Progressive Waves Vs Standing Waves
If wave phase propagates towards certain direction such as towards shoreline, it is
known as progressive wave. On the other hand, if the wave phase simply oscillates up and
down as in front of a vertical wall it is called a standing wave. For given wave height, H, and
wave period, T (or wave length, L), the water surface variation, rl, respect to the undisturbed
mean free surface for progressive waves is given expressed as,
H
rl = cos(kf oat) = a cos(F oat) (2.29)
2
where a = H/2 is defined as wave amplitude. The negative sign corresponds to waves
propagating towards positive xdirection whereas the positive sign corresponds to negative x
direction.
For standing wave, the expression becomes,
H
r = coskccosot (2.30)
2
Usually, we are more interested in progressive wave than standing wave. There are, of
course, exceptions such as the case of vertical walls or end of channels where waves bound to
reflect and creates standing waves in front of walls. In the following few sections, we only
address the properties under progressive waves. Their standing wave counterparts can be easily
expressed.
Water Particle Kinematics for Progressive Wave
Particle Velocity
The horizontal velocity under the wave is given by:
H coshk(h+z)
u= oC cos(krat) (2.31)
2 sinhkh
The local acceleration is then,
Hu H coshk(h+ z)) (
f 'i" sin(kxat) (2.32)
Ot 2 sinhkh
and the vertical velocity and local acceleration are:
H sinhk(h+z) sin( (2.33)
v2a sinFkxt) (2.33)
2 smhkh
Examining these equations one sees that:
av H sinhk(h+z)
2= h' cos(kxat) (2.34)
9t 2 sinhkh
1. Horizontal and vertical velocities are out of phase.
2. Velocity component and its acceleration component are out of phase.
3. All components have maximum values at the surface and decrease with depth and
follow hyperbolic function variations.
4. At bottom, the vertical velocity is zero and horizontal velocity is not.
The velocity components for four phase positions are shown in Fig 2.5.
Particle Displacement
.1
The water particle displacement with respect to a mean position, (x,, z2), is given by
H coshk(h+z;)
H= coshk(h + z) sin(krx o) (2.35)
2 sinhkh
for horizontal displacement and
H sinshk(h+ z,)
=H nh + cos(kx, at) (2.36)
2 sinh kh
for vertical displacement.
Pressure Field
For progressive wave the pressure field is,
H cosh k(h+ z)
p=pg+pg coshk cos(k at) (2.37)
where the first part is due to hydrostatic pressure and the second part is the wave dynamic
pressure.
Wave Energy and Wave Energy Flux
Wave energy contains two parts: the wave potential energy and wave kinetic energy,
the formal due to the disturbance of water surface from an equilibrium position whereas the
latter is the result of water particle motion beneath the surface. Therefore, it is convenient to
express wave energy density in terms of unit surface area. Since the free surface is periodically
energy varies from location to location. The spatially averaged energy, referred to as mean
wave energy, can be expressed as
1. Mean Potential Energy (with respect to the equilibrium surface) is equal to Ep = pg
H2/16.
2. Mean Kinetic Energy is also equal to E. = pgH2/16.
Therefore, the mean energy is equally partitioned. The total energy being equal to the
sum becomes E = pgH2/8.
Energy flux is the transmission of wave energy across a vertical plane from the mean
free surface to the bottom. It is expressed in energy transmission per unit crest surface. The
mean energy flux across a unit crest is the value averaged over a wave period. It is given by:
= E (2.38)
Here C, is known as the group velocity which is the speed energy is being transmitted by a
train of wave traveling as a group. In linear wave theory, it I related to the wave celerity by:
C, = nC (2.39)
where
1 2kh
n = (+ (2.40)
2 sinh 2kh
2.4 Wave Transformations in Coastal Water
Ocean waves entering coastal water will undergo various transformations due to bottom
effects and the presence of obstacles such as breakwaters, jetties and islands. Waves eventually
becomes unstable and break at a certain depth. In this section, wave transformations in coastal
water are briefly treated based mainly on small amplitude wave theory.
WAVE SHOALING AND REFRACTION
Wave shoaling and refraction are two of the most common shallow water effects
included in engineering computations. Shoaling is the increase in wave height in shoaling
water and refraction is the change in wave propagation direction over a sloping bottom.
The general wave energy transport equation is given by:
+ A*(CsE)= G, (2.41)
where E is wave energy, C, is wave energy transport velocity, also known as the group
velocity and G, are energy source and sink terms. For steady state case and ignoring the energy
dissipation and generation we have,
A*(C,E)=O (2.42)
For the case of straight and parallel bottom contours as shown in Fig.2.5, recognizing that
there is no energy flux across wave rays, the above equation gives
(EnC), Bi = (EnC),b (2.43)
Using linear wave theory to express E= pgH', we can solve for the wave height:
8
Depth contours   1 1 
Figure 2.6 Wave energy conservation between wave rays.
= V (2.44)
here the subscript 0 refers to deepwater condition.
For the case of waves propagate perpendicular to the contour, bo = b, the wave height
change is strictly due to shoaling. When waves propagates with an angle, the distance between
wave rays also changes because of changing wave angle. This latter effect is known as wave
refraction. To determine the wave refraction effect we resort to the property of wave number
irrotationality. We define the wave number vector k as the gradient of a potential, 0, or,
k = AQ (2.45)
Taking the curl of k gives
Axk=O (2.46)
Substituting the components k yields
(k sin 0) (k cosO)
=0 (2.47)
ay
The equation given above together with the wave dispersion relationship,
36
a = gktanhkh (2.48)
permit us to determine the wave number changes in shoaling water with given wave period and
input wave angle, 0, at the offshore boundary. For straight and parallel offshore contours, the
following Snell law is obtained,
sin sin 0o
(2.49)
C Co
From simple geometrical relationship we obtain,
be cosOo sin2o (/
b cos J1sin 0)
Therefore, Eq. (4) can be written as
Ho= I C, lT sin 00o 114 (2.5s n
H C( Isin'o fi = K,K, (2.51)
Ho C, 1sin 9
where K. is the shoaling coefficient and K, refraction coefficient.
For irregular bottom topography, the wave refraction an shoaling can be computed by a
variety of numerical techniques including wave ray tracing method, finite difference numerical
scheme with varying degrees of sophistication and accuracy.
WAVE DIFFRACTION DUE TO STRUCTURE
When incoming waves are interrupted by a barrier such as a breakwater or the tip of a groin,
the incident waves curve around the barrier and spread into the shadow zone such as shown in
Fig.2.6. This phenomenon is called wave diffraction which is a process of transferring wave
energy laterally along a wave crest. Without this process wave energy will approach infinity at
these singular points where wave rays cross each other. Assuming the velocity potential to
have the following time harmonic functional form,
f(x,y,z,t) = coshk(h+ z)F(x, y)e"' (2.52)
the Laplace equation yields the following Helmholtz equation,
d'F i'F
+ + k'F = O (2.53)
where F(x, y) is defined as a surface wave potential. This is the basic equation for computing
wave diffraction behind a structure. A limited number of theoretical solutions are available for
simple structural geometry, notably, semiinfinite breakwaters in constant water depth, small
gaps between breakwaters. Again, numerical models are available for more complex
geometry. Similar to shoaling a refraction, a diffraction coefficient can be defined as:
H
= Kd (2.54)
HI
Figure 2.7 shows the diffraction coefficient around a semiinfinite rigid breakwater.
Figure 2.6 Wave Diffraction around a Detached Breakwater (From SPM, 1984)
8 6 4 2 0 2 4 6
L
Figure 2.7 Wave Diffraction Coefficient in the Lee of a Semiinfinite Rigid Breakwater.
Design charts for different incident wave angles for semiinfinite breakwater and other
simple geometrical forms such as parallel breakwater with gaps, etc. can be found in a number
of coastal engineer design manuals.
COMBINED REFRACTION AND DIFFRACTION
For structures at locations with irregular topography, both wave refraction and
diffraction will occur. At present, there is no analytical solution even for simple geometry.
The socalled mild slope wave equation can be used to derive numerical solutions. The mild
slope equation was first proposed by Berkhoff(1972) which has the following form:
Ah (CC, A F) + a ( )F = 0 (2.55)
Numerous numerical sChemes have been developed for different applications.
WAVE REFLECTION
When waves impinging on a structure, part of wave energy will be reflected. The wave
reflection effect is usually localized but is rather difficult to compute except for very simple
geometry such as vertical wall in constant water depth. The effect of wave reflection is often
expressed by a reflection coefficient, KR, which is defined as the ratio of reflected wave height
to incident wave height. To balance wave energy flux across a structure, the following
equation is applicable:
K + K' + energy loss = 1 (2.56)
where KT is known as the wave transmission coefficient which is the ratio of transmitted wave
height to reflected wave height. In most instances, both KR and K, are determined by
laboratory experiments.
WAVE DAMPING DUE TO BOTTOM FRICTION
Waves propagating over shallow water will experience energy dissipation due to bottom
frictional effect and gradually attenuates. This effect usually is not pronounced in short
distance but it is cumulative over large distance. This energy dissipation is enipirically
estimated using the quadratic energy dissipation equation as follows:
1
= o*u= fp_u_uu (2.57)
where f is bottom friction coefficient. To compute mean wave energy loss, the above equation
needs to be time averaged. This is not as easy a task particularly if there is background mean
current.
WAVE BREAKING TRANSFORMATION IN SURF ZONE
On a slope, wave shoaling will eventually lead to wave breaking. After breaking, the
flow becomes turbulent and the wave height decays rapidly. At certain point close to the
shoreline, the wave totally collapses and the water mass runs up the beach in the form of a
bore. In addition to the primary oscillatory flow motion a mean circulation generally exists in
the surf zone. This mean surf zone circulation is, in general, three dimensional and is often
decomposed into crossshore and longshore flows. For complicated topography rip current
could appear intermittently. Under such circumstance, the longshore and crossshore
description might not be adequate. The interactions of incident waves and the topography
could also induce long period motions in the nearshore zone. Depending upon the modes of
oscillations, they are variously described as low frequency waves, infragravity waves, edge
waves, etc.
Types of Breaker
Breaking waves can assume different forms and appearances. They can be roughly
classified into four types.
Spilling breakers
Plunging breakers
Surging breakers
Collapsing Breakers.
Spilling breaker initiates as the slope of the forward face becomes unstable. A plume of
water and air mixture soon forms and slides down slope from the crest. The plume then travels
with the wave in the form often described as a surface roller. Spilling breaker is also the most
common type in deep water breaking due to wave instability.
For a plunging breaker the crest of the wave curves forward and plunges onto the
trough in the front. The jetlike water mass, upon impinging on the surface, continues to
penetrate into the water column producing coherent vortices, which can often reach the bottom
and then generate splashup.
In surging breaker, the wave front steepens without breaking. Turbulence eventually
appears at the toe. Wave then rushes upslope in a borelike motion with diminishing wave
crest and finally becomes a sheet of water runs up the beach.
Incipient Wave Breaking Criteria
Wave breaking is described by breaking wave height, Hb, breaking depth, hb, and
breaker types. All of them are empirically defined. A commonly used index is the ratio of Hb
to h, known as the "breaker depth index", Yb. The earliest breaker depth index was introduced
by McCowan (1894) based on the instability of solitary wave. The index is a simple constant
value of 0.78 (commonly referred as Miche's criterion). This criterion is still widely used.
Later on, a variety of breaking criteria have been proposed. A number of them are
summarized in Table 2.1.
Table 2.1 Summary of wave breaking index (yb)
Author yh note
McCowan (1894)
Galvin (1968)
Weggel (1972)4
Sunamura (1980)
Moore (1982)
Larson and Kraus
(1989)
Smith and Kraus
(1990)
y, = 0.78
b = 1.087me0.07
1
b1.40 6.85mm 0
Hb
Yb = b(m)a(m) 
a(m) = 43.8(1.0 e"')
b(m) = 1.56(1. 0 + e"s) 1
y = 1. i[ ]"'
(Ho/ Lo)l
H s
Yb =b(m)a(m)( )
21r Lo
a and b same as Weggel
m )
Yb = 114[ (Ho/L)
Lo)
yz =b~ma~)( .,
solitary
lab.
lab.
lab.
hybrid
1 1
80 10
Ho
0.007 < 0.0921
La
Although these criteria are very different in functional form, they do produce certain
common results such as, for a wave of given steepness, steeper beach slope yields larger y,
value. On the other hand, for a given beach slope, steeper wave produces smaller Yb value.
For random waves with an assumed Rayleigh wave height distribution, Battjes and
Janssen (1978) proposed the following expression derived from Miche's criterion:
0.88 .rTkh
H, tanh J
k 0.88)
(2.58)
where is a mean wave number and y is an empirical parameter given by Battjes and Stive
(1985) as:
S=0.5+0.4tanh 33 H (2.59)
\ L,o
where the subscript "eo" refers to equivalent deepwater values. Thornton and Guza(1983)
suggested a simple expression H,= yh with y, to be around 0.4 for inner surf zone.
Post Breaking Wave Decay
After incipient breaking, waves proceed towards the shoreline with rapid wave height
decay. The zone between the incipient breaking point and shoreline is defined as surf zone.
Surf zone can generally be divided into an outer zone of flow establishment and an inner zone
of broken waves. For spilling type breakers these two zones are more separable than the other
two types. The rate ofWave height decay are influenced by many factors; two of the important
factors are the breaker type and the bottom slope. Figure 2.8 for instance, shows the effect of
bottom slope on the decay of spillingtype breakers. The process of wave decay is extremely
complicated. Modeling efforts were mostly restricted to prescribe the gross flow properties of
highly simplified nature. At present, these models are largely restricted to application of
spilling breakers in the inner zone..
Wave decay models such mostly based on the energy conservation at steady state which
is given as:
a(F)
=(x) (2.60)
in which 6 is the rate of energy dissipation per unit surface area; F is energy flux, and x is the
horizontal coordinate. Depending upon the approaches of estimating F and 8, various model
were proposed. The simplest kind leads to wave height proportional to local water depth.
Shore Protection Structures
Shore protection structures addressed here are loosely defined as structures constructed in
the inshore zone for the purpose of reducing wave energy, mitigating shore and beach erosion,
improving or training intercoastal navigation channels and inlets. The traditional type structures built
with stones, steel, timber and concrete are now called hard structures to differentiate from such
measures as beach nourishment, vegetation and other schemes with little or permanent structural
components. The latter group is called soft structure. In this section, we deal mainly with hard
structures. Method of beach nourishment will be discussed in subsequent section. The main topics
covered here including breakwaters, groins and jetties.
1. Breakwaters
The main function of breakwaters is for wave energy reduction on the lee side such as harbors
and ports. They are also used for other purposes such as coastal protection or bank protection. There
are a great varieties of them in shapes, forms, materials and functions. In terms of design, they can
roughly categorized into two major groups: rubble mound and composite breakwaters. Floating
breakwaters also exist but of much smaller scale and limited application. The rubble mound type is
generally of trapezoidal shape containing a core of aggregates and layers of protective armors of
natural stones or manmade concrete blocks. The composite type is a monolithic vertical structure
on a rubble foundation. Depending upon the height of the rubble foundation, the composite type is
further differentiated as vertical type with low or below ground foundation, composite type and high
mound. Breakwaters can be shore detached, shore attached, subaerial or submerged.
Shoreconnected breakwaters are mainly for providing calm water for harbors and boat
protections. They are subaerial structures. Shoredetached breakwaters are usually shoreparallel
located in water depth between 2 to 10 meters although more ambitious projects in waters up to 20
meters have been contemplated. The structure can be continuous and segmented. Their function is
for providing calm water, shore protection or both. Structure for bank protection along open coast
sometimes called revetment in essence is a breakwater with no leeside slope. The design
considerations, however, are essentially the same.
1.2 Rubble Mound Structures
General Description and Types
A standard typical crosssection of rubble mound breakwater is shown in Fig. 1. This is the
recommendation from SPM. There are, of course, many variations depending upon local condition,
local code requirement and individual designer's performance.
tock Slta
Rock Sir Layer Gradtion (%)
d y Cover Layer
To Berm and First Underlaye
saoond Underler
Com and Bed ng Layer
125 to 75
130 to 70
150 to 50
17010 to 30
H Wmv Height
W Weiht of Indivdu Arnor Unit
r Avera Layer Thickne
,Crest Width
Recommended Threeloyer Section
Figure 1 SPM Recommended Standard Cross Section for Wave Exposure on Both Sides
with Moderate Overtopping Conditions
W
W/1O0
W/200
W/4000
This crosssection consists of three major layers: the outer most layer known as the armor layer; the
layer beneath known as the secondary layer and the inner core layer. This breakwater as
Recommended by SPM is exposed on both sides. Moderate overtopping is permitted. In general
suitable foundation material is required. The armor layer is typically built with largesize quarry stones
or concrete blocks. The secondary layer, in general, is made of quarry stones of lesser sizes. The
center core is commonly formed with quarry runs or crush stones. Although uncommon, on
occasions of smaller structures sand can be used.
As stated there are many design variations for special reasons such the limitation and
availability of construction material, equipment limitations, etc. From the structural functional point
of view there are two main reasons requiring design variations: for poor soil condition and for
reduction of wave force and overtopping. If the soil condition is poor, a substantial base must be
constructed to support the structure or employing soil improvement measures, or both. Figure. 2
shows an example the breakwater at the Hook of Holland built on soft base material. Here a
substantial base was constructed to combat the poor foundation material.
Various design variations are also sought to reduce the wave force on the main structure
and/or to reduce overtopping; these will result in reducing the size requirement of armor units and
the height of the structure. Recently, the socalled berm breakwater has gained some recognition and
will probably see more application in the future. A berm breakwater shown in Fig. 3 is a breakwater
with an extended flat section near the high water level. This extended flat section is aimed at
dissipating the wave energy by inducing early breaking as well as reducing runup and the subsequent
wave overtopping. Thus, the function of this flat section is much like the berm of a beach that the
nature creates to protect the backshore.
Breakwaters can also be used as direct means for shore and bank protections. A shore parallel
detached breakwater or a series of segmented detachedbreakwater built close to the shoreline are
examples of this kind. Their sole purpose is to build up the beaches behind them. Figures 4 and 5
show two examples. Figure 4 shows a single detached breakwater and Figure 5 shows a series of
segmented breakwater. We can see that a tombolo or a series oftombolos were formed behind the
detached breakwaters.
Rubble structures built directly against bank slopes are called as revetment or ripraps. They
could be substantial structures just like conventional breakwaters or lesser structures made of few
layers of stones and gravels. Figure 6, for instance, shows a rather substantial structure for land
retention whereas Figure 7 shows a simple type for bank protection. Structures designed for land
retention will be discussed later in a separate Section. Only conventional breakwater design is
addressed here.
Figure 2 Cross Section of Breakwater at the Hook of Holland on Soft Base Material
5.0 4.2 3.0 / 6.0
HVL +1.3
Sand Back Fill
Geotextle Fitter
Prefabricated Drain Area
28.5
Unlti n
Figure 3 An example of Berm Breakwater
7.4 ., A a0.O
I I1I1
*~t 9 r 4'
al.. At
Wk
Figure 4 A single detached Breakwater, Ratington Beach, Florida
Figure 5 Segmented breakwater,
'/
Wave
Reflector
Armor
Key
Figure 6 A typical crosssection of rubble structure for land protection
100 150
I
Figure 7 A simple Revetment structure
Design Procedures
The major steps in rubble mound breakwater design is outlined in the schematics shown in the
following:
Configuration
Design Condition
Cross Section
External Forces
Stability Analysis
Detailed Design
In preplanning stage, it is important to first gather and analyze information concerning
material cost and availability, equipment limitation and local labor situation.
Governing Design Parameters
The governing parameters related to the cross section are shown in Fig. 8. They are:
h : Water depth of structure relative to swl
h,: Peak crown elevation relative to design water level
h : Breakwater crest relative to design water level
h,: Depth of structural toe relative to swl
B :Crest width
a : Front slope
; : Back slope
t's: Layer thicknesses
Determination of these parameters is an iterative process involving structure and flow interaction
analysis. The design methodologies are discussed here.
Figure 8 Governing dimensions related to crosssection
Design Structural Elevation. Runun and Overtonning
The structural design elevation (peak crown elevation) is determined from design high water
plus wave setup, wave run up and freeboard. If overtopping is allowed, the freeboard is equal to zero
and the allowed overtopping is deducted from the design elevation. The selection of design high water
and design wave depends upon the importance of the structure. For major structure 100 yr return
period is a preferred choice.
For sloped structures, the wave setup is usually neglected. The wave runups are mainly
influenced by two parameters, the surf parameter, L, and core permeability. The surf parameter is
defined as
m=mtana/s= tana/ 2nH3/gT
(1)
Based on laboratory data, the following empirical formulas were proposed by Van der Meer (1981):
RUlHS = a m
R1/H =bc
UX
for m< 1.5
for >1.5
(2)
The runup for permeable structures is limited to a maximum:
Values of the coefficients for different levels of exceedence are shown in the following Table:
Table 1 Values of Coefficients a, b, c and d for different exceedence levels
0.1 1.12 1.34 0.55 2.58
1 1.01 1.24 0.48 2.15
2 0.96 1.17 0.46 1.97
5 0.86 1.05 0.44 1.68
10 0.77 0.94 0.42 1.45
Sign. 0.72 0.88 0.41 1.35
Mean 0.47 0.60 0.34 0.82
level (%) a b c d
Results of experiments and the equations are shown in Figs 9 and 10 and for example values of 1=2%
and significant, for impermeable and permeable core material
1.5
0 1 2 3 4
Cm
Figure 9 Relative 2% runup on rock slope
9
Figure 10 Relative significant runup on rock slopes
The above formulas were later was consolidated into one of the following form:
Ru2%/HS=1.5y with a maximum of 3.0y (4)
where C = the surf parameter based on the peak spectral period and y = a total reduction factor
given as the product of individual factors:
Y=nHiY (5)
Some of the important individual factors included berm, roughness, oblique waves and
overtopping. A equivalent slope method is suggested to estimate the reduction factor for berm. The
equivalent slope is simply a straight line between points on the slope 1.5 H, below and above the
berm. The reduction factor due to roughness is given in Table 2:
A simple approach has been suggested to treat overtopping as a negative free board.
Overtopping is defined as discharge per unit crown width, or, Q =m3 /sec/m. Figure shows the
dimensionless plot of overtopping vs reduced run up. The mean curve is represented by:
p(Q) =8.105 rgiexp[3.1(Ru2Rc)/Hs] (6)
The variation coefficient V is defined as o/p. The confident level is established by assuming that Q
follows lognormal distribution. With a given allowable overtopping the reduction of crown elevation
can be computed with the aid of Fig. 9 or from Eq.(6).
Table 2 Wave Run up Reduction Factors
Covering
reduction Factor, yf
Smooth, concrete, asphalt, 1.0
impermeable smooth block revetment
Grass 0.9 1.0
1 layer of rock 0.55 0.6
2 layers of rock 0.500.55
The effects of oblique waves and short crested waves are the general reduction in both run
up limit and overtopping discharge. Figure 11 provides the values of reduction factor as observed in
the laboratory. The reductions in run up and overtopping by short crested oblique waves, more similar
to nature, are far less than that of long crested waves.
The chart presented in Figure 12 provides a design guideline on the allowable overtopping.
Crown Width
The width of crest or crown depends greatly on the degree of allowable overtopping; where
there will be no overtopping, crest width is not critical. As a rough guideline for breakwaters with
a low degree of overtopping, the crown width should be about three armor units (n=3) (SPM) or
around 3 meters (Japan Harbor Design Manual). The SPM formula for crown width is:
B=nk (W/p) 1/3 (7)
where B = crown width,
n = number of stones (n=3 minimum)
kA = layer coefficient; generally can be taken as equal to 1.
W = weight of armor unit in primary cover layer
p,= density of armor unit.
2
5
'4
0
5
6 
.5
0 .5
(Ru2%Rc)/Hs
Figure 10 Wave overtopping over slopes
0 ." '
0 10 20 30 40 50 60 70 80 90D 
ongle of wove ottock p
Figure 11 Influence of oblique long and short crested waves on overtopping and runup
1000
100
10
10
0.1
0.01
Uncomfortable
but not
dangerous
Wet, but not
uncomfortable
Structural
damage
.. .
Minor damage
to fittings etc.
No damage
_~~~. _ '~r ~ ____ _ I
Damage even
if frly.
proteted'::
Damage if
back lope
not protected
Damage If
crest not
protected
No damage
Damage if
promenade
not paved
1000
100
10
1.0
Nodamage 10.1
Vehicles Pedestrians Buildings Embankment Revetment
seawalls seawalls
SAFE OVERTOPPING DISCHARGES
Figure 12 Guideline for save overtopping discharges
0.01
0.001
0.0001
Dangerous
0.0001 
Unsafe at any
speed
Unsafe at
high speed
Safe at all
speeds
  
0.001 
Armor Layer Stability
Armor stability is determined by ratio of the upsetting force exerted on the armor unit versus
the resistance offered by the unit. The main upsetting force is considered as due to waves and the
resistance is mainly due to the weight of the unit and the interlocking ability with its adjacent blocks.
Although there are a great variety of design formulas, the one proposed by Hudson (1958) is still the
most widely accepted. The original Hudson formula is given as:
psH3
W= P (8)
K53cota
where W = median weight of armor unit in primary cover layer, H is the design wave height, A is the
specific gravity of the submerged unit, a is the breakwater slope and KD is a stability coefficient. This
coefficient lumps the effects of all the unspecified and unknown factors, among them the more
important ones are: shape of armor units, method of placement, number of layers, wave forms and
the location of the unit whether on structural trunk or structural head. KD values suggested for design
could correspond to "no damage" up to 5% of the armor units displaced. Suggested KD values for
use in determining armor unit weight by SPM is given in Table 3.
Other alternative formulas for armor unit weight determination can be found in Brunn (1985)
and Van der Meer (1992).
Underlayers and Filter Design
In rubble mound breakwater design the structural stability against external forces is solely
provided by the armor layers. The function ofunderlayers is purely for preventing core material from
escaping. The basic design requirements are:
1. Prevent fine material (base material) from leaching out.
2. Allow for sufficient porosity to avoid excessive pore pressure buildup inside the
breakwater that could lead to instability or liquefaction in the extreme case.
These two requirements conflict each other and a proper engineering solution simply seeks
a middle ground.
Since armor layer(s) for breakwater is usually of fairly large size requirement (2) can usually
be satisfied. Size determination can be based on geometrical consideration aided by past experiment.
As shown in the sketch for idealized spherical shape the base material cannot escape the cover layer
if the diameter ratio of the cover material to the base material is less than six, or D/d <6.
Table 3. Suggested K, Values for use in determining armor unit weight'.
NoDamage Criteria and Minor Ovetopping
Armor Units n' Placement Structure Trunk Structure Head
K,' K, Slope
Breaking Nonbreaking Breaking Nonbreaking Cot 0
Wave Wave Wave Wave
Quarrystone
Smooth rounded 2 Random 1.2 2.4 1.1 1.9 1.5 to 3.0
Smooth rounded >3 Random 1.6 3.2 1.4 2.3 s
Rough angular 1 Random 4 2.9 4 2.3 5
1.9 3.2 1.5
Rough angular 2 Random 2.0 4.0 1.6 2.8 2.0
1.3 2.3 3.0
Rough angular >3 Random 2.2 4.5 2.1 4.2 s
Rough angular 2 Special' 5.8 7.0 5.3 6.4 '
Parallelepiped7 2 Special' 7.020.0 8.524.0 
Tetrapod 5.0 6.0 1.5
and 2 Random 7.0 8.0 4.5 5.5 2.0
Quadripod 3.5 4.0 3.0
8.3 9.0 1.5
Tribar 2 Random 9.0 10.0 7.8 8.5 2.0
6.0 6.5 3.0
Dolos 2 Random 15.8' 31.8' 8.0 16.0 2.09
7.0 14.0 3.0
Modified cube 2 Random 6.5 7.5  5.0 '
Hexapod 2 Random 8.0 9.5 5.0 7.0
Toskane 2 Random 11.0 22.0 
Tribar 1 Uniform 12.0 15.0 7.5 9.5 s
Quarystone (Ku)
Graded angular Random 2.2 2.5
'CAUTION: Those K, values shown in itaics are unsupported by test results and are only provided for preliminary
design purposes.
'Applicable to slopes ranging from 1 on 1.5 to 1 on 5.
'n is the number of units comprising the thickness of the armor layer.
4The use of single layer of quarrystone armor units is not recommended for structures subject to breaking waves, and
only under special conditions for structures subject to nonbreaking waves. When it is used, the stone should be
carefully placed.
'Until more information is available on the variation of KD value with slope, the use of KD should be limited to slopes
ranging from 1 on 1.5 to 1 on 3. Some armor units tested on a structure head indicate a K,slope dependence.
'Special placement with long axis of stone placed perpendicular to structure face.
TParallelpipedshaped stone: long slablike stone with the long dimension about 3 times the shortest dimension
(Markle and Davidson, 1979).
Refers to nodamage criteria (< 5 percent displacement, rocking, etc.); if no rocking (< 2 percent) is desired, reduce
K, 50 percent (Zwambom and Van Niekerk, 1982).
'Stability of dolosse on slopes steeper than 1 on 2 should be substantiated by sitespecific model tests.
For sorted material such as quarry stones the stability criterion from soil mechanics is given
<5
where the subscripts refer to percentage passing. Under dynamic load condition such as wave forces,
more strict geometrical rules are applied. For wellsorted stones, it is common to select
=2.5 to 3 (10)
which translates into weight ratio as,
W15 o25
15 to 25
(11)
The standard SPM breakwater cross section recommends the following underlayer sizes:
Layer Weight Ratio Equivalent Diameter Ratio
Primary Armor Layer W/1 1
First underlayer W/10 2.15
Second underlayer W/200 2.7
Base material W/4,000 2.7
Filter layer(s) usually refers to the layer between the structure and the foundation material
in the case of breakwaters or between the cover layer and the bank material for revetment type of
structures. The main function of filter layer is still in preventing base material from leaching out. Since
the base material is now much finer than stones, pore pressure build up in the base material becomes
an important concern. The design filter layer should also consider the hydraulic behavior of porous
flows. The following criteria are suggested for granular filter design:
(9)
(a):
<4 to 5
d5
(b):
>4 to 5 (13)
dl5
where D and d are the granular diameters of the filter and base material, respectively. Criterion 'a'
states 15% of the coarsest base material is prevented from entering the filter. This 15% of the material
further forms a layer beneath the filter to prevent the subbase material from leaching out. Criterion
'b' assures free seepage flow to prevent pore pressure build up. In addition to the above constraints,
the following conditions are often stipulated to maintain filter layer internal stability:
<10
40
Therefore, poorly sorted material (D60/D.0o20 or larger) is not suitable for filter.
The guideline for layer thickness is as follows:
a). 2 3 times the diameter for large stone.
b). 10 cm for coarse for coarse sand.
c). 20 cm for gravel.
Geotextile filter clothes are also common, particularly, for bank protection structures and for
structural toe protection. They have the advantage of uniform property and quality but could be
susceptible to weathering, tearing, clogging and flopping.
Toe Protection
Toe protection design represents one of the more difficult tasks partly because there is no
rigorous design criterion and very little research. Design is often carried out in an ad hoc manner or
based on laboratory testing. Therefore, toe failure was common in the past which sometimes led to
major structural failure. Toe structure serves two purposes: supporting the armor layer from sliding
and preventing scouring. To serve the first function the arrangement the armor layer should be
anchored by the toe. A number of different arrangement are illustrated in Fig. 13. To prevent
scouring, the size of the toe structure as well as individual stone should be adequate to resist current
and wave forces and to prevent leaching and failure of underlying material.
IF
Lot namrau s amr mw PtLt l tifLe
poDzJntaLtfEtor Bslor wntllbl SLtL
I
L..
M31aS.S*1
Fig 1I'. Examples of Toe Protection Structures
18
Usually one can assume that if the unit in the toe has the same size as the armor unit the toe
will be stable. For economic reason one likes to reduce the size in the toe. Based on limited laboratory
results, the depth of toe below water level, lh, has been identified as one of the important factors. It
is easy to reason that the larger the value the h, the smaller the stone size is needed. Figure 18 shows
the results from a CIAD report (1985) relating the nondimensional toe water depth to the armor
stability number (N) which is defined as H,,/L 0.
Toe stability
"1 00.3X DH
depth limited conditions x 310o OH
310X DOH
.A >20X OH
suggested design curve 0 o >20oX o
P.  P (Ha)
. SP (H10)

./ l0
1 2 3 4 8 7
H. /ADn5o
Figure 17 Toe Stability as a function of depth of toe
Empirical equations based on the data are given as:
ht; H.
0.22( (15)
h Ao
for mean value, and
h Ao5
for 90% confidence level.
The above equations can be used as a simple guideline for the selection of toe armor unit. If
underlayer(s) is required the design criterion is the same as filter layer design. The berm width of the
toe is recommended to be at least of three armor stones.
If no structural toe is used, the armor layer should penetrate below the maximum scouring
depth. While there are some limited guide lines for scouring under pure current field the knowledge
on wave induced scouring is very poor. Through dimensional analysis, the following nondimensional
groups can be established,
S f(, U, D ,) (17)
D /(s1)gd v
where S. = ultimate scouring depth.
D = diameter of cylinder
u* = shear velocity
U = free stream velocity
d = mean sand grain size
v = kinematic viscosity of water
S= sand internal friction angle
The above relation is often simplified to
S =f( ) =f(Ns) (18)
D /(s1)gd
The nondimensional parameter on the RHS, N,, is known as the sediment number. For median to
coarse granular bottom material it was found that there exists a critical sediment number that the
ultimate scouring depth reaches a maximum. For cylindrical body such as a bridge pier, the laboratory
test results are shown in Fig. 14. The scouring on the left hand side of the peak value is known as the
clearwater scouring whereas the scouring on the right hand side of the peak value is known as the live
bed scouring. Therefore, initially once the free stream velocity exceeds the incipient velocity scouring
increases rapidly with increasing velocity until a peak is reached. Beyond the peak the scouring depth
is compensated by live bed material. This is, of course, a very desirable property for engineering
purpose. The peak value is estimated to be about 2.3, or, for a cylindrical body the maximum scouring
is about 2.3 times the diameter. Most of the experiments, however, were limited to small cylinders.
Caution must be exercised to apply them to prototype.
If the results from the cylindrical body is adopted the peak scouring value can be corrected for
application to rectangular shapes by the following formula,
2.3KA,
D
where K, = shape correction factor with the following suggested values
Aspect ratio
1
2
3
and K. = angle of attack factor.
0
5
t
2.5
2.0
1.2
1.2
COARSE SEDIMENT
RIPPLEFORMING SEDIMENT
livebed scour
0.$
1.5 2.5 3.5
4.5  U/U,
Figure 14 Relative scouring depth for cylindrical body in uniform current
The presence of waves over a current field usually aids in the reduction of scouring provided
waves do not break over the structure and penetrates to the bottom since waves promotes live bed
scouring in addition to being oscillatory.
Scouring due to breaking waves is a different topic with very limited available information.
One of the major difficulties in the laboratory experiment is the equal importance of both Reynolds
number and Froude number effects. This can be illustrated in the following nondimensional analysis,
su H
 f(
H gTe
U* u*d
/(s1)gd V
The first term on the LHS is a wave steepness parameter and has to be simulated in accordance with
Froude similitude whereas the second and third parameters are mainly Reynold number effects. In
the breaking wave field they are of comparable importance. Experiments showed that scouring depth
increases rapidly with increasing wave steepness until to a maximum point is reached; further increase
in wave steepness causes a fast decrease in scouring depth. The rapid decrease in scouring is
UD D
HI
v H
associated with the presence of ripples. Therefore, the formation of ripples in front of the toe
structure is an indication the scouring has reached a stable stage. Based on experience, the wave
induced scouring is usually about 10 to 20 percent of the incident wave height. However, under
certain circumstances, the exact nature of which is still unknown, scouring depth could be
significantly larger reaching the same order as the incident breaking wave height. Therefore,
depending upon
the vulnerability of the structure the design scouring depth could be from 0.1 to 1.0 the incident wave
height.
2.9 Slope and Foundation Stability
The sliding failure of rubble mound slope as a whole is usually referred to as macrostability
to differentiate from the armor layer failures including armor layer sliding failure. This macro
instability can also involve foundation failure. Slope instability is usually a cumulative effect but often
triggered by a critical event such as heavy rain fall or storm waves. Some symptoms prior to major
include sudden structural settlement, foundation bulging at the toe or large lateral deformation,
propagating longitudinal cracks on the slope followed by lateral cracks, inordinate amount of local
seepage, etc.
The standard slope stability analysis is based on a two dimensional arc surface failure mode
such as shown in Fig. 15. The arc is then sliced into strips each of them is balanced by force diagram
as shown. The stability coefficient is expressed by,
ir
K= (21)
where M, is the moment of shear resistance and M, is the moment of sliding force. They are given by
M,= (Ecl+E + E cosoqctarn^) R (22)
and
M,= (EWisini) R (23)
here 1= arc length
W= net weight
c = soil cohesion strength
tan4 = soil internal friction angle
a = angle between the tangent of arc and horizontal
R = arc radius
The critical failure surface is then determined by trial and error. This task is best left to
computer. Various programs are commercially available.
HIGH SEA
NORMAL SEA
Figure 15 Slope Stability due to arc surface failure mode
1.2. Vertical Breakwaters and Composite Breakwaters
General Description and Design Procedures
Vertical breakwaters are free standing gravity structures with vertical face. They are mainly
in the form concrete caissons or keyed blocks on improved foundation. Two examples are shown in
Figures 16 and 17. Pure vertical breakwaters are rare to find. Most of them are of composite type.
Composite breakwaters are vertical gravity structures erected on rubble mound foundations
such as shown in Figures 18 and 19. The composite structure in Figure 18 is the normal type in that
rubble foundation is low and flat. The breakwater in Figure 19, on the other hand, has a high mound.
This type of structure is often found in region of soft foundation material.
The basic design procedures for vertical breakwater are similar to rubble mound type except
that analyses of wave forces and structural stabilities are different. For rubble mound breakwater
design, wave force on structure is of not directly computed as a design load but is for vertical wall.
For structural stability, the rubble mound breakwater concerns with armor stability and slope stability
whereas sliding and overturning are the main modes of stability failure.
The major design steps are outlined in Chart 2.
Wave Force Computation
1. Sainflou Formula for nonbreaking and no overtopping
The earliest wave pressure formula on vertical wall was by Sainflou (1920). The formula
based on total wave reflection and linear wave theory is still the durable one used for nonbreaking
Configuration
Design Condition
Cross Section
External Forces
I
Sliding Stability
I
Overtu
Overturning Stability
I
1
Stability Analysis
Detailed Design
Chart 2 Major Steps in Composite Breakwater Design
wave forces. The magnitudes of pressure at wave crest and trough defined in the pressure distribution
diagram shown in Fig.20 are given as,
A OWfI MwLe (OrMi Cter W Cieall
1   V, A.  
o     
STrouh of Clopotis / /
EllA, F" C, a \ 8
," / """
.wd \ IP,
F E t T, Pressure Diaogram Crest Pssure Di0ogram
I . P i 
t*<<p
Figure 20 Loading on Vertical Wall Due to nonbreaking waves
24
At wave crest:
p=iy+H[ cshk(y+d) sinhk(y+d) (24)
coshkd sinhkd
AT wave trough
p=i yH[ coshk(y+d) sinhk(y+d) (25)
coshkd sinhkd
At this moment there is no set rule to select the design wave height but H,3 or Hi10 are the
most common choice. Experience shows that Sainflou formula under estimates wave pressure in the
mean water level zone under storm conditions ifHm is selected as the design wave.
2. MicheRundgren Formula for Standing Waves
Miche (1944) and Rundgren (1958) modified Sainflou's formula by using second order wave
theory and a linear depthdependent pressure distribution (Fig.23b) below water line and proposed
the following simplified formula:
p=O (26)
at the surface and
yHi
p=y(lX) (27)
cosh(2nd/L)
at the bottom. The + and signs correspond to wave crest and wave trough respectively and x is the
wave reflection coefficient. For total reflection, X = 1 and the mean water level is increased by an
amount, ho, known as the clapostis:
H2 2nd
=ncoth (28)
L L
The corresponding resultant forces and moments about the base are given, respectively, for the
maximum crest level subscriptt e) and the maximum trough level subscriptt I) by the following
formulas:
(d+H+ho) (yd+P1) yd2
2 2
S(d+ho+) 2 (d+P1) yd3
6 6
yd2 (d+A B) (ydP9)
Fi
2 2
Syd3 (d+hH) 2(ydP)
=I "^^
6 6
The MicheRundgren formula is recommended in SPM (1984).
3. NonBreaking Waves with Overtopping
In this case, a force reduction factor, r, and a moment reduction factor, r, are applied,
respectively, for force and moment computations. The force reduction factor is given by:
b b b
r= (2) when 0.5<<1.0
Y Y Yc
and
rf=1.0 when 11.0
Ye
where b and y are defined in Fig.21. The values of these force reduction factors are given in Fig.22.
,Crest of Clapotis
SWL
Cwdi P, r
Figure 21 Pressure Distribution on wall of low crest
26
041
0.2 0.4 0.6 0.8
b
Y
(1rf)
0.1 and
(I rm)
(I rf)
and
(I rm)
0.6
.0.8
1.0
1.0
Figure 22 Force and Moment Reduction Factors for Low Crest Breakwater
Uc
4. Minikin's Formula for Breaking Waves
When waves breaks directly against a vertical surface, a short duration impact loading acting
near the region where wave crests hit the wall develops in addition to the slowly varying wave
loadings. Minikin (1955, 1963) developed a design procedure based on field observation and the tank
experiment by Bagnold (1939). His formula superimposes an impact loading near the water surface
to the slowly varying wave loading as given in Fig.23. The maximum pressure assumed to act at the
SWL is given by
wI. d,
P,=101 (d+d,)
Ld d
(32)
where d. = water depth at the toe or the mound foundation
d water depth one wave length from the wall
Ld = wave length at depth d
Hb
SOLID WALL LINE
S"Pm S.W.L.

\d
Pd "
Figure 23 Pressure distribution according to Minikin's formula
This impact pressure is assumed to decrease parabolically to zero at a distance of H/2,
p=P (2y )2
H
Thus, the force and moment due to this impact pressure then become:
(33)
1
PmHb
3
PmHbds
(34)
The contribution due to slowly varying components can be computed by the MicheLungren formula
as described previously. The combined total horizontal force can be expressed in the following non
dimensional form,
Fh H d. d 1 dd
33.7 [1+] +0.25+
wH Ld d H 2 H
(35)
The first part on the LHS is the impact loading whereas the second and third parts on LHS are due
to slowly varying wave loading.
The Minikin formula was derived based on limited laboratory results of a vertical breakwater
on small rubble mound and has a very narrow range of applicability. In most cases it yields
unrealistically large impact load. Therefore, although the formula is listed in SPM it is not widely
used.
5. Hiroi's Total Force Formula
Hiroi(1919) proposed his formula based purely on field observations and is widely used in
Japan for years. It includes consideration of breaking wave induced impact loading. The formula has
a uniform pressure distribution as shown in Fig.24.
II
R>125H
S777
/7
( HIGH CREST I
/777Fvr
R <1.25H
7 77
* M4
*,
* .4
;7777
,L S.W.L.
( LOW CREST )
Figure 24 Pressure distribution by Hiori formula
The formula is simply,
The total horizontal loading becomes,
Fh d
1.88+1.5
wR2 H
Significant wave height is recommended as the design wave. Equation (37) does not actually reflect
the pressure distribution and should not be used. Equation (38), on the other hand, is judged to be
quite suitable for shallow water breakwater applications for water depth at the toe of the vertical
section to be less than two times the significant wave height. It has a proven record of success in
Japan.
6. Russian Design Manual Formula
The Russian design manual formula has a pressure distribution shown in Fig. 25.
Figure 25 Pressure distribution due to breaking waves in Russian Formula
The pressure intensity at mean water line is
po=1.5wH
and at the bottom of the vertical wall is,
(38)
p= 1.5wH
(36)
(37)
wH
Pd 
2nh,
cosh
(39)
The nondimensional total horizontal force is,
F 1 d,
 =0.75+(0.75+ )
wH2 2nd, H
2cosh
(40)
7. Goda's Universal Pressure Formula
Goda (1974) argued that the separation of impact loading and slowingly varying wave
dynamic loading is unrealistic and proposed socalled universal pressure formula.His formula is
currently adopted as a standard design formula in Japan. It is claimed to be applicable to any water
depth for both breaking and nonbreaking environment. The shape of the pressure distribution and
the definitions are given in Fig. 26.
Figure 26 Wave pressure distribution by Goda's formulas
The various pressure intensities and elevations are given as follows,
n* = 0.75 (1+cosP) H (41)
where p is the wave incident angle. For normal incidence, rl' is 1.5 H and,
p_ =0.5 (1 +cosp) (ai +a2coSO) wH
P2 (42)
coshkd
p3 =aA
in which
a, =0.6+0.5[ 2kd ]2
sinh2kd
dbd, H 2d,
a2 =miri[ d] (H ) 2,. ) (43)
3db d, H
h1 1
a(=1(h) [1 1 ]
d coshkd
The coefficient a, takes the minimum value 0.6 for deepwater waves and the maximum value
1.1 for waves in very shallow water. It represents the effect of wave period on wave pressure
intensities. The coefficient a2 is introduced to express an increase of wave pressure intensities by the
presence of rubble mound foundation. The coefficient a3 is based on linear pressure distribution.
8. Broken Wave Force Wall Seaward of Still Water line
In the broken wave zone, the portion of the water mass above the waterline can be considered
to transform into a transitory motion with forward velocity equal to the wave celerity at the breaking
point. The kinetic energy associated with this water mass is assumed to completely converted to flow
energy (pressure) without influencing the water level change. This case as shown in Fig.27, gives rise
a dynamic pressure on the wall given by
P.= LpC2 (44)
2
Since the wave celerity at the breaking point can be approximated by the square root of gdb, we have
Pm i wdb (45)
2
SWL
Figure 27 Wave pressures from broken waves: wall seaward of still water line
then the dynamic component of the wave force and induced moment are given as
=h 1 c
h (46)
Mm=F (d +c)
The slowly varying loading is approximated by a linear pressure distribution given by:
Ps= w(ds+hc)
F=(d+h )2
M = (d +hc)3
6 6
The total force and moment are then the sum of the dynamic and slowly varying components,
therefore,
F = F + F (48)
and
Mt=b+M (49)
10. Broken Waves Wall Shoreward of Still Water Line
For this case, shown in Fig.28, the force and moment on the wall can be treated in a similar
manner as the previous case.
INSERT
Shoreline
Figure 28 Wave pressures from broken waves: wall landward of still water line
The velocity of the bore between the SWL and the point of maximum wave runup where the
velocity must be zero may be approximated by:
X X1
v=c (1)= =/b(1) (50)
X2 X2
and the bore height above the ground surface by:
X1
h=h (1) (51)
x2
where
x, = distance from SWL to the structure.
x, = distance from SWL to runup limit, or = 2 I /m.
An analysis similar to that for the structures located seaward of the SWL gives for the dynamic
pressure
WV2 Wb X1 ) 2
Po, (1 (52)
2g 2 x,
This dynamic pressure is assumed to act uniformly over the broken wave height at the structure toe
with bore height h, hence the force and moment are, respectively:
1h, l 3
Fmp h= c (1)
2 x,
2 2 (53)
t= F c (1 X)4
2 2 4 xz
The slowly varying part of the force and moment are given by:
2
h 2 h2 1 2
2 2 x,
3(5
h hh xl)3
M,= F w (1 )
3 6 x,
Sliding and Overturning Stabilities
To assess the sliding and overturning stabilities of the upright section, the buoyancy and the
wave induced uplift force need to be taking into consideration. The buoyancy is simply the weight
of water displaced by the volume. The dynamic uplift pressure is assumed to vary linearly from the
toe of the breakwater to the lee side such as shown in Fig. 26. The safety factors against sliding and
overturning are given by the following,
Against Sliding:
S.F. =p(W) /Fh (55)
Against Overturning
S.F. = (MMu)/i (56)
with W = net weight of the upright section.
U = uplift pressure induced by wave motion.
F, = horizontal wave force.
p = friction coefficient.
M's = corresponding moments.