short term wave

57
Chapter 3
Short-Term Wave Analysis
3.1 Introduction
In analysis of wave data, it is important to distinguish between short-
term and long-term wave analysis. Short-term analysis refers to analysis
of waves that occur within one wave train
1
or within one storm (Figs. 2.3
and 3.1). Long-term analysis refers to the derivation of statistical
distributions that cover many years. To emphasize the difference between
the two, they have been arranged into two separate chapters. Short-term
wave analysis is discussed here and long-term wave analysis in Ch. 4.
It was stated in Ch. 2 that the complex sea surface appears to defy
scientific analysis. A number of simplifying assumptions must be made to
describe short-term recordings of the water surface and research has
shown that a number of excellent approximations can be made (Goda,
1970, 1985, 2000). Because there are many sizes of waves in any wave
record we will need to resort to statistical analysis.
We define z as the instantaneous water level related to a datum, and η
as the difference between the instantaneous water and the mean water
level; the values of z and η are functions of location (x, y) and time (t). A
water level record such as shown in Fig. 3.1 therefore represents the
process z(t) at a specific location. Water level records are normally not
continuous, because they are recorded digitally. Thus z is only sampled
at sampling intervals of ∆t. A record of length t
R
then consists of N
samples z
j
taken at times j∆t, where 1 ≤ j ≤ N.

1
Series of waves.
58 Introduction to Coastal Engineering and Management


Fig. 3.1 Water level record

The water level record in Fig. 3.1 is one realization of the process z(t).
We will call this z
1
(t). To understand the relevant terminology, imagine a
basin of water with a wave generator at one end. We start up the wave
generator steered by a certain drive signal, and after 5 minutes we
measure the water level in the middle of the basin for 30 seconds. That
produces a short-term record z(t) as shown in Fig. 3.1. Now we shut off
the generator and wait for the water to become quiet. Then we start the
generator up again (with the same drive signal) and after 5 minutes we
measure another 30 second record at the same location. The second
record is a second realization, z
2
(t) of the same process. We could repeat
this many times to produce an infinite number of realizations (records) of
the process z(t), which in this case represents water surface fluctuations
in the middle of a basin, after 5 minutes of wave generation. Three
realizations of this process are shown in Fig. 3.2. The complete set of K
realizations z
k
(t) is called an ensemble.
We can take all the values of z at t = j∆t in the k realizations and
calculate statistical parameters such as ensemble mean
j
z and ensemble
standard deviation σ
j
, where

2
, ,
1 1
1 1
( )
K K
j k j j k j j
k k
z z and z z
K K
σ
= =
= = −
∑ ∑
(3.1)
Ensemble skewness and kurtosis can also be determined. If none of
these ensemble parameters vary in time, the process is called stationary;
if only the ensemble mean and standard deviation are constant, the system
- 0 . 0 4
- 0 . 0 3
- 0 . 0 2
- 0 . 0 1
0
0 . 0 1
0 . 0 2
0 . 0 3
0 . 0 4
0 5 1 0 1 5 2 0 2 5 3 0
T i m e ( s e c )
W
a
t
e
r

L
e
v
e
l

(
m
)

Chapter 3 Short-Term Wave Analysis 59































Fig. 3.2 Ensemble of three realizations of a stationary and ergodic process


-0.04
-0.03
-0.02
-0.01
0
0.01
0.02
0.03
0.04
0 5 10 15 20 25 30
Time (sec)
W
a
t
e
r

L
e
v
e
l

(
m
)
-0.04
-0.03
-0.02
-0.01
0
0.01
0.02
0.03
0.04
0 5 10 15 20 25 30
Time (sec)
W
a
t
e
r

L
e
v
e
l

(
m
)
-0.04
-0.03
-0.02
-0.01
0
0.01
0.02
0.03
0.04
0 5 10 15 20 25 30
Time (sec)
W
a
t
e
r

L
e
v
e
l

(
m
)

60 Introduction to Coastal Engineering and Management

is said to be weakly stationary. If the time average for each realization
) t ( z
k
is equal to the ensemble average
k
z , where
( )
k
z t
,
1 1 1
1 1 1
( )
N K N
k j k k j
j k j
z t z and z z
N K N
= = =
= =
∑ ∑∑
(3.2)
the process is called ergodic. For the ensemble in Fig. 3.2, the process is
stationary and ergodic. It is stationary because, for example,
j
z and σ
j
,
do not increase with time and ergodic because ) (t z
k
for each realization
is equal to
k
z , the ensemble average.
What does this mean in practice? A water level record is always only
a single realization of the process to be studied. We have no other
realizations. We cannot turn off and later “re-play” the same situation as
we could in the wave basin example. Therefore any record is only an
approximation of the process. Weak stationarity can only be inferred
from this single wave record if z and σ do not vary with time (there is no
trend in the mean water level and the wave heights). Finally, with only
one realization, we can never show that the process is ergodic; we simply
must assume ergodicity as also discussed in Kinsman (1965).
3.2 Short-Term Wave Height Distribution
To determine wave heights, it is necessary to use η, the difference
between the water level and the mean water level. It is usual to think of
η as a superposition of an infinite number of small waves, each generated
by its own wind eddies at different locations and at different times. The
resulting sea surface is, therefore, the sum of a large number of
statistically independent processes, and common sense would tell us that
it is impossible to predict the exact value of η at any time or location. In
other words, η is a random variable. The probability that η has a certain
value is called the Probability Density Function (PDF), p(η). The Central
Limit Theorem states that the PDF for a sum of many independent
variables is Gaussian, which means that p(η) can be described by the
Normal Distribution. The overall behavior of p(η) may be summarized
by its mean, , η standard deviation, σ, and some additional statistical
Chapter 3 Short-Term Wave Analysis 61

parameters such as skewness and kurtosis. Most often a two-parameter
normal distribution is used, defining p(η) by η and σ only. By
definition, η = 0 and therefore

(
(
¸
(


¸

2
exp
σ
η
π σ
η
2
-

2
1
= ) p(
2
(3.3)
where σ is the standard deviation of the process, η(t). It is equal to the
square root of the variance of η.

2 2 2
0
1
1
lim
R
R
N
t t
2
j t
t
j R
1
= dt
t N
σ η η η
=
→∞
=
=
= =
∑
∫
(3.4)
If the wave frequencies all occur within a narrow frequency band, (if
the wave periods do not vary greatly) it may be shown theoretically
(Longuet-Higgins, 1952; Cartwright and Longuet-Higgins, 1956;
Benjamin and Cornell, 1970) that the PDF of the maximum instantaneous
water levels is:

max max
max 2 2
exp
2
-
p( ) =
2
η η
η
σ σ
(
(
(
¸ ¸
(3.5)
If it is assumed that for waves of a narrow frequency band the wave
height H is equal to 2η
max
, then the PDF for H becomes:

2 2
exp
2
1 H -
H
p(H) =
4 8 σ σ
(
(
¸ ¸
(3.6)
Equations 3.5 and 3.6 are known as Rayleigh distributions. The
distribution of Eq. 3.6, multiplied by H is shown in Fig. 3.3.
To determine the Cumulative Distribution Function (CDF) of wave
heights, the Rayleigh distribution is integrated to yield the probability that
any individual wave of height H' is not higher than a specified wave
height H

2
0
exp
2
H
-
H
P(H < H) = p(H) dH = 1 -
8σ
(
′
(
¸ ¸
∫
(3.7)

62 Introduction to Coastal Engineering and Management













Fig. 3.3 Rayleigh distribution

The probability of exceedence, the probability that any individual
wave of height H' is greater than a specified wave height H may be
obtained as

2
exp
2
-
H
Q(H > H) = 1 - P(H < H) =
8σ
(
′ ′
(
¸ ¸
(3.8)
The functions P and Q are also shown in Fig. 3.3. Research has shown
that for practically all locations the wave height distribution is reasonably
close to a Rayleigh distribution. One exception is in shallow water when
the waves are about to break.
The wave height with a probability of exceedence Q, may be
determined from Eq. 3.8 as

2 2
ln ln ln
Q
1 1
= 8 (- Q) = 8 = 2 2
H
Q Q
σ σ σ
| | | |
| |
\ ¹ \ ¹
(3.9)
To determine
Q
H , the average height of all the waves that are larger
than H
Q
in a record or a storm

Q
H
Q
H p(H) dH
=
H
Q
∞
∫
(3.10)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 1 2 3 4 5 6 7 8
H/σ
(
H
)
.
p
(
H
)
,

P
,

Q
(H).p(H)
P
Q

Chapter 3 Short-Term Wave Analysis 63

Equation 3.9 and the numerical evaluation of Eq. 3.10 yield Table 3.1, in
which a number of common wave height definitions are related to σ.
Table 3.1 Commonly used wave height parameters
Symbol Description Value
H.01

Average of highest 1% of the waves
σ 6.67
H.01
Height, exceeded by 1% of the waves σ 6.07
H.02

Average of the highest 2% of the waves
σ 6.23
H.02
Height, exceeded by 2% of the waves σ 5.59
H0.1

Average of highest 10% of the waves
σ 5.09
H0.1
Height exceeded by 10% of the waves σ 4.29
3 / 1
H H
s
=
Significant wave height (Average height of the
highest 1/3 of the waves)
σ 4.0
H Average wave height
σ π 2
H0.5
Median wave height σ 2.35
Hmode
Most probable wave height σ 2.0
Hrms

N
H H H ....
2
3
2
2
2
1
+ + +
σ 2 2

Of all these definitions, Significant Wave Height (H
s
) is the most
important
2
. It is defined as the average of the highest 1/3 of the waves in
a wave train, 3 / 1 H . In terms of significant wave height, four commonly
used relationships based on the Rayleigh distribution are

0.1 0.01 1.27 ; 1.67
0.63 ; 0.707
s s
s rms s
H H H H
H H H H
= =
= =
(3.11)
The meaning of average wave height is self-explanatory. The modal
or most probable wave height is the wave height with the greatest
probability of occurrence. The median wave height has 50% probability,
i.e., half the waves in the wave train are higher and half the waves are

2
This wave height definition was historically chosen as “significant” because it comes
closest to the traditional estimates of average wave height made by experienced
observers before we had instruments to measure wave heights.
64 Introduction to Coastal Engineering and Management

lower than this wave height. The rms wave height is the constant wave
height that represents the wave energy.
The probability of exceedence for the average wave heights may now
be calculated from their values in Table 3.1, using Eq. 3.8. The results
of this calculation are shown in Table 3.2.
Table 3.2 Probabilities of exceedence of average wave heights
H
01 . 0
H
1 . 0
H H
s

H
H
mode
Q(H′>H) 0.004 0.039 0.136 0.456 0.606

The expected value of the maximum wave in a wave train of N
w

waves could be estimated by setting Q = 1/N
w
in Eq. 3.9. A more
accurate estimate is

-3/ 2 max H
= 2 ln N + + [(ln N ] )
2 2 ln N
w w
w
O
γ
µ
σ
(
(
¸ ¸
(3.12)
where µ(x) denotes “expected value” of x, γ is the Euler constant
(= 0.5772) and O(x) denotes terms of order greater than x, i.e. small
terms.


Example 3.1 Calculation of short-term wave heights

To analyze a wave record it must be stationary. Hence, it is normal
to record waves for relatively short time durations (10 to 20 minutes).
A longer record would not be stationary because wind and water
level variations would change the waves. Thus it is usual to record, for
example, 15 minutes every three hours. It is subsequently assumed that
the 15 min. record is representative of the complete three hour recording
interval.
Suppose the analysis of such a record yields
10 sec 1.0 T and m σ = = (3.13)
Chapter 3 Short-Term Wave Analysis 65

We want to calculate significant wave height H
s
, average wave height, H ,
average of the highest 1% of the waves
0.01
H , and the maximum wave
height in the record. From Table 3.1:
0.01 4 4.0 , 2 2.5 6.67 6.7
s
H m H m and H m σ πσ σ = = = = = =
With sec 10 = T , the average number of waves in the 15 min record N
w
=
90 and Eq. 3.12 yields

max
0.5772
2 ln (90) 3.00 0.19 3.19
2 2 ln (90)
H
m µ
σ
(
= + = + =
(
¸ ¸
(3.14)
or the expected value of H
max
= (3.19)(2)(1.0) = 6.4 m. This calculated
value of H
max
can be verified against the actual record. If the record is
representative of a 3 hour recording interval, then
0.01
, ,
s
T H H and H for
the 3 hours would be the same as above. However, H
max
would be larger
than 6.4 m. For the 3 hour recording interval, N
w
= 1080 and Eq. 3.14
yields H
max,3hrs
= 7.8 m.

3.3 Wave Period Distribution
In the above discussion the frequency bandwidth for the waves was
assumed to be small (the wave periods are more or less the same) and in
practice, wave period variability is often ignored. One attempt to define
wave period distribution has been made by Bretschneider (1959) who
postulated that the squares of the wave periods form a Rayleigh
distribution. His expression for the PDF for wave periods is
exp
4
3
4
T
T
p(T) = 2.7 -0.675
T
T
(
| |
(
|
\ ¹
(
¸ ¸
(3.15)
From this, by integration, the expression for probability of exceedence
of a certain wave period becomes
exp
4
T
T
Q(T >T) = p(T)dt = -0.675
T
∞
(
| |
′
(
|
\ ¹
(
¸ ¸
∫
(3.16)
66 Introduction to Coastal Engineering and Management

Wave periods are related to wave heights particularly in a growing,
locally-generated sea, where high wave heights are always accompanied
by long wave periods. A joint distribution of wave heights and periods
can be postulated and that is normally assumed to be a joint Rayleigh
distribution.
3.4 Time Domain Analysis of a Wave Record
Wave recordings are time series of water levels that typically look like
Fig. 3.4. They are discrete time series z(t), sampled at N short intervals
of ∆t. The water level recording must first be converted into a discrete
time series η(t), the fluctuation about mean water level by subtracting the
mean water level from the record.
Although the record is assumed to be stationary, there may be a small
change in mean water level with time, as is the case in Fig. 3.4, where the
water level drops slightly. This could be, for example a result of tides.
Because the record is short (20 min), the water level fluctuation is
assumed to be a linear function of time, . z a bt = + It is determined from
the record by regression analysis and then subtracted, so that
( ). z z z a bt η = − = − + Figure 3.5 presents the η time series for Fig. 3.4.
The bottom graph is for the first 120 seconds and shows more detail.











Fig. 3.4 Water level record
1.5
1.7
1.9
2.1
2.3
2.5
2.7
2.9
3.1
3.3
3.5
0 200 400 600 800 1000 1200
Time (sec)
W
a
t
e
r

L
e
v
e
l

-

z

(
m
)

Chapter 3 Short-Term Wave Analysis 67

As an initial analysis of a record, we could simply calculate σ, using
Eq. 3.4. The wave analysis program WAVAN
⊗
was used
3
and σ was
found to be 0.28 m for the record in Fig. 3.5. If a Rayleigh distribution of
wave heights is assumed (Tables 3.1 and 3.2), we can determine the
values for the important wave heights for Fig. 3.5.
0.1
1.12 , 0.70 , 0.79 1.43
s rms
H m H m H m and H m = = = =
(3.17)























Fig. 3.5 Wave record for Fig. 3.4


3
An alternate program ZCA
⊗
was also provided with the software.
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
0 200 400 600 800 1000 1200
Time (sec)
η

(
m
)

-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
0 20 40 60 80 100 120
Time (sec)
η

(
m
)

68 Introduction to Coastal Engineering and Management

This initial analysis does not tell us anything about wave period, the
other important wave parameter.
The actual distributions of H and T may be obtained from the record
by analysis of individual waves. Figure 3.6 shows a very short segment
of Fig. 3.5. The earliest definition for wave height H is the vertical
distance between a wave crest and the preceding trough (Fig. 3.6) where
crest and trough are defined as a local maximum and a local minimum in
the record. That definition would result in all the small ripples being
identified as waves. How many waves are there between t = 20 and 60
sec in Fig. 3.6?
To define wave height more realistically, zero down-crossing wave
height, H
d
, is defined as the vertical distance between the maximum
and minimum water levels that lie between two subsequent zero
down-crossings (in which η crosses zero on the way down). Similarly
zero up-crossing wave height, H
u
, is the difference between maximum
and minimum water levels between two subsequent zero up-crossings.
These definitions are also shown in Fig. 3.6. They disregard the small
ripples that do not cross the mean water level. Example 3.2 presents the
zero crossing analysis of the wave record of Fig. 3.5 and compares the
results with the values in Eq. 3.17.













Fig. 3.6 Wave height definitions
-1.0
-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
20 30 40 50 60
Time (sec)
E
t
a

(
m
)
H
H
u
H
d
x
xx
x x x

η

(
m
)

Chapter 3 Short-Term Wave Analysis 69


Example 3.2 Zero crossing analysis of Fig. 3.5

The zero up-and down crossing wave heights in Fig. 3.5 were
determined using WAVAN
⊗
or ZCA
⊗
. The two estimates were virtually
identical. The following wave statistics were obtained by averaging the
up- and down-crossing results
0.1
1.05 , 0.68 , 0.76 1.30
s rms
H m H m H m and H m = = = =
(3.18)
When these calculated values are compared with Eq. 3.17 it is seen
that Eq. 3.17 overpredicts for this record.
The wave heights were also grouped into 10 bins and the histogram
of the wave heights is shown in Fig. 3.7a. This distribution can be
compared with the Rayleigh distribution. It would also be possible to
plot the cumulative distribution function (P, as in Eq. 3.7) or the
probability of exceedence (Q, as in Eq. 3.8). However, since the wave
height distribution is expected to be Rayleigh, it is best to compare wave
heights directly with this theoretically expected distribution. Equation
3.8 may be re-written as

2
ln( )
2 2
H
Q
σ
 
− =
 
 
(3.19)
Thus,

1/ 2
{ ln( )} (2 2 )
2 2
H
R Q or H R σ
σ
= − = = (3.20)
where R is called the Rayleigh parameter. A true Rayleigh distribution
would plot as a straight line with zero intercept and a slope of 2 2σ on a
graph of H versus R. Values of R were calculated for each wave
height bin and H versus R was plotted in Fig. 3.7b. The solid line in
Fig. 3.7b represents the initial analysis, combining σ = 0.28 m with
Tables 3.1 and 3.2. This analysis clearly overpredicts the actually
measured values of H.

70 Introduction to Coastal Engineering and Management









a) b)

Fig. 3.7 Histogram and Rayleigh distribution

The best fit line through the measured values has a slope of 0.69 and
from this; using Eq. 3.20, another estimate for σ may be computed —
z
=0.69 / (2 2 )=0.24m σ . This version is called σ
z
since it was
determined by zero-crossing analysis. An estimate of average wave
period may be obtained by dividing the record length (t
R
= 1200 seconds)
by the number of waves (N
w
= 197) to find T 6.1 = seconds.
Finally H
max
was found to be 1.56 m from the record. This can
be compared to the theoretical value of H
max
= 1.92m, obtained from
Eq. 3.12, using σ = 0.28 m. Using σ
z
= 0.24 m yields a more reasonable
value of H
max
= 1.65 m.

3.5 Frequency Domain Analysis of a Wave Record
A completely different type of analysis, based on wave frequencies, is
called wave spectrum analysis. We use the statistical assumptions that
the wave record is both stationary and ergodic. Although these
assumptions are necessary to perform a statistically correct analysis, in
practice we have no choice but to assume that records are short enough to
be both stationary and ergodic.
To express the time signal η in terms of frequency, we can use a
Fourier series summation for each value of η
j

0
5
10
15
20
25
30
35
40
0.08 0.23 0.39 0.55 0.70 0.86 1.01 1.17 1.33 1.48
Wave Height - H (m)
N
u
m
b
e
r

o
f

W
a
v
e
s
y = 0.6854x
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
0.0 0.5 1.0 1.5 2.0
Rayleigh Parameter - R
W
a
v
e

H
e
i
g
h
t

-

H

(
m
)
H=0.69R

Chapter 3 Short-Term Wave Analysis 71


1 1
cos(2 ) sin (2 )
j o n n j n n j
n n
a a f t b f t η π π
∞ ∞
= =
= + +
∑ ∑
(3.21)
Using Euler’s relationship
cos sin
i
e i
ψ
ψ ψ = + (3.22)
Eq. 3.21 becomes

2 ( )
n j
i f t
j n
n
C e
π
η
∞
 
 
=−∞
=
∑
(3.23)
where C
n
is a complex coefficient

1 1
; ( ); ( ) *
2 2
o o n n n n n n n
C a C a ib C a ib C
−
= = − = + = (3.24)
Equation 3.23 expresses the time signal η in terms of discrete
frequency components and is known as a Discrete Fourier Transform
(DFT). The series in Eqs. 3.21 and 3.23 are infinite. However, a wave
record, such as in Fig. 3.5 is neither infinitely long, nor continuous. The
water level is sampled only at N specific times, ∆t apart. There are N
values of (η
j
) at times t
j
= j∆t, where 1 ≤ j ≤ N. As a result, the frequency
domain is also not continuous. The smallest frequency that can be
defined from a record of length t
R
is f
min
= 1/t
R
. To provide the most
accurate representation in the frequency domain (best resolution) we will
use this smallest possible frequency as the frequency increment.
Therefore ∆f = 1/t
R
and we define f
n
= n∆f. The highest frequency that
can be defined from a time series with increments ∆t is the Nyquist
frequency

1
2 2 2
N
r
N N
f f
t t
= = = ∆
∆
(3.25)
This results in the finite discrete Fourier transform (FDFT)

2
[2 ( )]
1
2
n j
N
i f t
j n
N
n
F e
π
η
=− +
=
∑
(3.26)

72 Introduction to Coastal Engineering and Management

Because F
n
= F
-n
*, Eq. 3.26 may also be written as:

1
[2 ( )]
0
n j
N
i f t
j n
n
F e
π
η
−
=
=
∑
(3.27)
The inverse of Eq. 3.27 is

[2 ( )]
1
1
n j
N
i f t
n j
j
F e
N
π
η
−
=
=
∑
(3.28)
Equations 3.27 and 3.28 form a FDFT pair that permits us to switch
between the time and frequency domains. Equation 3.28 allows us to
calculate the complex frequency function F
n
from a real time function η
j

and Eq. 3.27 permits calculation of the real time function η
j
from the
complex frequency function F
n
. The complex variable F
n
may also be
expressed as

n
i
n n
F F e
θ −
= (3.29)
where

2 2 1
1
tan
2
n
n n n n
n
b
F a b and
a
θ
−
 
= + = −
 
 
(3.30)
Substitution of Eq. 3.29 into Eq. 3.27 results in

1
[2 ( ) ]
0
n j n
N
i f t
j n
n
F e
π θ
η
−
−
=
=
∑
(3.31)
In practice waves have only positive frequencies, only frequencies
lower than f
N
can be defined, and η
j
is real. Therefore, using Eq. 3.22,
we can rewrite Eq. 3.31 as

1
0
/ 2 / 2
0 0
cos(2 )
2 cos(2 ) cos(2 )
N
j n n
n
N N
n n n n
n n
F ft
F ft A ft
η π θ
π θ π θ
−
=
= =
= −
= − = −
∑
∑ ∑
(3.32)
where
2
n n
A F = (3.33)
Chapter 3 Short-Term Wave Analysis 73

Here
n
A is called the amplitude spectrum and θ
n
the phase
spectrum. In standard wave spectrum analysis only the amplitude
spectrum
n
A is calculated. In effect, θ
n
is assumed to be a random
variable -π < θ
n
< π resulting in the random phase model. This
unfortunate assumption loses all phase relationships between the N terms,
which means for example, that wave groups are not reproduced when
calculating η
j
from
n
A using random θ
n
. Resonance and reflection
patterns are also not reproduced.
Parseval’s theorem can be used to calculate σ from the amplitude
spectrum
n
A because

1 / 2
2 2 2
2 2
0 0
1
2
N N
n n n
n n n
C F A σ η
∞ −
=−∞ = =
= = = =
∑ ∑ ∑
(3.34)
Thus the variance at any frequency can be expressed as

2 2 1 1
( ) ( )
2 2
n n n n
S f df A or S f A
df
= = (3.35)
where S(f ) is known as the wave variance spectral density function or
wave spectrum. Variance is a statistical term. In physical terms, wave
energy density is

2
E = g ρ σ (3.36)
and hence wave energy distribution as a function of frequency is
( ) ( ) E f = g S f ρ (3.37)
Wave spectra for Fig. 3.5 were computed using WAVAN
⊗
and are
shown in Fig. 3.8
4
. Because we always have only one realization of the
process and the record length (t
R
) is finite, resulting in finite increments
of frequency (∆f ), the calculated value of S(f ) is always an estimate of
the true S(f ).
The wave spectrum for the record in Fig. 3.5, using Eqs. 3.32 to 3.35,
is shown in Fig. 3.8a. This spectrum gives the maximum possible
resolution and distinguishes between frequencies that are ∆f = 1/t
R
=
1/1200 = 0.00083 Hz apart. It contains many closely spaced spikes of

4
An alternate program WSPEC
⊗
was also provided with the software.
74 Introduction to Coastal Engineering and Management








a) b)








c) d)

Fig. 3.8 Wave spectra of the wave record in Fig. 3.5

wave energy. Physically, such very local energy peaks are not possible.
They are a result of the uncertainties in our estimates and therefore the
wave spectrum is smoothed.
Smoothing can be done by averaging S(f ) over frequency ranges
longer than ∆f so that

/ 2
2
/ 2
1
( )
m M
n m f
m M
S f A where df M f
df
=
+
=−
′ = = ∆
′
∑
(3.38)
where df ′ is the resolution of the spectrum and M
f
denotes how many
values of ∆f are averaged. The results for M
f
= 6 and 12 (df ′ = 0.005 and
0.01 Hz) are shown in Figs. 3.8b and c. Smoothing produces a more
regular spectrum. The amount of smoothing to be used depends on the
purpose of the analysis. If a general impression of a wave field is needed,
Fig. 3.8c is most useful. If specific frequencies need to be identified, then

0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40
Frequency (Hz)
S
(
f
)

(
m
2
/
H
z
)
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40
Frequency (Hz)
S
(
f
)

(
m
2
/
H
z
)
0.0
0.5
1.0
1.5
2.0
2.5
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40
Frequency (Hz)
S
(
f
)

(
m
2
/
H
z
)
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40
Frequency (Hz)
S
(
f
)

(
m
2
/
H
z
)
Chapter 3 Short-Term Wave Analysis 75

less smoothing such as in Fig. 3.8b may be more appropriate. Another
method used for smoothing the spectrum is to divide the record into
shorter sections, compute the spectrum for each section and then average
the results. Figure 3.8d, presents the average of 4 spectra, each for ¼ t
R
.
The resolution is now 4/t
R
= 0.0033 and averaging the four spectra
introduces some further smoothing.
The value of σ, as computed from the frequency analysis by
integrating the spectrum is denoted as σ
f
because it was derived by
frequency analysis. Because Eqs. 3.4 and 3.34 both integrate η
2
, σ
f
for
all four spectra in Fig. 3.8, as well as σ found using Eq. 3.4, are the same
and equal to 0.28.
Only the basic principles of wave spectrum are presented here. There
are other methods of computing the wave spectra and dealing with
smoothing of the spectra. Further details may be found in the literature,
such as Bendat and Piersol (1966) and ASCE (1974) and Janssen
(1999).
Frequencies that exceed the Nyquist frequency (Eq. 3.25) cannot be
defined as separate frequencies. The energy in these high frequency
waves is superimposed on the spectrum by a process known as aliasing.
Figure 3.9a demonstrates aliasing in the time domain. A wave of
frequency 1.1 Hz is sampled at ∆t = 1.0, for which f
N
= 0.5 Hz. It is seen
that the sampled signal (square points) does not have a frequency of
1.1 Hz, but of 0.1 Hz. The energy of such a wave component would
therefore become added at 0.1 Hz in a wave spectrum. Aliasing in the
frequency domain is depicted in Fig. 3.9b.
Aliasing can be prevented by filtering frequencies greater than f
N

out of the signal. Alternately, if f
c
is the highest frequency that must
be computed correctly (without aliasing), then it is reasonable to assume
that f
c
= f
N
/2. That defines the necessary sampling interval for the record
as ∆t ≤ 1/(2f
N
) = 1/(4f
c
). For wind waves, if we wish to define
the spectrum correctly for all frequencies f < 2Hz, ∆t should be less than
1/8 sec.


76 Introduction to Coastal Engineering and Management



Fig. 3.9 Aliasing of a wave spectrum by high frequency components
3.6 Parameters Derived from the Wave Spectrum
The moments of the wave spectrum are defined as

0
= ( )
f
h
h
f
S f df f
m
=∞
=
∫
(3.39)
The zero moment (n = 0) is therefore the area under the spectrum

2
0
= ( ) =
f
o f
f
S f df
m σ
=∞
=
∫
(3.40)
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
0 0.1 0.2 0.3 0.4 0.5 0.6
Frequency - f (Hz)
S
(
f
)


(
m
2
/
H
z
)
Correct
Spectrum
Aliased
Spectrum


-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
0 2 4 6 8 10 12 14
t
η
a)
b)
Chapter 3 Short-Term Wave Analysis 77

From the area under the wave spectrum, assuming the wave height
distribution to be Rayleigh, the various wave heights of Table 3.1 may be
estimated as in Eq. 3.17. To distinguish between significant wave height
derived from time domain analysis and its counterpart derived from
frequency analysis, the latter is called the Characteristic Wave Height or
Zero Moment Wave Height.

f ch mo
H = H = 4
σ
(3.41)
and for Fig. 3.5, H
mo
= (4)(0.28) = 1.12 m.
The representation of the wave energy distribution with frequency is
an improvement over the time-domain analysis methods discussed earlier.
With this information we can study resonant systems such as the
response of drilling rigs, ships’ moorings, etc. to wave action, since it is
now known in which frequency bands the forcing energy is concentrated.
It is also possible to separate sea and swell when both occur
simultaneously (Fig. 3.10).













Fig. 3.10 Wave spectrum with sea and swell

The moments of the wave spectrum also define spectral bandwidth

2
2 2
4
1
o
m
=
m m
ε
¦ ¹
¦ ¦
−
´ `
¦ ¦
¹ )
(3.42)
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
0 0.1 0.2 0.3 0.4 0.5 0.6
Frequency - f (Hz)
S
(
f
)


(
m
2
/
H
z
)
Swell
Sea

78 Introduction to Coastal Engineering and Management

Cartwright and Longuet-Higgins (1956) show that for a narrow
bandwidth (ε → 0) all wave periods in a wave train are almost the same
and the distribution of η is purely Rayleigh
5
. For ε → 1, the distribution
of η is random. This would obviously affect the wave height definitions
used in Tables 3.1 and 3.2. For the record in Fig. 3.5, ε
2
was calculated
to be 0.65.
Since there are many wave frequencies (or wave periods) represented
in the spectrum it is usual to characterize a wave spectrum by its peak
frequency f
p
, the frequency at which the spectrum displays its largest
variance (or energy). The peak period is then defined as

1
p
p
T
f
=
(3.43)
Other spectrum-based definitions of wave period found in the
literature are

0 0
1 2
1 2
;
m m
T T
m m
= = (3.44)
Theoretically T
2
is approximately equal to , T as obtained by zero
crossing analysis. For the spectra in Fig. 3.8, f
p
= 1.3 Hz, if the narrow
peak in Fig. 3.8a is discounted. Thus T
p
= 7.6 seconds and T
1
= 6.4
seconds and T
2
= 6.1 seconds. T was also 6.1 seconds in Ex. 3.2. In
Fig. 3.10 we can distinguish peak f
p
and T
p
values for both the sea and
the swell. Rye (1977) indicates that the moments for the spectrum are
functions of the cutoff frequency (the highest frequency considered in the
analysis) and thus ε, T
1
and T
2
should be viewed with caution.

Sometimes the angular frequency, ω, is used to define the frequencies
in the wave spectrum. The total variance for the S(ω) spectrum is

0
1
( )
2
2
= S d
ω
ω
ω
ω ω
σ
π
=∞
=
∫
(3.45)
The results of the three methods of analysis for the waves of Fig. 3.5
are compared in Table 3.3.

5
The Rayleigh distribution is in fact based on the assumption that ε → 0 (Sec. 3.2).
Chapter 3 Short-Term Wave Analysis 79

Table 3.3 Comparison of analysis methods
Analysis Initial Zero Crossing Frequency
σ (m) σ = 0.28 σ
z
= 0.24 σ
f
= 0.28
N 197
T (sec)
T = 6.1
T
p
= 7.6
T
1
= 6,4
T
2
= 6.1
H
s
(m) 1.12 1.05 H
mo
= 1.11
H (m)
0.70 0.68 0.70
H
rms
(m) 0.79 0.76 0.79
0.1
H (m)
1.43 1.30 1.43
H
max
(m) 1.92 1.56 1.92
ε
2
0.65
3.7 Uncertainties in Wave Measurements
At this point, we reflect upon how well we know the basic wave
parameters: wave height, wave period and wave angle. First we define
uncertainty. It quantifies the combination of errors, randomness and
general lack of physical understanding. For most physical quantities,
errors increase with the magnitude of the quantity. For example, the
absolute error in measuring a wave height of 0.5 m will be less than the
absolute error in measuring a wave height of 5 m. For this reason we
normally use a relative error to define the accuracy of our quantities. The
errors in a quantity such as wave height H are assumed to have a normal
distribution with H as its mean value and σ
H
as its standard deviation.
The uncertainty in H is defined as its coefficient of variation
'
H
H
H
σ
σ = (3.46)
More detail about uncertainties may be found in Thoft-Christensen
and Baker (1982), Ang and Tang (1984), Madsen, Krenk and Lind
(1986), Pilarczyk (1990), Burcharth (1992) and PIANC (1992). From
the definition of standard deviation, H is between (1 ' )
H
H σ ± 68% of the
80 Introduction to Coastal Engineering and Management

time, there is a 95% probability that H is between (1 2 ' ),
H
H σ ± and
virtually all values lie between (1 3 ' ).
H
H σ ±
Wave heights are based on measurements of instantaneous water
levels, usually measured offshore, at frequent intervals (e.g. 10 Hz) over
a recording period (e.g. 10 to 20 Minutes). Zero Crossing or Wave
Spectrum Analysis is then used to reduce the instantaneous water level
measurements to one single wave height value (H
s
or H
mo
) to represent
the complete recording period. Along an exposed coast H
s
= 1 m would
be typical. Even for very carefully measured instantaneous water levels,
using the latest equipment H
s
= 1 m would contain an absolute error
(standard deviation) σ
H,Measured
= 0.05 to 0.1 m (say 0.075 m). The
uncertainty in a 1 m wave height would therefore be σ′
H,Measured
= 0.075.
The errors in measuring smaller waves would be less and for larger waves
they would be greater. Therefore an uncertainty σ′
H,Measured
= 0.075 would
not be unreasonable for wave height measurements. This means that for
H = 1 m, there is a 68% probability that 0.92 < H < 1.08 m, a 95%
probability that 0.85 < H < 1.15 m and that almost all values of H lie
between 0.78 < H < 1.22.
Such relatively accurate offshore wave height measurements are
subjected to several conversions before they can be considered useful for
subsequent computations. So far, the value of H
s
= 1 m represents 10 to
20 minutes of record. For the 1 m wave height to represent a complete
recording interval of 3 to 6 hours, it must be remembered that the
environmental parameters such as wind speed and direction, water levels,
etc. are not constant over the recording interval. This increases the
uncertainty of the representative wave height values. The additional
uncertainty depends on the variability of the conditions over the recording
interval, but in most cases it would be reasonable to expect the
uncertainty to double so that σ′
H,Interval
= 0.15.
Uncertainty of measured wave periods T is known to be greater
than for H and reasonable estimates would be σ′
T,Measured
= 0.1 and
σ′
T,Interval
= 0.2. Wave direction (α) is notoriously poorly measured. Even
the best directional instrumentation has difficulty to produce wave
directions within ± 3° for large, well-formed waves and may be as much
Chapter 3 Short-Term Wave Analysis 81

as 10° wrong for smaller, more irregular waves. Estimates of wave
direction by means other than directional measurement are much worse.
Assuming the values of 3° and 10° to be maximum values of the errors in
angle (assuming these values to be 3σ
α
removed from the mean), σ
α
can
be estimated as 1° in the first case and 3.3° in the second. Thus an
average value of standard deviation is σ
α,Measured
= 2°. This value of σ
α
is
independent of the incident wave angle and hence we cannot define σ
α
′.
However, in order to complete our subsequent discussion about
uncertainties, we will relate σ
α
to an incident wave angle of 10° with
respect to the shoreline. In that case, σ′
α,Measured
becomes 0.2. Many
times the incident wave angle on a sandy shore is much smaller than 10°,
which would result in much higher uncertainty values. When the wave
angle with the shoreline approaches 0° (as is often the case), the
uncertainty for wave angle approaches infinity and the whole discussion
about uncertainty loses its meaning. For the longer interval and α = 10°,
a reasonable estimate is σ′
α,Interval
= 4°.

These uncertainty values are only general indications. They are
heavily influenced by assumed average values for wave heights and
periods, and particularly angles of breaking. The actual values are not as
important, however, as the fact that they clearly indicate that the basic
coastal data — wave heights, wave periods and incident wave angles —
contain large uncertainties. Since these wave quantities are basic to all
coastal design calculations, the effects of these uncertainties will pervade
all subsequent calculations. The awareness of uncertainties is basic to
our understanding of the fundamental issues of coastal engineering and
management. For example, it explains why we can use small amplitude
wave theory successfully for most design calculations. The discussion
and evaluation of uncertainties will be extended in later chapters.
3.8 Common Parametric Expressions for Wave Spectra
Since the measured spectra show considerable similarity (they
basically consist of a peak and two curves decreasing toward f = 0 and
f = ∞), attempts have been made to formulate parametric expressions.
82 Introduction to Coastal Engineering and Management

Only the most common expressions will be presented here. Phillips
(1958) postulated that for the “equilibrium range” (for f > f
p
) the spectral
shape S(f ) is proportional to f
-5
. He quantified his results as

2 5
4
(2 )
-
P
P
= g f
α
π
Φ
(3.47)
where Φ
p
denotes the Phillips Function and the “Phillips constant” is
0.0074
P
= α (3.48)
Pierson and Moskowitz (1964), added a low frequency filter to extend
the Phillips expression over the complete frequency range
( )
P PM PM
f =
S Φ Φ
(3.49)
where:

4
5
exp
4
PM
p
f
=
f
−
(
| |
(
− |
Φ
|
(
\ ¹
¸ ¸
(3.50)
The Pierson-Moskowitz spectrum is therefore

4
2
( )
4 5
( )
(2 )
PM / f
PM
g
f =
S e
f
β
α
π
−
(3.51)
Commonly used expressions for α and β are:

4
5
0.0081;
4
PM p
= = f α β (3.52)
The quantity β was also related to wind speed U so that this spectrum can
be used to hindcast waves from wind data (Ch. 5)

4
0.74
2
g
=
U
β
π
| |
|
\ ¹
(3.53)
where U is the wind speed. The Pierson-Moskowitz spectrum is valid
for a fully developed sea condition. For developing seas the Jonswap
Spectrum was proposed by Hasselmann et al (1973). It is essentially an
enhanced Pierson-Moskowitz spectrum as shown in Fig. 3.11.

Chapter 3 Short-Term Wave Analysis 83

Fig. 3.11 Jonswap and Pierson-Moskowitz spectra

A developing seas filter, Φ
J
, can be assumed so that the Jonswap
spectrum is
( )
P PM J J
f =
S
⋅ ⋅
Φ Φ Φ
(3.54)
where

a
e
J
= γ
Φ
(3.55)
and

2
2
2
( )
p
p
f f
a =
f 2
δ
(
− −
(
(
¸ ¸
(3.56)
Typical values of δ are

0.07
0.09
p
p
= for f f
= for f f
δ
δ
≤
>
(3.57)
The Jonswap expression is therefore

4
2
5
4
4 5
( ) ( )
(2 )
a
a
p
e
f
e J
f J PM
g
f = f =
S S e
f
α γ
γ
π
−
| |
| −
|
\ ¹
(3.58)


0
0.0002
0.0004
0.0006
0.0008
0.001
0.0012
0.0014
0.0016
0.0018
0 0.5 1 1.5 2
Frequency - f (Hz)
S
(
f
)


(
m
2
/
H
z
)
Jonswap
Pierson-
Moskowitz
γ

84 Introduction to Coastal Engineering and Management

The coefficient α can be related to the wave generating conditions

0.22
2
0.076
J
gF
=
U
α
−
| |
|
\ ¹
(3.59)
where F is fetch length. Alternately, Mitsuyasu (1980) states

2/7
2
10
0.0817
M
gF
=
U
α
−
| |
|
\ ¹
(3.60)
and

0.33
2
2.84
p
gF
= f
U
−
| |
|
\ ¹
(3.61)
The peak enhancement factor

( )
( )
J
p
PM
p
f
S
=
f
S
γ (3.62)
has an average value of 3.3 and typically lies between 1 and 7.
Mitsuyasu (1980) postulates

1/7
2
10
7.0
gF
=
U
γ
−
| |
|
\ ¹
(3.63)
The above development traces the derivation of a parametric
expression for a wave spectrum from the equilibrium spectrum (Phillips)
through the fully developed sea spectrum (PM) to the developing sea
spectrum (Jonswap). But waves have a limiting steepness. Thus, any
wave in a wave train that reaches a limiting steepness will break. This is
known as Spectral Saturation. Bouws et al (1985) modify the Jonswap
spectrum to take spectral saturation into account and produce the TMA
spectrum.

P J d TMA PM
( f,d) =
S
⋅ Φ ⋅ ⋅
Φ Φ Φ
(3.64)
where

tanh
2
d
1 2 d
=
2n L
π
Φ
(3.65)
Chapter 3 Short-Term Wave Analysis 85

In deep water, the value of Φ
d
is one. In other words, the Jonswap
spectrum through its own derivation takes into account the deep water
wave steepness limitation and Eq. 3.63 modifies the Jonswap spectrum
for wave breaking induced in shallow water.
3.9 Directional Wave Spectra
Until now η has been considered to be a function of time at a single
location and we learned to calculate S(f ) from such a time series. We
also discussed some parametric expressions for such wave spectra.
However, η is also a function of direction (of x and y). Measurement of
wave direction involves correlating spectra for several synoptic, adjacent
records of water levels, pressures and/or velocities. Some discussion may
be found in Sec. 2.3.1, but detailed discussion is beyond the scope of this
book. An example of a Directional Wave Spectrum (a function of both
wave frequency and direction) is shown in Fig. 3.12 and a good
description of directional wave spectra may be found in Goda (1985).
To describe such a spectrum, the simplest approach is
( ) ( ) ( ) S f, = S f G θ θ (3.66)
G is called the Directional Spreading Function and θ is measured
counter-clockwise from the wave direction. A necessary condition is
obviously
( ) 1 G d =
θ π
θ π
θ θ
=
=−
∫
(3.67)
Two common directional spreading functions used are the
Cos-Squared function

2
2
( ) cos
2
( ) 0
G = for <
G for all other values of
π
θ θ θ
π
θ θ =
(3.68)
and the Cos-Power function (Mitsuyasu, 1980; Goda, 1985).
86 Introduction to Coastal Engineering and Management


Fig. 3.12 Directional wave spectrum
θ θθ θ
f
S
η ηη η
(f,θ θθ θ)

Chapter 3 Short-Term Wave Analysis 87