GUIDE WAVE ANALYSIS AND FORECASTING - WMO

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for instance, a frequency of 0.16 Hz is considered to be a
mean value in an interval which could be 0.155 to
0.165 Hz. This value, divided by the width of the interval,
is a measure for the energy density and expressed in
units of m 2 /Hz (again omitting the factor ρ wg). In fact
the wave spectrum is often referred to as the energydensity
spectrum.
Thus, this method of analysing wave measurements
yields a distribution of the energy of the various
wave components, E(f,θ). It was noted in Section 1.3.2
that wave energy travels at the group velocity c g, and
from Equation 1.15 we see that this is a function of
both frequency and direction (or the wavenumber
vector) and possibly water depth. The energy in each
spectral component therefore propagates at the associated
group velocity. Hence it is possible to deduce how
wave energy in the local wave field disperses across the
ocean.
It is important to note that a wave record and the
spectrum derived from it are only samples of the sea
state (see Section 1.3.4). As with all statistical estimates,
we must be interested in how good our estimate is,
and how well it is likely to indicate the true state. There
is a reasonably complete statistical theory to describe
this. We will not give details in this Guide but refer the
interested reader to a text such as Jenkins and Watts
(1968). Suffice to say that the validity of a spectral
estimate depends to a large extent on the length of the
record, which in turn depends on the consistency of the
sea state or statistical stationarity (i.e. not too rapidly
evolving). The spectral estimates can be shown to have
the statistical distribution called a χ 2 distribution for
which the expected spread of estimates is measured by
a number called the “degrees of freedom”. The larger
the degrees of freedom, the better the estimate is likely
to be.
1.3.8 Wave parameters derived from the
spectrum
A wave spectrum is the distribution of wave energy (or
variance of the sea surface) over frequency (or wavelength
or frequency and direction, etc.). Thus, as a
statistical distribution, many of the parameters derived
from the spectrum parallel similar parameters from any
statistical distribution. Hence, the form of a wave spectrum
is usually expressed in terms of the moments* of
the distribution (spectrum). The nth-order moment, mn, of the spectrum is defined by:
n
m n = f E( f ) df (1.22)
(sometimes ω = 2πf is preferred to f). In this formula,
E(f) denotes the variance density at frequency, f, as in
∞
∫0 _________
GUIDE TO WAVE ANALYSIS AND FORECASTING
2 Figure 1.16, so that E(f) df represents the variance ai /2
contained in the ith interval between f and f + df. In practice,
the integration in Equation 1.22 is approximated by
a finite sum, with fi = i df:
m f (1.22a)
From the definition of mn it follows that the
moment of zero-order, m0, represents the area under the
spectral curve. In finite form this is:
a
n i n
N 2
i = ∑ .
i = 0 2
which is the total variance of the wave record obtained
by the sum of the variances of the individual spectral
components. The area under the spectral curve therefore
has a physical meaning which is used in practical applications
for the definition of wave-height parameters
derived from the spectrum. Recalling that for a simple
wave (Section 1.2.4) the wave energy (per unit area), E,
was related to the wave height by:
then, if one replaces the actual sea state by a single sinusoidal
wave having the same energy, its equivalent height
would be given by:
8E
Hrms
= ,
ρ g
the so-called root-mean-square wave height. E now
represents the total energy (per unit area) of the sea
state.
We would like a parameter derived from the spectrum
and corresponding as closely as possible to the
significant wave height H – 1/3 (as derived directly from
the wave record) and, equally, the characteristic wave
height H c (as observed visually). It has been shown that
H rms should be multiplied by the factor √2 in order to
arrive at the required value. Thus, the spectral wave
height parameter commonly used can be calculated
from the measured area, m 0, under the spectral curve as
follows:
H
m
0
N
a a
= ∑ =
2 2
i ,
i = 0
E = gH ,
1 2
ρw 8
8E
= 2 = 4
m .
m0
0
ρwg Note that we sometimes refer to the total variance
of the sea state (m 0) as the total energy, but we must be
mindful here that the total energy E is really ρ wgm 0. In
theory, the correspondence between H m0 and H – 1/3 is valid
only for very narrow spectra which do not occur often in
nature. However, the difference is relatively small in
* The first moment of a distribution of N observations X 1, X 2, …, X n is defined as the average of the deviations x 1, x 2, …, x n
from the given value X 0. The second moment is the average of the squares of the deviations about X 0; the third moment is the
average of the cubes of the deviations, and so forth. When X 0 is the mean of all observations, the first moment is obviously
zero, the second moment is then known as the “variance” of X and its square root is termed the “standard deviation”.
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w
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