EM 1110-2-1100 (Part II)
30 Apr 02
E [η(t) η(t%τ)]
will be equal to 1.
For two different random signals η1 and η2, the cross-correlation coefficient R may be defined as
m& τ 1
R ' E [η1(t) η2(t%δt)] '
η (t) η2(t%δt) dt
which measures the degree of correlation between two signals. This concept is useful for example in relating
wave velocities and pressures obtained at two separate locations during wave gauge measurements in coastal
projects. Note that the process η(t) is stationary if η and ση are constant for all values of t, and that R is a
function only of τ = t2 - t1.
(j) Assuming that the water surface elevation η(t) is a stationary random process, representing a sea
state over the duration of several hours, we will next focus our attention on defining the probabilistic
properties of ocean waves. The probabilistic representation of sea state is useful in practice for two reasons.
First, it allows the designer to choose wave parameters within a limit that will yield an acceptable level of
risk. Second, a probabilistic-based design criterion may result in substantial cost savings by considering
uncertainties in the wave estimates. Therefore, an overview of the probability laws and distributions for
ocean waves follows.
(6) Probability distributions for a sea state.
(a) As noted earlier, irregular sea states are random signals. For engineers to effectively use irregular
waves in design, properties of the individual wave records must follow some probability laws so that wave
statistics can readily be obtained analytically. Rice (1944-1945) developed the statistical theory of random
signals for electrical noise analysis. Longuet-Higgins (1952) applied this theory to the random water surface
elevation of ocean waves to describe their statistics using certain simplified assumptions. Longuet-Higgins
found that the parameters of a random wave signal follow known probability laws.
(b) The probability distribution P(x) is the fraction of events that a particular event is not exceeded.
It can be obtained directly from a plot of the proportion of values less than a particular value versus the
particular value of the variable x0, and is given by
P(x) ' prob x # x0
(c) The probability density p(x) is the fraction of events that a particular event is expected to occur and
thus, it represents the rate of change of a distribution and may be obtained by simply differentiating P(x) with
respect to its argument x.
(d) The two most commonly used probability distributions in the study of random ocean waves are the
Gaussian (Figure II-1-28) and Rayleigh distributions (Figure II-1-29). The Gaussian distribution is
particularly suited for describing the short-term probabilities of the sea surface elevation η. Its probability
density is given by
Water Wave Mechanics