ρη '

(II-1-122)

'

will be equal to 1.

(i)

For two different random signals *η*1 and *η*2, the *cross-correlation coefficient R *may be defined as

τ

1 2

m& τ 1

η (*t*) η2(*t*%δ*t*) *dt*

(II-1-123)

τ

2

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 *σ*η are constant for all values of *t*, and that *R *is a

function only of *τ = t*2 - 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 *x*0, and is given by

(II-1-124)

(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

particularly suited for describing the short-term probabilities of the sea surface elevation η. Its probability

density is given by

II-1-70

Water Wave Mechanics

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