(c) First, it would be necessary to assume that the process described by the wave record (i.e., a sea

state), say *η(t)*, is *stationary*, which means that the statistical properties of *η(t) *are independent of the origin

of time measurement. Since the statistics of stationary processes are time-invariant, there is no drift with time

in the statistical behavior of *η(t)*. The stationarity requirement is necessary as we shall see later for

developing a *probability distribution *for waves, which is the fraction or percentage of time an *event *or

distribution may be obtained by taking *η*1(t1), η2(t1), η3(t1),..., as variables, independent of the instant *t*1. If in

addition, *η(t) *can be measured at different locations and the properties of *η(t) *are invariant or do not depend

on location of measurements, the process may then be assumed *homogenous*. In reality, *η(t) *may be assumed

stationary and homogenous only for a limited duration at the location data are gathered. Wind waves may

be considered approximately stationary for only a few hours (3 hr or less), beyond which their properties are

expected to change.

(d) Second, the process *η(t) *is assumed to be *ergodic*, which means that any measured record of the

process say *η*1(t) is typical of all other possible realizations, and therefore, the average of a single record in

an ensemble is the same as the average across the ensemble. For an ergodic process, the sample mean from

the ensemble approaches the real mean , and the sample variance approaches the variance *σ *of the process

(sea state). The ergodicity of *η(t) *implies that the measured realization of *η(t)*, say *η*1(t1) is typical of all other

possible realizations *η*2(t1), η3(t1), ...., all measured at one instant *t*1. The concept of ergodicity permits

derivation of various useful statistical information from a single record, eliminating the need for multiple

recordings at different sites. The assumptions of stationarity and ergodicity are the backbones of developing

wave statistics from wave measurements. It is implicitly assumed that such hypotheses exist in reality, and

are valid, particularly for the sea state.

(e) To apply these concepts to ocean waves, consider an ensemble of records representing the sea state

by *η(t) *over a finite time *T*. The *mean *or *expected value *of the sea state, denoted by *η*, or η, or *E[η]*, is

defined as

τ

1 2

τ m& τ

η ' *E *[η(*t*)] '

η(*t*) *dt*

(II-1-119)

2

where the symbol *E *denotes the expected value of *η(t)*. Similarly, the mean-square of *η *corresponds to the

second moment of *η*, denoted by *E[η*2]. The standard deviation *σ*η or the root-mean-square value of the

process is the square root of this. The *variance *of *η*, represented by *σ*η2 may be expressed in terms of the

variance of the process *V *as

2

2

ση ' *V *[η(*t*)] ' *E *[η2] & η

(II-1-120)

(f) The *standard deviation σ*η is the square root of the variance, also called the second central moment

of *η(t)*. The standard deviation characterizes the spread in the values of *η(t) *about its mean.

(g) The *autocorrelation *or *autocovariance function *of the sea state is denoted by *R*η, relating the value

of *η *at time *t *to its value at a later time *t+τ*. This is defined as

(II-1-121)

(h) The value of *R*η gives an indication of the correlation of the signal with itself for various time lags

itself for zero lag *τ*, its autocorrelation coefficient, defined as

Water Wave Mechanics

II-1-69

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