(7) Wave height distribution.

(a) The heights of individual waves may be regarded as a stochastic variable represented by a

probability distribution function. From an observed wave record, such a function can be obtained from a

histogram of wave heights normalized with the mean heights in several wave records measured at a point

(Figure II-1-30). Thompson (1977) indicated how well coastal wave records follow the Rayleigh distribution.

If wave energy is concentrated in a very narrow range of wave period, the maxima of the wave profile will

coincide with the wave crests and the minima with the troughs. This is termed a *narrow-band condition*.

Under the narrow-band condition, wave heights are represented by the following Rayleigh distribution

(Longuet-Higgins 1952, 1975b, 1983)

2*H*

exp &

2

2

(II-1-130)

2

(b) The significant wave height *H*1/3 is the centroid of the area for *H *$ H* under the density function

where *H > H** corresponds to waves in the highest one-third range as shown in Figure II-1-29, that is

2

&

1

2

(II-1-131)

'1&*e*

3

from which we find *H** = 1.05Hrms. Various estimates of wave heights may then be obtained upon integration

of the above equation using certain mathematical properties of the Error function (Abramowitz and Stegun

1965). We find

(II-1-132)

(*for *1000 *wave cycles in the record*)

(c) The *most probable maximum wave height *in a record containing *N *waves is related to the rms wave

height (Longuet-Higgins 1952) by

0.2886

0.247

log*N *%

(II-1-133)

&

(log*N*)3/2

log*N*

(d) The value of Hmax obtained in this manner can be projected to a longer period of time by adjusting

the value of *N *based on the mean zero-upcrossing period (Tucker 1963).

II-1-74

Water Wave Mechanics

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