(e) The fact that the statistics of wave height for wave records in general follows a Rayleigh distribution

is of great significance in coastal engineering. For instance, an engineer may have information from a

hindcast (see Part II-2) that the significant height for a storm is 10 m. Assuming that the Rayleigh distribution

describes the wave record, the engineer can estimate that the 10-percent wave will be 12.7 m and that the Hmax

(assuming 1,000 waves in the record) will be 18.6 m. Often measured ocean wave records are analyzed

spectrally (see "Spectral Analysis" section later in this chapter) by the instrument package and only

condensed information is reported via satellite to a data bank, with no other information retained. The

inherent assumption made is that the Rayleigh distribution is adequate.

(f) Theoretical relationships derived from the Rayleigh distribution generally agree well with the values

determined directly from the records. The Rayleigh probability distribution density function is compared

with a histogram of the measured deepwater wave heights in Figure II-1-30 (Chakrabarti 1987). Clearly the

Rayleigh distribution fits this data well, even though the frequency spectra of ocean waves may not always

be narrow-banded as assumed in the Rayleigh distribution. Field measurements sometimes deviate from the

Rayleigh distribution, and the deviation appears to increase with increasing wave heights, and decrease as

the wave spectrum becomes sharply peaked. The effect of bandwidth on wave height distribution has been

accounted for theoretically (Tayfun 1983).

(g) Deepwater wave height measurements from different oceans have been found to closely obey a

Rayleigh distribution (Tayfun 1983a,b; Forristall 1984; Myrhaug and Kjeldsen 1986). This is not true for

shallow-water waves, which are strongly modulated by the bathymetric effects combined with the amplitude

nonlinearities. The wave energy spectrum of the shallow-water waves is not narrow-banded and may

substantially deviate from the Rayleigh distribution especially for high frequencies. In general, the Rayleigh

distribution tends to overpredict the larger wave heights in all depths.

(h) In summary, the Rayleigh distribution is generally adequate, except for near-coastal wave records

in which it may overestimate the number of large waves. Investigations of shallow-water wave records from

numerous studies indicate that the distribution deviates from the Rayleigh, and other distributions have been

shown to fit individual observations better (SPM 1984). The primary cause for the deviation is that the large

waves suggested in the Rayleigh distribution break in shallow water. Unfortunately, there is no universally

accepted distribution for waves in shallow water. As a result, the Rayleigh is frequently used with the

knowledge that the large waves are not likely.

(8) Wave period distribution.

(a) Longuet-Higgins (1962) and Bretschneider (1969) derived the wave period distribution function

assuming the wave period squared follows a Rayleigh distribution. This distribution is very similar to the

normal distribution with a mean period given by

(II-1-134)

where the moments are defined in terms of cyclic frequency (i.e., Hertz). The probability density of wave

period *T *is given by (Bretschneider 1969)

Water Wave Mechanics

II-1-75

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