EM 1110-2-1100 (Part II)
30 Apr 02
(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
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