EM 1110-2-1100 (Part II)
30 Apr 02
(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)
P(H) ' 1 & exp &
(b) The significant wave height H1/3 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
P(H() ' 1 &
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
H1/3 . 4.00 m0 ' 1.416 Hrms
H1/10 ' 1.27 H1/3 ' 1.80 Hrms ' 5.091 m0
H1/100 ' 1.67 H1/3 ' 2.36 Hrms ' 6.672 m0
Hmax ' 1.86 H1/3
(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
(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).
Water Wave Mechanics