exp [&0.675τ4]

(II-1-135)

τ'

(b) A different probability density distribution of the wave period has been derived by Longuet-Higgins

(1962). This is given by

1

2(1 % τ2) 3/2

(II-1-136)

2

τ'

;

ν'

υ*T*0,1

2

where *ν *is the *spectral width parameter *and *m*0, *m*1, and *m*2 are *moments *of the wave spectrum, which will

be defined later. This probability density function is symmetric about *τ *= 0 where it is maximum, and is

similar to the normal distribution with a mean equal to *T*0,1. This distribution fits field measurements

reasonably well, and is often used in offshore design. In general, probability density for the wave period is

narrower than that of wave height, and the spread lies mainly in the range 0.5 to 2.0 times the mean wave

period.

(c) Various characteristic wave periods are related. This relationship may be stated in a general way

as

(II-1-137)

where the coefficient *C *varies between 1.1 and 1.3.

(9) Joint distribution of wave heights and periods.

(a) If there were no relation between wave height and wave period, then the joint distribution between

wave height and wave period can simply be obtained from the individual probability distributions of the

height and period by

(II-1-138)

(b) The distribution p(H,T) so obtained is inappropriate for ocean waves, since their heights and periods

are correlated. For the joint distribution of wave height-period pairs, Longuet-Higgins (1975b) considered

wave heights and periods also representable by a narrow-band spectrum. He derived the joint distribution

assuming wave heights and periods are correlated, a more suitable assumption for real sea states.

(c) The probability density function of wave period may be obtained directly from the joint distribution,

provided that a measure of the spectrum width is included in the latter. Under this condition, the distribution

of wave period is simply the marginal probability density function of the joint distribution of *H *and *T*. This

is done by integrating *p(H,T) *for the full range of *H *from 0 to 4. Likewise, the distribution for wave heights

may be obtained by integrating *p(H,T) *for the full range of periods. The joint distribution derived by

Longuet-Higgins (1975b) was later modified (Longuet-Higgins 1983), and is given by

II-1-76

Water Wave Mechanics

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