(f)

It therefore follows from the above last two equations that *G(f,θ) *must satisfy

m&π

π

(II-1-164)

(g) The functional form of *G(f,θ) *has no universal shape and several proposed formulas are available.

In the most convenient simplification of *G(f,θ)*, it is customary to consider *G *to be independent of frequency

2

cos2θ *for **θ* < 90E

(II-1-165)

π

(h) This cosine-squared distribution is due to St. Denis and Pierson (1953), and testing with field data

shows that it reproduces the directional distribution of wave energy. Longuet-Higgins (1962) found the

cosine-power form

θ&

θ

2

(II-1-166)

π

Γ (*s *% 1)

2π

1

Γ *s*%

2

where *θ *is the principal (central) direction for the spectrum, *s *is a controlling parameter for the angular

distribution that determines the peakedness of the directional spreading, *C(s) *is a constant satisfying the

normalization condition, *θ *is a counterclockwise measured angle from the principal wave direction, and *Γ*

is the Gamma function.

(i) Mitsuyasu et al. (1975), Goda and Suzuki (1976), and Holthuijsen (1983) have shown that for wind

waves, the parameter *s *varies with wave frequency and is related to the stage of wave development (i.e., wind

speed and fetch) by

5

(II-1-167)

&2.5

where *s*max and *f*p are defined as

&2.5

2π*f*pU

(II-1-168)

2π*f*pU

&0.33

' 18.8

(j) In the above equations, *U *is the wind speed at the 10-m elevation above the sea surface and *F *is the

fetch length. These equations remain to be validated with field data for wind waves. The parameter *s *for

shallow-water waves may also vary spatially during wave transformation. This is due to refraction. A large

value greater that 50, may be necessary if dependence of *s*max on refraction is of concern. For deepwater

II-1-94

Water Wave Mechanics

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