EM 1110-2-1100 (Part II)
30 Apr 02
(f)
It therefore follows from the above last two equations that G(f,θ) must satisfy
m&π
π
G(f,θ) dθ ' 1
(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
f such that we have
2
cos2θ for *θ* < 90E
G(θ) '
(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
G(θ) ' C(s) cos 2s
2
(II-1-166)
π
Γ (s % 1)
C(s) '
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
f
for f # fp
smax
fp
s'
(II-1-167)
&2.5
f
smax
for f > fp
fp
where smax and fp are defined as
&2.5
2πfpU
smax ' 11.5
g
(II-1-168)
2πfpU
&0.33
gF
' 18.8
g
U2
(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 smax on refraction is of concern. For deepwater
II-1-94
Water Wave Mechanics
Previous Page
