The directional spectrum is calculated as follows (Vincent and Briggs 1989),
( jσm )2 cos j θ -
1 j
1
D(θ) =
(
θm )
+ ∑ exp-
2π π j =1
2
where θm = mean wave direction = 0
j
= number of terms in the series (20 in numerical calculations)
σm = spreading parameter
The same functions E(f) and D(θ) were also used in the laboratory study by
Vincent and Briggs (1989) to construct a number of frequency and directional spectra (for
details, see Vincent and Briggs 1989 or Panchang et al. 1990). One spectral case, B1,
consisted of a broad frequency spectrum; the other case, N1, had a narrow frequency
spectrum. The input data are summarized in Table 2. Two monochromatic and two
spectral cases are shown. The incident wave height is 2[2E(f)D(θ)∆f∆θ]1/2.
Table 2. Model Input Data
α
γ
σm
Input
Case ID
Period
Significant
(sec)
Wave
Height (m)
Monochromatic
M1
1.3
.0550
--
--
--
M2
1.3
.0254
--
--
--
30
Broad-directional
B1
1.3
.0775
.01440
2
10
Narrow-directional
N1
1.3
.0775
.01440
2
The discretization of the directional spectrum is summarized in Table 3. Other
researchers have used the following discretizations.
Panchang et al. (1990) used a
directional bin width (∆θ) of 4.39in a range from 45to +45 resulting in 39 directional
,
components for all spectra. Zhao and Anastasiou (1993) found that as many as 63
components were needed for the B1 and N1 cases. Li et al. (1993) produced results that
suggest that an adequate solution can be obtained with as few as five directional
components and five frequency components. In light of the results of Li et al. (1993), it
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