4.1 Non-Linear Energy Transfer in Wind Wave Spectrum
89
where (31i = ;::i. In case~= 2 ' :n or~= ~;:,, the expression ( 4.10) is valid for
Cm(f3) = Tm_ 1 ((3)+am cos(m(3), where Tm-1 is a trigonometrical polynomial
with the power of m - 1. In order to obtain results with an error less than
1-2 per cent, a sufficiently large number of ordinates, i.e. m;::: 90, should be
taken.
The problem is that the latter integration over (31 is not optimal. A large
number of ordinates is to be used because the function F(a1 , (31) includes
singularities as well. The function F(a1, (31) becomes infinite when Ea =
2ka/a~ = 1. The most effective integration can be achieved by transforming the variables. In this case the Jacobi functions can be used to obtain the
cubature formulas. Thus, the function ( 4.10) can be written as:
7(
7(
J(a1) = J F(a1,(3I) df31 = J F\a1,(3I)/Jicos((3- !31)- AI d(31, (4.11a)
-7(
-7(
where
F = PJ1cos((3- !31)- AI,
A= ((a+a1) 4 -4(af+o.4))j(8a 2 oD.
The function A:::; 1 takes its maximum value at the point a= aa.
In order to introduce a new variable x = cos((3- (31), the integral (4.11a)
can be written in the following form:
(4.11b)
where f3t = (3 ± ~rccos(x).
The function F(a1 ,x) is smooth enough. The integral (4.12) includes the
same singularities as the first-order elliptical integral. Numerical results show
that it is enough to use 6-8 ordinates to obtain rather good accuracy.
The last integration over a 1 can be carried out effectively taking into
account that the function J(ai) is approximated as J(ai) "' a1 6 for large
values of a 1, and as J(al) "' d 5725 for small values of a 1. This allows the
use of the traditional cubature method of integration. It should be noted that
in order to speed up the computation, the part of the function in ( 4. 7), not
depending on the spectral value, is computed using the symmetric quantity
(Hasselmann & Hasselmann, 1981).
It should be noted that the main advantage of the algorithm is that the
integration is based on a relatively small number of grid points compared
to the usual methods (Masuda 1981; Polnikov 1988). This speeds up the
computation by at least two orders of magnitude (Lavrenov, 1998).
89
where (31i = ;::i. In case~= 2 ' :n or~= ~;:,, the expression ( 4.10) is valid for
Cm(f3) = Tm_ 1 ((3)+am cos(m(3), where Tm-1 is a trigonometrical polynomial
with the power of m - 1. In order to obtain results with an error less than
1-2 per cent, a sufficiently large number of ordinates, i.e. m;::: 90, should be
taken.
The problem is that the latter integration over (31 is not optimal. A large
number of ordinates is to be used because the function F(a1 , (31) includes
singularities as well. The function F(a1, (31) becomes infinite when Ea =
2ka/a~ = 1. The most effective integration can be achieved by transforming the variables. In this case the Jacobi functions can be used to obtain the
cubature formulas. Thus, the function ( 4.10) can be written as:
7(
7(
J(a1) = J F(a1,(3I) df31 = J F\a1,(3I)/Jicos((3- !31)- AI d(31, (4.11a)
-7(
-7(
where
F = PJ1cos((3- !31)- AI,
A= ((a+a1) 4 -4(af+o.4))j(8a 2 oD.
The function A:::; 1 takes its maximum value at the point a= aa.
In order to introduce a new variable x = cos((3- (31), the integral (4.11a)
can be written in the following form:
(4.11b)
where f3t = (3 ± ~rccos(x).
The function F(a1 ,x) is smooth enough. The integral (4.12) includes the
same singularities as the first-order elliptical integral. Numerical results show
that it is enough to use 6-8 ordinates to obtain rather good accuracy.
The last integration over a 1 can be carried out effectively taking into
account that the function J(ai) is approximated as J(ai) "' a1 6 for large
values of a 1, and as J(al) "' d 5725 for small values of a 1. This allows the
use of the traditional cubature method of integration. It should be noted that
in order to speed up the computation, the part of the function in ( 4. 7), not
depending on the spectral value, is computed using the symmetric quantity
(Hasselmann & Hasselmann, 1981).
It should be noted that the main advantage of the algorithm is that the
integration is based on a relatively small number of grid points compared
to the usual methods (Masuda 1981; Polnikov 1988). This speeds up the
computation by at least two orders of magnitude (Lavrenov, 1998).