88
4 Physical Mechanisms Forming the Wave Spectrum in Deep Water
Unf
0.35
O.;JO
0.20
0.10
f il
0
' I
---- -----0.10
-0.20
-0.30
-0.40
- - - - 1
·-·-·2
-0.50
_.,_..,... J
-o--o- ~
- - 5
-0.60
·••••••• 6
-0.70
-0.80
0.6
1.0"
1.~
1.8
2.2
2.5
J.O
J.~G/Gmax
Fig. 4.3. Computations of non-linear energy transfer function with different number of points n in (4.9) and (4.10): 1- n = 2; 2- n = 3; 3- n = 4; 4- n = 5;
5 - n = 7; 6 - n = 8
where Aj are weight coefficients; 0"2j are function ordinates; and f2(0"2 , 0"1 , (31)
is a function without singularity at the point 0"2 = a. As shown by the
numerical results (see Fig. 4.3) it is enough to use n = 7 in order to obtain
good results with a relative error less than 1-2 per cent. It is enough to use
n = 4 for many practical computations.
The following integration is carried out with respect to (31 assuming that
the function F2 (0"1 , (31) is periodic. It is known (Krylov & Shulgina, 1966) that
the numerical integration algorithm of the highest precision is the ordinary
rectangular method:
(4.10)