86
4 Physical Mechanisms Forming the Wave Spectrum in Deep Water
Using the symmetry of the variables k 2 and k3 the integral (4.1) can be
written as:
(4.5)
The first integration of (4.5) is carried out over the variable k 3 . Transforming
the variables from ki = {kxi, kyi} to Ui and to the angles f3i =arctan (kyi/kxi)
and from the wave action N(k) to the wave energy S(w,O), the expression
N(k) = (w 2 l2g 4 )S(w, 0) is obtained.
The integral (4.5) is written in the form:
Using the function 6(!7) the following expression is obtained:
= +n
GnJ(O", {3) = 4 L f f f T { s sl (820"~ + S3!7~)- s2 S3(Bt7t + Slu 4 )}
{32 0 -7{ <72
where:
fJ2 = f3a ±arccos ( (k~ + u~- !7~) I (2kat7~))
{33 = f3a =f arccos ( (k~ + !7~- w~) I (2kat7~))
f3a = arccos ( ( u 2 cos {3 + uf cos fJ1) I ka) sign( k1y + ky) .
The function B = B(t71, 0"2, fJ1) is written as
(4.7)
It should be noted that the function B = B(0"1,u2,f31) becomes zero at
some points (see Fig. 4.2). It produces a singularity and causes difficulties in
integrating ( 4. 7) numerically.