42
where
{\
Radder
i Boij
(3.66)
and where P2 = Pi + j for both approximations. Kirby chose the approximation with
Pi = 0 since for his derivation which was for weakly nonlinear waves it should correspond
to the information contained in the nonlinear Schrodinger equation. With Pi = 0 Eq. (3.65)
becomes
1
k(p u2)fc + {fc(p u2)}x 4>+ = ik2(p U2)j>+ + '-M4>+
(3.67)
where M+ is defined by Eq. (3.57). An equation for the complex amplitude A is then
obtained by making the substitution
(3.68)
which after dropping squares of the components of the mean current results in
(C, + U)A, + \ 'A + (£)VA, + \ (7)J A
+i [cc' () J + ?A = 0 (3'69)
Kirby then further modified the equation by a shift to a reference phase function using the
relation
A = A'ei(i-fkdx) (3.70)
where k can be taken as an averaged wave number, resulting in
(Ct + U)A'X + i(ic k)(C¡ + U)A' + I A'
The shift to a reference phase function is intended to incorporate the phase information
in the complex amplitude so that the instantaneous water surface can be easily recovered
in the form
q(x,y) = 3i{AV/£d1}
(3.72)