40
where £ is an as yet unknown operator on . Substituting into Eq. (3.45) and neglecting
derivatives of ( alone leads to the result
4>zz +
<¡>Z + = 0
(3.50)
The elliptic equation (3.37) is now put into the form of Eq. (3.50). Letting p = CCg
for notational convenience and expanding some of the terms of Eq. (3.37) in their spatial
coordinates gives
[p4>z)z + (p4>y)v + k7p4> + (u>2 a2 + iw(Vfc U))4> + iaw4>
~{U2k)z ~ {V2iv)v (UV]>z)v (UV4>v)z + 2jU -Vj = 0 (3.51)
Booij chose to neglect terms involving the square of the mean current components assuming
that they are smaller than terms involving p and then arranged the above equation as
4>zz + P l[Pz + 2iuU)4>x + k2 ^1 + Jj-'j 4> Q (3.52)
where
M4> = (w2 a2 + tw(Vfc ) + 0w)< + 2iuV<¡>v + [p4>v)v (3.53)
Kirby points out that this choice by Booij leads to his inability to derive the correct wave
action equation. Kirby instead does not drop any of the squares of the mean current
components until after the splitting is complete. By assuming that the waves are oriented
in the x direction the dispersion relation is written as
a = oj kU
(3.54)
leading to the result
(jj2 o2 2ukU {kU)2
(3.55)
Using this relationship Kirby then arranged Eq. (3.51) as
h, + (p v')-'(p U*)J, +(1 + ^ ) i = 0
(3.56)