1.3 Wave Action Conservation Law
19
and te = Et are the slowly changing horizontal coordinates and time; and a is
a small amplitude parameter. As soon as Vo = Vo(re, z, te) it follows from
the continuity equation ( 1. 2) that W 0 "' c IV 0 1- The slowness of the bottom
surface variation H = H(re) is also assumed.
By substituting the expression (1.30) into (1.1)-(1.3) the main equations
of the "background" motion can be derived:
avo
1
ate + (Vo\7)Vo = -p\7rPo;
\7Vo = 0;
a Po
gp = - az .
(1.31)
(1.32)
(1.33)
The boundary conditions for the system (1.31)-(1.33) coincide with the
expressions (1.28), (1.29), assigning a "0" index to all values. The solution of
the equation is derived by the WKB expansion for the disturbance :
(1.34)
Substituting the expansion (1.34) into the equations for disturbances and
equaling values of the order a in the expansion (1.30), the equation and
boundary conditions for the vertical velocity of the first-order disturbance wl
can be obtained (the index (1) is omitted below):
W" + (-a~' -k 2 ) W = 0 .
(1.35)
( w)' __ 9 k 2
a
a 3
with z=r]o, W=O with z=-H(re),
(1.36)
where a = w- (k, V) is the Doppler frequency depending on the vertical
coordinate z. A prime means a derivative with respect to z.
The boundary problem produces a set of dispersion ratios for different
modes:
W = F(k, Te, t)
(1.37)
and the eigenfunctions W = W(re, z, te), dependent on re and te, are given
with the help of the marginal problem (1.35), (1.36).
The other wave values are expressed using W by the formulas:
V = ik a (W)' _ (iW) av 0 ;
k 2
a
a
az
P = ia (w)'
k 2
a
'
.W
T) = 1 - .
a
(1.38)