45
Details of the algebra and assumptions involved in obtaining Eq. (3.79) are given in Ap
pendix D. For computational purposes it will be assumed that cos# = cos#.
From Eq. (3.82) the correct form of the conservation of wave action can be obtained.
For the case of no y variation of the wave number Eq. (3.82) reduces to
(Cf cos # + U)A'X + L
Cgcos # + U
A' + VA[
l+i/
(3.83)
Multiplying Eq. (3.83) by the complex conjugate of A and adding to it A times the complex
conjugate equation of (3.83) yields
(C, cos# + U)
+
\V\A\2]
a
X
a
= 0
(3.84)
which is a closer approximation to the exact expression which should have a (C, sin + V)
term instead of V in the y derivative term.
3.3 Energy Dissipation Due to Wave Breaking
Kirby (1983) found a method of relating the dissipation coefficient w to the work of
Dally (1980) who proposed that the decay of energy flux in the surf zone was proportional
to the amount of excess energy flux. The excess energy flux is the difference between the
actual flux and a stable energy flux. In mathematical terms this relationship is expressed
as
(ECg)x = -^[ECg-(ECg)t] (3.85)
where
EC,
E
c,
K
h
the energy flux
wave energy = ^pgH2
d(T
the rate of energy propagation which is
ok
constant
the water depth