47
Since R and 0 are real, the real and imaginary parts of (3.91) must individually equal zero.
Taking the imaginary part and multiplying by R gives
V* (kCCg)R2 awR2 =
(3.92)
which using the substitution kCCa = aCt is written as
{(rR2)t + (aR2(0 + Cy)) = -awR2
Using (3.88) R2 may be replaced by
Ri = = 2p
a2 a \a J
which yields the wave action conservation equation
Assuming a time steady wave field and neglecting currents, (3.95) becomes
(3.93)
(3.94)
(3.95)
[ECt)x = -w E
(3.96)
Comparing (3.96) with (3.85) and noting that (Cg)t = Cg, w is written as
(3.97)
where H is the local wave height given by 2|A|.
In obtaining Eq. (3.97) Kirby neglected the effects of the currents. However, the same
expression (3.97) is obtained when currents are included in the derivation. Dally (1987)
extended his work to investigate wave breaking on currents and proposed the following
equation:
+ U) (f )]i = lEC* ~ (3 98)
For the case of a steady state wave field propagating in the x direction the conservation of
wave action Eq. (3.95) reduces to
(3.99)