36
3.2 A Parabolic Approximation
Equation (3.33) is a hyperbolic wave equation. It can be altered to an elliptic equation
by assuming that any time dependency is solely in the wave phase. Before doing this an
energy dissipation term DD is introduced as follows. It is assumed that Eq. (3.33) holds
for the case of no energy dissipation; then for the case with dissipation the left hand side is
balanced by any energy that is dissipated. The governing equation is then written as
+ (V* V*(CC,Vfc) + (
(3.35)
where w is an undefined coefficient indicating the strength of the dissipation. Eventually
the w term will be related to the energy dissipation due to wave breaking following the work
of Dally, Dean and Dalrymple (1984).
Using Eq. (3.35) in Eq. (3.34) and making the assumption that the only time dependency
is in the phase, i.e.
dd>
= tu)* (3.36)
reduces the hyperbolic equation (3.34) to the elliptic form
-2tuff + 0 Vh( vj) + (Vfc V)(U Vfc) v (cc,vfc)
+{[Vk U)} = rw<£ (3.37)
where only the phase contribution to the horizontal derivative of ^ is retained in obtaining
the term on the right hand side of (3.37).
Before deriving a parabolic approximation to Eq. (3.37), the need for such an approxi
mation should be discussed. There are two major computational drawbacks to numerically
*