29
are shown in Appendix B:
c P9 L (dBdB*) 1 f 2kh \
4 | ydxa dxp J k2 \ sinh2fch/
ap
IBI
2 kh
sinh 2 kh
+
dxa dxp
(|VhB|2 *2|B|2)(2*hcoth2*h 1)] } (2.57)
where B* denotes the complex conjugate of B. In obtaining Eq. (2.57) use was made of the
flat bottom Helmholtz equation
Vj;£ + Jfc2£ = 0 (2.58)
The general complex amplitude function B is related to the complex amplitude A which
is solved for in the wave model by
B = Ae^flco,idz^
(2.59)
There are some problems with Eq. (2.57). One may question the assumption of a
flat bottom; however the inclusion of the slope and gradients of the wave number and the
intrinsic frequency leads to intractable mathematics. The use of the localized flat bottom
is also implicit in the derivation of Longuet-Higgins and Stewart to obtain Eqs. (2.28- 2.30)
since in carrying out the integrations involved it is assumed that horizontal derivatives of
the velocity potential are limited to derivatives of the phase function. For plane waves
using a spilling breaker assumption, results using Eq. (2.57) differ little from those using
Eqs. (2.28-2.30); however in diffraction zones Eq. (2.57) is superior for reasons mentioned
previously.
The major problem with the use of Eq. (2.57) is also a major problem associated with
the use of Longuet-Higginss formulation. This problem is that in both formulations the
initiation of wave breaking implies the dissipation of energy and thus the forcing of set
up and longshore currents starting at the breaker line. The physical reality is somewhat
different in that in the initial stages of breaking, energy is not immediately dissipated but
rather is transformed from organized wave energy to turbulent energy and to a surface roller
(Svendsen 1984). There is a space lag between the initiation of breaking and the dissipation