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Berkhoff (1972) derived an equation governing combined refraction and diffraction with
the assumption of a mild slope. Radder (1979) applied the splitting method of Tappert
(1977) to obtain a parabolic equation governing the propagation of the forward propagat
ing component of the wave field. Booij (1981) extended the work of Berkhoff to include
the effects of currents and also obtained a parabolic approximation. Kirby (1984) made
corrections to the equation of Booij.
Parabolic wave equations have recently been used in wave-induced circulation models.
Liu and Mei (1975) solved for the wave field by matching conditions between the shadow
zones and the illuminated zone in the lee of a structure. The wave information was then
used to force a circulation model which formulated the governing depth integrated time
averaged momentum equations in terms of a stream function. However this model did not
have current feedback in the wave equation. Yan (1987) and the present study both include
a current feedback for the parabolic wave equation and solve directly for the mean currents.
Prior to the use of parabolic wave equations to obtain the wave forcing in circulation
models, refraction wave models using a numerical scheme derived by Noda et al. (1974)
were used. Birkemeier and Dairymple (1976) produced an explicit numerical solution of
the depth averaged equations of momentum and continuity using the wave model of Noda
et al. (1974). Their model did not include the nonlinear advective acceleration terms
or any mixing. Ebersole and Dalrymple (1980) included the advective acceleration terms
and lateral mixing in an explicit model which also used the refraction scheme of Noda
et al. Vemulakonda (1984) Produced an implicit model which also was based upon the
wave refraction scheme of Noda et al. Yan (1987) used a parabolic approximation to the
mild slope equation which accounts for combined refraction and diffraction. The present
model Improves upon Yans model in numerical efficiency, by introducing structures and
most importantly by obtaining the radiation stress terms directly from the complex wave
amplitude.