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producing wave rays. Pierson (1951) showed that the method using wave fronts introduced
more error than the wave ray method. These methods proceeded upon the assumption that
the contours are locally straight. The wave rays obtained were important in that the wave
height was determined to vary inversely as the square root of the spacing between the wave
rays. Using these ray construction techniques OBrien (1950) was able to demonstrate that
the April 1930 destruction of the Long Beach Harbor breakwater on a seemingly calm day
most probably resulted from the focusing of long wave energy from a South Pacific storm.
Munk and Arthur (1952) showed a method to obtain the wave height directly from the
curvature of the ray and the rate of change of the depth contours. The wave ray tracing
techniques did not include the effects of currents and had the disadvantage (from a numerical
modeling point of view) that wave heights and directions were obtained at positions along
a ray and not at pre-specified grid positions. Noda et al. (1974) solved this problem by
devising a numerical scheme that solved a wave energy equation and a statement for the
irrotationality of the wave number to obtain wave height and direction at grid points. This
scheme also included the effects of current refraction but neglected diffraction.
Numerical models of wave induced currents were developed by Noda et al. (1974) in
which time dependent and advective acceleration terms were ignored. Birkemeier and Dai
ry mple (1976) wrote a time-marching explicit model which neglected lateral mixing and
advective acceleration. Ebersole and Dalrymple (1979) added lateral mixing and the non
linear advective acceleration terms. Vemulakonda (1984) wrote an implicit model which
included lateral mixing and advective acceleration terms. All of the above used the numer
ical scheme of Noda et al. (1974) to obtain wave heights and wave angles. Each of these
models neglected diffraction.
Diffraction of water waves represents some very difficult mathematical problems. His
torically most solutions were based upon the assumption of a flat bottom, since to quote
Newman (1972),
The restriction to constant depth h is important.... Problems involving wave
propagation in a fluid of variable depth are of great interest, and the resulting