6
diffraction effects are among the most important of all ... but these problems
are less tractable than those involving a fluid of constant depth, (page 2)
Penney and Price (1944) showed that Sommerfelds solution of the optical diffraction prob
lem as described by Bateman (1944) is also a solution of the constant depth water wave
diffraction problem. Penney and Price (1952) solved the constant depth diffraction problem
for breakwaters assuming a semi-infinitely long, infinitesimally thin barrier. The solution
is in terms of Fresnel integrals which need to be evaluated numerically. The Shore Protec
tion Manual (SPM,1977) produced many graphical solutions based upon Penney and Price
(1952) that give the wave height and direction in the lee of breakwaters.
The problem of combined refraction and diffraction according to the SPM could be
handled by assuming a constant depth for a few wavelengths so as to obtain the diffraction
effects and then continue with the refraction effects. Perlin and Dean (1983) used essentially
the same procedure in a numerical model which calculated the longshore thrust in terms of
the breaking wave height and angle.
Liu and Mei (1975) solved for the wave field around shore parallel and shore perpendicu
lar breakwaters by using a curvilinear coordinate system where one coordinate corresponded
to the wave ray and the other to a curve of constant phase. Their solution was for an am
plitude modulation factor D(£,f) which accounts for diffraction (page 72), where £ and f
are the two coordinates. An equation for D(£,f) was obtained and a boundary value prob
lem was established by matching conditions in the incident zone and the shadow zones. For
constant depth the solution for £>(£,f) is in terms of Fresnel integrals. Upon obtaining the
wave field the radiations stresses were obtained and used to force a circulation model that
solved for the stream function and mean water surface level while ignoring the non-linear
advective acceleration terms and the lateral diffusion of current momentum. Although Liu
and Mei produced results for the wave field, currents, and set-up resulting from the diffrac
tion of waves by breakwaters, their study is not a wave-current interaction model since
there are no current terms in the wave equations and no feedback of current terms from the
circulation model.