Some problems of edge waves and standing waves on beachesare examined.
The nonlinear interaction of a wave normally incident on asloping beach with a subharmonic edge wave is studied. A two-timingexpansion is used in the full nonlinear theory to obtain themodulation equations which describe the evolution of the waves. It isshown how large amplitude edge waves are produced; and the resultsof the theory are compared with some recent laboratory experiments.
Traveling edge waves are considered in two situations. First,the full linear theory is examined to find the finite depth effect on theedge waves produced by a moving pressure disturbance. In the secondsituation, a Stokes' expansion is used to discuss the nonlinear effectsin shallow water edge waves traveling over a bottom of arbitraryshape. The results are compared with the ones of the fulltheory for a uniformly sloping bottom.
The finite amplitude effects for waves incident on a slopingbeach, with perfect reflection, are considered.A Stokes' expansionis used in the full nonlinear theory to find the corrections to the dispersionrelation for the cases of normal and oblique incidence.
Finally, an abstract formulation of the linear water wavesproblem is given in terms of a self adjoint but nonlocal operator. Theappropriate spectral representations are developed for two particularcases.
