Nonlinear Water Wave Theory Via Pressure Formulation

Abstract

It is well-recognized that the main difficulties of water wave theory in the classical formulation are due to the surface boundary conditions which are nonlinear and have to be satisfied at an unknown boundary. A new formulation is given in which the pressure is regarded as an independent variable and the continuity equation replaced by two stream functions. It is shown that in the cases of three-dimensional steady wave motion and two-dimensional unsteady wave motion, the free surface boundary conditions become linear and need be satisfied at a fixed boundary. The governing equations become more complex. However, they are amenable to symbolic computation in conjunction with a singular perturbation method. In particular, both the standing wave solution of Penney and Price and the Stokes wave solution are reproduced analytically to very high orders. Furthermore, the study of the time evolution of a sinusoidal wave train reveals the almost periodic behaviour of the waves, for which the correlation function and the energy spectrum are in good agreement with some hitherto unexplained observations of waves in Lake Ontario.