NUMERICAL-SIMULATION OF GRAVITY-WAVES

We present a new spectral method to simulate numerically the waterwave problem in a channel for a fluid of finite or infinite depth. It is formulated in terms of the free surface elevation η and the velocity potential φ{symbol}. The numerical method is based on the reduction of this problem to a lower-dimensional computation involving surface variables alone. To accomplish this, we describe the Taylor expansion of the Dirichlet Neumann operator in homogeneous powers of the surface elevation η. Each term is a concatenation of Fourier multipliers with powers of η and its derivatives and is valid uniformly in wavenumber. These are easily calculated using the fast Fourier transform. The method is illustrated by computing the long time evolution of modulated wave packets and of approximations to the Stokes steady wave train. By imposing a surface pressure we observe surface steepening in large amplitude evolution, and wake and bow wave development for flows with a close to critical Froude number. Finally, we give an example of nonlinear evolution of the distribution of energy among normal modes. © 1993 Academic Press. All rights reserved.