Computation of Stokes waves using conformal mapping

Abstract

In this paper we study the time dependent water wave problem for Stokes waves on water of infinite and finite depth using numerical conformal mapping techniques. We begin by briefly reviewing Fornberg's numerical conformal mapping method and then briefly develop the theory for periodic cubic splines which are necessary in our later applications. Since this study involves the dynamics of water waves, we derive the hydrodynamics equations describing such motion. It is here that we find that the velocity potential of an incompressible, irrotational fluid flow satisfies Laplace's equation throughout the fluid and that all motion on the surface of a fluid depends on what happens on the interior. We then develop a numerical representation for Stokes waves and use this representation to study the time dependent wave problem on water with infinite and finite depth. It is here that conformal mapping techniques become important because solutions to Laplace's equation are invariant under conformal mapping, allowing us to solve the problem for the velocity potential in more ideal domains, namely the unit disk and the annulus with ρ≤|z|≤1 rather than the physical plane. We then develop a numerical approach to both problems utilizing Euler's method to solve the differential equations describing the free surface conditions of the wave.