On the Local Uniqueness Problem for Periodic Surface Waves of Permanent Type in a Channel of Infinite Depth

Abstract

We shall be interested in studying wave motions at the surface of a heavy2 liquid in a channel bounded by parallel vertical walls. we assume that the motion is the same in each vertical plane parallel to the walls, so that the problems considered are two dimensional, and that the flow is incompressible, non-viscous, and irrotational. There are two characteristic difficulties of such problems. First, the wave profile (surface) is an unknown of the problem so that even the region of flow is unknown in advance, Secondly, one of the conditions to be fulfilled along this unknown or boundary is non-linear in character. It is obtained from Bernoulli's law involving the square of the velocity, the particle ordinate and the pressure, by stipulating that the pressure is constant along the free boundary (E.g., at an air-water interface the pressurewould be atmospheric pressure)., In most discussions of such problems the standard procedure is to linearize the equations by considering only small motions and neglecting second order quantities. One also replaces the (unknown) free boundary by a fixed straight line (equilibrium surface of liquid) in stipulating the linearized boundary condition. The resulting linear problem for a known region constitutes only a first order approximation to the original problem, but is comparatively easy to solve.2 One of the simplest types of such wave motion is a traveling wave of Idpermanent type:' This denotes a d-isturbance which travels down 1. This paper represents results obtained under contract Nonr 22809 (NR 041/152) with the Office of Naval Research. 2. The adjective heavy signifies that we do not neglect the force of gravi ty. 3. Seee.g., Lamb's Hydrodynamics, chapter 9.