Exact solution for capillary waves on the surface of a liquid of finite depth

Abstract

Using the Schwartz function method, we have obtained a new exact solution for the problem of stationary capillary waves of finite amplitude on the surface of a liquid that has a finite depth. The reliability of the solution was confirmed by the results of numerical verification of the main boundary equation. The obtained solution of the problem is general in the sense that for any Weber number one can find the corresponding wave configuration. Parametric analysis showed a nonmonotonic dependence of the wave-length and its amplitude on the Weber number. The fact that the problem has one more branch of the solution (the trivial solution) indicates the possibility of the existence of other branches. The Schwartz function method cannot guarantee finding all solutions of the problem even from the specified class of functions. Therefore, the question of reproducing the known exact solution of W. Kimmersley for this problem and its reliability remains open. Note that for the parameter ß included in the main boundary equation, W. Kimmersley preliminarily laid down assumption ß = 1. The found exact solution has the property that ß > 1 and cannot coincide with W. Kimmersley’s solution.