Variational principles in water-wave problems (wave radiation conditions and their variational treatment)

Abstract

To discuss water-wave problems in unlimited waters, it is important to know what type of wave radiation condition should be placed on a virtual surface corresponding to infinity. For this kind of problem, the Sommerfeld radiation condition is well known. In this article, the condition is extended to treat a case with an incident wave. Furthermore, a more general wave radiation condition is introduced from a different point of view. The above-mentioned wave radiation conditions are introduced into the variational principles of the Kelvin, Hellinnger–Reissner, and Dirichlet type. The Dirichlet-type variational principles are then used in numerical calculations for bending waves in a bar, and the effectiveness of the wave radiation conditions and the variational principles is shown. The numerical results for one-dimensional water-wave problems are then given. As expected, the region required for the numerical solution is reduced drastically compared with that required by the Sommerfeld-type formulation. Furthermore, the amplitude of the diverging wave is obtained in the process of reaching the variational solution. Finally, two-dimensional water-wave problems are briefly discussed.