Theoretical aspects of gravity–capillary waves in non-rectangular channels

Abstract

This article reports the results of theoretical research concerning linear waves propagating on the surface of water in a uniform horizontal channel of arbitrary crosssection. Three different versions of the problem are considered. The first is the hydrodynamic problem when surface tension is neglected. The second and third include capillary effects, necessitating the use of edge conditions at the points of contact of the free edges and the channel walls. Two sets of edge constraints are used: pinned edges, where the lines of contact are fixed, and free edges, where the surface meets locally vertical walls orthogonally. These choices are physically realistic and have certain advantages for mathematical analysis.The hydrodynamic problems are shown to have a Hamiltonian structure in which the non-local operators inherent in the water-wave problem are explicitly exhibited. The existence, properties and applications of normal-mode solutions are discussed, and a qualitative comparison of those obtained for each problem is given. Explicit and numerical calculations of the dispersion relations for the normal modes are also carried out. A long-wave theory based upon a decomposition of the hydrodynamic problems in Fourier-transform space is developed. Finally a bifurcation theory for linear travelling waves is discussed, a potential application of which is the construction of an existence theory for periodic travelling-wave solutions of the corresponding nonlinear problems.