Abstract
Generalized expansion formulae for the velocity potentials associated with plane gravity-wave problems in the presence of surface tension and interfacial tension are derived in both the cases of finite and infinite water depths in two-layer fluids. As a part of the expansion formulae, orthogonal mode-coupling relations, associated with the eigenfunctions of the velocity potential, are derived. The dispersion relations are analyzed to determine the characteristics of the two propagating modes in the presence of surface and interfacial tension in both the cases of deep-water and shallow-water waves. The expansion formulae are then generalized to deal with boundary-value problems satisfying higher-order boundary conditions at the free surface and interface. As applications of the expansion formulae, the solutions associated with the source potential, forced oscillation and reflection of capillary–gravity waves in the presence of interfacial tension are derived.
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