Interfacial periodic waves of permanent form with free-surface boundary conditions

In a two-fluid system where the upper surface of the upper fluid is free, there are two independent modes of oscillation about the state of equilibrium, an ‘internal’ mode and an ‘external’ mode, which are described by two distinct dispersion curves. An efficient numerical scheme based on Fourier series expansions is used to calculate periodic waves of permanent form and of finite amplitude. Three kinds of waves are calculated: combination waves resulting from the interaction between an ‘internal’ mode and an ‘external’ mode with the same phase speed but wavelengths in a ratio of 2 (1:2 resonance), combination waves resulting from the interaction between a long ‘internal’ mode and a short ‘external’ mode with the same phase speed, and pure ‘external’ waves. It is shown that the 1:2 resonance, which is well-known for capillary – gravity surface waves and can profoundly affect wave field evolution, can affect pure gravity waves in a two-fluid system, but not in oceanic conditions. On the other hand, it is shown that the long/short wave resonance can occur in ocean-type conditions. Finally it is confirmed that pure external waves of finite amplitude behave like surface waves.