On the interaction of surface and internal gravity waves: uniformly valid solution by extended stationary phase

Using simple Fourier transform techniques and extensions to stationary-phase methods, the behaviour of surface gravity waves is determined near triply coalescing roots of the dispersion relation. It is shown that the amplitude of the surface wave is proportional to (∂U/∂x)−¼ at the location of the triple root. Far from the triple root it satisfies conservation of action. The internal wave is modelled simply by its surface current U. Asymptotic orders of magnitude are also given for the case ∂U/∂x = 0 at the triple root.