Computing unstable periodic waves at the interface of two inviscid fluids in uniform vertical flow

Periodic waves at the interface of two immiscible fluids are considered. Each fluid is inviscid and incompressible, and is moving vertically with constant speed. The upper fluid is more dense than the lower one, and the interface between them is thus unstable to small perturbations. A linearized solution, valid for waves of small amplitude, is reviewed and a novel numerical method is presented for computing waves of moderate amplitude. The technique uses a Fourier–Galerkin approach, and converts the governing equations into a system of ordinary differential equations for the Fourier coefficients. It is then shown how the method may be modified to allow for the evolution of overhanging waves, using a novel time-dependent arclength formulation.