Steady Periodic Hydroelastic Waves

This is a study of steady periodic travelling waves on the surface of an infinitely deep irrotational ocean when the top streamline is in contact with a light frictionless membrane that strongly resists stretching and bending and the pressure in the air above is constant. It is shown that this is a free-boundary problem for the domain of a harmonic function (the stream function) which is zero on the boundary and at which its normal derivative is determined by the boundary geometry. With the wavelength fixed at 2 π , we find travelling waves with arbitrarily large speeds for a significant class of membranes. Our approach is based on Zakharov’s Hamiltonian theory of water waves to which elastic effects at the surface have been added. However we avoid the Hamiltonian machinery by first defining a Lagrangian in terms of kinetic and potential energies using physical variables. A conformal transformation then yields an equivalent Lagrangian in which the unknown function is the wave height. Once critical points of that Lagrangian have been shown to correspond to the physical problem, the existence of hydroelastic waves for a class of membranes is established by maximization. Hardy spaces on the unit disc and the Hilbert transform on the unit circle play a role in the analysis.