Surface waves on two-dimensional rough Neumann surfaces

•A dispersion relation is obtained and solved for surface waves on a doubly periodic Neumann surface.•A dispersion relation is obtained and solved for surface waves on a two-dimensional randomly rough Neumann surface.•These results are applicable to a system of a liquid in contact with a rough hard substrate.•These are the first studies of surface waves in such systems.

A planar surface on which a scalar wave satisfies the Neumann boundary conditions does not support a surface wave. However, a structured Neumann surface can support such a wave. By means of a Rayleigh equation for a scalar field in the region above the two-dimensional rough surface of a semi-infinite medium on which the Neumann boundary condition is satisfied, we derive the dispersion relation for surface waves on both doubly periodic and randomly rough surfaces. Dispersion curves for these waves on doubly periodic surfaces with three forms of the surface profile function are presented together with dispersion curves for surface waves on a two-dimensional randomly rough surface.