Internal wave reflexion from a sinusoidally corrugated surface

The reflexion of an internal gravity wave with stream function \[ \psi = \exp\{i(kx + lz - \omega t)\} \] from a corrugated surface of the form z = a cos px is investigated using an extension of Rayleigh's method for an inviscid non-diffusive fluid. It is found that the method converges for many wall slopes ap and incoming-wave phase propagation incidence angles θ = tan⁻¹l/k but that the appropriate series solution is only asymptotic in other cases. The accuracy of the calculations is assured by requiring that the solution satisfy the boundary condition at the wall using a least-squares error minimization technique. The accuracy is then verified through the conservation of energy flux. It is found that, as the surface slope is increased for constant θ, less energy appears in specular reflexions and more is either back-scattered or redistributed into other forward-scattered modes. As the horizontal internal wavelength is decreased to become comparable to the corrugation wavelength of the wall, substantially less energy appears in specular reflexion, but of the order of 95% of the incoming energy is specularly reflected for 30° < θ < 60° when the two wavelengths are equal. In contrast to this, it is found that the general level of the ratio of back-scattered to forward-scattered energy is reduced by O(10⁻³) to O(10⁻²) as the incident horizontal internal wavelength becomes smaller than the corrugation wavelength. The results are compared with the linear theory of Baines (1971); agreement is good for forward-scattered energy and excellent for the back-scattered flux when p ≥ k.