An asymptotic theory for water waves on beaches of mild slope

A new technique of asymptotic expansion is applied to water wave propagation over an uneven bottom that has straight and parallel contours. Attention is focused on the establishment of a rigorous scheme of successive approximation for higher-order corrections. The bottom depth is assumed to vary slowly within a wavelength. By introducing a compressed coordinate in the direction normal to the contours and by assuming an expansion of the WKB type, the weakness of the depth variation in the normal direction is incorporated in the mathematical formalism. The conventional linearized theory of wave refraction is obtained as the first-order solution without the explicit assumption of Snell's law. In the second order, a steady-state depression of the mean water surface is found for the general case where the incident wave approaches the contours obliquely. Reflection is neglected.