The mild-slope equations

Abstract

In its original form the mild-slope equation, which approximates the motion of linear water waves over undulating topography, is a simplified version of the more recently derived modified mild-slope equation. However, the reduced equation does not deal adequately with rapidly varying small-amplitude perturbations about an otherwise slowly varying bedform and it does not produce free-surface profiles that inherit slope discontinuities from the topography, an intrinsic feature of the approximation on which both equations are based. The inconsistency between the two equations is rectified by the derivation of an alternative form of the mild-slope equation, having the simplicity of the standard form and yet containing all of the essential features of the full equation. In the process, a more transparent version of the modified mild-slope equation is identified. The standard and revised mild-slope equations are compared analytically in the context of two-dimensional plane wave scattering and it is found that they lead to values of the reflected wave amplitude that differ at lowest order in the mild-slope parameter, for a general topography. It is also confirmed that the revised mild-slope equation gives the dominant contribution in the solution of the new form of the modified mild-slope equation. Indeed, the two equations differ only by a term that is virtually negligible.