Extensions of the mild-slope equation

Abstract

The use of the mild-slope approximation, which is invoked to simplify the problem of linear water wave diffraction-refraction by bed undulations, is reassessed by using a variational method. It is found that smooth approximations to the free surface elevation obtained by using the long-standing mild-slope equation are not consistent with the continuity of mass flow at locations where the bed slope is discontinuous. The use of interfacial jump conditions at such locations significantly improves the accuracy of approximations generated by the mild-slope equation and by the recently derived modified mild-slope equation. The variational principle is also used to produce a generalization of these equations and of the associated jump condition. Numerical results are presented to illustrate the main points of the theory.