Differential equations for long-period gravity waves on fluid of rapidly varying depth

The conventional long-wave equations for waves propagating over fluid of variable depth depend for their formal derivation on a Taylor series expansion of the velocity potential about the bottom. The expansion, however, is not possible if the depth is not an analytic function of the horizontal co-ordinates and it is a necessary condition for its rapid convergence that the depth is also slowly varying. We show that if in the case of two-dimensional motions the undisturbed fluid is first mapped conformally onto a uniform strip, before the Taylor expansion is made, the analytic condition is removed and the approximations implied in the lowest-order equations are much improved. In the limit of infinitesimal waves of very long period, consideration of the form of the error suggests that by modifying the coefficients of the reformulated equation we may find an equation exact for arbitrary depth profiles. We are thus able to calculate the reflexion coefficients for long-period waves incident on a step change in depth and a half-depth barrier. The forms of the coefficients of the exact equation are not simple; however, for these particular cases, comparison with the coefficients of the reformulated long-wave equation suggests that in most cases the latter may be adequate. This opens up the possibility of beginning to study finite amplitude and frequency effects on regions of rapidly varying depth.