Wave parameter tuning for the application of the mild-slope equation on steep beaches and in shallow water

The linear mild-slope equation (MSE) is examined in the limit or very shall ow water. This is done by means of a series comparison with the more 'exact ' linear classical theory (E) valid over arbitrary uniform slopes and known to have a "minimum norm" solution basis pair, respectively, regular and lo garithmically singular at the shore line. It is shown that the agreement be tween E and MSE is exact for the first three terms for the regular wave and the first two for the singular wave. It is further demonstrated, by applic ation of this example, that the MSE represents a better approximation than does the classical linearised shallow water equation (SWE) in the case of e xtremely small depth. In particular, if solutions to each are tuned to the same finite wave height at the shoreline, then MSE predicts the correct cur vature of wave height there whereas SWE does not. The work of Booij (Booij, N,A., 1983. A note on the accuracy of the Mild-Sl ept: Equation. Coastal Engineering 7, 191-203,) is supported and varied to allow performance on very steep beds to be tested against exact values rath er than those of numerical simulation. Those tests are carried out both as Boundary Value Problems, BVP (Scheme A) and Initial Value Problems, IVP (Sc heme B) with matching, results on global er I or. Methods are found of spec ifying phase and group velocity, which are consistent with linear wave beac h theory and lead to improvements in solving the MSE over steep flat beache s. The improvements are found generally superior, in the case considered, t o those of some recently developed 'modified' and 'extended' MSEs. Finally, it is demonstrated, and confirmed by both asymptotic theory and calculatio n, that the addition of evanescent modes constitutes improvement only in in termediate depths and is not recommended in depths of the order of only a w avelength on a steep (e.g. 45 degrees) beach. (C) 2001 Elsevier Science B.V . All rights reserved.