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

Abstract

The linear mild-slope equation (MSE) is examined in the limit of very shallow 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 logarithmically singular at the shore line. It is shown that the agreement between 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 application of this example, that the MSE represents a better approximation than does the classical linearised shallow water equation (SWE) in the case of extremely small depth. In particular, if solutions to each are tuned to the same finite wave height at the shoreline, then MSE predicts the correct curvature of wave height there whereas SWE does not. The work of Booij (Booij, N.A., 1983. A note on the accuracy of the Mild-Slope Equation. Coastal Engineering 7, 191–203.) is supported and varied to allow performance on very steep beds to be tested against exact values rather than those of numerical simulation. Those tests are carried out both as Boundary Value Problems, BVP (Scheme A) and Initial Value Problems, IVP (Scheme B) with matching results on global error. Methods are found of specifying phase and group velocity, which are consistent with linear wave beach theory and lead to improvements in solving the MSE over steep flat beaches. The improvements are found generally superior, in the case considered, to those of some recently developed ‘modified’ and ‘extended’ MSEs. Finally, it is demonstrated, and confirmed by both asymptotic theory and calculation, that the addition of evanescent modes constitutes improvement only in intermediate depths and is not recommended in depths of the order of only a wavelength on a steep (e.g. 45°) beach.