Explicit modified mild-slope equation for wave scattering by piecewise monotonic and piecewise smooth bathymetries

Abstract

The mild-slope equation and modified mild-slope equation (MMSE) have played an important role in modeling the interaction between waves and bathymetries or structures. However, they are implicit equations whose coefficients are defined by an implicit dispersion relation. In this paper, for both two-dimensional bathymetries and three-dimensional axisymmetric bathymetries with piecewise monotonicity and piecewise second-order smoothness, we are able to transform the implicit MMSE into an explicit equation by introducing a new independent variable. Besides, an alternative form of the implicit MMSE is also transformed into an explicit equation under the same assumptions. These explicit equations make it much easier to obtain analytic solutions to the MMSE. As practical examples, two analytic solutions to the present explicit MMSE for wave reflection by a single linear slope and for wave scattering by a submerged circular truncated shoal are constructed.