Novel Approximation of the Modified Mild Slope Equation

TL;DR

This paper introduces a simplified version of the modified mild-slope equation that retains key features for modeling water wave propagation over complex seabed topographies, making it more tractable for practical use.

Contribution

A drastically simplified modified mild-slope equation is derived, closely matching the original's predictions while reducing complexity for easier application.

Findings

The simplified equation agrees with the original at leading orders.

It accurately predicts wave scattering over various topographies.

It performs well under higher order resonant conditions.

Abstract

The mild-slope equation and its various modifications aim to model, with varying degrees of success, linear water wave propagation over sloping or undulating seabed topography. However, despite multiple modifications and attempted simplifications, the different variants of the equation include multiple higher order terms involving the nonlinear water wave dispersion relation and thus remain analytic intractable. To further facilitate its use, we derive a drastically simplified alternative version of the modified mild-slope equation that bears striking resemblance to the linear shallow water equation while retaining all critical features of the original equation that enable it to be valid for a wide range of wave numbers and water depths. Direct comparison of the modified mild-slope equation and our simplified formulation indicates that the simplified equations agree with the modified mild slope equation at leading orders in the forcing frequency and local depth. Validations with multiple sets of benchmark wave scattering problems demonstrate that despite the clearly reduced complexity, the simplified equations were able to replicate the predictions of the modified mild slope equations over a wide range of wave numbers and surface topographies, including higher order resonant conditions.