A three-dimensional long-wave steep-slope equation

A complete three-dimensional long-wave polar–Cartesian equation is developed in the frequency domain. This development employs an auxiliary axis system oriented locally in the bottom gradient direction. The long-wave limit of the two-dimensional polar–Cartesian steep-slope equation is also derived. An approximate explicit expression of the coefficients is developed without restrictions on bed steepness. This is achieved by utilising a rational function approximation of the $\arctan$ function, which arises from the formulation of the vertical profile of the flow parameters. Additionally, long-wave equations in both two and three dimensions are developed in the time domain. Simulations of the long-wave equations are compared with those of the extended shallow-water equation for two-dimensional test cases, as well as for the quasi-three-dimensional scenario of oblique incidence. Our equations exhibit better agreement with the exact solutions in the majority of the test cases.