On the connection between step approximations and depth-averaged models for wave scattering by variable bathymetry

Two popular and computationally-inexpensive class of methods for approximating the propagation of surface waves over two-dimensional variable bathymetry are “step approximations” and “depth-averaged models”. In the former, the bathymetry is discretised into short sections of constant depth connected by vertical steps. Scattering across the bathymetry is calculated from the product of 2 × 2 transfer matrices whose entries encode scattering properties at each vertical step taken in isolation from all others. In the latter, a separable depth dependence is assumed in the underlying velocity field and a vertical averaging process is implemented leading to a 2nd order Ordinary Differential Equation (ODE). In this paper the step approximation is revisited and shown to be equivalent to an ODE describing a depth-averaged model in the limit of zero-step length. The ODE depends on how the solution to the canonical vertical step problem is approximated. If a shallow-water approximation is used, then the well-known linear shallow water equation results. If a plane-wave variational approximation is used, then a new variant of the Mild-Slope Equations is recovered.