An extended linear shallow water equation

An extension to the classical shallow water equation (SWE) is derived which exactly satisfies the bed condition and can be regarded as an approximation to wave scattering at the next order in the small parameter $(h/\lambda)^2$ (depth to wavelength ratio squared.) In the frequency domain, the extended SWE shares the same simple structure as the standard SWE with coefficients modified by terms relating to the bed variation. In three dimensions the governing equation demonstrates that variable topography gives rise to anisotropic effects on wave scattering not present in the standard SWE with consequences for the design of water wave metamaterials. Numerical examples illustrate that approximations to wave scattering using the extended SWE are significantly improved in comparison with the standard SWE.