Extended mild-slope equations for compressible fluids

In this paper we derive new forms of the mildslope equation (MSE) for water waves in a weakly compressible fluid on a slowly varying bathymetry, with surface and bottom disturbances. The MSE is a powerful tool to model the refraction-diffraction dynamics of water waves propagating on a variable bathymetry [1]. Traditionally, mild-slope models are derived by assuming that the wave steepness is small, the fluid is inviscid and incompressible and the motion is irrotational. Furthermore, no disturbances are normally considered both on the free surface and at the bottom of the fluid domain [2]. In this paper we shall find new expressions of the MSE by relaxing the incompressibility hypothesis and considering both surface and bottom disturbances. We shall name the set of new formulae as the extended acoustic-gravity mild-slope equations (EAG-MSE). Such a system of equations can be implemented in numerical models for the early detection of coastal flooding based on the hydro-acoustic precursors of surface gravity waves (see [3]–[5]).