Composite equations of water waves over uneven and porous seabed

The composite equations for water waves propagating over aporous uneven bottoms are derived from Green's second identity, whichincorporates the effects of porous medium in the nearshore region andconsiders the advances in models of water waves propagation over rigidbottoms. Assuming that both water depth and thickness of the porous layerconsist of two kind of components: The slowly varying component whosehorizontal length scale is longer than the surface wave length, and the fastvarying component with the horizontal length scale as the surface wavelength. The amplitude of the fast varying component is, however, smallerthan the surface wave length. In addition, the fast varying component of thelower boundary surface of the porous layer is one order of magnitude smallerthan that of the water depth. By Green's second identity and satisfying thecontinuous conditions at the interface for the pressure and the verticaldischarge velocity the composite equations are given for both water layerand porous layer, which can fully consider the general continuity of thevariation of wave number and include some well-known extended mild-slopeequations.