A coupled-mode study on weakly viscous Bragg scattering of surface gravity waves

The two-dimensional scattering of weakly viscous waves over a finite region of undulating topography linking a series of small abrupt steps is studied. The free surface is assumed to be irrotational on the stationary condition. The limitation is for the case of a weak viscous wave propagating over an arbitrary topography. The viscous eigenfunction matching method (VEMM) is proposed by including the evanescent modes. And the resulted system matrix is solved efficiently by the sparse matrix solver of the Super LU. The improvement of wave reflection is obvious for waves propagating over a strongly undulating topography or a series of bars with steep side-slopes. It is noteworthy that the shift of reflection coefficient exits mainly at primary resonance and second-order resonance because of the combined effect of molecular viscosity and bottom condition. Numerical results show that the shear stress of viscous fluid leads to the energy loss. Moreover and it becomes severe for a shorter wave propagating over an undulated bottom.