Mild-slope equation for water waves propagating over non-uniform currents and uneven bottoms

A time-dependent mild-slope equation for the extension of the classic mild-slope equation of Berkhoff is developed for the interactions of large ambient currents and waves propagating over an uneven bottom,using a Hamiltonian formulation for irrotational motions.The bottom topography consists of two compon}ts:the slowly varying component which satisfies the mild-slope approximation,and the fast varying component with wavelengths on the order of the surface wavelength but amplitudes which scale as a small parameter describing the mild-slope condition.The theory is more widely applicable and contains as special cases the following famous mild-slope type equations:the classical mild-slope equation,Kirby's extended mild-slope equation with current,and Dingemans's mild-slope equation for rippled bed.Finally,good agreement between the classic experimental data concerning Bragg reflection and the present numerical results is observed.