Two different time-dependent hyperbolic mild-slope equations each with a dissipation term for wave propagation on non-uniform currents are transformed into wave action and eikonal equations. By considering their respective performance, the mathematical formulation is selected that is more rigorous and complete with regard to intrinsic frequency and wave number. Using a perturbation method, a parabolic form of the time-dependent mild-slope equation is derived from the selected hyperbolic version, and solved using the alternating direction implicit method. The resulting numerical model is applicable to wave propagation on non-uniform currents and depth. Making comparisons between the numerical solutions with the theoretical solutions of collinear waves and current, the results show that the numerical solutions are in good agreement with the exact ones. Calculating the interactions between incident wave and current on a sloping beach(Arthur,1950), the differences of wave number vector between refraction and combined refraction-diffraction of waves are discussed quantitatively, while the effects of different methods of calculating wave number vector on numerical results are shown.
In the past, Several researchers (e.g. Longuet-Higgins & Stewart,1961; Bretherton & Garrett,1969) studied the wave-current interaction using the conventional wave-action equation, which is equivalent to the geometrical ray theory and cannot be applied to the regions where wave diffraction becomes important. To overcome the shortcomings of wave-action equation, many researchers (e.g. Booij, 1981; Liu, 1983; Kirby, 1984; Kirby, 1986; Hong, 1996) proposed different time dependent hyperbolic mild slope equations for wave propagation on non-uniform currents. Kirby(1984)'s equation is not only different from Booij(1981)'s one but also different from Liu (1983)'s one. Kirby (1984) pointed out that Booij (1981) didn't use the correct form of the dynamic free surface boundary condition and the error of Liu (1983)'s equation was due to the limitations in the procedure deriving the equation.