Refraction—diffraction model for linear surface water waves

Based on the parabolic approximation, a refraction—diffraction model for linear water waves is developed. With the assumption that the water depth (refraction index) is slowly varying, the model equation describes the forward-scattered wavefield. Two examples are considered in particular: (i) wave diffraction by a long thin barrier on a uniform slope, and (ii) wave convergence over a semicircular step shoal. For the former problem, a similarity solution in terms of Fresnel integrals is obtained for the wavefield in the neighbourhood of the shadow boundary. For the latter problem, the resulting Schrödinger equation is solved numerically. The wavefield near the caustics as well as in the shadow region is obtained and compared with experimental data.