Asymptotic Solution for the Reflection of Long Water Waves by Asymmetric Convergent/Divergent Harbours

In this work, formulas for the reflection and transmission coefficients of linear long water waves propagating along a slender harbour composed of an asymmetric convergent/divergent region connected to uniform inlet and outlet regions are obtained. The governing equation is solved using the Wentzel-Kramers-Brillouin singular perturbation technique. A better sheltering function of the harbour is found for a divergent harbour with asymmetric variations in width. For convergent harbours, the largest values of the reflection coefficient are obtained for the limit in which the length of the harbour is of the same order of magnitude as the wavelength. A zero-reflection phenomenon occurs for a convergent symmetric harbour with a linearly varying width and a parabolic depth profile. The results reveal that in the presence of a linear transition in either width or depth, the reflection coefficients exhibit oscillating behaviour. The present mathematical model is compared with a simple numerical solution and with another analytical solution expressed in terms of Bessel functions. The formulas are also checked by considering an energy identity, which is satisfied to very good accuracy. Therefore, the deduced formulas can serve as a preliminary means of identifying which harbour geometries can significantly amplify or attenuate the amplitudes of water waves.