Discussion of “Analytic solution of long wave propagation over a submerged hump” by Niu and Yu (2011)

In this discussion, we would like to introduce a technique of variable transform to construct a new analytical solution for the general hump studied by Niu and Yu (2011), which can converge in the whole hump region without any additional restriction to the hump submergence except for the assumption of infinitesimal amplitudes which is originally required by the linear long-wave theory as we will discuss in Section 1.2. Finally, the current analytical solution for the limited case when the power exponent s approaches infinite is linked to the classical analytical solution for wave scattering by a submerged circular cylinder given by Longuet-Higgins (1967).