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

Abstract

Recently, Niu and Yu (2011) presented an analytical solution of the long wave refraction by a submerged circular hump. The geometry of the hump was assumed to be axi-symmetric and the water depth over the hump region was described by a positive constant plus a power function of the radial distance with an arbitrary value of the power exponent, i.e., h = h 1 + βr s , where h 1 is the water depth at the crest of the hump. Their general hump is an extension of the paraboloidal hump (i.e., s = 2) studied by Zhang and Zhu (1994) and Zhu and Harun (2009). Because of this extension in the topography of the hump, the problem to seek a general analytical solution to the long-wave equation becomes much more complicated and the solution technique need to be more skillful, especially for the case with the exponent s being a rational, see Eq. (17) in Niu and Yu (2011). However, the solution presented by Niu and Yu (2011) is valid only when the circular hump is sufficiently submerged, i.e., h 1/ h 0 > 0.5, see Eq. (27) in Niu and Yu (2011), where h 0 is the water depth over the flat bottom. It is no doubt that this restriction on the hump submergence will narrow down the application range of their general analytical solution. 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).