Water-Wave Green’s Function For a 3D Uneven-Bottom Problem with Different Depths at x → +∞ and x → −∞

Abstract

The interaction of floating and/or immersed bodies with the uneven bottom topography in the coastal environment is a mathematically difficult problem that, recently, has also become practically important, as the dimensions of large floating obstacles increase. That means that the characteristic lengths: horizontal dimension of the body L, wave length. λ, bottom variation length.λ _b, and average depth h may be all comparable and, thus, the ratio h/λ may be neither large enough (> 0.5) nor small enough (< 0.07) so that the deep or the shallow water models to be applicable. Furthermore, all existing shallow water models, developed for coastal environment applications, are based on the assumption of small or mild bottom slope. See, e.g., the relevant surveys by [Mei, 1983] and [Massel, 1989, 1993]. Thus, it seems timely to investigate water wave problems in a marine environment such that λ, λ _b and h are all of the same order of magnitude, being also non-constant because of a bottom topography variation. Although the non-linearity of water waves is of great importance, especially in the case of shallow water, it seems unavoidable to start studying a linearised version of the problem.