Three-dimensional Green's function for harmonic water waves over a bottom topography with different depths at infinity

Abstract

The three-dimensional Green's function of water waves in variable-bathymetry regions, associated with the problem of propagation of water waves emitted from a monochromatic point source, is derived and studied. The solution is of interest in its own right but also provides useful information for the formulation and treatment of complex wave–body–seabed interaction problems in variable-bathymetry regions, especially as regards the hydrodynamic characteristics of large structures installed in the nearshore and coastal environment. Assuming a parallel-contour bathymetry, with a continuous depth function of the form $h(x,y) \,{=}\, h(x)$, attaining constant, but possibly different, values at the semi-infinite regions $x \,{<}\, a$ and $x \,{>}\, b$, the problem is reduced to a two-dimensional one, by using Fourier transform. The transformed problem is treated by applying domain decomposition and reformulating it as a transmission problem in the finite domain containing the bottom irregularity. An appropriate decomposition of the wave potential is introduced, permitting the singular part to be solved analytically, and the problem for the regular part to be reformulated as a variational problem. An enhanced local-mode series representation is used for the regular wave potential in the variable-bathymetry region, including the propagating mode, the sloping-bottom mode (see Athanassoulis & Belibassakis 1999), and a number of evanescent modes. Using this representation, in conjunction with the variational principle, a forced system of horizontal coupled-mode equations is derived for the determination of the complex modal-amplitude functions of the regular wave potential. This system is numerically solved by using a second-order central finite-difference scheme. The source-generated water-wave potential is, finally, obtained by an efficient numerical Fourier inversion based on FFT. Numerical results are presented and discussed for various bottom topographies, including smooth but steep underwater trenches and ridges, putting emphasis on the identification of the important features of the near- and far-field patterns on the horizontal plane and on the vertical plane containing the point source. Characteristic patterns of trapped (well-localized) wave propagation over ridges are predicted and discussed.