The singularity expansion method and near-trapping of linear water waves

Abstract

AbstractThe problem of near-trapping of linear water waves in the time domain for rigid bodies or variations in bathymetry is considered. The singularity expansion method (SEM) is used to give an approximation of the solution as a projection onto a basis of modes. This requires a modification of the method so that the modes, which grow towards infinity, can be correctly normalized. A time-dependent solution, which allows for possible trapped modes, is introduced through the generalized eigenfunction method. The expression for the trapped mode and the expression for the near-trapped mode given by the SEM are shown to be closely connected. A numerical method that allows the SEM to be implemented is also presented. This method combines the boundary element method with an eigenfunction expansion, which allows the solution to be extended analytically to complex frequencies. The technique is illustrated by numerical simulations for geometries that support near-trapping.