Time-dependent linear water-wave scattering in two dimensions by a generalized eigenfunction expansion

We consider the solution in the time domain of the two-dimensional water-wave scattering by fixed bodies, which may or may not intersect with the free surface. We show how the problem with arbitrary initial conditions can be found from the single-frequency solutions using a generalized eigenfunction expansion, required because the operator has a continuous spectrum. From this expansion we derive simple formulas for the evolution in time of the initial surface conditions, and we present some examples of numerical calculations.

Time-Dependent Solution for Linear Water Waves by Expansion in the Single-Frequency Solutions

We consider the solution in the time-domain of water wave scattering by fixed bodies (which may or may not intersect with the free surface). We show how the the problem with arbitrary initial conditions can be found using the single-frequency solutions. This result relies on a special inner product and is known as a generalized eigenfunction expansion (because the operator has a continuous spectrum). We also show how this expansion should be modified when trapped modes are present.