The time-dependent motion of a floating elastic or rigid body in two dimensions

The linear time-dependent motion of a floating elastic or rigid body, subject to some initial displacement, which subsequently evolves freely is considered. The solution is derived by a Fourier transform and by the generalized eigenfunction method. Compared to other solutions, such as the Cummins method, the present solution requires neither time-stepping nor high-frequency calculations. A series of new identities for the frequency-domain problem are also presented. The Fourier transform solution allows an approximate solution to be calculated by an expansion over the complex resonances known as the singularity expansion method. Simple expressions for the singularity expansion method approximation are given. The method is illustrated with a series of numerical calculations.