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

• Efficient formulas for the time dependent motion of a floating rigid or elastic body, which require only the frequency domain solution and no time-stepping, are given.
• A series of new identities which are satisfied by the general floating rigid or elastic body are given.
• Simplified formulas for the singularity expansion method are given.
• The method is illustrated through numerical results for rigid and elastic floating bodies.

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.