Diffraction of a directionally spread wave group by a cylinder

The problem of diffraction of a directionally spread focused wave group by a bottom-seated circular cylinder is considered from the viewpoint of second-order perturbation theory. After applying the time Fourier transform and separation of vertical variable the resulting two-dimensional non-homogeneous Helmholtz equations are solved numerically using finite differences. The detailed formulation of the second-order radiation condition is presented. Numerical solutions of the problem are obtained for JONSWAP amplitude spectra for the incoming wave group with various types of directional spreading. The results are compared with the corresponding results for a unidirectional wave group of the same amplitude spectrum. Finally we discuss the applicability of the averaged spreading angle concept for practical applications.