Local and far-field surface elevation around a vertical cylinder in unidirectional steep wave groups

The problem of diffraction of a unidirectional incident wave group by a bottom-seated cylinder is considered. We assume the amplitude of the incoming wave to be small in comparison with other linear scales of the problem, and develop the corresponding second-order perturbation theory. We use the Fourier transform to treat time variation and separate spatial variables when solving the non-homogeneous second-order problem. The resulting set of non-homogeneous Bessel equations is solved numerically. Solutions for various types of incoming wave spectrum are obtained including the Gaussian spectrum and the Pierson-Moskowitz spectrum. To validate the method, problems with gradually decreasing bandwidth of Gaussian spectrum are solved and it is shown that the corresponding solution approaches that for the monochromatic case. The Pierson-Moskowitz spectrum with a set of realistic physical parameters is used as an example of extreme wave interaction with an offshore structure. The corresponding first- and second-order solutions are obtained and the effect of non-linearity on the solution is discussed with the emphasis on the growth of maximum free-surface elevation on the cylinder's surface and generation of high frequency free radiated waves.