Sum-and Difference-Frequency Wave Loads on a Body in Unidirectional Gaussian Seas

When nonlinear effects are included in the diffraction of waves by a large body, there are, at second order, interactions at the sums and differences of the component frequencies of the incident waves. In this paper, the complete deterministic and stochastic solutions of second-order (sum-and difference-frequency) wave loads in unidirectional Gaussian waves are considered. The deterministic result, namely the wave force quadratic transfer function (QTF) in bichromatic incident waves, is complete in the context of second-order diffraction theory in that all the relevant components including those due to the exact second-order potentials are obtained. Statistical properties of second-order wave excitations are then investigated using the QTF results and a two-term Volterra series model. For illustration, the exact second-order force spectra and probability distributions for the simple geometry of a truncated vertical cylinder are obtained and compared with those based on a number of existing approximation methods. It is found that second-order exciting force variances and probability of extreme values may be significantly underestimated by existing approximation methods.