Semi-analytical solution to the second-order wave loads on a vertical cylinder in bi-chromatic bi-directional waves

A complete solution is presented for the second-order wave loads experienced by a uniform vertical cylinder in bi-chromatic bi-directional waves. The solution is obtained based on the introduction of an assisting radiation potential without explicitly evaluating the second-order diffraction potential. The semi-analytical formulation for calculating the wave loads is provided and an efficient numerical technique is developed to treat the oscillatory free-surface integral that appears in the force formulation. After validating the present solution by comparing with the predictions based on other methods, numerical studies are conducted for different combinations of incident wave frequencies and wave headings, and the influence of frequencies and headings of dual waves on the second-order wave loads is investigated. In addition, by expressing the second-order wave loads in a power expansion with respect to the wave frequency difference and wave heading difference which are both assumed to be small, approximations on the calculation of wave loads are developed. The accuracy of different approximations is assessed by comparing the approximate results with those based on the complete solution.