Nonlinear Superposition Methods Applied to Continuous Ocean Wave Spectra

The present paper addresses the challenges associated with applying weakly nonlinear mode-coupled solutions for wave interaction problems to irregular waves with continuous spectra. Unlike the linear solution, the nonlinear solutions will be strongly dependent on cut-off frequency for problems such as the wave elevation itself or loads on a slender cylinder used together with typical ocean wave spectra. It is found that the divergence of the solutions with respect to the cut-off frequency is related to the nonlinear interaction between waves with very different frequencies. This is, in turn, linked to a long standing discussion about the ability of mode-coupled methods to describe the modulation of a short wave due to the presence of a long wave. In cases where nonlinear properties associated with a measured or assumed history of the surface elevation is sought, it is not necessary to calculate accurately the nonlinear evolution of the wave field in space and time. For such cases it is shown that results which are independent of frequency cut-off may be obtained by introducing a maximum bandwidth in frequency between waves which are allowed to interact. It is shown that a suitable bandwidth can be found by applying this method to the problem of back-calculating a linear wave profile from a measured wave profile. In order to verify that this choice of bandwidth is suitable for second and third order terms, nonlinear loads on a slender vertical cylinder are calculated using the FNV method of Faltinsen, Newman, and Vinje (1995, “Nonlinear Wave Loads on a Slender, Vertical Cylinder,” J. Fluid Mech., 289, pp. 179–198). The method is used to compare loads calculated based on measured surface elevations with measurements of loads on two cylinders with different diameters. This comparison indicates that the bandwidth formulation is suitable and that the FNV solution gives a reasonable estimate of loading on slender cylinders. There are, however, loading mechanisms that the FNV solution does not describe, notably the secondary loading cycle first observed by Grue et al. (1993, Higher Harmonic Wave Exciting Forces on a Vertical Cylinder, Institute of Mathematics, University of Oslo, Preprint No. 2). Finally, the method is employed to calculate the ringing response on a large concrete gravity base platform. The base moment response is calculated using the FNV loading on the shafts and linear loads from a standard diffraction code, together with a structural finite element beam model. Comparison with results from a recent model testing campaign shows a remarkable agreement between the present method and the measured response.