This paper describes a new laboratory study in which a large number of waves, of varying frequency and propagating in different directions, were focused at one point in space and time to produce a large transient wave group. A focusing event of this type is believed to be representative of the evolution of an extreme ocean wave in deep water. Measurements of the water-surface elevation and the underlying waterparticle kinematics are compared with both a linear solution and a second-order solution based on the sum of the interactions first identified by Longuet-Higgins & Stewart. Comparisons between these data confirm that the directionality of the wavefield has a profound effect upon the nonlinearity of a large wave event. If the sum of the wave amplitudes generated at the wave paddles is held constant, an increase in the directional spread of the wavefield leads to lower maximum crest elevations. Conversely, if the generated wave amplitudes are increased until the onset of wave breaking, at or near the focal position, an increase in the directional spread allows larger limiting waves to evolve.An explanation of these results lies in the redistribution of the wave energy within the frequency domain. In the most nonlinear wave cases, neither the water-surface elevation nor the water-particle kinematics can be explained in terms of the free waves generated at the wave paddles and their associated bound waves. Indeed, the laboratory data suggest that there is a rapid widening of the free-wave regime in the vicinity of a large wave event. For a constant input-amplitude sum, these important spectral changes are shown to be strongly dependent upon the directionality of the wavefield. These findings explain the very large water-surface elevations recorded in previous unidirectional wave studies and the apparent contrast between unidirectional results and recent field data in which large directionally spread waves were shown to be much less nonlinear. The present study clearly demonstrates the need to incorporate the directionality of a wavefield if extreme ocean waves are to be accurately modelled and their physical characteristics explained.

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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.