A laboratory study of the focusing of transient and directionally spread surface water waves

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 water-particle 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.