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

Abstract

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.

The set-down and set-up of directionally spread and crossing surface gravity wave groups

“…Consistently, the down-shift becomes less pronounced for increasing degrees of directional spreading, as the effects of spreading serve to decrease the overall degree of nonlinearity (see also Johannessen & Swan 2001). There are no significant trends in the estimated bandwidth k .…”

“…Recently, Herbers & Janssen (2016) have shown that the set-down associated with (unidirectional) groups can appear as a significant set-up in Lagrangian buoy records, emphasizing the need to carefully distinguish between Lagrangian and Eulerian field observations. Experimentally, Johannessen & Swan (2001) examined the evolution and focusing of moderately directionally spread focused wave groups and found that the directionality of the wave groups serves to reduce the overall nonlinearity of the waves, affecting the onset of breaking and nonlinear modification of the free waves. Onorato et al (2009) and Toffoli et al (2010) performed experiments and numerical analysis of irregular crossing waves, observing a direct relationship between crossing angle and kurtosis, an indicator of the probability of freak wave occurrence.…”

Second-order wave maker theory using force-feedback control. Part II: An experimental verification of regular wave generation

“…The nonlinear wave-wave interactions associated with the evolution of the largest or most extreme wave events (Baldock et al (1996), Johannessen & Swan (2001), (2003)). …”

Importance of second-order wave generation for focused wave group run-up and overtopping

“…A symmetrybased method for separation of harmonics from the calculated free surface time series is used to identify the second-order error waves (for examples of the application of this method see e.g. Baldock et al (1996), Jonathan and Taylor (1997), Johannessen and Swan (2001), Hunt et al (2004) and Borthwick et al (2006b)). The effectiveness of second-order paddle signals in eliminating the contaminating error waves is also examined.…”

“…Pioneering laboratory experiments on focused wave groups have been carried out in water of uniform depth (Rapp and Melville (1990), Baldock et al (1996) and Johannessen and Swan (2001)), demonstrating that ocean waves are dispersive and can evolve into transient, localised but energetic groups that focus in shallow coastal waters (Baldock (2006)). It is also plausible that similar focused-wave analysis could be useful in assessing storm-induced wave run-up maxima at beaches and overtopping volumes at coastal defences.…”