Note on wavefront dislocation in surface water waves

Abstract

At singular points of a wave field, where the amplitude vanishes, the phase may become singular and wavefront dislocation may occur. In this Letter we investigate for wave fields in one spatial dimension the appearance of these essentially linear phenomena. We introduce the Chu-Mei quotient as it is known to appear in the 'nonlinear dispersion relation' for wave groups as a consequence of the nonlinear transformation of the complex amplitude to real phase-amplitude variables. We show that unboundedness of this quotient at a singular point, related to unboundedness of the local wavenumber and frequency, is a generic property and that it is necessary for the occurrence of phase singularity and wavefront dislocation, while these phenomena are generic too. We also show that the 'soliton on finite background', an explicit solution of the NLS equation and a model for modulational instability leading to extreme waves, possesses wavefront dislocations and unboundedness of the Chu-Mei quotient.

Qualitative comparisons of experimental results on deterministic freak wave generation based on modulational instability

“…The theoretical SFB signal shows phase singularity at the extreme position, a phenomenon closely related to wavefront dislocation (Karjanto, 2006;Karjanto and van Groesen, 2007). This phenomenon occurs when the wave envelope vanishes, i.e.…”

Displaced phase-amplitude variables for waves on finite background

“…These correspond to wave dislocations of the spatial-temporal wave field, wave splitting and merging, and happens when the complex amplitude vanishes. This is the case (only at the extreme position) for sufficiently small values of ν, explicitly 0 < ν < √ 3/2; see [8,27].…”

“…Till, at a certain position, called the extreme position (in scaled variables taken to be at x = 0), the largest wave appears, after which the reverse process sets in the decay towards the asymptotic harmonic wave train (with some phase change). Defining the amplification factor of the whole process as the quotient Q of the highest crest and the background amplitude, the amplification is larger for smaller ν, maximal 3 (obtained in the limit ν → 0), as follows from the explicit expression given explicitly by [1,24] Q = 1 + 2 1 − ν2 /2. Note that the local amplification factor near the extreme position can actually be much larger, since near the extreme position, the extreme wave is locally surrounded by waves of much smaller amplitude, as if the total energy in one wavegroup is conserved but with the energy redistributed between waves.…”

Peregrine Soliton as a Limiting Behavior of the Kuznetsov-Ma and Akhmediev Breathers

“…Experimental attempts on deterministic rogue wave generation using the Akhmediev solitons suggested that the symmetric structure is not preserved and the wave spectrum experiences frequency downshift even though wavefront dislocation and phase singularity are visible [43,[110][111][112][113][114]. A numerical calculation of rogue wave composition can be described in the form of the collision of Akhmediev breathers [115].…”