Phase convergence and crest enhancement of modulated wave trains

TL;DR

This paper investigates the physical mechanisms behind crest enhancement in modulated wave trains, revealing that spectral broadening, bound-wave production, and phase convergence contribute to crest heights exceeding classical predictions, supported by numerical and experimental results.

Contribution

It identifies the combined roles of spectral broadening, bound-wave production, and phase convergence in crest amplification beyond the Akhmediev breather solution, using advanced numerical and experimental methods.

Findings

Crest heights can exceed the AB solution due to nonlinear effects.

Spectral broadening enhances bound-wave production at high wavenumbers.

Phase convergence occurs in nonlinear wave evolution, amplifying crest heights.

Abstract

The Akhmediev breather (AB) solution of the nonlinear Schr$\"{o}$dinger equation (NLSE) shows that the maximum crest height of modulated wave trains reaches triple the initial amplitude as a consequence of nonlinear long-term evolution. Several fully nonlinear numerical studies have indicated that the amplification can exceed 3, but its physical mechanism has not been clarified. This study shows that spectral broadening, bound-wave production, and phase convergence are essential to crest enhancement beyond the AB solution. The free-wave spectrum of modulated wave trains broadens owing to nonlinear quasi-resonant interaction. This enhances bound-wave production at high wavenumbers. The phases of all the wave components nearly coincide at peak modulation and enhance amplification. We find that the phase convergence observed in linear-focusing waves can also occur due to nonlinear wave evolution. These findings are obtained by numerically investigating the modulated wave trains using the higher-order spectral method (HOSM) up to the fifth order, which allows investigations of nonlinearity and spectral bandwidth beyond the NLSE framework. Moreover, we confirm the crest enhancement through a tank experiment wherein waves are generated in the transition region from non-breaking to breaking. Owing to strong nonlinearity, the maximum crest height observed in the tank begins to exceed the HOSM prediction at an initial wave steepness of 0.10.