Experimental and numerical investigations of temporally and spatially periodic modulated wave trains

Abstract

A number of studies on steep nonlinear waves were conducted experimentally with the temporally periodic and spatially evolving (TPSE) wave trains and numerically with the spatially periodic and temporally evolving (SPTE) ones. The present study revealed that, in the vicinity of their maximum crest height, the wave profiles of TPSE and SPTE modulated wave trains resemble each other. From the investigation of the Akhmediev-breather solution of the nonlinear Schrödinger equation (NLSE), it is revealed that the dispersion relation deviated from the quadratic dependence of frequency on wavenumber and became linearly dependent instead. Accordingly, the wave profiles of TPSE and SPTE breathers agree. The range of this agreement is within the order of one wave group of the maximum crest height and persists during the long-term evolution. The findings extend well beyond the NLSE regime and can be applied to modulated wave trains that are highly nonlinear and broad-banded. This was demonstrated from the numerical wave tank simulations with a fully nonlinear potential flow solver based on the boundary element method, in combination with the nonlinear wave generation method based on the prior simulation with the higher-order spectral model. The numerical wave tank results were confirmed experimentally in a physical wave tank. The findings of this study unravel the fundamental nature of the nonlinear wave evolution. The deviation of the dispersion relation of the modulated wave trains occurs because of the nonlinear phase variation due to quasi-resonant interaction, and consequently, the wave geometry of temporally and spatially periodic modulated wave trains coincides.

Phase Convergence and Crest Enhancement of Modulated Wave Trains

“…The estimated error in the crest height of regular waves with a wavelength of 3 m and wave height between 10 and 20 cm is less than 4% in this stereo-imaging scheme [36]. Note that the standard deviation of the wave-maker motion was found to be 1.065 times larger than the given signal owing to a problem with controlling the mechanical wave maker [31]. Therefore, the experimental results presented in Section 3 are corrected by a factor of 1.065 for comparison with the HOSM simulation.…”

“…We numerically simulated the temporal evolution of spatially periodic deep-water modulated wave trains using the HOSM [28,29] in the same manner as in our preceding studies [31,32]. The HOSM numerically solves Laplace's equation (∇ 𝜙 = 0) subject to nonlinear kinematic and dynamic free-surface boundary conditions with respect to the surface elevation 𝜁 and velocity potential on the free surface, 𝛷 = 𝜙| [22]:…”

Drifting breathers and Fermi–Pasta–Ulam paradox for water waves

“…In the cases of marginal drift speed, the wave packets remain indeed symmetric at all stages of evolution. Moreover, the shape of the unstable envelopes in the latter cases resembles the propagation of the ABs within the framework of the space-NLS (the framework evolving the wave packets along the temporal co-ordinate t) [35,36,37].…”

On an eddy viscosity model for energetic deep-water surface gravity wave breaking

“…It was concluded that in some cases it is inappropriate to use the linear dispersion relation for the post-processing of experimental data. Houtani et al (2018a) and Houtani, Waseda & Tanizawa (2018b) have demonstrated that the dispersion characteristics of the Akhmediev breather solution to the nonlinear Schrödinger equation in deep water deviates significantly from the linear relationship (2.13). It was also shown that these findings extend beyond the applicability of nonlinear Schrödinger equation.…”