Energy spectra of \textbf{2D} gravity and capillary waves with narrow frequency band excitation

TL;DR

This paper introduces the chain equation method (CEM) for calculating quantized energy spectra in 2D weakly nonlinear wave systems with narrow band excitation, demonstrated on gravity and capillary waves.

Contribution

The paper presents the CEM as a new explicit computational approach for quantized energy spectra in 2D nonlinear waves, applicable across various physical systems.

Findings

CEM effectively computes energy spectra for gravity and capillary waves.

The method accounts for different nonlinearity levels, showing versatility.

Demonstrations confirm the physical mechanism of modulation instability in cascade generation.

Abstract

In this Letter we present a new method, called chain equation method (CEM), for computing a cascade of distinct modes in a two-dimensional weakly nonlinear wave system generated by narrow frequency band excitation. The CEM is a means for computing the quantized energy spectrum as an explicit function of frequency $\o_0$ and stationary amplitude $A_0$ of excitation. The physical mechanism behind the generation of the quantized cascade is modulation instability. The CEM can be used in numerous \textbf{2D} weakly nonlinear wave systems with narrow frequency band excitation appearing in hydrodynamics, nonlinear optics, electrodynamics, convection theory etc. In this Letter the CEM is demonstrated with examples of gravity and capillary waves with dispersion functions $\o(k) \sim k^{1/2}$ and $\o(k) \sim k^{3/2}$ respectively, and for two different levels of nonlinearity $\eps=A_0k_0$: small ($\eps\sim 0.1$ to 0.25) and moderate ($\eps\sim 0.25$ to 0.4).