Phenomenological model for predicting stationary and non-stationary spectra of wave turbulence in vibrating plates

TL;DR

This paper introduces a phenomenological model for the power spectrum of vibrating plates in wave turbulence, capturing both stationary and non-stationary behaviors, and analyzing the effects of damping laws on turbulence spectra.

Contribution

The paper presents a new phenomenological model that describes the time-frequency dependence of wave turbulence spectra in vibrating plates, including non-stationary solutions and damping effects.

Findings

Derived self-similar universal solutions relating power spectrum to injected power and damping law.

Analyzed effects of frequency-dependent damping on stationary turbulence spectra.

Provided a framework for understanding both stationary and non-stationary wave turbulence in plates.

Abstract

A phenomenological model describing the time-frequency dependence of the power spectrum of thin plates vibrating in a wave turbulence regime, is introduced. The model equation contains as basic solutions the Rayleigh-Jeans equipartition of energy, as well as the Kolmogorov-Zakharov spectrum of wave turbulence. In the Wave Turbulence Theory framework, the model is used to investigate the self-similar, non-stationary solutions of forced and free turbulent vibrations. Frequency-dependent damping laws can easily be accounted for. Their effects on the characteristics of the stationary spectra of turbulence are then investigated. Thanks to this analysis, self-similar universal solutions are given, relating the power spectrum to both the injected power and the damping law.