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

• A phenomenological model has been derived in the case of elastic vibrating plates.
• Without damping, self-similar dynamics for forced and free turbulence are retrieved.
• In the framework of damped wave turbulence, self-similar universal solutions are given.
• An equation that links power spectra, damping law and injected power has been found.
• The agreement of the model with experiments and simulations is found to be good.

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.