Can we hear Kolmogorov spectra?

TL;DR

This paper investigates the evolution of wave spectra in a thin elastic plate, deriving a kinetic equation for energy transfer, confirming the approach to Kolmogorov-Zakharov spectra through numerical simulations.

Contribution

It derives a kinetic equation for wave spectral transfer in elastic plates and confirms the formation of Kolmogorov-Zakharov spectra via numerical simulations.

Findings

Derivation of a kinetic equation for spectral transfer.

Confirmation of energy cascade towards Kolmogorov-Zakharov spectrum.

Numerical simulations support theoretical predictions.

Abstract

We study the long-time evolution of waves of a thin elastic plate in the limit of small deformation so that modes of oscillations interact weakly. According to the theory of weak turbulence a nonlinear wave system evolves in long-time creating a slow transfer of energy from one mode to another. We derive this kinetic equation for the spectral transfer in terms of the second order moment. We show explicitly that such a non-equilibrium theory describes the approach to an equilibrium wave spectrum, and describes also an energy cascade, often called the Kolmogorov-Zakharov spectrum. We perform numerical simulations confirming this scenario.