# An exact result in strong wave turbulence of thin elastic plates

## TL;DR

This paper derives an exact relation for energy transfer in strong wave turbulence of thin elastic plates, extending turbulence theory to elastic systems and confirming the relation with numerical data.

## Contribution

It introduces an exact third-order structure function relation for elastic plates, analogous to Kolmogorov's law, applicable to both weak and strong wave turbulence regimes.

## Key findings

- Derived an exact Kármán-Howarth-Monin relation for elastic plates.
- Confirmed the third-order structure function law numerically.
- Defined and validated energy fluxes in Fourier space.

## Abstract

An exact result concerning the energy transfers between non-linear waves of thin elastic plate is derived. Following Kolmogorov's original ideas in hydrodynamical turbulence, but applied to the F\"oppl-von K\'arm\'an equation for thin plates, the corresponding K\'arm\'an-Howarth-Monin relation and an equivalent of the $\frac{4}{5}$-Kolmogorov's law is derived. A third-order structure function involving increments of the amplitude, velocity and the Airy stress function of a plate, is proven to be equal to $-\varepsilon\, \ell$, where $\ell$ is a length scale in the inertial range at which the increments are evaluated and $\varepsilon$ the energy dissipation rate. Numerical data confirm this law. In addition, a useful definition of the energy fluxes in Fourier space is introduced and proven numerically to be flat in the inertial range. The exact results derived in this Letter are valid for both, weak and strong wave-turbulence. They could be used as a theoretical benchmark of new wave-turbulence theories and to develop further analogies with hydrodynamical turbulence.