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An exact result concerning the energy transfers between non-linear waves ofthin elastic plate is derived. Following Kolmogorov's original ideas inhydrodynamical turbulence, but applied to the F\"oppl-von K\'arm\'an equationfor thin plates, the corresponding K\'arm\'an-Howarth-Monin relation and anequivalent of the $\frac{4}{5}$-Kolmogorov's law is derived. A third-orderstructure function involving increments of the amplitude, velocity and the Airystress 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 incrementsare evaluated and $\varepsilon$ the energy dissipation rate. Numerical dataconfirm this law. In addition, a useful definition of the energy fluxes inFourier space is introduced and proven numerically to be flat in the inertialrange. The exact results derived in this Letter are valid for both, weak andstrong wave-turbulence. They could be used as a theoretical benchmark of newwave-turbulence theories and to develop further analogies with hydrodynamicalturbulence.

An exact result in strong wave turbulence of thin elastic plates