  The Weak Turbulence Theory has been applied to waves in thin elastic plates
obeying the F\"oppl-Von K\'arm\'an dynamical equations. Subsequent experiments
have shown a strong discrepancy between the theoretical predictions and the
measurements. Both the dynamical equations and the Weak Turbulence Theory
treatment require some restrictive hypotheses. Here a direct numerical
simulation of the F\"oppl-Von K\'arm\'an equations is performed and reproduces
qualitatively and quantitatively the experimental results when the
experimentally measured damping rate of waves $\gamma_\mathbf{k}= a + bk^2$ is
used. This confirms that the F\"oppl-Von K\'arm\'an equations are a valid
theoretical framework to describe our experiments. When we progressively tune
the dissipation so that to localize it at the smallest scales, we observe a
gradual transition between the experimental spectrum and the
Kolmogorov-Zakharov prediction. Thus it is shown dissipation has a major
influence on the scaling properties stationary solutions of weakly non linear
wave turbulence.
