Wave turbulence theory aims at describing the long time behaviour of weakly nonlinear and out-of-equilibrium systems. For thin vibrating plates, this framework allows predicting a Kolmogorov-Zakharov spectrum with an energy flux from the injection to the dissipative scales. Discrepancies between measured and theoretical velocity spectra have been mainly attributed to damping –acting at all scales and precluding the existence of a transparency window– as well as other effects such as the anisotropy brought by the forcing. As the complete derivation of a wave turbulence theory including the presence of a given damping law is still missing, an alternative approach is used in this contribution. A phenomenological equation for the energy spectrum is established from the general solutions (Rayleigh-Jeans and Kolmogorov-Zakharov) of the kinetic equation. This model equation is then used in order to establish the self-similar solutions of the system in the case of non-stationary turbulence. The phenomenological model is then enriched by adding a damping law close to experimental measurements. In that case of stationary wave turbulence, the model also predicts power spectra that are in close agreement with the measurements.