Non-linear Processes and Stimulated Hawking Radiation in Hydrodynamics for Decelerating Subcritical Free Surface Flows with a Subluminal Dispersion Relation

In \cite{PRL2011}, the authors have {\it "conducted experiments in order to verify the thermal nature of the stimulated Hawking process at a white hole horizon in a fluid analogue gravity system"} namely the linear mode conversion giving rise to negative energy waves i.e. the classical ingredient at the root of the Hawking effect in astrophysics. However, here we show that these experiments in Vancouver operated in a weakly non-linear regime that obscure them as was the case for the seminal experiments in Nice within a stronger non-linear regime \cite{NJP2008}. We finally shed some light on these matters by demonstrating that the linear conversion of water waves on a counter-current takes place with or without a dispersive white hole horizon as anticipated in the Nice experiment no matter the frequency is conserved within the entire process provided there is an absence of wave breaking during the wave-current interaction. The main novelty is the role of free (and not forced as usual in non-linear effects) harmonics generation in the interpretation of both the Nice (for at least the stimulating mode) and Vancouver (for the converted modes only) experiments. Unfortunately, the thermality of the spectrum is not demonstrated in the Poitiers reproduction of the Vancouver experiments based on the analysis of the scattering coefficients themselves and not just of their ratio as was done previously.