Mode mixing in sub- and trans-critical flows over an obstacle: When should Hawking's predictions be recovered?

Abstract

We reexamine the scattering coefficients of shallow water waves blocked by a stationary counter current over an obstacle. By considering series of background flows, we show that the most relevant parameter is $F_{\rm max}$, the maximal value of the ratio of the flow velocity over the speed of low frequency waves. For subcritical flows, i.e., $F_{\rm max} < 1$, there is no analogue Killing horizon and the mode amplification is strongly suppressed. Instead, when $F_{\rm max} \gtrsim 1.1$, the amplification is enhanced at low frequency and the spectrum closely follows Hawking's prediction. We further study subcritical flows close to that used in the Vancouver experiment. Our numerical analysis suggests that their observation of the "thermal nature of the mode conversion" is due to the relatively steep slope on the upstream side and the narrowness of the obstacle.