background density $n_{\rm bf}$ also grows in the cavity. Now the saturated amplitude ($n_k$) of a BCR standing wave should be proportional to both the strength of the obstacle ($V_{\rm ob}$), and the density of the background flow ($n_{\rm bf}$) [@Carusotto; @pethick] \[see Fig.\[fig:exp\](e)\]. The “obstacle" in the present case has a strength proportional to the BEC density to the left of the WH, which grows similarly to that on the right, so it follows that the saturated wave amplitude $n_k$ should grow as $n_{\rm bf}^2$.

The very close agreement with this scaling relation displayed in Fig.\[fig:GP\_growth\](a), gives further compelling evidence that the standing wave observed inside the supersonic cavity is in fact BCR, and indicates that its growth results from the increase of background density, rather than from a black hole laser instability. Moreover, effects due to classical or quantum fluctuations, not included in our simulation, could not remove this robust, large BCR wave, but rather would have to appear in addition to it.

The absence of fluctuations in our simulation here implies that we are unable to capture the behavior of the density-density correlation function. That correlation function was measured in the experiment [@nphys3104], and displays a checkerboard 