at $x=-12$ $\mu$m with window width $D=6.5$ $\mu$m. Inset (b) shows the windowed wavevector spectrum at $\text{t=120 ms}$.](steinhauer_evolution_growth_072417){width="3.in"}

Reference [@nphys3104] reported exponential growth of the oscillatory density pattern in the BH-WH cavity, and suggested that it results from the black hole laser effect. Our simulations exhibit similar growth, but lead us to attribute it to a different mechanism. Figure \[fig:GP\_growth\] displays the growth of the background flow density $n_{\rm bf}$ and of a standing wave, $n_k$, defined by the peaks of the WFT of the density at $x_{\rm BH}-12 \mu$m, as shown in the inset. Note that this spectrum is different from the one in Fig. \[fig:exp\](b), which is the squared modulus of the spatial WFT of the wavefunction. Over 120 ms the standing wave density grows by $\sim \exp(4.4)$. Figure \[fig:GP\_growth\] shows that $n_k$ grows in proportion to $n_{\rm bf}^2$. The two oscillation features superimposed on the growth curve coincide with the variations seen in Fig. \[fig:stimulated\_HR\](b), and will be discussed in Sec. \[sec:comparison\].

To understand this quadratic relationship between the standing wave amplitude and the background flow density, we begin by noting that the step moves toward the region of higher BEC density \[see Fig. \[fig:exp\](a)\], so the 