rest frame of the WH horizon, so we expect that the BCR production process should conserve energy. One way to view it is that incoming atoms reflect from the flow transition at the horizon. The velocity of the WH horizon is quite low compared to the flow velocity upstream, so to a good approximation energy conservation in the WH frame implies that the laboratory frame wavevector should simply reverse sign. A more precise account of this given in Sec. \[sec:M2\] below, using a linearized mode analysis.

Another consequence of the approximate local time independence in the WH frame is that the frequency of the standing wave generated there should have zero frequency in that frame. Indeed it does, but we postpone the demonstration of that to later in this section.

### Growth of the standing wave {#comparison}

![\[fig:GP\_growth\] (a) Simulated growth of the standing-wave pattern in the supersonic region for $U_{s}/k =$ 6 nK. Solid green: normalized standing-wave amplitude $\bar{n}_k(t)$, $\bar{n}_{k}(t)=n_{k}(t)/n_{k}(0)$, for which $\ln[\bar{n}_{k}(120)]\sim 4.4$. Dashed black: the square of background density, $\bar{n}_{\rm bf}(t)$, scaled to match the final standing-wave amplitude, $\bar{n}_{\rm bf}^2(t)=n_{\rm bf}^2(t)[\bar{n}_{k}(120)/n_{\rm bf}^2(120)]$. The growth of $n_{\rm bf}$ and $n_{k}$ is determined from a spatial WFT of $n(x)$ 