mode slightly modulates the standing wave.

As illustrated in Fig.\[fig:exp\](d), an obstacle in a stationary supersonic flow produces an upstream, Bogoliubov-Čerenkov standing wave [@Leboeuf2001; @Carusotto], analogous to a bow wave on water [@Carusotto:2012fy]. It was observed in Ref. [@Mayoral2011] that such a standing wave is generated at a WH, triggered by an incident wavepacket on the stationary flow, and saturating at an amplitude determined by nonlinear effects. A similar standing wave, generated by inhomogeneity at a WH horizon, can be seen in Fig. 4 of Ref. [@PhysRevA.91.053603]. In our case, the subsonic component to the left of the WH serves as an obstacle in the supersonic flow, generating a Bogoliubov-Čerenkov wave.

### Wavevector spectrum {#sec:kspectrum}

In the WFT spectrum in Fig. \[fig:exp\](b), we observe in the cavity region an excitation mode at $k\sim1.4$ $\mu$m$^{-1}$ coming from the WH, which is roughly the reflection of the supersonic flow $\Psi_{\rm bf}$ with $k\sim-1.4$ $\mu$m$^{-1}$. The interference of the two results in the standing wave in the density profile shown in Fig. \[fig:GP1\](e), which has a wavevector with twice the above value, $k\sim3$ $\mu$m$^{-1}$.

The relation $k_{\rm BCR} \sim-k_{\rm bf}$ is expected from energy conservation: the flow structure is approximately time-independent in the 