Water-Wave Horizons

A water-wave horizon is a blocking region in a nonuniform channel flow: surface waves attempting to propagate against the current reach zero laboratory group velocity and convert to other dispersive branches. These experiments provide direct, visible tests of stimulated Hawking-mode conversion. The waves and their ambient noise are classical, so correlations alone do not demonstrate quantum vacuum emission.

Water-wave construction and status

Property	Statement
Object class	Analogue system
Medium	Free surface of a finite-depth, flowing liquid
Perturbations	Linear surface-gravity and capillary waves
Horizon	Blocking point for a selected dispersive branch
Typical configuration	White-hole analogue for waves sent upstream
Physical status	Classical laboratory analogue of horizon scattering

For a locally uniform current $U(x)$, depth $h(x)$, density $\rho$, and surface tension $\gamma$, define $q=\lvert k\rvert$. The linear dispersion relation is

$$(\omega-U k)^2 =\left(g_\oplus q+\frac{\gamma}{\rho}q^3\right)\tanh(qh).$$

The intrinsic frequency $\Omega=\omega-Uk$ determines the wave-energy sign in the idealized conservative theory. A blocking point satisfies $d\omega/dk=0$ on the incident branch, not simply $U=\sqrt{g_\oplus h}$ once finite-depth and capillary dispersion matter.

Controlled limits and mode conversion

In the shallow-water, gravity-dominated limit $qh\ll1$ and $\gamma q^2\ll\rho g_\oplus$,

$$\Omega^2\simeq g_\oplus h\,k^2, \qquad c_{\rm shallow}=\sqrt{g_\oplus h}.$$

This recovers the nondispersive acoustic horizon. In deep water, $\tanh(qh)\to1$ and the group velocity depends strongly on frequency; at still shorter wavelengths the capillary term creates additional branches. Those are two controlled limits of the same measured dispersion, not corrections that may be ignored after the fact.

A stationary inhomogeneous flow conserves $\omega$ and mixes roots at that frequency. Negative-$\Omega$ roots carry negative wave energy relative to the background flow. Their appearance with a positive-energy partner is the classical counterpart of the Bogoliubov scattering channel. Rousseaux et al. 2008, §§2–6 directly observed negative-frequency waves in a tank.

The thermal diagnostic is a ratio of norm-corrected scattering amplitudes, not a fit to surface-height power alone. Group velocity converts squared Fourier amplitude into conserved wave-action flux. Therefore the wavemaker spectrum, spatial window, branch-dependent group velocities, and sensor response must be unfolded before comparing $|\beta_\omega/\alpha_\omega|^2$ with an exponential. Measuring several incident frequencies and all outgoing roots tests a scattering law; selecting only a visibly amplified branch does not.

Strongest experimental result

Weinfurtner et al. 2011, §§II–V sent a controlled long-wave input toward a white-hole blocking region and measured the amplitudes of converted short-wave branches, finding a thermal relation over the accessible range. This is stimulated Hawking emission: the wavemaker supplies the incident field.

Euvé et al. 2016, Eqs. (1)–(4) and Figs. 1–5 measured correlations between opposite-energy modes in the fluctuating free surface and compared the noise analysis with waves produced by a wavemaker. The result is a strong classical demonstration that ambient fluctuations are organized by the same scattering matrix. Because the water surface is not prepared in a quantum vacuum and the measured variables are classical elevations, “noise-correlated” does not mean spontaneous quantum Hawking radiation.

At the 20 August 2026 evidence cutoff, the strongest established claim is therefore quantitative, dispersive, stimulated mode conversion—including opposite-energy correlations—in a calibrated classical fluid.

Inference systematics and adverse checks

The background $U(x)$ and $h(x)$ must be reconstructed rather than inferred only from pump settings. Viscosity, wave breaking, turbulence, transverse channel modes, reflections, obstacle wakes, capillarity, and nonlinear harmonics can spoil one-dimensional conservative scattering. A long stationary undulation near a white-hole horizon can itself scatter incoming fluctuations and change the low-frequency spectrum, as analysed by Coutant and Parentani 2014, §§II.C–III.C and IV.

A major systematic uncertainty is the background reconstruction; finite spatial windows, branch overlap in Fourier space, attenuation correction, and correlations already present in the incident noise add further inference systematics. Each changes the inferred norm-corrected coefficient ratio and must be varied in the reported analysis.

Useful adverse controls include removing the blocking point while keeping the wavemaker and detector unchanged; reversing flow or incidence; varying obstacle height and input amplitude; measuring all open outgoing branches; and checking flux balance after attenuation corrections. Curvature Scattering and Tails provides the canonical comparison for wave scattering by nonuniform effective geometry, without equating the media.

Principal obstruction and limitations

The principal obstruction to a quantum claim is state preparation. At laboratory temperature, the populated surface modes are classical; bath noise and mechanical forcing are many orders of interpretation away from a vacuum state. A thermal ratio follows from the scattering coefficients and can appear in stimulated data. It neither measures entanglement nor establishes particle creation from vacuum.

There is also a geometric limitation. A blocking point can move or split with frequency because the dispersion is not relativistic. One should therefore report a frequency-dependent turning region rather than a single universal horizon position. This dispersive structure is precisely what makes the experiment useful, but it prevents a literal identification with the null boundary of one spacetime metric.

The analogy is also kinematic. Water flow is governed by Navier–Stokes and free-surface boundary conditions, not Einstein dynamics. Backreaction is ordinary wave–mean-flow coupling, and no gravitational entropy or semiclassical stress tensor follows. The appropriate comparison to quantum particle creation is Bogoliubov Coefficients and Pair Creation, with the classical state boundary kept explicit.

References

• Coutant, Antonin, and Renaud Parentani. “Undulations from Amplified Low Frequency Surface Waves.” Physics of Fluids 26 (2014), 044106. doi:10.1063/1.4872025.
• Euvé, Léo-Paul, Florent Michel, Renaud Parentani, Thomas G. Philbin, and Germain Rousseaux. “Observation of Noise Correlated by the Hawking Effect in a Water Tank.” Physical Review Letters 117 (2016), 121301. doi:10.1103/PhysRevLett.117.121301.
• Rousseaux, Germain, Christian Mathis, Philippe Maïssa, Thomas G. Philbin, and Ulf Leonhardt. “Observation of Negative-Frequency Waves in a Water Tank: A Classical Analogue to the Hawking Effect?” New Journal of Physics 10 (2008), 053015. doi:10.1088/1367-2630/10/5/053015.
• Weinfurtner, Silke, Edmund W. Tedford, Matthew C. J. Penrice, William G. Unruh, and Gregory A. Lawrence. “Measurement of Stimulated Hawking Emission in an Analogue System.” Physical Review Letters 106 (2011), 021302. doi:10.1103/PhysRevLett.106.021302.