In the 2008 water-tank experiment of Rousseaux, Mathis, Maïssa, Philbin and Leonhardt, a counter-flowing water current blocked incident surface waves and converted them into outgoing modes with negative co-moving frequency, the classical analogue of Hawking mode conversion at a horizon. The measured conversion was strongly enhanced relative to the prediction of the one-dimensional linear dispersive model, and the authors could not identify the responsible mechanism. The open problem is to identify and quantify the physical processes (nonlinear wave steepening and harmonic generation, vorticity creation at steepening crests, transverse curvature of crests and other three-dimensional effects, turbulence, depth-dependent velocity profiles) that produce this enhancement, to extend the models accordingly until they agree quantitatively with experiment, and to decide whether a genuinely novel fluid-mechanics phenomenon is involved or whether extended wave models suffice. Later work has resolved parts of this in small-amplitude subcritical regimes (notably the role of the background undulation and of free-harmonic generation), but no quantitative, mechanism-decomposed account of the original strongly nonlinear, breaking regime exists, and the field's own 2025 review still lists key ingredients as unknown.

Background

Analogue gravity studies wave propagation on moving media in which an effective spacetime metric emerges: for water waves, the surface gravity-wave dispersion relation in a flowing channel mimics wave propagation on a curved background with a Lorentz-like horizon, called a white-hole or black-hole horizon, wherever the flow speed matches the wave speed. At such a horizon, incident waves cannot propagate upstream and undergo mode conversion: part of the energy is converted into outgoing waves whose co-moving frequency (the frequency measured in the frame moving with the water) is negative, which is negative-energy as seen from the laboratory frame — the classical counterpart of the mode mixing behind Hawking radiation, the quantum thermal emission of black holes. A negative co-moving-frequency mode carries negative energy (its wave action has the opposite sign), so converting an incoming positive-energy wave into such a mode is the classical mechanism whose quantum counterpart yields Hawking emission. The one-dimensional linear reference model is that of Schützhold and Unruh (Phys. Rev. D 66, 044019 (2002)), which numerically integrates the linear wave equation with the correct capillary-gravity dispersion relation on a prescribed transcritical background flow (a so-called gravity-wave analogue of a black hole). In the 2008 experiment (New J. Phys. 10, 053015 (2008)), a flume with a contracting counter-current blocked incident waves of a few-centimetre scale and clearly observed the negative-frequency converted branch, but the measured degree of mode conversion was significantly higher than the theory predicted. The authors' conclusions explicitly list candidate mechanisms: nonlinear processes such as wave steepening and harmonic generation near crests, local vorticity creation at steepening crests, transverse curvature of the crests (three-dimensional effects), turbulence and depth-dependent velocity profiles, and possibly other uncontrolled experimental factors, and state that it would be highly desirable to find out exactly what happens to water waves at horizons. Subsequent developments define the neighboring results a reader needs: Weinfurtner and co-workers' 2011 Vancouver flume experiment measured, at small incident amplitudes, the amplitude ratios of the converted outgoing modes and found them to follow a Boltzmann (thermal) law, with conversion scaling linearly in the incident amplitude; the Poitiers experiments (Euvé et al. 2015, 2016) achieved quantitative agreement between measured transmission and scattering coefficients and theory in subcritical flows (flows whose Froude number — the ratio of the flow speed to the long-surface-wave speed — stays below one everywhere, so the flow does not truly cross the wave speed) once the static undulation of the free surface downstream of the obstacle — a standing deformation spontaneously established by the background flow, whose low-frequency amplification is described by Coutant and Parentani (Phys. Fluids 26, 044106 (2014)) — was included in the model; transverse (three-dimensional) effects were treated theoretically by Coutant, Fabbri, Parentani, Balbinot and Anderson (Phys. Rev. D 86, 064022 (2012)), who showed that transverse wave numbers act as an effective mass and can suppress transmission; and the nonlinear generation of harmonics from the converted modes was demonstrated by Euvé and Rousseaux. This is the question the 2008 authors left open, at its original generality.

Problem Statement

For gravity-capillary water waves blocked at a horizon on a counter-flowing current, as realized in the water-tank experiment of Rousseaux et al. (New J. Phys. 10, 053015 (2008)) — in which the observed conversion of incident waves into negative co-moving-frequency modes is strongly enhanced relative to the prediction of the one-dimensional linear dispersive model of Schützhold and Unruh — determine, at the original generality and in quantitative form: (i) which physical processes cause the enhancement, among the candidates acknowledged by the authors (nonlinear wave steepening and harmonic generation near crests, local vorticity creation at steepening crests, transverse curvature of wave crests and other three-dimensional effects, turbulence, depth-dependent velocity profiles, and other uncontrolled experimental factors), stating which candidates contribute and which are excluded, and what the relative quantitative contribution of each contributing process is, including the dependence of the conversion amplitude on local slope, nonlinearity level and transverse structure; (ii) determine whether extended numerical models incorporating nonlinearity, turbulence and three-dimensional flow structure can achieve quantitative agreement, within its experimental uncertainty, with the measured conversion amplitudes of the Rousseaux et al. (2008) experiment; and (iii) decide whether the enhancement is a genuinely novel fluid-mechanics phenomenon not captured by extended wave models, is reproduced within extended wave models, or reflects uncontrolled experimental factors.

References

  1. Germain Rousseaux, Christian Mathis, Philippe Maïssa, Thomas G. Philbin, Ulf Leonhardt, 'Observation of negative-frequency waves in a water tank: a classical analogue to the Hawking effect?', New J. Phys. 10, 053015 (2008), DOI 10.1088/1367-2630/10/5/053015, https://doi.org/10.1088/1367-2630/10/5/053015; also arXiv:0711.4767, https://arxiv.org/abs/0711.4767
  2. Ralf Schützhold, William G. Unruh, 'Gravity wave analogues of black holes', Phys. Rev. D 66, 044019 (2002), DOI 10.1103/PhysRevD.66.044019, https://doi.org/10.1103/PhysRevD.66.044019 (the one-dimensional linear dispersive reference model, cited as [13] in the 2008 paper)
  3. Silke Weinfurtner, Edmund W. Tedford, Matthew C. J. Penrice, William G. Unruh, Gregory A. Lawrence, 'Measurement of Stimulated Hawking Emission in an Analogue System', Phys. Rev. Lett. 106, 021302 (2011), DOI 10.1103/PhysRevLett.106.021302, https://doi.org/10.1103/PhysRevLett.106.021302; also arXiv:1008.1911, https://arxiv.org/abs/1008.1911
  4. Léo-Paul Euvé, Florent Michel, Renaud Parentani, Germain Rousseaux, 'Wave blocking and partial transmission in subcritical flows over an obstacle', Phys. Rev. D 91, 024020 (2015), DOI 10.1103/PhysRevD.91.024020, https://doi.org/10.1103/PhysRevD.91.024020
  5. L.-P. Euvé, F. Michel, R. Parentani, T. G. Philbin, G. Rousseaux, 'Observation of Noise Correlated by the Hawking Effect in a Water Tank', Phys. Rev. Lett. 117, 121301 (2016), DOI 10.1103/PhysRevLett.117.121301, https://doi.org/10.1103/PhysRevLett.117.121301; also arXiv:1511.08145, https://arxiv.org/abs/1511.08145
  6. Antonin Coutant, Renaud Parentani, 'Undulations from amplified low frequency surface waves', Phys. Fluids 26, 044106 (2014), DOI 10.1063/1.4872025, https://doi.org/10.1063/1.4872025
  7. Antonin Coutant, Alessandro Fabbri, Renaud Parentani, Roberto Balbinot, Paul R. Anderson, 'Hawking radiation of massive modes and undulations', Phys. Rev. D 86, 064022 (2012), DOI 10.1103/PhysRevD.86.064022, https://doi.org/10.1103/PhysRevD.86.064022
  8. Florent Michel, Renaud Parentani, Scott Robertson, 'Gravity waves on modulated flows downstream from an obstacle: The transcritical case', Phys. Rev. D 97, 065018 (2018), DOI 10.1103/PhysRevD.97.065018, https://doi.org/10.1103/PhysRevD.97.065018
  9. Florent Michel, Renaud Parentani, 'Probing the thermal character of analogue Hawking radiation for shallow water waves?', Phys. Rev. D 90, 044033 (2014), DOI 10.1103/PhysRevD.90.044033, https://doi.org/10.1103/PhysRevD.90.044033
  10. Germain Rousseaux, Philippe Maïssa, Christian Mathis, Pierre Coullet, Thomas G. Philbin, Ulf Leonhardt, 'Horizon effects with surface waves on moving water', New J. Phys. 12, 095018 (2010), DOI 10.1088/1367-2630/12/9/095018, https://doi.org/10.1088/1367-2630/12/9/095018
  11. Léo-Paul Euvé, Germain Rousseaux, 'Génération non-linéaire d'harmoniques après une conversion linéaire en interaction houle-courant' (Nonlinear generation of harmonics after a linear conversion in wave-current interaction), XIVèmes Journées Nationales Génie Côtier – Génie Civil, Toulon, pp. 29-38 (2016), DOI 10.5150/JNGCGC.2016.004, https://doi.org/10.5150/JNGCGC.2016.004
  12. Léo-Paul Euvé, Germain Rousseaux, 'Non-linear Processes and Stimulated Hawking Radiation in Hydrodynamics for Decelerating Subcritical Free Surface Flows with a Subluminal Dispersion Relation', arXiv:2112.12504 (2021, preprint), https://arxiv.org/abs/2112.12504
  13. Scott Robertson, Germain Rousseaux, 'Viscous dissipation of surface waves and its relevance to analogue gravity experiments', arXiv:1706.05255 (2017), https://arxiv.org/abs/1706.05255
  14. Alexis Bossard, Nicolas James, Valentin Jules, Johan Fourdrinoy, Scott Robertson, Germain Rousseaux, 'On the art of designing effective space-times with free surface flows in Analogue Gravity', C. R. Physique 25, 457-511 (2025), DOI 10.5802/crphys.215, https://doi.org/10.5802/crphys.215; also arXiv:2408.15629, https://arxiv.org/abs/2408.15629