Scattering of gravity waves in subcritical flows over an obstacle

Abstract

We numerically study the scattering coefficients of linear water waves on stationary flows above a localized obstacle. We compare the scattering on trans-and subcritical flows, and then focus on the latter which have been used in recent analog gravity experiments. The main difference concerns the magnitude of the mode amplification: whereas transcritical flows display a large amplification (which is generally in good agreement with the Hawking prediction), this effect is heavily suppressed in subcritical flows. This is due to the transmission across the obstacle for frequencies less than some critical value. As a result, subcritical flows display high-and low-frequency behaviors separated by a narrow band around the critical frequency. In the low-frequency regime, transmission of long wavelengths is accompanied by non-adiabatic scattering into short wavelengths, whose spectrum is approximately linear in frequency. By contrast, in the high-frequency regime, no simple description seems to exist. In particular, for obstacles similar to those recently used, we observe that the upstream slope still affects the scattering on the downstream side because of some residual transmission.

Wave scattering in spatially inhomogeneous currents

“…Notice in the conclusion that the wave-current interaction in recent years became a very hot topic due to applications both to the natural processes occurring in the oceans and as a model of physical phenomena closely related with the Hawking radiation in astrophysics [1][2][3][4][5][9][10][11]. The influence of high-momentum dissipation on the Hawking radiation was considered in astrophysical application [22] (see also [23] where the dissipative fields in de Sitter and black hole spacetimes metrics were studied with application to the quantum entanglement due to pair production and dissipation).…”

“…We have shown, in particular, that in the case of accelerating transcritical current both the reflection coefficient of positive-energy wave and transmission coefficient of negative-energy wave decrease asymptotically with the frequency as jRj ∼ T n ∼ω −1 . This can be presented in terms of the Hawking temperature T H ¼ ð1=2πÞðdU=dxÞ (see, e.g., [3,10]) and dimensional frequency ω as jRj ∼ T n ∼ 2πT H =ω.…”

“…In contrast to other papers also dealing with the surface waves on a spatially varying current (see, e.g., [2][3][4][5]), we consider here the case of shallow-water waves when there is no dispersion, assuming that the wavelengths λ ≫ h.…”

“…We will bear in mind such a model, although we do not pretend here to consider a current in a real duct, but rather to investigate an idealized hydrodynamic model which is described by the equation analogous to that appearing in the context of black hole evaporation due to Hocking radiation [1][2][3][4][5][9][10][11].…”

“…Since 1981 when Unruh established the analogy in wave transformation occurring at the horizon of a black hole and at a critical point of a hydrodynamic flow [1], there have been many attempts to calculate the transformation coefficients and find the analytical expression for the excitation coefficient of a negativeenergy mode (see, for instance, [2][3][4][5] and references therein). In parallel with theoretical study there were several attempts to model the wave scattering in spatially inhomogeneous currents experimentally and determine this coefficient through the measurement data [6,7] (similar experiments were performed or suggested in other media, for example, in the atomic Bose-Einstein condensate-see [8] and numerous references therein).…”