Rotational superradiant scattering in a vortex flow

Abstract

When an incident wave scatters o of an obstacle, it is partially reflected and partially transmitted. In theory, if the obstacle is rotating, waves can be amplified in the process, extracting energy from the scatterer. Here we describe in detail the first laboratory detection of this phenomenon, known as superradiance [1][2][3][4] . We observed that waves propagating on the surface of water can be amplified after being scattered by a draining vortex. The maximum amplification measured was 14% ± 8%, obtained for 3.70 Hz waves, in a 6.25-cm-deep fluid, consistent with the superradiant scattering caused by rapid rotation. We expect our experimental findings to be relevant to black-hole physics, since shallow water waves scattering on a draining fluid constitute an analogue of a black hole [5][6][7][8][9][10] , as well as to hydrodynamics, due to the close relation to over-reflection instabilities [11][12][13] . In water, perturbations of the free surface manifest themselves by a small change ξ(t, x) of the water height. On a flat bottom, and in the absence of flow, linear perturbations are well described by superpositions of plane waves of definite frequency f (Hz) and wavevector k (rad m −1 ). When surface waves propagate on a changing flow, the surface elevation is generally described by the sum of two contributions ξ = ξ I + ξ S , where ξ I is the incident wave produced by a source, for example, a wave generator, while ξ S is the scattered wave, generated by the interaction between the incident wave and the background flow. In this work, we are interested on the properties of this scattering on a draining vortex flow which is assumed to be axisymmetric and stationary. At the free surface, the velocity field is given in cylindrical coordinates by v = v r e r + v θ e θ + v z e z .Due to the symmetry, it is appropriate to describe ξ I and ξ S using polar coordinates (r, θ). Any wave ξ(t, r, θ ) can be decomposed into partial waves 10,14 ,where m ∈ Z is the azimuthal wavenumber and ϕ f ,m (r) denotes the radial part of the wave. Each component of this decomposition has a fixed angular momentum proportional to m, instead of a fixed wavevector k. (To simplify notation, we drop the indices f ,m in the following.) Since the background is stationary and axisymmetric, waves with different f and m propagate independently. Far from the centre of the vortex, the flow is very slow, and the radial part ϕ(r) becomes a sum of oscillatory solutions,where k = ||k|| 2 is the wavevector norm. This describes the superposition of an inward wave of (complex) amplitude A in propagating towards the vortex, and an outward wave propagating away from it with amplitude A out . These coefficients are not independent. The A in values, one for each f and m component, are fixed by the incident part ξ I . If the incident wave is a plane wave ξ = ξ 0 e −2iπft+ik·x , then the partial amplitudes are given by A in = ξ 0 e imπ+iπ/4 / √ 2πk. In other words, a plane wave is a superposition containing all azimuthal waves, something that we have exploited...

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