Rotational superradiant scattering in a vortex flow
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
(rad m−1). When surface waves propagate on a changing flow, the surface elevation is generally described by the sum of two contributions ξ = ξI + ξS, where ξI is the incident wave produced by a source, for example, a wave generator, while ξS 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
.Due to the symmetry, it is appropriate to describe ξI and ξS using polar coordinates (r, θ). Any wave ξ(t, r, θ) can be decomposed into partial waves10, 14,
where
is the azimuthal wavenumber and φf, 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
. (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
is the wavevector norm. This describes the superposition of an inward wave of (complex) amplitude Ain propagating towards the vortex, and an outward wave propagating away from it with amplitude Aout. These coefficients are not independent. The Ain values, one for each f and m component, are fixed by the incident part ξI. If the incident wave is a plane wave
, then the partial amplitudes are given by
. In other words, a plane wave is a superposition containing all azimuthal waves, something that we have exploited in our experiment. In contrast, Aout depends on the scattered part ξS, and how precisely the waves propagate in the centre and interact with the background vortex flow. In the limit of small amplitudes, there is a linear relation between the Ain and Aout val
-Abstract Truncated-
Publisher URL: http://dx.doi.org/10.1038/nphys4151
DOI: 10.1038/nphys4151
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