# Rotational superradiant scattering in a vortex flow

*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*=

*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 .

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
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
. (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 *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
, 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, *A*_{out} depends on the scattered part *ξ*_{S}, 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 *A*_{in} and *A*_{out} val

-Abstract Truncated-

Publisher URL: http://dx.doi.org/10.1038/nphys4151

DOI: 10.1038/nphys4151

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