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We study the collective excitations in a relativistic fluid with an anomalous $U(1)$ current. In $3+1$ dimensions at zero chemical potential, in addition to ordinary sound modes we find two propagating modes in presence of an external magnetic field.
The first one which is a transverse degenerate mode, propagates with a velocity proportional to the coefficient of gravitational anomaly; this is in fact the Chiral Alfv\'en wave recently found in \cite{Yamamoto:2015ria}. Another one is a wave of density perturbation, namely a chiral magnetic wave (CMW). The velocity dependence of CMW on the chiral anomaly coefficient is well known. We compute the dependence of CMW's velocity on the coefficient of gravitational anomaly as well. We also show that the dissipation splits the degeneracy of CAW.
At finite chiral charge density we show that in general there may exist five chiral hydrodynamic waves. Of these five waves, one is the CMW while the other four are mixed Modified Sound-Alfv\'en waves. It turns out that in propagation transverse to the magnetic field no anomaly effect appear while in parallel to the magnetic field we find sound waves become dispersive due to anomaly. }
% In 1+1 dimensions we find only one propagating mode associated with the anomalous effects. We explicitly compute the velocity of this wave and show that in contrast to $3+1$ dimensions, no external field is needed for this mode to propagate.
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