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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 toordinary sound modes we find two propagating modes in presence of an externalmagnetic field. The first one which is a transverse degenerate mode, propagateswith a velocity proportional to the coefficient of gravitational anomaly; thisis 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 wellknown. We compute the dependence of CMW's velocity on the coefficient ofgravitational anomaly as well. We also show that the dissipation splits thedegeneracy of CAW. At finite chiral charge density we show that in generalthere may exist five chiral hydrodynamic waves. Of these five waves, one is theCMW while the other four are mixed Modified Sound-Alfv\'en waves. It turns outthat in propagation transverse to the magnetic field no anomaly effect appearwhile in parallel to the magnetic field we find sound waves become dispersivedue to anomaly.

Hydrodynamic Waves in an Anomalous Charged Fluid