The general theory of anelastic substances is simplified for the case of an anelastic fluid. It is shown that slow steady motions and the diffusion of vorticity obey the same laws as in the Stokes-Navier theory of a viscous fluid. A criterion for the validity of these equations is derived.The propagation and absorption of longitudinal isentropic waves exhibits new phenomena, however. There is hysteresis in the response of the density to the pressure. This hysteresis vanishes for very low and very high frequencies. The compressibility for these two limiting cases approaches different, real, values. For intermediate values the compressibility is a complex number, corresponding to the hysteresis. The velocity of propagation is also a complex number, which is a function of the frequency. The roots of this function are two in number, and determine two relaxation times. One of these is associated with the hysteresis mentioned above, the other with viscosity.