Wave equation for sound in fluids with unsteady inhomogeneous flow

An approximate wave equation is derived for sound propagation in an inhomogeneous fluid with ambient properties and flow that vary both with position and time. The derivation assumes that the characteristic length scale and characteristic time scale for the ambient medium are larger than the corresponding scales for the acoustic disturbance. For such a circumstance, it is argued that accumulative effects of inhomogeneities and the ambient unsteadiness are satisfactorily taken into account by a wave equation that is correct to first order in the derivatives of the ambient quantities. A derivation that consistently neglects second- and higher-order terms leads to a concise wave equation similar to the familiar ordinary wave equation of acoustics. The wide applicability of this equation is established by showing that it reduces to previously known wave equations for special cases and by showing, with the eikonal approximation, that it yields the geometrical acoustics equations for ray propagation in moving inhomogeneous media.