 Linearized Euler equations

The linearized Euler equations retain only first-order perturbations about a specified inviscid fluid state. Together with the linearized continuity equation and a pressure-density constitutive relation, they describe small-amplitude acoustic waves. About a uniform resting state, ${\rho }_0{\partial }_t{\mathbf{u}}^{\prime }=-\nabla p^{\prime }$, ${\partial }_t{\rho }^{\prime }+{\rho }_0\nabla \cdot {\mathbf{u}}^{\prime }=0$, and an isentropic pressure perturbation has $p^{\prime }=c_0^2{\rho }^{\prime }$. Gradients and advection of a moving background must be retained when its variation is not negligible.