The general equations for a small displacement superimposed on a finite deformation of a perfectly elastic material of arbitrary symmetry are derived anew. These equations are the basis of an analysis of plane waves of small amplitude propagating in an initially deformed and stressed elastic material. Certain restrictions on the energy function of finite elasticity theory are determined. These restrictions, which provide necessary and sufficient conditions that a homogeneously deformed material admit plane waves, are then compared with other restrictions obtained from themostatic equilibrium considerations. Compatibility conditions, which are necessary and sufficient that data on sound wave propagation in elastic materials be compatible with classical elasticity theory upon suitable assignment of the material symmetry, are derived. Finally, it is shown how the variation of sound speeds with initial stress and the measured magnitude of the acoustoelastic effect can be used to determine the third‐order elastic constants of an isotropic material and as a partial confirmation and experimental check of the theory.