A relatively new type of constitutive equation for elastic bodies, wherein the linearized strain tensor is a function of the Cauchy stress tensor, is studied for the particular case the stresses are divided into two parts. One is an initial time-independent stress to which is added an incremental (small in comparison with that initial stress) time-dependent stress. Incremental equations are derived that can be used to study the propagation of small-amplitude waves. The behaviour of such waves is studied for three boundary value problems, where the distributions of the initial stresses are inhomogeneous. The problems are: the inflation of a hollow sphere, the inflation of an infinitely long cylindrical annulus, and the circumferential shear and inflation of a cylindrical annulus. The behaviour of the small-amplitude waves is compared for different external loads, investigating how the initial stress affects such waves. A nonlinear model for a class of isotropic dry elastic rock is considered.