We study the propagation of circumferential stress waves in an annular cylindrical region comprised of a new class of isotropic homogeneous elastic bodies. The class of elastic bodies under consideration does not belong to either the classical Cauchy elastic or Green elastic bodies. Unlike the case of classical elasticity theory that reduces to the classical linearized elasticity theory when the displacement gradients are sufficiently small, it is possible that in the new class the linearized strain is related non-linearly to the stress. This possibility leads to a non-linear system of equations that admits the propagation of circumferential shear stress waves in the body.