The reflection and transmission of plane P- and S-waves by a laterally homogeneous band is discussed. A dyadic representation of a ‘plane wave Green's tensor’ is derived, which is used to describe the reflection and transmission of plane waves by a thin homogeneous layer in the first Born approximation. From this, the reflection and transmission by an arbitrarily thick continuously stratified band is derived using invariant imbedding. We derive an exact set of matrix Ricatti equations which describe the reflection and transmission of plane waves by the laterally homogeneous band. These equations remain regular at turning points, and incorporate both homogeneous and inhomogeneous waves within the heterogeneity. It is not necessary for the band to be stratified; the density and the elasticity tensor of the band may have an arbitrary depth dependence. It is shown that in case the band is a smooth heterogeneity without turning points, its only effect is a phase shift of the transmitted wave. In a numerical example for the analogue case of 1-D scattering in quantum mechanics the behaviour of homogeneous and inhomogeneous (tunneling) waves is illustrated.