Anti-plane shear waves in an elastic strip rigidly attached to an elastic half-space

We consider the anti-plane shear waves in a domain consisting of an infinite layer with a thin coating lying on an elastic half-space. The elastic properties of the coating, layer, and half-space are assumed to be different. On the free upper surface we assume the compatibility condition within the Gurtin–Murdoch surface elasticity, whereas at the plane interface we consider perfect contact. For this problem there exist two possible regimes related to waves exponentially decaying in the half-space. The first one, called transversely exponential–transversely exponential (TE–TE) regime, is related to waves described by exponential in transverse direction functions; the second, transversely harmonic–transversely exponential (TH–TE) regime, corresponds to waves in the upper layer which have the harmonic behaviour in the transverse direction. Detailed analysis of the derived dispersion equations for both regimes is provided. In particular, the effects of surface stresses, the layer thickness as well as of the ratio of shear moduli of the upper layer and half-space on the dispersion curves is analysed.