Steady waves propagating in an anisotropic elastic layer that is attached to an anisotropic elastic half-space is studied. By attached we mean that the interface between the layer and the half-space can be perfectly bonded (b) or in sliding contact (s). The other surface of the layer can be traction-free (F), a rigid surface (R) or a slippery surface (S). We also study steady waves in an anisotropic elastic layer that is attached between two different anisotropic elastic half-spaces. The two interfaces between the layer and the half-spaces can be both perfectly bonded (b/b), one of the interfaces is perfectly bonded while the other is a slippery surface (b/s or s/b) or both interfaces are slippery surfaces (s/s). In the derivation the thickness h of the layer is assumed to be small, and the solution is in the form of an infinite series in the power of h from which an approximate solution can be obtained by keeping the terms up to O(h n) for any n. However, the infinite series has a closed-form expression so that the thickness h of the layer need not be small.
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