The Stroh formalism for elastic surface waves of general profile

Abstract

The Stroh formalism is widely used in the study of surface waves on anisotropic elastic half-spaces, to analyse existence and for calculating the resultant wave speed. Normally, the formalism treats complex exponential solutions. However, since waves are non-dispersive, a generalization to waves having general waveform exists and is here found by various techniques. Fourier superposition yields a description in which displacements are expressed in terms of three copies of a single pair of conjugate harmonic functions. An equivalent representation involving just one analytic function also is deduced. Both these show that at the traction-free boundary just one component (typically the normal component) of displacement may be specified arbitrarily, the others then being specific combinations of it and its Hilbert transform. The algebra is closely related to that used for complex exponential waves, although the surface impedance matrix is replaced by a transfer matrix, which better embodies the scale invariance properties of the waves. Using the scale-invariance property of the boundary-value problem, a further derivation is presented in terms of real quantities.