This paper is the first in a series of articles describing the refraction and propagation of infinitesimal disturbances in a 'coarse grained' inhomogeneous anisotropic material which is fused to an isotropic substrate. Here, the basic constitutive law for the material is motivated by applications to the non-destructive evaluation of austenitic steel welds, although it is clear that the phenomena described and the mathematical analysis used is also of interest in geophysics, the study of composite materials and several other areas of continuum mechanics. This work is concerned with the refraction of a horizontally polarized shear wave source at the fusion interface between a homogeneous isotropic material and transversely isotropic material. The latter is inhomogeneous by virtue of the fact that the zonal axis or axis of symmetry of the crystals varies in direction with the distance from the interface. The mathematical boundary-value problem is solved exactly, and, in the high-frequency limit, a uniform asymptotic expansion for the displacement vector is found. It is shown that in this limit, and for a wide range of material constants, the refracted energy which penetrates certain regions of the 'weld material' is totally internally reflected. This conclusion is highly significant in the design of inspection procedures for structurally important welds.