Progress in materials’ science and engineering has lead to the development of low-density composite materials exhibiting high strength and toughness. Those structural materials like fiber composites and metal-matrix composites (MMC) have gained an important industrial role, being widely applicated e.g. in aerospace industries. The need for their proper testing in view of delaminations, inclusions and other defects has correspondingly stimulated the interest in studying wave propagation in such anisotropic media. In this study, the fundamental mathematical formulation of Huygens’ principle is employed to derive an integral representation for transducer-generated wavefields. Use is made of the dyadic and triadic Green’s functions, which have been derived recently in form of their 2d-space-time spectral representations for the case of transversely isotropic symmetry [1], which is characteristic for unidirectional fiber composites and extruded MMCs, but also for ideally fiber-textured columnar-grained steels. A particularly interesting outcome of this analysis is the generalized formulation of the transversely isotropic Rayleigh-function describing the Rayleigh-wavefronts propagating at the free surface. Since the material’s spatial orientation is included as an additional parameter, a coordinate-free approach is used to reduce the inherent complexity of this analysis as far as possible. The use of the derived relationships for modeling transducer radiation into these media, possible ways of evaluation and the significance of the generalized Rayleigh-function are discussed.