An intrinsic trait of the plane inhomogeneous waves in elastic media is a particular parametrization duality, stemming from two components characterizing the complex wave vector (bivector) k with respect to the reference frame of orthonormal vectors. Correspondingly, the characteristic condition associated with the equation of motion involves two parameters p and λ, where p is the eigenvalue of the Stroh matrix N (λ) and λ is the eigenvalue of the complex Christoffel matrix λ (p). One of the consequences is a rather elaborate layout of the complex-valued degeneracy types implying repeated p, or λ, or both p and λ. Different types of degeneracy are characterized by dissimilar analytical, algebraic and topological features. The general theory, which has been established for an arbitrary anisotropic medium, is developed in the present paper into an explicit form for a transversely isotropic continuum. By reducing the number of geometrical parameters and enabling factorization of the characteristic polynomial, transverse isotropy fosters explicit resolving of the equations which define different degeneracy types. A particular thrust of the paper is concerned with the algebraic analysis of the conditions, under which the degenerate inhomogeneous modes, having linearly varying or/and circularly polarized amplitude, can be excited on a traction-free boundary by means of reflection or within the Rayleigh wave for, respectively, two- and one-parameter manifolds in the space of orientations of the free surface and propagation direction. Various settings are identified which ensure reflected and surface-localized wave packets comprising a degenerate inhomogeneous mode.
