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The Frobenius Power Series Solution for Cylindrically Anisotropic Radially Inhomogeneous Elastic Materials
( 1 Division of Theoretical Mechanics, University of Nottingham, Nottingham NG7 2RD )
Solution in the form of Frobenius power series to the system of equations of three-dimensional elasticity, describing propagation of cylindrical waves, is constructed and studied. A general case of radially inhomogeneous materials with arbitrary cylindrical anisotropy is considered. The governing ordinary differential system is taken in the form of six first-order equations. The intrinsic algebraic symmetry of the matrix of coefficients reveals basic properties of the indicial equation and ensuing features of the fundamental solution. On this basis, complemented by energy considerations, it is shown that partial solutions in the form of the Frobenius series satisfy specific orthogonality. Its physical meaning is that the corresponding wave modes do not carry energy flux across cylindrical surfaces. The limiting behaviour of the partial solutions near the cylinder axis (r 
0) is analysed. It is proved that they always partition into two triplets, one with converging displacement amplitudes tending to zero or to a rigid-body translation at r 0, and the other with displacements diverging at r 0. Analogies with elasticity of rectilinearly anisotropic media are discussed.
Received 7 January 2002. Revised 16 September 2002.