Elastic waves in rods and clad rods

Abstract

Clad rods have been investigated for use as long delay lines because they offer isolation of the signal from the surface and low dispersion. In addition, single-mode propagation is achieved with a larger (and hence more conveniently transduced) cross section than is possible with a homogeneous rod at the same frequency. This paper describes the modes that have a steady-state sin(ωt−βz) dependence on (t,z), with emphasis on the modes and ranges of parameters that are of interest for delay lines. Only rods of circular cross section, and isotropic, linear elastic materials are considered. Attention is drawn to correspondences with homogeneous rods and with the hypothetical case of infinite thickness cladding, which is most useful as a model for understanding the behavior of corresponding modes in an actual clad rod. Written for the nonspecialist, the paper includes a tutorial review of the concepts and results needed to understand wave propagation in rods and clad rods. In addition, the following new results are reported: (1) for homogeneous rods, a representation of the lowest flexural mode dispersion curve that is for practical purposes independent of Poisson’s ratio; (2) for homogeneous rods, displacement distributions of the first three high-frequency shear modes, i.e., the ’’flexural’’ modes that are asymptotic to the shear velocity (it was found that the distribution previously attributed to the lowest such mode actually belongs to the next); (3) for clad rods, the first demonstration of interface modes of the Stoneley type for nonaxially symmetric waves; (4) classification of clad rods according to the nature and ordering of their asymptotic velocities showing that there are eight types of clad rod, rather than four as stated in the previous literature; (5) for infinitely clad rods having the same shear modulus, proof that the dispersion of torsional waves and their penetration into cladding are universal functions of f/fc∞. The ratio of the shear velocities of the two materials affects the cutoff frequency fc∞, but not the universal penetration and dispersion functions. Subjects on which significant tutorial or descriptive material is given include typical waveguide dispersion, characteristic velocities of an isotropic elastic material, the effect of coupling of dilatational and shear waves at a boundary, waves in homogeneous rods, the connection of isolation to total internal reflection (with Love waves and SH waves in a clad plate as an example), Stoneley waves at a plane interface, and previous results on the clad rod.