Torsional waves in lossy cylinders

Abstract

The pure shear problem is one of relative mathematical simplicity and includes the essential physics common to more complicated cases, where multiple and coupled deformations occur. In this sense, the analysis of torsional waves serves as a pilot problem for investigating the influence of anisotropy and/or anelasticity on solution behavior. We obtain the kinematic and dynamic properties of torsional axially symmetric harmonic waves propagating in an infinitely long circular cylinder. The medium is transversely isotropic and dissipative, with its symmetry axis coincident with the axial axis of the cylinder. For an elastic cylinder each mode has a cutoff frequency and below that frequency there is no propagation. For tubes made of quartz and aluminum Lucite, we found that the existence of the cutoff frequencies depend on the degree of anisotropic attenuation, i.e., if the axial quality factor is greater than the transverse quality factor, the modes propagate at all frequencies. In contrast to the elastic case, the Poynting vector and the energy velocity have a component along the radial direction, whose values depend on the transverse attenuation. The presence of intrinsic attenuation confines the energy near the (elastic) cutoff frequencies while the radial distribution of the energy is governed by the geometrical features of the cylinder.