In this study the relation between frequency and complex wave number of axisymmetric wave modes in an isotropic, thin-walled, cylindrical shell containing a linear viscoelastic medium is derived. Shell wall bending and longitudinal motion are coupled in an empty cylindrical shell. When a viscoelastic medium is enclosed, the shell motion is affected by the complex bulk and shear modulus, as well as by the density of the medium enclosed. A Maxwell model is used for both complex Lamé constants and to describe the constitutive equations of the medium. By varying the complex moduli, the medium can be modeled as an inviscid fluid, an elastic material, or anything between these two extremes. The interaction of the thin-walled linear elastic shell and the viscoelastic medium is discussed numerically by calculating the complex dispersion relation. Numerical results are presented for an empty shell and a shell filled with three types of core material: an inviscid fluid, a shear dissipative fluid, and a shear elastic fluid. In a companion paper ͓J. Vollmann et al., J. Acoust. Soc. Am. 102, 909-920 ͑1997͔͒, the experimental setup and the signal processing used to perform the high-resolution measurement of the dispersion relation are described in detail. Theoretical and experimental results are compared.

The complex frequency spectrum of axisymmetric wave modes in a circular cylindrical shell containing various viscoelastic media is measured. A new measurement technique has been developed for this purpose by combining a high-resolution laser interferometer with modern spectrum estimation methods. To decompose the complex wave-number dependence, a complex spectrum estimation method has been implemented. Up to 40 dispersion curves of traveling, axisymmetric modes are decomposed simultaneously in a frequency range between 1 kHz and 2 MHz. The guided structural waves are excited by piezoelectric transducers. Linear elasticity can be considered as an extreme case of viscoelasticity (long relaxation times compared with the deformation periods). To ascertain the validity of the theory, dispersion curves are calculated for a shell containing a viscoelastic material behaving like the elastic shell and are compared with the measured curves of an isotropic aluminium rod. The phenomenon of “backward wave propagation,” in which the group velocity and the phase velocity of one mode have opposite signs, is clearly measured. Excellent agreement between experimental and theoretical results, which are also presented in a corresponding paper [J. Vollmann and J. Dual, J. Acoust. Soc. Am. 102, 896–908 (1997)], is found over a wide parameter range, including the case of a linear elastic rod.

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