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This paper presents analytical models of wave propagation in single- and double-walled carbon nanotubes, as well as nanotubes embedded in an elastic matrix. The nanotube structures are treated within the multilayer thin shell approximation with the elastic properties taken to be those of the graphene sheet. The double-walled nanotubes are coupled together through the van der Waals force between the inner and outer nanotubes. For carbon nanotubes embedded in an elastic matrix, the surrounding elastic medium can be described by a Winkler model. Tube wave propagation of both symmetrical and asymmetrical modes can be analyzed based on the present elastic continuum model. It is found that the asymmetrical wave behavior of single- and double-walled nanotubes is significantly different. The behavior is also different from that in the surrounding elastic medium.