The asymptotic solutions of elasticity theory equations for long and short-waves being propagated along the generator of a closed circular cylindrical shell, the normal waves, are examined. Two problems are investigated: Th Cauchy problem for an infinite shell and the problem of edge perturbation propagation in a semi-infinite shell. A general asymptotic solution is constructed for both problems. The asymptotic parameter is the shell wall thickness or the number of waves across this thickness. In the first problem, the frequency spectrum is determined, and in the second, the wave spectrum. Operators of the fundamental solutions are constructed for the problems considered. On the whole, the solutions for shells are close to the solution of an analogous problem for a solid cylinder /1–3/, and especially to the results of /3/ in which quadratic bundles of operators generated by the problem were investigated. Below the asymptotic solution is constructed by analogy with /4/, but the order of the approximation is higher. The results represented differ from the extensively utilized solutions based on the Timoshenko equations /5,6/. This refers expecially to the problem of edge perturbation propagation. By using asymptotic representations, the quadratic bundle of operators analogous to /3/, are reduced to quadratic bundles of matrices whose spectral properties are investigated by using algebraic methods and perturbation theory /7,8/.