Analytical expressions are found for the wavenumbers in an in vacuo, infinite, flexible, orthotropic cylindrical shell using asymptotic methods. These expressions are valid for arbitrary circumferential orders n as they are obtained by treating n as a parameter in the derivation. The Donnell-Mushtari shell theory is used to model the dynamics of the cylindrical shell. Initially, an isotropic cylindrical shell is considered and expressions for all the wavenumbers (bending, near-field bending, longitudinal and torsional) are found. Subsequently, defining a suitable orthotropy parameter, the problem of wave-propagation in an orthotropic shell is posed as a perturbation on the corresponding problem for an isotropic shell. Asymptotic expressions for these wavenumbers in the orthotropic shell are then obtained in the limit of small ?. Wherever necessary, a frequency-scaling approach is used to find elegant expansions in the different frequency regimes following the method of Matched Asymptotic Expansions (MAE). The asymptotic expansions are compared with numerical solutions in each of the cases and the match is found to be good. The main contribution of this work lies in the extension of the existing literature by developing closed-form expressions for wavenumbers with arbitrary circumferential orders n in the case of both, isotropic and orthotropic shells. A secondary contribution is in illustrating a convenient approach to study the problem of wave-propagation in anisotropic shells in general, starting with orthotropy in this case. The advantage of using asymptotic methods is that it is possible to track the continuous transition of the wavenumber as a chosen parameter is varied within the limits of validity of the asymptotic expansion. This adds to the physical understanding of the problem and is not easily possible in a numerical study.