Guided wave propagation in slowly-varying elastic bars is of increasing importance in the non-destructive evaluation of structures. This paper investigates the simpler case of an acoustic tube by extending the concept of quasi-modes developed by Gridin~\& Craster (2003, 2004) for two-dimensional geometries to a three-dimensional guide of constant radius and slowly-varying orientation. Quasi-modes are a generalisation of the normal modes of a straight guide to the weakly-curved case. Their properties depend on three lengthscales: bulk wavelength, scale of variation of orientation, and guide radius. Different asymptotic regimes are encountered when the relative sizes of these lengthscales change. Asymptotic expressions for the quasi-modes are derived for two regimes in which the bulk wavelength is comparable to guide radius: one regime applies when the frequency is not close to a cutoff frequency of the corresponding straight guide; the other when it is close to cutoff. A numerical scheme to solve the governing equations directly, which does not depend on the various lengthscales, is also developed to confirm the accuracy of the asymptotic expressions.
Co-authored with Dmitri Gridin and Alex Adamou