Acoustic quasi-modes in slowly varying cylindrical tubes

• Published in: Quarterly journal of mechanics and applied mathematics Vol. 58; no. 3; pp. 419 - 438
• Main Authors: Adamou, A. T. I., Gridin, D., Craster, R. V.
• Format: Journal Article
• Language: English
• Published: Oxford Oxford University Press 01.08.2005; Oxford Publishing Limited (England)
• ISSN: 0033-5614; 1464-3855
• DOI: 10.1093/qjmam/hbi019

Guided wave propagation in slowly varying elastic bars is of increasing importance in the non-destructive evaluation of structures. This paper develops an asymptotic theory for the simpler case of an acoustic tube by extending the quasi-mode theory developed for two-dimensional geometries by Gridin and Craster to a three-dimensional guide of constant radius and slowly varying orientation. Quasi-modes are a generalization of the normal modes of a straight guide to the weakly curved case. Their properties depend on three lengthscales: the wavelength, the guide radius, and the length over which the guide's orientation typically varies. 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 wavelength and guide radius are of similar size. The first regime holds when the wavelength is small compared to the scale of variation or, equivalently, the frequency is far from the cutoff frequency. The second applies when the wavelength is comparable to the scale of variation, or the frequency is close to the cutoff. A numerical scheme to solve the governing equations directly, which does not depend on the various lengthscales, is also developed to demonstrate the accuracy of the asymptotic expressions.