Geodesic
High-order accurate equations describing vibrations of thin bars
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 46 (2006) no. 3, pp. 457-472 Cet article a éte moissonné depuis la source Math-Net.Ru

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A method for deriving one-dimensional wave propagation equations in thin inhomogeneous anisotropic bars based on the mathematical homogenization theory for periodic media is used to obtain equations governing the longitudinal and transverse vibrations of a homogeneous circular bar. The equations are derived up to $O(\varepsilon^8)$ terms and take into account variable body forces and surface loads. Here, $\varepsilon$ is the ratio of the bar's typical thickness to the typical wavelength. 
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N. S. Bakhvalov; M. È. Èglit. High-order accurate equations describing vibrations of thin bars. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 46 (2006) no. 3, pp. 457-472. http://geodesic.mathdoc.fr/item/ZVMMF_2006_46_3_a9/

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