**10. References**


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24 Will-be-set-by-IN-TECH

The support of the Italian "Gruppo Nazionale per la Fisica Matematica" of the "Istituto Nazionale di Alta Matematica" and of the Department of Mechanics and Materials of the

*Dipartimento di Meccanica e Materiali - Università degli Studi "Mediterranea", Via Graziella 1,*

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"Mediterranean" University of Reggio Calabria (Italy) is gratefully acknowledged.

micro-mode does not occurr, in general.

*Località Feo di Vito, I-89122 Reggio Calabria, Italy*

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**Acknowledgements**

**Author details** Pasquale Giovine

**10. References**

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Supplement.

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At the end, for generalized transverse macro-acceleration waves, only the extensional


© 2012 Liu et al., licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

© 2012 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution,

distribution, and reproduction in any medium, provided the original work is properly cited.

and reproduction in any medium, provided the original work is properly cited.

**Dispersion Relations and Modal Patterns of** 

The dispersion relation of a circular cylindrical shell had been a subject of great interest for several decades [1-3]. In some earlier works, numerical procedures were used exclusively in the computation of dispersion relations. Recently, Karczub [4] obtained an analytical expression for the dispersion relations from Flügge shell theory by using a symbolic algebra package, Mathematica. Karczub's main concern was to check the agreement between his analytical results and results obtained previously from numerical methods, and limited to the harmonic orders *n*≧1. The agreement was excellent. The axisymmetric waves (*n*=0) are important in the transmission of longitudinal waves, and are particularly important in acoustics due to their high radiation efficiency. In order to have complete analytical solutions for the shell dispersion

The solutions of the modal patterns for each of the propagating and non-propagating modes are also important and are not discussed in Karczub's paper. These solutions are crucial to determine the vibration of a finite shell under various admissible boundary conditions and arbitrary external forces. The dispersion relations and the associated eigenvectors are also the means by which to construct transfer matrices for vibroacoustic transmission in cylindrical shell structures or pipe-hose systems [5]. The traditional and standard method of finding eigenvectors was given in detail by Leissa [6]; the eigenvector was normalized in such a way that its radial component of the displacement was unity, while the longitudinal and circumferential components were expressed as displacement ratios relative to the radial displacement. In some cases, these ratios could become exceedingly large and the eigenvector thus could deviate significantly from the eigenvector commonly used in mathematical physics. In order to improve this problem for finding the normalized eigenvectors with norms equal to unity, an alternative method by using the built-in eigenvalue problem in any numerical package is proposed. A continuous variation of mode patterns for all kind of wave types in

relations, the solutions for the *n*=0 case are included and discussed in this paper.

**Wave in a Cylindrical Shell** 

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/50477

**1. Introduction** 

Yu-Cheng Liu, Yun-Fan Hwang and Jin-Huang Huang
