**4. Conclusion**

98 Wave Processes in Classical and New Solids

**Figure 8.** The normalized eigenvectors of *n*=2 circular harmonic

(c) Branch 3 roots

U

V

W

0 0.5 1 1.5 2 2.5 3 3.5 4

0 0.5 1 1.5 2 2.5 3 3.5 4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

> 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

E igenvector U

U V W

V

W

E igenvector

(a) Branch 1 roots (b) Branch 2 roots

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

E igenvector

E igenvector

0 0.5 1 1.5 2 2.5 3 3.5 4

U

W

V

U V W

U V W

W

0 0.5 1 1.5 2 2.5 3 3.5 4

U V

(d) Branch 4 roots

U V W

> The classic problem of the dispersion of waves in a cylindrical shell had been re-examined with analytical solutions obtained by using a symbolic algebra package. The previous work by Karczub did not include the analytical solutions for the dispersion relations of axisymmetric waves (of circular harmonic order n=0). An axisymmetric wave contains both longitudinal and transverse components; the former are important in the transmission of longitudinal vibration, and the latter are particularly important in acoustics due to their higher acoustic radiation efficiency than the corresponding waves of higher circular harmonic orders (*n*>0). In this way, a complete set of analytical solutions which based on Flügge shell theory is now available for all orders of circular harmonics, *n*=0, 1, 2, …, ∞.

> A considerable effort has been expended on the solutions for the modal patterns of all propagating and non-propagating modes, because a complete set of properly normalized eigenvectors is crucial to the solution of the vibration problem of a finite shell under various admissible boundary conditions. The eigenvectors obtained by the conventional method are not conveniently normalized as are those commonly used in mathematical physics. A new alternative method to find normalized eigenvectors with norms equal to unity has been proposed and discussed.

> Use of analytical solutions has demonstrated the capability of a straightforward continuous tracking on the frequency-dependent changes of the root types and of the corresponding modal patterns for each branch of the dispersion curves associated with a particular analytical root. Therefore, a parallel display of the dispersion curves and of the associated modal patterns used in this paper has provided more insight about the wave phenomena in a cylindrical shell.
