There are two alternative approaches. The wavelength, and thus the wave number kZ, can be considered as a parameter. Then,
ω can be computed by an eigenfrequency analysis for the 2D cross section with all three displacement components taken as dependent variables. As a result, one obtains
Alternatively, the frequency f (and thus
ω) can be taken as a parameter. Then, the solution can be computed via eigenvalue analysis with respect to the wave number
kZ using the 2D cross section geometry. Hence,
where m is a
circumferential mode number (or
azimuthal mode number) that can only have integer values to obey the axially symmetric nature of the corresponding 3D problem. Thus,
The displacement vector u1 can have nonzero values in all three components, which are functions of the radial and axial positions. For a given circumferential mode number
m, it can be found using an eigenfrequency analysis in a 2D axially symmetric geometry. Hence,