Axially Symmetric Waves in Hollow, Elastic Rods. Part II

In Part I of this study [‐H.D. McNiven, A. H. Shah, and J. L. Sackman, J. Acoust. Soc. Am. 40, 784–792 (1966)], an approximate theory was developed that governs the relationship between the frequency and the propagation constant for axisymmetric waves in hollow rods, and comparison was made for the lowest three branches contained in the theory with the comparable branches from the exact theory when the propagation constant is real. In this paper, the relationship is explored when the propagation constant is purely imaginary and when it is complex. Frequency spectra are formed from the roots of the frequency equation of the approximate theory, and these roots form a complex branch and a loop on the imaginary plane connecting the second and third spectral lines on the real plane. The imaginary loop and the complex branch are compared to the comparable branches from the exact, three‐dimensional theory. Some general properties of all of the complex branches that appear in the exact theory are established.