Abstract
As in solid rods, it is found that a train of extensional waves striking the traction-free end of a hollow rod will reflect as a pure extensional wave and two secondary waves with complex conjugate wave numbers. The two secondary waves account for an end mode in which the motions are large at the end of the rod and diminish exponentially away from the rod end. The approximate theory used predicts that resonance for the end mode will be excited over a very narrow band of frequencies but that within this band the motions will be very large. The resonant frequencies for finite, hollow rods are found and are displayed by means of spectra in which frequency is plotted against the length of the rod. The spectra show the influences of the lowest radial mode, the lowest axial shear mode and the end mode. As might be expected, the end mode will be excited in a rod of any length. The frequency of the end mode is appreciably lower for a hollow rod than for a solid rod with the same outside diameter.
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