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Scientific Investigations Report 2009–5142

High-Frequency Normal Mode Propagation in Aluminum Cylinders

By Myung W. Lee and William F. Waite


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Acoustic measurements made using compressional-wave (P-wave) and shear-wave (S-wave) transducers in aluminum cylinders reveal waveform features with high amplitudes and with velocities that depend on the feature's dominant frequency. In a given waveform, high-frequency features generally arrive earlier than low-frequency features, typical for normal mode propagation. To analyze these waveforms, the elastic equation is solved in a cylindrical coordinate system for the high-frequency case in which the acoustic wavelength is small compared to the cylinder geometry, and the surrounding medium is air. Dispersive P- and S-wave normal mode propagations are predicted to exist, but owing to complex interference patterns inside a cylinder, the phase and group velocities are not smooth functions of frequency. To assess the normal mode group velocities and relative amplitudes, approximate dispersion relations are derived using Bessel functions. The utility of the normal mode theory and approximations from a theoretical and experimental standpoint are demonstrated by showing how the sequence of P- and S-wave normal mode arrivals can vary between samples of different size, and how fundamental normal modes can be mistaken for the faster, but significantly smaller amplitude, P- and S-body waves from which P- and S-wave speeds are calculated.

First posted July 2, 2009

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Suggested citation:

Lee, M.W., and Waite, W.F., 2009, High-frequency normal mode propagation in aluminum cylinders: U.S. Geological Survey Scientific Investigations Report 2009–5142, 16 p.




Measuring Waveforms in Aluminum Cylinders


Wave Equation Solution

Phase Velocity

Group Velocity

Normal Mode Amplitude

Amplitude of P-Body Wave

Normal Mode and Body-Wave Amplitudes for P-Waves

S-Wave Amplitude


P-Wave Group Velocity

S-Wave Group Velocity

P-Wave Amplitudes

S-Wave Amplitudes

P-Wave Group Velocity: Dependence on Cylinder Radius



References Cited

Appendix A. Low-Frequency and Large Diameter Approximation

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