The longitudinal¹ normal modes of vibration of rods are similar to the modes seen in pipes open at both ends. A maximum of particle displacement exists at both ends and an integral number (n) of half wavelengths fit into the rod length. The frequencies f_n of the normal modes is given by Eq. (1), where L is the rod length and V is the wave velocity:

Many methods have been used to measure the velocity of these waves. The Kundt's tube method commonly used in student labs will not be discussed here. A simpler related method has been described by Nicklin.² Kluk³ measured velocities in a wide range of materials using a frequency counter and microphone to study sounds produced by impacts. Several earlier methods^4,5 used phonograph cartridges complete with needles to detect vibrations in excited rods. A recent interesting experiment⁶ used wave-induced changes in magnetization produced in an iron rod by striking one end. The travel time, measured as the impulsive wave reflects back and forth, gave the wave velocity for the iron rod. In the method described here, a small magnet is attached to the rod with epoxy, and vibrations are detected using the current induced in a few loops of wire. The experiment is simple and yields very accurate velocity values.

Figure 1 shows the apparatus used in these measurements. In part (a), a small disk-shaped NdFeB magnet⁷ is shown attached to one end of a rod with epoxy. The detector coil (5 to 10 turns) is positioned very close to the magnet. Longitudinal motions produced by vibrations move the magnet relative to the loosely attached coil, producing an induced current. This signal was amplified using the LF411 JFET amplifier shown in Fig. 1(b). Amplifier output was recorded using a LabPro interface and Logger Pro software.⁸ Finite Fourier transforms (ffts) of the data were calculated with Logger Pro to determine the frequencies of normal mode vibrations excited in the rod.

Figure 1.

Figure 2 shows the fft of data produced by tapping an aluminum rod (0.95-cm diameter, 1.045 m long) that I held with two fingers at the center. In this experiment I collected 10,000 data points at 50,000 points/s. Even n modes are seen to make a smaller contribution than the next higher n odd modes. At the rod center, odd n modes have nodes while even n modes have maximum displacement. The pattern of peaks seen in Fig. 2 results because holding the rod at the center damps even n modes more quickly than odd n modes. More technical details of the experiments are discussed in the Appendix.

Figure 2.

Figure 3 plots normal mode frequency versus mode order n for the data seen in Fig. 2. A best-fit straight line is plotted with the data points. This best fit gives V/(2L) = 2411 ± 3 Hz and V = 5040 ± 6 m/s. This agrees well with the tabulated result⁹ 5000 m/s for very thin aluminum rods. At this level of difference, wave velocity depends upon the details of composition and material processing. This method works well for a wide range of materials. For example I obtained V = 1798 ± 4 m/s for a nylon rod 2.54 cm in diameter and 1.002 m long. This compares well with the standard value⁹ 1800 m/s. For materials with much stronger damping, like wood, fewer and broader normal mode peaks are produced, and less accurate velocity values are obtained.

Figure 3.