The Singing Rod (in the Modern Age)
B. Lasby, J. M. O’Meara, and M. Williams, University of Guelph, Ontario, Canada
This is a classic classroom demonstration of resonance, nodes, anti-nodes, and standing waves that has been described elsewhere.1,2 The modern age twist that we are advocating is the coupling of this classic demo with free (or relatively inexpensive) sound analysis software, thereby allowing for quantitative analysis of resonance while experimenting with a number of important variables.
Figure 1 shows one of the authors and the apparatus. In this photo we are using an aluminum rod that is about 1 m long and approximately 1/8 in in diameter. Holding the rod at the midpoint, slide a damp sponge or piece of resin from the midpoint to the end repeatedly. Make sure that your movements are as smooth and continuous as possible and always in the same direction. Don’t give up! As with any musical instrument, this demonstration will take a little practice to perfect. Try varying the speed and pressure of the hand holding the sponge if you are having difficulty getting the rod to sing. An alternate means of causing the rod to sing is to strike it sharply on the ground on its end while holding it upright at the midpoint. This Sick Science video demonstrates the activity.
There are a number of free sound analysis software packages available for download that can be used on a PC or Mac in conjunction with a microphone3 (e.g., Seventh String Tuner, Frequency Analyzer by Relisoft, and Raven Lite). All of these packages have a frequency analyzer that will allow you to find the fundamental frequency and to determine the orders of the harmonics observed in the resonance of the rod. There is also a very useful iPhone/iPad app4 called Audio Kit that allows for similar experimentation. Figures 2(a) and (b) show the screen shots collected via iPad and the Audio Kit app of the frequencies emitted by a ½-in diameter, 1-m long hollow aluminum rod made to sing. In Fig. 2(a) the rod is held at the halfway point along its length while vibrating. In Fig. 2(b) the rod is held at a point that is ¼ of the length along the rod while it vibrates. From Fig. 2(a), we see the strongest frequency emission is 2520 Hz, with the next most intense peaks observed at three times and five times this frequency. There are also smaller peaks at two times and four times the fundamental frequency. When we hold the rod at ¼ of the length, the even harmonics become stronger, as seen in Fig. 2(b).
These observations compare nicely with what we would predict from theory. The speed of sound in a thin rod of aluminum can be determined by L where Y= 6.931010 Pa is the Young’s modulus and r = 2700 kg/m3 is the bulk density of aluminium,5 giving a value of 5055 m/s. The fundamental resonant mode corresponds to a wavelength that is twice the length of the rod. We therefore expect the fundamental resonant mode to have a wavelength of 2 m in this rod. From the wave equation, we can then calculate the expected fundamental frequency as: f1 = vL/l1 or f1 = (5055 m/s)/(2 m) = 2530 Hz, which is consistent with the observed fundamental frequency [Fig. 2 (a)] of 2520 Hz. We then observe the higher order harmonics as multiples of this fundamental frequency as expected, with the odd harmonics appearing more strongly than the even harmonics. This is also consistent with theory, since the odd harmonics have a node at the midpoint of the rod, whereas the even harmonics have an anti-node at the midpoint where we are holding the rod and therefore preventing vibrations. When we move our fingers to a node for the even harmonics, ¼ of the way along the rod, we can clearly see the f2, f4, and f6 frequencies at approximately 5000 Hz, 10,000 Hz, and 15,000 Hz, respectively. We do not see the odd harmonics in Fig. 2(b) since the ¼-length position is an anti-node for these frequencies. The Audio Kit app also has the ability to view an oscilloscope trace of the sound being emitted by the rod. Please note that the Audio Kit app costs $6.99 from iTunes at the time of writing this article.
Some of the variables that you can experiment with in the classroom include:
- speed and pressure of the hand holding the sponge/resin.
- rods of different lengths, different materials, solid versus hollow, etc.
- holding the rod at different nodal points.
- generating transverse waves in the rod by holding it at a point that is ¼ of its length from one end and then striking the rod in the middle from the side.
- generating both longitudinal and transverse waves at the same time by striking the rod with a solid object close to the end with a slightly diagonal blow.
- demonstrating the Doppler effect by getting the rod to sing and then twirling it above your head in a horizontal circle.
As one example, in a recent exercise we used three different lengths of aluminum rod, all of the same diameter and cross-sectional shape. The longest rod, approximately 2 m in length, had a fundamental frequency of 1210 Hz; the medium length rod, approximately 1.6 m in length, had a fundamental frequency of 1507 Hz; whereas the shortest rod, approximately 1 m long, had a fundamental frequency of 2508 Hz. As expected, with decreasing length, the fundamental resonant frequency increases. Plotting these values as frequency versus inverse rod length, we can determine the speed of sound as twice the resulting slope. From our data, we determine the speed of longitudinal waves in a thin rod of aluminum to be 5.22 3103 m/s, which is within 5% of the calculated value of 5055 m/s, as discussed above.
Students could design many different experiments with such a setup, controlling different variables to explore the behavior of resonance in a rigorous and careful study. The sky is the limit and, with some free (or relatively inexpensive) software, a classic classroom demo becomes a quantitative hands-on learning experience for your students in a modernage makeover!
- http://www.oapt.ca/resources/democorner.html. Look for “Sound Demos with Rods and Tubes” by R. Meisel, originally published in September 2001.
- D. Oliver, J. Underwood, D. Marotta, J. Kane, and M. Scott, “Four free software packages related to the physics of sound,” Phys. Teach. 51, 101–104 (Feb. 2013).
- J. Kuhn and P. Vogt, “Analyzing acoustic phenomena with a smartphone microphone,” Phys. Teach. 51, 118–119 (Feb. 2013).
- E. D. Blodgett, “Comments on ‘Speed of sound in metal pipes: An inexpensive lab,’” Phys. Teach. 47, 31–32 (Jan. 2009).
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