A Fresh Look at Longitudinal Standing Waves on a

A Fresh Look at Longitudinal Standing Waves on a Spring Casey Rutherford Citation: Phys. Teach. 51, 22 (2013); doi: 10.1119/1.4772032 View online: http://dx.doi.org/10.1119/1.4772032 View Table of Contents: http://tpt.aapt.org/resource/1/PHTEAH/v51/i1 Published by the American Association of Physics Teachers

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A Fresh Look at Longitudinal Standing Waves on a Spring Casey Rutherford, Shakopee High School, Shakopee, MN

T

ransverse standing waves produced on a string, as shown in Fig. 1, are a common demonstration of standing wave patterns that have nodes at both ends. Longitudinal standing waves can be produced on a helical spring that is mounted vertically and attached to a speaker, as shown in Fig. 2, and used to produce both node-node (NN) and node-antinode (NA) standing waves. The resonant frequencies of the two standing wave patterns are related with theory that is accessible to students in algebra-based introductory physics courses, and actual measurements show good agreement with theoretical predictions. A helical spring can serve as a visual and mathematical analogy for open-open and open-closed pipes that have a pressure antinode at closed ends and a pressure node at open ends.1 The nodes on a spring standing wave are visible as a single stationary coil, as shown in Fig. 3. The stationary na-

Fig. 1. Transverse standing waves on a string produced by a speaker and a function generator.

ture of the nodes can be demonstrated by slipping a strip of paper into the node, as seen in Fig. 4. The same paper when pushed against an antinode will vibrate significantly.2 The derivation of the harmonics of a longitudinally vibrating spring is similar to the standard derivation for soundproducing pipes found in introductory physics textbooks.3 The primary difference is that while the speed of sound waves is constant for a constant temperature, longitudinal waves produced in helical springs have a speed v given by4 (1) where L is the vibrating length of the spring, k is the spring constant, and ms is the mass of the spring. NN standing waves in any medium form integer harmonics with the nth frequency given by (2) Combining these two equations yields (3) such that the fundamental f1 is5 (4) and subsequent harmonic frequencies fn given by fn = nf1. (5) For NA standing waves only the odd harmonics are produced with frequencies given by

Fig. 2. Apparatus for demonstrating longitudinal standing waves in a spring. Nodes are barely visible at this distance and are labeled N.

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Fig. 3. A single node on the spring, marked N.

The Physics Teacher ◆ Vol. 51, January 2013

Fig. 4. A slip of paper may be inserted to validate a node.

DOI: 10.1119/1.4772032


Fig. 6. Part of a node-antinode pattern produced with transverse waves on a string, where the speaker acts as the antinode (A) and a node (N) is produced nearby. Table I. Predictions and observed resonant frequencies for a helical spring.

Node-Node (NN) Standing Waves Fig. 5. The end node of a node-node standing wave is visible slightly above the bottom of the spring.

(6) where the prime () will be used to indicate the equation is for the node-antinode pattern. Combining this with Eq. (1) yields with a fundamental of

(7)

and harmonics

(8)

(9)

or

(10) Recall that in both Eqs. (9) and (10), n must be odd. While Eqs. (5) and (9) are usually covered in introductory physics courses for sound waves in open-open and open-closed pipes, the relationship of the two patterns, as shown in Eq. (10), is not. This could be because of the lack of examples where both patterns emerge in the same situation. It is interesting to note that both derivations reveal a lack of dependence on the length L of the spring for the frequency of the harmonics. Simply adjusting the stretched length of the spring verifies this result.5 Data were collected using a spring with an experimentally determined6 spring constant of 1.14 N/m and mass 11.6 g, driven by a Vernier Power Amplifier Accessory Speaker powered by a Vernier Power Amplifier.7 Table I shows values of the predicted and measured harmonic frequencies for both the NN and NA standing waves. The predicted and measured frequencies of the NA waves agree within the uncertainty of the equipment.8 For the NN waves, however, they do not. A similar systematic pattern

Node-Antinode (NA) Standing Waves

nth harmonic

Predicted resonant frequency (Hz)

Observed resonant frequency (Hz)

Predicted resonant frequency (Hz)

Observed resonant frequency (Hz)

1

4.96

5.125a

2.48

not observableb

2

9.92

10.25

3

14.9

15.25

7.44

7.5

12.4

12.5

17.4

17.5

4

19.8

20.25

5

24.8

25.25

6

29.8

31.0

7

34.7

36.0

a.

The Labquest function generator increases in steps of 0.125 Hz from 0-10 Hz, 0.25 Hz from 10-30 Hz, and by 0.5 Hz for the values tested thereafter. b. Harmonics are identified by the presence of nodes, and thus the fundamental frequency for the NA pattern is difficult to observe due to the lack of nodes.

was noticed in a previous analysis of the node-node standing waves9 that was attributed to subjectivity in measurement. Despite this, the accuracy of the predictions for the NA waves and the fact that the lowest nodes for the NN waves are actually produced slightly above the bottom of the spring (Fig. 5) indicate the possibility that some other factor contributes to these slight discrepancies. A preliminary experiment testing the effect of changing the effective length of the spring in Eq. (2) to match the observed bottom node location did not yield results that match the actual resonant frequencies. Further investigation into the nature of this effect is, however, beyond the scope of this paper. It is also worth mentioning the same relationships in Eqs. (5), (9), and (10) can be found for transverse waves on a string, though the effect is less pronounced due to the small amplitude of the antinodes. Figure 6 shows a node near the speaker, with the speaker itself acting as the antinode. Once the fundamental frequency of the standard NN pattern is produced, the fundamental



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of the NA pattern is known, and subsequent harmonics for both patterns emerge that follow Eqs. (5) and (9).

Conclusions Demonstrating longitudinal standing waves on a spring visually enhances teaching of standing waves in addition to revealing the relationship between the harmonics of the two standing wave types. The relationship between open-open pipes and open-closed pipes becomes more apparent, with further benefit from the accessibility of student comparisons between theory and measured values. References 1.

Standing waves in pipes can be confusing for students due to terminology. Though an open end of a pipe is a pressure node, it is an air-displacement antinode. Pressure is considered here due to similarity to the spring system, as commonly analyzed open-open pipes are considered a node-node system in terms of pressure. 2. A video depiction of this process and more is available at goo. gl/axso1. 3. For examples see D.C. Giancoli, Physics, 6th ed. (Pearson Prentice Hall, Upper Saddle River, NJ, 2005), pp. 331-334 or R.A. Serway, College Physics, 7th ed. (Thomson Learning, Belmont, CA, 2006), pp. 480-481.

4. Elisha Huggins, “Speed of wave pulses in Hooke’s law media,” Phys. Teach. 41, 142–146 (March 2008). 5. P. J. Ouseph and Thomas Poothackanal, “Longitudinal and transverse waves in a spring,” Phys. Teach. 32, 285–286 (May 1994). 6. The spring was attached to a stationary force detector and pulled toward a reversed motion detector. The slope of the resulting position-versus-time graph yields the spring constant. 7. Available from Vernier Software & Technology, www.vernier. com. 8. The overriding uncertainty is that the built-in function generator on a Vernier Labquest changes frequency in steps of 0.125 Hz from 0-10 Hz, steps of 0.25 Hz from 10-30 Hz, and steps of 0.5 Hz for the values tested thereafter. 9. Richard A.Young, “Longitudinal standing waves on a vertically suspended slinky,” Am. J. Phys. 61, 353–360 (April 1993). Casey Rutherford has been teaching physics and math for nine years and also works with teachers to effectively integrate technology into their classes. He has a BA in physics and mathematics from St. Olaf College and an MEd in science education from the University of Minnesota. He currently teaches general level physics as well as Introduction to College Physics, an algebra-based physics course articulated through the University of Minnesota such that students earn both high school and college credit. Shakopee High School, Shakopee, MN; [email protected]. mn.us; www.learningandphysics.wordpress.com

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