# A new framework to study the wave motion of flexible strings in the undergraduate classroom using linear elastic theory

### Abstract

When deriving the equation describing the transverse motion of a one-dimensional vibrating elastic string, introductory physics textbooks often assume constant tension and/or small amplitude vibrations. However, these simplifying assumptions are not only unnecessary, but they overlook the elastic nature of the tension and yield an inconsistent derivation of the potential energy density. Because of these assumptions, the derivation of the wave equation and the potential energy density use two different levels of mathematical approximation. In addition, students often get confused as of how a string can carry elastic potential energy if the underlying assumption is that the tension is constant.In this work, we present a mathematically consistent derivation of the wave equation and potential energy density for the vibrating string. Our approach is adequate for physics and engineering introductory courses. We emphasize throughout the derivations the role of elasticity and we propose a simple experiment where students can use wave theory to predict the elastic properties of strings. We also use our framework to illustrate under which conditions longitudinal waves can be neglected for strings that obey Hooke’s law of elasticity. We show that a small transverse amplitude vibration does not immediately justify neglecting longitudinal motion.

### References

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Antman, S. S. (1980). The Equations for Large Vibrations of Strings. American Mathematical Monthly, 87(5), 359-370.

Caamano Withall, Z., & Krysl, P. (2016). Taut String Model: Getting the Right Energy versus Getting the Energy the right Way. World Journal of Mechanics, 6, 24-33. doi:10.4236/wjm.2016.62004

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Garret, S. L. (2020). String Theory. In Understanding Acoustics. Graduate Text In Physics. (pp. 133-178). Springer, Cham. doi:https://doi.org/10.1007/978-3-030-44787-8_3

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hyperphysics. (2022). Energy in a String Wave. Retrieved 24 8, 2021, from http://hyperphysics.phy-astr.gsu.edu/hbase/Waves/powstr.html

Katz, D. M. (2014). Physics for Scientists and Engineers: Foundations and Connections. Cengage Learning.

Keller, J. B. (1959). Large Amplitude Motion of a String. American Journal of Physics, 27(8), 584-586.

Knight, R. (2016). Physics for Scientists and Engineers (4th Edition). Pearson Education (US).

Lee, R. B., & W., G. a. (1988). Partial Differential Equations of Mathematical Physics and Integral Equations. Prentice Hall.

Ling, S. J., Sanny, J., & Moebs, W. (2016). University Physics Volume 1: OpenStax.

Luke, J. C. (1992). The motion of a stretched string with zero relaxed length in a gravitational field. American Journal of Physics, 60(6), 529-532

Myint U., T., & Debnath, L. (2007). Linear Partial Differential Equations for Scientists And Engineers (4th ed.). Birkhauser Boston: Springer.

Ng, C. (2010). Energy in a String Wave. The Physics Teacher, 48(1), 46-47.

O'Reilly, O. M. (2017). Modelling Nonlinear Problems in the Mechanics of Strings and Rods. The role of the balance laws. Berlekey: Springer.

Ryan, M., Neary, E., RInaldo, J., & Olivia, W. (2019). Introductory Physics: Building Models to Describe Our World. doi:10.24908/BPWM9859

Salsa, S. (2008). Partial Differential Equations in Action (From Modelling to Theory). Milano: Springer-Verlag Italia.

Serway, R. A., & Jewett, J. W. (2019). Physics for scientists and engineers with Modern Physics. Cengage.

Shankar, R. (2014). Fundamentals of Physics: Mechanics, Relativity, and Thermodynamics. New Haven and London.

Tipler, P., & Mosca, G. (2007). Physics for Scientists and Engineers (6th Edition). Macmillan Higher Education.

Walker, J., Halliday, D., & Resnick, R. (2018). Fundamentals of Physics, 11th edition. Wiley.

Wolfson, & Richard. (2011). Essential University Physics, 2nd Edition. Addison-Wesley Pearson.

Yong, D. H. (2006). Strings, Chains, and Ropes. Siam Review, 48(4), 771-781.

Young, H. D., & Freedman, R. A. (2016). University Physics: with Modern Physics, 14th Edition. Pearson.

Alonso, M., & Finn, E. (1983). Fundamental university physics vol. 2 : fields and waves. Addison-Wesley.

Antman, S. S. (1980). The Equations for Large Vibrations of Strings. American Mathematical Monthly, 87(5), 359-370.

Caamano Withall, Z., & Krysl, P. (2016). Taut String Model: Getting the Right Energy versus Getting the Energy the right Way. World Journal of Mechanics, 6, 24-33. doi:10.4236/wjm.2016.62004

Clelland, J. N., & Vassiliou, P. J. (2013). Strings Attached: New Light on an old problem. arXiv e-prints, math.AP(arXiv:1302.6672).

Fowler, M. (n.d.). Analyzing Waves on a String. Retrieved 8 24, 2021, from http://galileo.phys.virginia.edu/classes/152.mf1i.spring02/AnalyzingWaves.htm

French, A. (2003). Vibrations and Waves. Norton.

Garret, S. L. (2020). String Theory. In Understanding Acoustics. Graduate Text In Physics. (pp. 133-178). Springer, Cham. doi:https://doi.org/10.1007/978-3-030-44787-8_3

Giancoli, D. C. (2020). Physics for Scientists and Engineers with Modern Physics (5th Edition). Pearson Education (US).

hyperphysics. (2022). Energy in a String Wave. Retrieved 24 8, 2021, from http://hyperphysics.phy-astr.gsu.edu/hbase/Waves/powstr.html

Katz, D. M. (2014). Physics for Scientists and Engineers: Foundations and Connections. Cengage Learning.

Keller, J. B. (1959). Large Amplitude Motion of a String. American Journal of Physics, 27(8), 584-586.

Knight, R. (2016). Physics for Scientists and Engineers (4th Edition). Pearson Education (US).

Lee, R. B., & W., G. a. (1988). Partial Differential Equations of Mathematical Physics and Integral Equations. Prentice Hall.

Ling, S. J., Sanny, J., & Moebs, W. (2016). University Physics Volume 1: OpenStax.

Luke, J. C. (1992). The motion of a stretched string with zero relaxed length in a gravitational field. American Journal of Physics, 60(6), 529-532

Myint U., T., & Debnath, L. (2007). Linear Partial Differential Equations for Scientists And Engineers (4th ed.). Birkhauser Boston: Springer.

Ng, C. (2010). Energy in a String Wave. The Physics Teacher, 48(1), 46-47.

O'Reilly, O. M. (2017). Modelling Nonlinear Problems in the Mechanics of Strings and Rods. The role of the balance laws. Berlekey: Springer.

Ryan, M., Neary, E., RInaldo, J., & Olivia, W. (2019). Introductory Physics: Building Models to Describe Our World. doi:10.24908/BPWM9859

Salsa, S. (2008). Partial Differential Equations in Action (From Modelling to Theory). Milano: Springer-Verlag Italia.

Serway, R. A., & Jewett, J. W. (2019). Physics for scientists and engineers with Modern Physics. Cengage.

Shankar, R. (2014). Fundamentals of Physics: Mechanics, Relativity, and Thermodynamics. New Haven and London.

Tipler, P., & Mosca, G. (2007). Physics for Scientists and Engineers (6th Edition). Macmillan Higher Education.

Walker, J., Halliday, D., & Resnick, R. (2018). Fundamentals of Physics, 11th edition. Wiley.

Wolfson, & Richard. (2011). Essential University Physics, 2nd Edition. Addison-Wesley Pearson.

Yong, D. H. (2006). Strings, Chains, and Ropes. Siam Review, 48(4), 771-781.

Young, H. D., & Freedman, R. A. (2016). University Physics: with Modern Physics, 14th Edition. Pearson.

Published

2022-06-27

How to Cite

ARGUDO, David; OH, Talise.
A new framework to study the wave motion of flexible strings in the undergraduate classroom using linear elastic theory.

**European Journal of Physics Education**, [S.l.], v. 13, n. 2, p. 10-29, june 2022. ISSN 1309-7202. Available at: <http://eu-journal.org/index.php/EJPE/article/view/331>. Date accessed: 30 sep. 2023.
Issue

Section

Classroom Physics

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