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The surprising influence of longitudinal motion in vibrating strings: Comment on “Video-based spatial portraits of a nonlinear vibrating string” [Am. J. Phys. 80
(10), 862–869 (2012)]
1. U. Hassan, Z. Usman, and M. Sabieh Anwar, “ Video-based spatial portraits of a nonlinear vibrating string,” Am. J. Phys. 80(10), 862–869 (2012).
2. S. B. Whitfield and K. B. Flesch, “ An experimental analysis of a vibrating guitar string using high-speed photography,” Am. J. Phys. 82(2), 102–109 (2014).
3. G. S. S. Murthy and B. S. Ramakrishna, “ Nonlinear character of resonance in stretched strings,” J. Acoust. Soc. Am. 38(3), 461–471 (1965).
8. T. C. Molteno and N. B. Tufillaro, “ An experimental investigation into the dynamics of a string,” Am. J. Phys. 72(9), 1157–1169 (2004).
9. Note that in Ref. 1, ω0 rather than ωn is given in Eq. (4) and their definition of ϵ, but this is inconsistent with using in ϵ. To use , k would have to be and their equations would then only apply to the fundamental mode.
10. D. R. Rowland, “ Small amplitude transverse waves on taut strings: Exploring the significant effects of longitudinal motion on wave energy location and propagation,” Eur. J. Phys. 34(2), 225–245 (2013).
12. The result follows from because the equilibrium strain of a metal wire is necessarily very small compared to unity. Here, Lr is the relaxed (i.e., unstretched) length of the string, while L is its length when stretched to obtain the tension T0 but with no waves present.
13. R. Resnick, K. S. Krane, and D. Halliday, Physics, 4th ed. ( Wiley, New York, 1992), Sec. 19-6.
14. Note that it is more usually stated in the literature that , but this formula for tension uses apparent rather than true strain (see p. 137 of Ref. 21, and more straightforwardly, p. 297 of Ref. 22), and hence is only an approximation to Hooke's law (Ref. 23) valid for very small equilibrium strains.
16. was obtained by using to solve Eq. (8) together with the boundary conditions . This solution leads to approximately the fundamental mode of oscillation as given on p. 861 of Ref. 17.
17. P. M. Morse and K. U. Ingard, Theoretical Acoustics ( McGraw-Hill, New York, 1968).
18. When T0 is not negligible with respect to , as can be the case for rubber-like strings, then the second term in Eq. (13) needs to be multiplied by a factor of .
19. That a chain of 40 point masses provides a sufficiently accurate approximation to the continuum result was confirmed in the following ways. After solving the coupled differential equations numerically using mathematica 8's numerical differential equation solver with PrecisionGoal and AccuracyGoal , the position of the center mass in the chain was plotted as a function of time and the FindRoot function was used to find the first and 21st roots, with the difference divided by 10 giving an estimate of the length of a single period. At the low amplitude limit, for , the numerically predicted period for a chain of 40 points differed from the theoretical period of linear oscillations by only 0.007%. At the large amplitude limit, for , the period of oscillation for a chain of 80 points differed from that of a chain of 40 points by only −0.029%, while for a chain of 120 points the difference was only −0.034%. For and a chain of 40 masses, the time from the first to the third roots only differed from 1/10th of the time from the first to the 21st roots by 0.004%, illustrating the stability of the oscillations.
20. A. H. Nayfeh, Perturbation Methods ( Wiley, New York, 1973), p. 60.
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An error in the quoted nonlinear coefficient that is commonly found in the literature is identified. The subtle origin of this error is identified as the neglect of longitudinal displacements of points in the string, which leads to a nonlinear coefficient that is a factor of 3/2 too large. A correct derivation is outlined and numerical simulations verify the correction.
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