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Understanding nonlinear effects on wave shapes: Comment on “An experimental analysis of a vibrating guitar string using high-speed photography” [Am. J. Phys. 82
(2), 102–109 (2014)]
2. G. S. S. Murthy and B. S. Ramakrishna, “ Nonlinear character of resonance in stretched strings,” J. Acoust. Soc. Am. 38(3), 461–471 (1965).
7. T. C. Molteno and N. B. Tufillaro, “ An experimental investigation into the dynamics of a string,” Am. J. Phys. 72(9), 1157–1169 (2004).
8. R. J. Hanson, H. K. Macomber, A. C. Morrison, and M. A. Boucher, “ Primarily nonlinear effects observed in a driven asymmetrical vibrating wire,” J. Acoust. Soc. Am. 117(1), 400–412 (2005).
9. 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).
10. 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).
13. G. F. Carrier, “ On the non-linear vibration problem of the elastic string,” Quart. Appl. Math. 3, 157–165 (1945).
14.Material coordinates are fixed at the equilibrium positions of points in the string and so act as labels of these points. Thus, if the point P of the string has the material coordinates [X,0,0], then its longitudinal position at time t is given by the dynamic spatial coordinate , where ξ is the longitudinal displacement of P from its equilibrium position. Elementary treatments of linear transverse waves do not distinguish between the longitudinal material and spatial coordinates because is considered to be negligible. This approximation cannot be made in the derivation of the Kirchhoff-Carrier wave equation, however, and so the fact that the wave equation is a function of the material coordinate X and not the spatial coordinate x is being explicitly made in this Comment.
16. A. H. Nayfeh and D. T. Mook, Nonlinear Oscillations ( John Wiley & Sons, New York, 1979).
20. A. Chaigne and A. Askenfelt, “ Numerical simulations of piano strings. I. A physical model for a struck string using finite difference methods,” J. Acoust. Soc. Am. 95(2), 1112–1118 (1994).
22. D. W. Jordan and P. Smith, Nonlinear Ordinary Differential Equations ( Clarendon Press, Oxford, 1977).
23. C. Gough, “ Musical acoustics,” in the Springer Handbook of Acoustics, edited by T. D. Rossing ( Springer, New York, 2007), p. 563.
24.The parameters L0 and f1 are provided in the original paper, while the diameter of the core and ρ0 were provided by Scott Whitfield by personal correspondence on April 3, 2015 and May 19, 2015, respectively.
25.Institute of Metals and Materials Australasia, IMMA Handbook of Engineering Materials ( Institute of Metals and Materials Australasia, Parkville, Vic, 1997), p. A.1.
26. S. B. Whitfield, personal communications, April 3 and June 5, 2015.
27.The reason for the approximately equal sign in e0 is that for the numerical simulations, Lr/L0 rather than e0 was the defined parameter, with e0= (Lr/L0)−1−1. The reason for using Lr/L0 to be the stretching parameter was to be consistent with the numerical modeling of a string as a chain of point masses developed in Ref. 29.
30. P. M. Morse and K. U. Ingard, Theoretical Acoustics ( McGraw-Hill, New York, 1968).
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In a recent paper, Whitfield and Flesh found unusual bowing behavior in the waveform of a guitar string for large amplitude plucks. This Comment discusses the theory needed to understand this nonlinear
effect, and it is shown that this theory provides reasonably good qualitative agreement with the observed wave form. This theory is interesting because: (i) it allows one to quantify the boundary between linear and nonlinear behavior in terms of key physical parameters; (ii) it reveals the importance of taking into account longitudinal displacements even when they are much smaller than the associated transverse displacements; and (iii) it reveals that dispersion due to tension changes and dispersion due to flexural rigidity have very similar functional forms, which leads to the question of when one effect can be neglected in comparison to the other.
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