Small amplitude transverse waves on taut strings: exploring the significant effects of longitudinal motion on wave energy location and propagation

Rowland, David R.

doi:10.1088/0143-0807/34/2/225

Cited by 8 publications

(6 citation statements)

References 26 publications

(124 reference statements)

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“…Butikov [18] criticized these conclusions, but also stated that the potential energy of the string resulted from the elastic stretching (ELONG argument). Rowland [19] concluded that Morse and Feshbach were correct, stating that the potential energy of deformation defined using the curvature was non-unique, and on that basis rejected the energy density defined using curvature (the CURV argument); the more fundamental problem with rigid body translation was not realized in [19] though.…”

Section: Discussionmentioning

confidence: 99%

Taut String Model: Getting the Right Energy versus Getting the Energy the Right Way

Caamaño-Withall¹,

Krysl²

2016

WJM

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The initial boundary value problem of the transverse vibration of a taut string is a classic that can be found in many vibration and acoustics textbooks. It is often used as the basis for derivations of elementary numerical models, for instance finite element or finite difference schemes. The model of axial vibration of a prismatic elastic bar also serves in this capacity, often times side-by-side with the first model. The stored (potential) energy for these two models is derived in the literature in two distinct ways. We find the potential energy in the taut string model to be derived from a second-order expression of the change of the length of the string. This is very different in nature from the corresponding expression for the elastic bar, which is predictably based on the work of the internal forces. The two models are mathematically equivalent in that the equations of one can be obtained from the equations of the other by substitution of symbols such as the primary variable, the resisting force and the coefficient of the stiffness. The solutions also have equivalent meanings, such as propagation of waves and standing waves of free vibration. Consequently, the analogy between the two models can and should be exploited, which the present paper successfully undertakes. The potential energy of deformation of the string was attributed to the seminal work of Morse and Feshbach of 1953. This book was also the source of a misunderstanding as to the correct expression for the density of the energy of deformation. The present paper strives to settle this question.

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Section: Discussionmentioning

confidence: 99%

Taut String Model: Getting the Right Energy versus Getting the Energy the Right Way

Caamaño-Withall¹,

Krysl²

2016

WJM

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“…The corresponding flow is The potential energy density (7c) has been obtained before using a force times distance argument of an infinitesimal string segment evaluated for a continuous quasistatic transition of the string [4, 8, p. 126]. However, only recently a continuity equation of the form (6) has been stated [9]. In order to elucidate the contributions to the flow, write the wave velocity in the restricted case of a positive direction traveling wave…”

Section: Complementary Fields Conservation Equationmentioning

confidence: 99%

“…The longitudinal displacement of the segments has been neglected in this simple model. The relevance of the longitudinal coupling in more realistic models has been addressed by different authors [1,9,22]. Here, the problem is restricted solely to the gradient in the y transverse direction, F y = −(∇V ) y = − y V .…”

Section: Force Derived From % Pot Potentialmentioning

confidence: 99%

“…Nonetheless, it has also been argued, in particular regarding counter-propagating waves, that the form involving second-order derivatives provides a better description of the potential energy spatial distribution within a mode [8]. The potential energy density ambiguity has triggered the more ample discussion of where the potential energy resides [9]. The energy in mechanical wave phenomena is of great importance for extracting energy for human oriented purposes.…”

Section: Introductionmentioning

confidence: 99%

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Energy density and flow distribution in a vibrating string

Fernández‐Guasti

2020

SN Appl. Sci.

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The kinetic and potential energy of a vibrating string is considered in the first-order approximation of purely transverse small amplitude linear oscillations. The energy continuity equation is obtained for an energy density having a potential that depends on second-order spatial derivatives of the perturbation. The concomitant flow involves two terms − t z and z t that will be shown to correspond to the kinetic and potential energy flows respectively. The string's transverse force is consistent with its derivation from minus the gradient of this potential. In contrast, the widely accepted potential energy of a string depends on the perturbation's first-order derivative squared. However, this expression does not yield the appropriate force required to satisfy a wave equation. The potential energy controversy is thus resolved in favor of the pot = − 1 2 T 2 z potential function. Contrary to what is usually recognized, the potential energy spatial distribution is shown to be uniquely determined. These results have far-reaching consequences pertaining the wave energy distribution in other mechanical systems.

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“…Thus it can be seen that x¢ is of the same order of magnitude as ¢ ( )y 1 2 2 , and so longitudinal displacements play a non-negligible role in determining the local tension in a vibrating string (see also [3]). Note that this conclusion is true regardless of whether the string is vibrating 1 This analysis reveals a problem with using the notation commonly used to describe linear transverse vibrations in a string when one is analysing nonlinear motion: the position of a point in a string is not given by ( ( ) ( ) x y x t z x t , , , , as is a reasonable approximation in a linear analysis, but by…”

Section: On the 3/2 Discrepancy In The Nonlinear Coefficientmentioning

confidence: 99%

Comment on ‘Direct determination of the nonlinear connection between tension and transverse amplitude for a vibrating string’

Rowland

2019

Eur. J. Phys.

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In a recent paper, Pedersen and Knudsen (2017 Eur. J. Phys. 38 045003) investigated nonlinear effects in a harmonically driven copper wire string. Although quite good agreements between theory and experiment were obtained, there were a few unresolved issues that are worth following up on. Using both theoretical analyses and a numerical simulation of a string modelled as a chain of point masses joined by massless Hookean springs, it is shown that the unexpected variation in tension at the driving frequency, in addition to the expected variation at twice the driving frequency, can be largely if not wholly explained by the small amount of sag in the wire due to gravity and the low equilibrium tension used. The collapse of the vertical component of vibration during free decay observed experimentally could not also be definitively ascribed to the asymmetry resulting from sag, so could be due to effects not included in the model. In addition, the reason some authors incorrectly obtain a nonlinear coefficient 3/2 times that used in this paper is identified as being due to their explicitly (and incorrectly) assuming that longitudinal displacements of points in the string can be neglected. Furthermore, differences in the tension-strain formulas seen in the literature, and their consequent impacts on the forms of the nonlinear equations of motion, are identified as being due to using true or engineering stress, or using the equilibrium length rather than the unstretched length of the string as the reference length in a supposed definition of Hooke’s law. Additional minor issues are also addressed.

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Small amplitude transverse waves on taut strings: exploring the significant effects of longitudinal motion on wave energy location and propagation

Cited by 8 publications

References 26 publications

Taut String Model: Getting the Right Energy versus Getting the Energy the Right Way

Taut String Model: Getting the Right Energy versus Getting the Energy the Right Way

Energy density and flow distribution in a vibrating string

Comment on ‘Direct determination of the nonlinear connection between tension and transverse amplitude for a vibrating string’

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