Energy in a one-dimensional small amplitude mechanical wave

Mathews, W. N.

doi:10.1119/1.14405

Cited by 7 publications

(13 citation statements)

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“…The matter of the energy of deformation of the taut string has also been subject to some critical examinations over the years. Matthews [16] derived the CURV equation, and also struggled with the ensuing non-uniqueness; the author did not realize that the expression for the density was wrong. Burko [17] derived the density of energy of deformation to arrive at the two versions of the energy density of [5], and then concluded that both expressions CURV and ELONG for the energy density were correct and that the ambiguity of the energy definitions did not matter.…”

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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“…One approach is to multiply the wave equation by the local string velocity and then use a differential identity to convert the resulting equation into an equation of the form of a continuity equation (e.g. section V of [12]). While such an approach can be generalized to take into account fully three-dimensional motion of elements of a string and not just purely transverse motion (section 5 of [7]), such derivations gloss over the fact that the differential identity one can use to obtain an equation of the right general form is not unique (see subsection 2.2), and so further physical reasoning is needed to determine which identity leads to the correct final equation.…”

Section: Mathematically Simple and Physically Transparent Derivation mentioning

confidence: 99%

“…Before leaving this section though, the following question might arise in readers' minds. Assuming that the potential energy density in a linearly elastic string is uniquely definable and that it is given by (8), one might wonder if the approach indicated by (12), of the three given above, does reliably give the right potential energy density. To see that it does not do so in all cases, consider purely longitudinal motion on a linearly elastic taut string for which the wave equation (10) simplifies to recalling that ξ (X, t ) represents the longitudinal displacement of a point in the string whose equilibrium x-coordinate is X. Multiplying 17byξ and then using the differential identity,…”

Section: The Problem With Trying To Derive the Energy Continuity Equamentioning

confidence: 99%

“…To understand where the Morse and Feshbach [8] and virtual displacement [12] approaches have gone wrong, note that neither approach directly takes into account the fact that as an element of string moves, in general its orientation and hence length changes. Both approaches only indirectly take this change in orientation into account by virtue of the fact that neighbouring elements in general move slightly different amounts and so strictly speaking the final string configuration these approaches end up with is shown in figure 2 To understand what these alternative approaches are measuring, note that there are two senses in which an element of string can have a potential energy.…”

Section: On Why the Potential Energy Density Must Be Uniquely Definabmentioning

confidence: 99%

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

Rowland

2013

Eur. J. Phys.

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Introductory discussions of energy transport due to transverse waves on taut strings universally assume that the effects of longitudinal motion can be neglected, but this assumption is not even approximately valid unless the string is idealized to have a zero relaxed length, a requirement approximately met by the slinky spring. While making this additional idealization is probably the best approach to take when discussing waves on strings at the introductory level, for intermediate to advanced undergraduate classes in continuum mechanics and general wave phenomena where somewhat more realistic models of strings can be investigated, this paper makes the following contributions. First, various approaches to deriving the general energy continuity equation are critiqued and it is argued that the standard continuum mechanics approach to deriving such equations is the best because it leads to a conceptually clear, relatively simple derivation which provides a unique answer of greatest generality. In addition, a straightforward algorithm for calculating the transverse and longitudinal waves generated when a string is driven at one end is presented and used to investigate a cos 2 transverse pulse. This example illustrates much important physics regarding energy transport in strings and allows the 'attack waves' observed when strings in musical instruments are struck or plucked to be approximately modelled and analysed algebraically. Regarding the ongoing debate as to whether the potential energy density in a string can be uniquely defined, it is shown by coupling an external energy source to a string that a suggested alternative formula for potential energy density requires an unphysical potential energy to be ascribed to the source for overall energy to be conserved and so cannot be considered to be physically valid.

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“…In this situation, the energy density per unit length at a point x is the sum of a kinetic and of an elastic term, 12,20 U͑x,t ͒ϭ 1 2…”

Section: ͑5͒mentioning

confidence: 99%

What happens to energy and momentum when two oppositely-moving wave pulses overlap?

Gauthier

2003

American Journal of Physics

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The overlap of two wave pulses that are moving in opposite directions along the same line in a linear, nondissipative and nondispersive medium is used to discuss the compatibility of wave superposition and of energy-momentum conservation. What happens to the energy and to the momentum when the pulses overlap in such a situation is examined. The treatment is applicable to electromagnetic waves or to small-amplitude linear mechanical waves on an ideal string. It is argued that introductory-to-intermediate textbooks and the pedagogical literature have neglected these questions, particularly the one relating to the conservation of linear momentum.

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Energy in a one-dimensional small amplitude mechanical wave

Cited by 7 publications

References 0 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

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

What happens to energy and momentum when two oppositely-moving wave pulses overlap?

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