The missing wave momentum mystery

Rowland, David R.; Pask, C.

doi:10.1119/1.19272

Cited by 19 publications

(38 citation statements)

References 18 publications

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“…An algorithm for finding solutions to the nonlinear coupled transverse-longitudinal wave equations when the string is driven at one end is derived and exact algebraic solutions for a cos 2 transverse wave pulse coupled to a longitudinal wave are then obtained. These solutions provide algebraic support for the approximations made earlier in this paper and an algebraic demonstration of similar results obtained numerically by simulating a string as a chain of point masses joined by massless Hookean springs [5]. The simplifying approximation that the potential energy is a constant across the whole string is also briefly discussed, as is a possible approach to deriving the previously obtained standing wave solutions to the coupled transverse-longitudinal wave equations.…”

Section: Introductionsupporting

confidence: 73%

“…with ξ T satisfying (25) and ξ L satisfying ρ 0ξ ≈ (τ 0 + SY )ξ [5]. Substituting (26) into (24) verifies this assumption.…”

Section: Travelling Wave Pulse On a Semi-infinite Stringmentioning

confidence: 63%

“…the previously obtained energy continuity equation, (5), is obtained. As the above approach seems much more efficient than the approach outlined in section 2.1, why is this approach not to be preferred?…”

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

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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Section: Introductionsupporting

confidence: 73%

“…with ξ T satisfying (25) and ξ L satisfying ρ 0ξ ≈ (τ 0 + SY )ξ [5]. Substituting (26) into (24) verifies this assumption.…”

Section: Travelling Wave Pulse On a Semi-infinite Stringmentioning

confidence: 63%

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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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“…Let η ∈ sec T 2 0 M be the metric on the cotangent bundle associated to η ∈ sec T 0 2 M . The Clifford bundle of differential forms Cℓ(M, η) is the bundle of algebras, i.e., Cℓ(M, η) = ∪ x∈M Cℓ(T * x M ), 8 We observe that the same problem occur for all linear field theories, and indeed in reference ( [11,12,21]) we have some discussion of the problem for sound (and other elastic) waves.…”

Section: A Clifford Bundlesmentioning

confidence: 95%

Superposition principle and the problem of additivity of the energies and momenta of distinct electromagnetic fields

Notte-Cuello

Rodrigues

2008

Reports on Mathematical Physics

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In this paper we prove in a rigorous mathematical way (using the Clifford bundle formalism) that the energies and momenta of two distinct and arbitrary free Maxwell fields (of finite energies and momenta) that are superposed are additive and thus that there is no incompatibility between the principle of superposition of fields and the principle of energy-momentum conservation, contrary to some recent claims. Our proof depends on a noticeable formula for the energy-momentum densities, namely, Riesz formula ⋆T a = 1 2 ⋆ (F θ aF ), which is valid for any electromagnetic field configuration F satisfying Maxwell equation ∂F = 0.

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“…20. By using a string of 25 point masses ͑this number was chosen to be as small as possible to keep computational demands reasonable while still achieving smooth looking resultant waveforms͒, it was found for the parameter sets shown in Table I that a sine wave of the form sin(k n x) was still the dominant mode excited.…”

Section: ӷ1 ͑41͒mentioning

confidence: 99%

Parametric resonance and nonlinear string vibrations

Rowland

2004

American Journal of Physics

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Periodic changes in the tension of a taut string parametrically excite transverse motion in the string when the driving frequency is close to twice the natural frequency of any transverse normal mode of the string. The literature on this phenomenon is synthesized and extended to include the effects of damping as well as nonlinearity. It is shown that it is nonlinearity rather than damping that limits the growth of a resonantly excited mode, although damping is needed for steady-state oscillations to occur. The validity of the usual approximation that the string tension depends only on time and not on space is checked by modeling a string as point masses joined by massless linear springs. It is found that although this approximation is likely to be violated in practice, the violation does not have a significant effect on the results. The source of the disagreement in the literature for the speed of longitudinal waves in a stretched string is identified.

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The missing wave momentum mystery

Cited by 19 publications

References 18 publications

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

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

Superposition principle and the problem of additivity of the energies and momenta of distinct electromagnetic fields

Parametric resonance and nonlinear string vibrations

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