David R. Rowland

Parametric resonance and nonlinear string vibrations

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.

Student interpretations of the terms in first-order ordinary differential equations in modelling contexts

A study of first-year undergraduate students′ interpretational difficulties with first-order ordinary differential equations (ODEs) in modelling contexts was conducted using a diagnostic quiz, exam questions and follow-up interviews. These investigations indicate that when thinking about such ODEs, many students muddle thinking about the function that gives the quantity to be determined and the equation for the quantitys rate of change, and at least some seem unaware of the need for unit consistency in the terms of an ODE. It appears that shifting from amount-type thinking to rates-of-change-type thinking is difficult for many students. Suggestions for pedagogical change based on our results are made.

The missing wave momentum mystery

The usual suggestion for the longitudinally propagating momentum carried by a transverse wave on a string is shown to lead to paradoxes. Numerical simulations provide clues for resolving these paradoxes. The usual formula for wave momentum should be changed by a factor of 2 and the involvement of the cogenerated longitudinal waves is shown to be of crucial importance.

The potential energy density in transverse string waves depends critically on longitudinal motion

The question of the correct formula for the potential energy density in transverse waves on a taut string continues to attract attention (e.g. Burko 2010 Eur. J. Phys. 31 L71), and at least three different formulae can be found in the literature, with the classic text by Morse and Feshbach (Methods of Theoretical Physics pp 126–127) stating that the formula is inherently ambiguous. The purpose of this paper is to demonstrate that neither the standard expression nor the alternative proposed by Burko can be considered to be physically consistent, and that to obtain a formula free of physical inconsistencies and which also removes the ambiguity of Morse and Feshbach, the longitudinal motion of elements of the string needs to be taken into account, even though such motion can be neglected when deriving the linear transverse wave equation. Two derivations of the correct formula are sketched, one proceeding from a consideration of the amount of energy required to stretch a small segment of string when longitudinal displacements are considered, and the other from the full wave equation. The limits of the validity of the derived formulae are also discussed in detail.

On claims that general relativity differs from Newtonian physics for self-gravitating dusts in the low velocity, weak field limit

Galaxy rotation curves are generally analyzed theoretically using Newtonian physics; however, two groups of authors have claimed that for self-gravitating dusts, general relativity (GR) makes significantly different predictions to Newtonian physics, even in the weak field, low velocity limit. One group has even gone so far as to claim that nonlinear general relativistic effects can explain flat galactic rotation curves without the need for cold dark matter. These claims seem to contradict the well-known fact that the weak field, low velocity, low pressure correspondence limit of GR is Newtonian gravity, as evidenced by solar system tests. Both groups of authors claim that their conclusions do not contradict this fact, with Cooperstock and Tieu arguing that the reason is that for the solar system, we have test particles orbiting a central gravitating body, whereas for a galaxy, each star is both an orbiting body and a contributor to the net gravitational field, and this supposedly makes a difference due to nonlinear general relativistic effects. Given the significance of these claims for analyses of the flat galactic rotation curve problem, this article compares the predictions of GR and Newtonian gravity for three cases of self-gravitating dusts for which the exact general relativistic solutions are known. These investigations reveal that GR and Newtonian gravity are in excellent agreement in the appropriate limits, thus supporting the conventional use of Newtonian physics to analyze galactic rotation curves. These analyses also reveal some sources of error in the referred to works.

On the value of geometric algebra for spacetime analyses using an investigation of the form of the self-force on an accelerating charged particle as a case study

The ability to treat vectors in classical mechanics and classical electromagnetism as single geometric objects rather than as a set of components facilitates physical understanding and theoretical analysis. To do the same in four-dimensional spacetime calculations requires a generalization of the vector cross product. Geometric algebra provides such a generalization and is much less abstract than exterior forms. It is shown that many results from geometric algebra are useful for spacetime calculations and can be presented as simple extensions of conventional vector algebra. As an example, it is shown that geometric algebra tightly constrains the possible forms of the self-force that an accelerating charged particle experiences and predicts the Lorentz–Abraham–Dirac equation of motion up to a constant of proportionality. Geometric algebra also makes the important physical content of the Lorentz–Abraham–Dirac equation more transparent than does the standard tensor form of this equation, thus allowing a prop...

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)]

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.

Comment on: ‘Peculiarities in the energy transfer by waves on strained strings’ (Phys. Scr. 88 065402)

Contrary to what is conventionally assumed, to determine the location and flow of potential energy in vibrating strings, the effects of longitudinal displacements need to be taken into account, even when these displacements are much smaller than the associated transverse displacements. In a recent paper, Butikov (2013 Phys. Scr. 88 065402) investigated the implications of this fact and found an infinitely long transverse travelling wave with associated longitudinal disturbance where the associated potential energy density apparently travelled along with the transverse wave at the transverse wave speed. This finding contradicts previous results with transverse pulses which show that in the presence of a unidirectional transverse travelling wave, potential energy propagates at the longitudinal not transverse wave speed. This apparent contradiction is reconciled in this paper by considering a finite length of Butikovs infinitely long travelling wave. It is shown that the energy flow found by Butikov is the net effect of potential energy propagating in both directions at the longitudinal wave speed, a fact that is hidden when a wave of infinite length is considered. This conclusion is supported by a general theoretical result. In addition, it is shown that Butikovs claim that the part of the potential energy density proportional to the first spatial derivative of the longitudinal displacement represents a relocation of the pre-existing potential energy which the string has due to the initial stretching required to create a tension in the string is only valid in a limited number of situations. In most of the examples considered, the term represents a relocation of the potential energy of the wave, though in general it can have many different physical interpretations.

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)]

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.

Geodesics without differential equations: general relativistic calculations for introductory modern physics classes

Introductory courses covering modem physics sometimes introduce some elementary ideas from general relativity, though the idea of a geodesic is generally limited to shortest Euclidean length on a curved surface of two spatial dimensions rather than extremal aging in spacetime. It is shown that Epstein charts provide a simple geometric picture of geodesics in one space and one time dimension and that for a hypothetical uniform gravitational field, geodesics are straight lines on a planar diagram. This means that the properties of geodesics in a uniform field can be calculated with only a knowledge of elementary geometry and trigonometry, thus making the calculation of some basic results of general relativity accessible to students even in an algebra-based survey course on physics.