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1. You know the classical wave equation:
You know the classical wave equation, right?
Suppose I ask you that.
What’s there in it? Just:
or, OK, to make it more general…
Something like that might have passed in your mind, first. When someone says “wave”, people think the water waves, say the ocean waves. Or, they think of the light or sound waves, the interference experiments, even the wave-particle duality. Yet, curiously, when you say “wave equation”, people tend to think of the SHM (simple harmonic motion)—the oscillations of a point-mass, but not the waves in continua.
So, to make it clear, suppose I ask you:
How about the space part?
You might then reply:
Ah, I see what you mean. Pretty simple, too. But now it makes sense to get into a little bit of the complex algebra:
You are likely to continue…
…Remember the Euler identity? The minus sign, because we want to have a wave that travels to the right? Oops, in the positive -direction…
That might be your reply.
I would have to say at this juncture,
the wave equation? I mean, the differential equation. The linear one!
To which, you are likely to retort back
What a ridiculous question! Of course I know it!
OK, it goes like this…
You might then proceed to jot down the following equation in a hurried manner, more or less to get done and be over with my questioning:
Yeah, of course, so you do seem to know it. That’s what I was saying!
You studied the topic as early as in XI or XII standard (if not in your high-school). You had mastered it—right back then. You aced your exams, always. You then went to a great engineering school, and studied waves that were a lot more complicated. Like, may be, the EM waves radiated by a radio antenna, or may be, the vibrations in the machinery and cars, whatever …. You have even mastered the simulation techniques for them. Not just FDM but also FEM, BEM, pseudo-spectral methods, and all that.
Or, may be, you weren’t driven by the lowly commercial considerations. You were really interested in the fundamentals. So, you were interested in physics.
“Fundamentals”, you remember you had said some time ago in a distant past, as if to just once again re-affirm your conviction, all in the silence of your mind. And so, obviously, it would have to be physics! It couldn’t possibly have been chemistry for you! And that’s how, you went ahead and attended a great university. Just to pursue physics.
You calculated a lot of quantum wavefunctions but only while you were in UG years—and only in order to clear those stupid exams. But you already knew that fundamental physics is where your focus really was. Real physics. Mathematical physics. Maths!
That’s why, you zipped past that ridiculously simple stage of those mere wavefunctions. You still remember that way before your formal coursework covered it, you had mastered the Dirac notation, the Heisenberg formulation (where operators are time-dependent, not the stupid wavefunction, you had announced to your stupid class-mates), the Uncertainty Principle (uh!), the Poisson brackets, and all that… You had studied it all completely on your own. Then, you had gone into the relativistic QM as well—the Klein-Gordon equation, Dirac’s equation, Feynman’s path integral formulation… All of that. Also GR. Even QFT… May be you even landed into the string theory right while you still were a high-school or UG student.
… It was long ago that you had left those idiotic wavefunctions and all way behind you. They were best left for others to look after, you just knew. That’s what you had thought, and that’s how you’d come to that conclusion.
2. Will you be able to explain its derivation, now?:
So, whether you are an engineer or a physicist, now, it indeed seems that it’s been a long time since you studied the wave equation. That’s why, if someone now asks you to explain the derivation of the wave equation, you might perhaps narrow your eyes a bit. The reason is, unless you’ve been teaching courses to UG students in the recent times, you may not be able to do it immediately. You may have to take a look at the text-book, perhaps just the Wiki? … The Wiki may not be reliable, but since your grasp has been so solid, it wouldn’t take much to mentally go on correctingt the Wiki even as you are reading through it. …Yes, it might take a little bit of time now, but not much. May be a few minutes? Half an hour at the most? May be. But that’s only because you are going to explain it to someone else…
All the same, you are super-duper-damn confident that given the derivation in the text-books (those XII standard or UG level text-books), you are going to zip through it.
Given a brilliant school-kid, you would obviously be able to explain him the derivation all the way through: each and every step of it, and all the assumptions behind them, and even the mathematical reasonability of all those assumptions, too, in turn. You could easily get it all back right in a moment—or half an hour. … “It’s high-school classical physics, damnit”—that’s what you are likely to exclaim! And, following Feynman, you think you are going to enjoy it too…
You are right, of course. After all, it’s been more than 200 years that the wave equation was first formulated and solved. It has become an inseparable part of the very intuition of the physicist. The great physicists of the day like d’Alembert and Euler were involved in it—in analyzing the wave phenomena, formulating the equation and inventing the solution techniques. Their thought processes were, say, a cut above the rest. They couldn’t overlook something non-trivial, could they? especially Euler? Wasn’t he the one who had first written down that neat identity which goes by his name? one of the most beautiful equations ever?
That’s what you think.
Euler, Lagrange, Hamilton, … , Morse and Feschback, Feynman…
They all said the same thing, and they all couldn’t possibly be careless. And you had fully understood their derivations once upon a time.
So, the derivation is going to be a cake-walk for you now. Each and every part of it.
Well, someone did decide to take a second look at it—the derivation of the classical wave equation. Then, the following is what unfolded.
3. A second look at the derivation. Then the third. Then the fourth. …:
3.1. Lior Burko (University of Alabama at Huntsville, AL, USA) found some problems with the derivation of the transverse wave equation. So, he wrote a paper:
Abstract: “We consider the energy density and energy transfer in small amplitude, one-dimensional waves on a string and find that the common expressions used in textbooks for the introductory physics with calculus course give wrong results for some cases, including standing waves. We discuss the origin of the problem, and how it can be corrected in a way appropriate for the introductory calculus-based physics course.”
In this abstract and all the ones which follow, the emphasis in italicized bold is mine.
3.2. Eugene Butikov (St. Petersburg State University, St. Petersburg, Russia) found issues with Burko’s arguments. So, he wrote a paper (a communication) by way of a reply in the same journal:
Abstract: “In this communication we comment on numerous erroneous statements in a recent letter to this journal by Burko (Eur. J. Phys. 2010 31 L71–7) concerning the energy transferred by transverse waves in a stretched string.”
3.3. C. E. Repetto, A. Roatta, and R. J. Welti (Vibration and Wave Laboratory, Physics Department, Faculty of Exact Sciences, Engineering and Surveying [per Google Translate] (UNR), Rosario Santa Fe, Argentina, and Institute of Physics, Rosario, Argentina) also found issues with Burko’s paper, and so, they too wrote another paper, which appeared in the same issue as Butikov’s:
Abstract: “This work is based on propagation phenomena that conform to the classical wave equation. General expressions of power, the energy conservation equation in continuous media and densities of the kinetic and potential energies are presented. As an example, we study the waves in a string and focused attention on the case of standing waves. The treatment is applicable to introductory science textbooks.”
Though they didn’t mention Burko’s paper in the abstract, the opening line made it clear that this was a comment on the latter.
3.4. Burko, the original author, replied back to both these comments. All the three were published in the same issue of the same journal:
Abstract: “In this reply we respond to comments made by Repetto et al and by Butokov on our letter (Burko 2010 Eur. J. Phys. 31 L71–7), in which we discussed two different results for the elastic potential energy of a string element. One derived from the restoring force on a stretched string element and the other from the work done to bring the string to a certain distorted configuration. We argue that one cannot prefer from fundamental principles the former over the latter (or vice versa), and therefore one may apply either expression to situations in which their use contributes to insight. The two expressions are different by a boundary term which has a clear physical interpretation. For the case of standing waves, we argue that the latter approach has conceptual clarity that may contribute to physical understanding.”
3.5. David Rowland (University of Queensland, Brisbane, Australia) also wrote a reply, which too was published in the same issue of the same journal.
Rowland, David R. (2011) “The potential energy density in transverse string waves depends critically on longitudinal motion,” European Journal of Physics, Volume 31, Number 6. doi: [^]. The author’s pre-print (pre-publication version) is available here, [^].
Abstract: “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.”
3.6. Butikov wrote another paper, a year later, now in Physica Scripta.
Abstract: “The localization of the elastic potential energy associated with transverse and longitudinal waves in a stretched string is discussed. Some misunderstandings about different expressions for the density of potential energy encountered in the literature are clarified. The widespread opinion regarding the inherent ambiguity of the density of elastic potential energy is criticized.”
3.7. Rowland, too, seems to have continued with the topic even after the initial bout of papers. He published another paper in 2013, continuing in the same journal where earlier papers had appeared:
Rowland, David R. (2013) “Small amplitude transverse waves on taut strings: exploring the significant effects of longitudinal motion on wave energy location and propagation,” European Journal of Physics, Volume 34, Number 2. doi: [^] . PDF ePrint is available here [^].
Abstract: “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 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.”
3.8. Caamaño-Withall and Krysl (University of California, San Diego, CA, USA) aimed for settling everything. They brought in a computational engineer’s perspective too:
Caamaño-Withall, Zach and Krysl, Petr (2016) “Taut string model: getting the right energy versus getting the energy the right way,” World Journal of Mechanics, Volume 6, Number 2. doi: [^]. This being an open-access article, the PDF is available right from the doi.
Abstract: “The initial boundary value problem of the transverse vibration of a taut stringisa 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 baralso 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 equivalentin 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 Feshbachof 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.”
4. A standard reference:
Oh, BTW, for a mainstream view prevalent before Burko’s paper, check out a c. 1985 paper by Mathews, Jr. (Georgetown University):
Mathews Jr., W. N. (1985) “Energy in a one‐dimensional small amplitude mechanical wave,” American Journal of Physics, Volume 53, 974. doi: [^].
Abstract: We present a discussion of the energy associated with a one‐dimensional mechanical wave which has a small amplitude but is otherwise general. We consider the kinetic energy only briefly because the standard treatments are adequate. However, our treatment of the potential energy is substantially more general and complete than the treatments which appear in introductory and intermediate undergraduate level physics textbooks. Specifically, we present three different derivations of the potential energy density associated with a one‐dimensional, small amplitude mechanical wave. The first is based on the ‘‘virtual displacement’’ concept. The second is based on the ideas of stress and strain as they are generally used in dealing with the macroscopic elastic properties of matter. The third is based on the principle of conservation of energy, and also leads to an expression for the energy flux of the wave. We also present an intuitive and physical discussion based on the analogy between our system and a spring.
I could not access it, but it was quoted by most (all?) of the papers cited above (which I could).
5. Is it a settled matter, now?:
Have these last few papers settled all the issues that were raised?
Ummm… Why don’t you read the papers and decide by yourself?
6. Why bother?
“But why did you get into all this exasperating thing / stupidity / mess, when all engineers have anyway been using the wave equation to design everything from radios, TVs, Internet router hardware to cars, washing machines, and what not?”
Many of you are likely to phrase your question that way.
My answer is: Well, simply because I ran into these papers while thinking something else about the wave equation and waves. I got puzzled a bit about one very simple and stupid physical idea that had struck me. Far, far simpler than what’s discussed in the above papers. Even just a conceptual analysis of my stupid-simple idea seemed pretty funny to me. So, I’d googled on the related topics just in order to know if any one had thought of along the same lines. Which then led me to the above selection of papers.
What was that idea?
Not very important. Let me mention it some other time. I think there is much more than enough material already in this post!
In the meanwhile, browse through these papers and see if you get all the subtle arguments—all of them being accessible to engineers too, not just to physicists or mathematicians.
Come to think of it, it might be a good idea to post a shortened version of this entry at iMechanica too. … May be a few days later…
In the meanwhile, take care and bye for now…
A song I like:
(Western, Pop): “karma chamelion”
Band: Culture Club
[A Western song that is also hummable! … As always, I couldn’t (and still can’t!) make out words, though today I did browse the lyrics [^] and the Wiki on the song [^]. Back in the 1980s, it used to be quite popular in Pune. Also in IIT Madras. … I like this song for its “hummability” / “musicality” / ”lyricality” / melody or so. Also, the “texture” of the sound—the bass and the rhythm blends really well with the voices and other instrumentals. A pretty neat listen…]