# An interesting problem from the classical mechanics of vibrations

Update on 18 June 2017:
Added three diagrams depicting the mathematical abstraction of the problem; see near the end of the post. Also added one more consideration by way of an additional question.

TL;DR: A very brief version of this post is now posted at iMechanica; see here [^].

How I happened to come to formulate this problem:

As mentioned in my last post, I had started writing down my answers to the conceptual questions from Eisberg and Resnick’s QM text. However, as soon as I began doing that (typing out my answer to the first question from the first chapter), almost predictably, something else happened.

Since it anyway was QM that I was engaged with, somehow, another issue from QM—one which I had thought about a bit some time ago—happened to now just surface up in my mind. And it was an interesting issue. Back then, I had not thought of reaching an answer, and even now, I realized, I had not very satisfactory answer to it, not even in just conceptual terms. Naturally, my mind remained engaged in thinking about this second QM problem for a while.

In trying to come to terms with this QM problem (of my own making, not E&R’s), I now tried to think of some simple model problem from classical mechanics that might capture at least some aspects of this QM issue. Thinking a bit about it, I realized that I had not read anything about this classical mechanics problem during my [very] limited studies of the classical mechanics.

But since it appeared simple enough—heck, it was just classical mechanics—I now tried to reason through it. I thought I “got” it. But then, right the next day, I began doubting my own answer—with very good reasons.

… By now, I had no option but to keep aside the more scholarly task of writing down answers to the E&R questions. The classical problem of my own making had begun becoming all interesting by itself. Naturally, even though I was not procrastinating, I still got away from E&R—I got diverted.

I made some false starts even in the classical version of the problem, but finally, today, I could find some way through it—one which I think is satisfactory. In this post, I am going to share this classical problem. See if it interests you.

Background:

Consider an idealized string tautly held between two fixed end supports that are a distance $L$ apart; see the figure below. The string can be put into a state of vibrations by plucking it. There is a third support exactly at the middle; it can be removed at will.

Assume all the ideal conditions. For instance, assume perfectly rigid and unyielding supports, and a string that is massive (i.e., one which has a lineal mass density; for simplicity, assume this density to be constant over the entire string length) but having zero thickness. The string also is perfectly elastic and having zero internal friction of any sort. Assume that the string is surrounded by the vacuum (so that the vibrational energy of the string does not leak outside the system). Assume the absence of any other forces such as gravitational, electrical, etc. Also assume that the middle support, when it remains touching the string, does not allow any leakage of the vibrational energy from one part of the string to the other. Feel free to make further suitable assumptions as necessary.

The overall system here consists of the string (sans the supports, whose only role is to provide the necessary boundary conditions).

Initially, the string is stationary. Then, with the middle support touching the string, the left-half of the string is made to undergo oscillations by plucking it somewhere in the left-half only, and immediately releasing it. Denote the instant of the release as, say $t_R$. After the lapse of a sufficiently long time period, assume that the left-half of the system settles down into a steady-state standing wave pattern. Given our assumptions, the right-half of the system continues to remain perfectly stationary.

The internal energy of the system at $t_0$ is $0$. Energy is put into the system only once, at $t_R$, and never again. Thus, for all times $t > t_R$, the system behaves as a thermodynamically isolated system.

For simplicity, assume that the standing waves in the left-half form the fundamental mode for that portion (i.e. for the length $L/2$). Denote the frequency of this fundamental mode as $\nu_H$, and its max. amplitude (measured from the central line) as $A_H$.

Next, at some instant of time $t = t_1$, suppose that the support in the middle is suddenly removed, taking care not to disturb the string in any way in the process. That is to say, we  neither put in any more energy in the system nor take out of it, in the process of removing the middle support.

Once the support is thus removed, the waves from the left-half can now travel to the right-half, get reflected from the right end-support, travel all the way to the left end-support, get reflected there, etc. Thus, they will travel back and forth, in both the directions.

Modeled as a two-point BV/IC problem, assume that the system settles down into a steadily repeating pattern of some kind of standing waves.

The question now is:

What would be the pattern of the standing waves formed in the system at a time $t_F \gg t_1$?

The theory suggests that there is no unique answer!:

Since the support in the middle was exactly at the midpoint, removing it has the effect of suddenly doubling the length for the string.

Now, simple maths of the normal modes tells you that the string can vibrate in the fundamental mode for the entire length, which means: the system should show standing waves of the frequency $\nu_F = \nu_H/2$.

However, there also are other, theoretically conceivable, answers.

For instance, it is also possible that the system gets settled into the first higher-harmonic mode. In the very first higher-harmonic mode, it will maintain the same frequency as earlier, i.e., $\nu_F = \nu_H$, but being an isolated system, it has to conserve its energy, and so, in this higher harmonic mode, it must vibrate with a lower max. amplitude $A_F < A_H$. Thermodynamically speaking, since the energy is conserved also in such a mode, it also should certainly be possible.

In fact, you can take the argument further, and say that any one or all of the higher harmonics (potentially an infinity of them) would be possible. After all, the system does not have to maintain a constant frequency or a constant max. amplitude; it only has to maintain the same energy.

OK. That was the idealized model and its maths. Now let’s turn to reality.

Relevant empirical observations show that only a certain answer gets selected:

What do you actually observe in reality for systems that come close enough to the above mentioned idealized description? Let’s take a range of examples to get an idea of what kind of a show the real world puts up….

Consider, say, a violinist’s performance. He can continuously alter the length of the vibrations with his finger, and thereby produce a continuous spectrum of frequencies. However, at any instant, for any given length for the vibrating part, the most dominant of all such frequencies is, actually, only the fundamental mode for that length.

A real violin does not come very close to our idealized example above. A flute is better, because its spectrum happens to be the purest among all musical instruments. What do we mean by a “pure” tone here? It means this: When a flutist plays a certain tone, say the middle “saa” (i.e. the middle “C”), the sound actually produced by the instrument does not significantly carry any higher harmonics. That is to say, when a flutist plays the middle  “saa,” unlike the other musical instruments, the flute does not inadvertently go on to produce also the “saa”s from any of the higher octaves. Its energy remains very strongly concentrated in only a single tone, here, the middle “saa”. Thus, it is said to be a “pure” tone; it is not “contaminated” by any of the higher harmonics. (As to the lower harmonics for a given length, well, they are ruled out because of the basic physics and maths.)

Now, if you take a flute of a variable length (something like a trumpet) and try very suddenly doubling the length of the vibrating air column, you will find that instead of producing a fainter sound of the same middle “saa”, the flute instead produces the next lower “saa”. (If you want, you can try it out more systematically in the laboratory by taking a telescopic assembly of cylinders and a tuning fork.)

Of course, really speaking, despite its pure tones, even the flute does not come close enough to our idealized description above. For instance, notice that in our idealized description, energy is put into the system only once, at $t_R$, and never again. On the other hand, in playing a violin or a flute we are continuously pumping in some energy; the system is also continuously dissipating its energy to its environment via the sound waves produced in the air. A flute, thus, is an open system; it is not an isolated system. Yet, despite the additional complexity introduced because of an open system, and therefore, perhaps, a greater chance of being drawn into higher harmonic(s), in reality, a variable length flute is always observed to “select” only the fundamental harmonic for a given length.

How about an actual guitar? Same thing. In fact, the guitar comes closest to our idealized description. And if you try out plucking the string once and then, after a while, suddenly removing the finger from a fret, you will find that the guitar too “prefers” to immediately settle down rather in the fundamental harmonic for the new length. (Take an electric guitar so that even as the sound turns fainter and still fainter due to damping, you could still easily make out the change in the dominant tone.)

OK. Enough of empirical observations. Back to the connection of these observations with the theory of physics (and maths).

The question:

Thermodynamically, an infinity of tones are perfectly possible. Maths tells you that these infinity of tones are nothing but the set of the higher harmonics (and nothing else). Yet, in reality, only one tone gets selected. What gives?

What is the missing physics which makes the system get settled into one and only one option—indeed an extreme option—out of an infinity of them of which are, energetically speaking, equally possible?

Update on 18 June 2017:

Here is a statement of the problem in certain essential mathematical terms. See the three figures below:

The initial state of the string is what the following figure (Case 1) depicts. The max. amplitude is 1.0. Though the quiescent part looks longer than half the length, it’s just an illusion of perception.:

Case 1: Fundamental tone for the half length, extended over a half-length

The following figure (Case 2) is the mathematical idealization of the state in which an actual guitar string tends to settle in. Note that the max. amplitude is greater (it’s $\sqrt{2}$) so  as to have the energy of this state the same as that of Case 1.

Case 2: Fundamental tone for the full length, extended over the full length

The following figure (Case 3) depicts what mathematically is also possible for the final system state. However, it’s not observed with actual guitars. Note, here, the frequency is half of that in the Case 1, and the wavelength is doubled. The max. amplitude for this state is less than 1.0 (it’s $\dfrac{1}{\sqrt{2}}$) so as to have this state too carry exactly the same energy as in Case 1.

Case 3: The first overtone for the full length, extended over the full length

Thus, the problem, in short is:

The transition observed in reality is: $T1:$ Case 1 $\rightarrow$ Case 2.

However, the transition $T2:$ Case 1 $\rightarrow$ Case 3 also is possible by the mathematics of standing waves and thermodynamics (or more basically, by that bedrock on which all modern physics rests, viz., the calculus of variations). Yet, it is not observed.

Why does only $T1$ occur? why not $T2$? or even a linear combination of both? That’s the problem, in essence.

While attempting to answer it, also consider this : Can an isolated system like the one depicted in the Case 1 at all undergo a transition of modes?

Enjoy!

Update on 18th June 2017 is over.

That was the classical mechanics problem I said I happened to think of, recently. (And it was the one which took me away from the program of answering the E&R questions.)

Find it interesting? Want to give it a try?

If you do give it a try and if you reach an answer that seems satisfactory to you, then please do drop me a line. We can then cross-check our notes.

And of course, if you find this problem (or something similar) already solved somewhere, then my request to you would be stronger: do let me know about the reference!

In the meanwhile, I will try to go back to (or at least towards) completing the task of answering the E&R questions. [I do, however, also plan to post a slightly edited version of this post at iMechanica.]

Update History:

07 June 2017: Published on this blog

8 June 2017, 12:25 PM, IST: Added the figure and the section headings.

8 June 2017, 15:30 hrs, IST: Added the link to the brief version posted at iMechanica.

18 June 2017, 12:10 hrs, IST: Added the diagrams depicting the mathematical abstraction of the problem.

A Song I Like:

(Marathi) “olyaa saanj veli…”
Music: Avinash-Vishwajeet
Singers: Swapnil Bandodkar, Bela Shende
Lyrics: Ashwini Shende

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