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A ``small'' but interesting riddle from the theory of vibrations

Here is a ``small'' riddle from classical physics which I recently happened to think of, in connection with my studies of QM. See if it interests you.

See the figure below.

 An ideal vibrating string with a removable support at the mid-point

There is an idealized string tautly held between two fixed end-supports that are a distance L apart. The string can be put into a state of vibrations by plucking it. There is a third support exactly at the mid-point; it can be removed at will. When touching the string, the middle support does not permit vibrations to pass through it.

Initially, the middle support is touching the string.

At time t_0, the left-half carries a standing wave pattern in the fundamental normal mode (i.e. it is the fundamental mode for the half part on the left hand-side, i.e., its half-wavelength is L/2). Denote its frequency as \nu_1. At this time, the right-half is perfectly quiscent. Thus, energy is present only in the left-half of the system.

At time t_1, the middle support is suddenly removed. Now, disturbances from any of the two halves can freely propagate into the other half.

Assume that at a time t_F >> t_1, the system reaches a steady-state pattern of standing waves.

The issue of interest is:

What is/are the frequency/frequencies of the standing waves now present over the entire length L?

Mathematically, the fundamental mode for the entire length L as well as any and all of its overtones are possible, provided that their individual amplitudes are such that the law of energy conservation would not get violated.

Practically speaking, however, only the fundamental mode for the total length (L) is observed.

In short:

Thermodynamically, an infinity of tones are perfectly possible. Yet, in reality, only one tone of them gets selected, and that too is always only the fundamental mode (for the new length). What gives?

What precisely is the reason that the system gets settled into one and only one option—indeed an extreme option—out of an infinity of them, all of which are, energetically speaking, equally possible?

Comments are welcome!

A very verbose version of this problem was posted yesterday at my personal blog, here: [^]

PS: If there is a useful reference where this problem already appears, please do drop a line; thanks in advance.

Best,

--Ajit

 

 

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