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Per wikipedia: natural frequency, also known as eigenfrequency, is the frequency at which a system tends to oscillate in the absence of any driving force.

Let's take a wine glass as an example.

The wine glass sits on a table, it is not visibly moving. But since natural frequency exists, therefore it is oscillate on some minute scale. Where does the oscillation come from?

Is it from the electrons and protons that are whizzing inside of the glass and getting excited due to stochastic heat? Is it because protons are hitting the surface of the glass? Or is it oscillating due to the minor tremor in the earth? Or perhaps the minute attractive force exerted upon it by all other objects?

In each case, it is clear that there is a driving force: heat, proton's impact force, tremor. So clearly natural frequency must be associated with a type of oscillation that's not any one of the above.

So where does the oscillation come from instead? It seems that given any source of this oscillation, you can identify a driving force. Hence such driving force is never absent, hence there is no natural frequency.

Are all objects naturally oscillatory and that natural frequency is like a fundamental aspect of an object, such as its 'mass'? If so, can we identify this natural frequency in some way? E.g., I want to know my body's natural frequency at this moment.

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    $\begingroup$ I think what is missing in the link is how enough energy is supplied for the vibrations. For a pendulum one has to spend energy to lift it, and then let it go, for the natural frequency to manifest. For the glass of wine, a sound? may provide enough energy for the vibration to start, or a tap. $\endgroup$
    – anna v
    Commented Sep 30, 2023 at 6:58
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    $\begingroup$ Kick the wine glass with your finger, and it will ring for several seconds even though your finger was only in contact with the glass for a moment. The wine glass is therefore oscillating in the absence of the driving force (your finger). That is a natural frequency oscillation. $\endgroup$
    – Jakob KS
    Commented Sep 30, 2023 at 7:08
  • $\begingroup$ @JakobKS The frequency is diminishing down to zero in this example. Hence we cannot say there is a natural frequency. There are infinite amount of frequencies from the moment the impulsive force impinges upon the glass till it is visibly not vibrating. But even then, it is still vibrating due to all other minute forces acting upon it (a tiny tremor deep underneath the crust). Even with all this, this concept is still kind of poorly defined. A gust of wind blowing on the glass due to someone walking by, by extension, could be thought of as a finger hitting the glass. $\endgroup$
    – Fraïssé
    Commented Sep 30, 2023 at 7:23
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    $\begingroup$ @Fraïssé "There are infinite amount of frequencies from the moment the impulsive force impinges upon the glass till it is visibly not vibrating" . No, there are not. The impulse might be modeled with a continuum large number of frequencies, but the glass responds to its specific frequency(s) (the tone might be around a wavepacket ( hyperphysics.phy-astr.gsu.edu/hbase/Waves/wpack.html ) $\endgroup$
    – anna v
    Commented Sep 30, 2023 at 7:54
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    $\begingroup$ In the real world you might hear some change in tone because different harmonics of the natural frequency decay at different rates, and a real wine glass might have multiple natural frequencies (which decay at different rates). $\endgroup$
    – The Photon
    Commented Sep 30, 2023 at 16:06

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The Wikipedia definition of natural frequency is a poor one, since (as you point out) it implies that a system can only oscillate at its natural frequency if there is no driving force. This is incorrect - if the driving force is periodic and its frequency is equal to the natural frequency then the system will oscillate strongly at its natural frequency - this is called "resonance".

A better definition (taken from here) is "The frequency or frequencies at which an object tends to vibrate when hit, struck, plucked, strummed or somehow disturbed".

Are all objects naturally oscillatory and that natural frequency is like a fundamental aspect of an object, such as its 'mass' ?

Yes. Every system will have some natural frequency or a spectrum of natural frequencies, and this is a fundamental attribute of the system.

If so, can we identify this natural frequency in some way ?

For a simple system it may be possible to calculate its natural frequency or frequencies from first principles. For example, the natural frequency of a plucked string depends on its length, its tension and its mass per unit length following relationships known as Mersenne's laws. For more complex systems, natural frequencies can be determined experimentally by subjecting the system to driving forces with a range of frequencies and looking for signs of resonance.

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The problem with an eigen-frequency is that it is exact, and the phase space to hit an exact frequency is 0, but if you do hit it, the response diverges...classically.

In quantum mechanics, the eigen frequency corresponds to a stationary state that never changes and never interacts.

That's obviously not how reality works. IRL, the exact states are approximation, and there is some interaction that broadens the response (in frequency space).

Classically, that is dissipation (see: $Q$-factor).

In quantum mechanics, it is some perturbing interaction. For example: the background electromagnetic field, which couples atomic orbitals (which are no longer exact solutions to Schrodinger's equation...but approximately, good enough). In atomic physics, see: "natural line width" (in: https://en.wikipedia.org/wiki/Spectral_line)

Much of this is covered in the Fokker-Planck flucation dissipation theory (start from: https://en.wikipedia.org/wiki/Fokker–Planck_equation), which is simply too huge a topic to cover here.

A complete answer would contain a large portion of Thorne and Blanford, see 6.8: http://www.pmaweb.caltech.edu/Courses/ph136/yr2012/1206.1.K.pdf

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