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On every source and in every example, it seems that sympathetic vibrations are quite directly linked to or are the same as resonance.

The classic example of sympathetic vibrations is of two similar tuning forks (same frequency). When one of these tuning forks is set into vibration and brought close to the other at rest, the latter starts to vibrate. Here, air is forced into vibration at the same frequency as itself by the first tuning fork, and these vibrations carry to the second tuning fork which vibrates readily (resonance).

My textbook, however, defines sympathetic vibrations as -

The amplitude of forced vibrations of a body does not remain constant due to the presence of damping forces of the surrounding medium. However, it is possible to keep the amplitude of vibrations constant by applying an external periodic force such that the external periodic force compensates for the loss of energy in each vibration due to the damping forces. The vibrations of the body are then called sympathetic vibrations.

I am aware that resonance means that bodies having natural frequencies or higher harmonics similar to the frequency of external periodic forces acting on them will vibrate readily with heightened amplitude.

If I'm understanding correctly:

Vibrations can carry from an external periodic force to a body even if the body does not resonate with the external force, the amplitude in such a case is small(compared to resonance) and the body vibrates with the frequency of the driving force and not its natural frequency.

Even in such a case - It is possible to keep the amplitude of vibrations of the body constant by applying an external periodic force which does not resonate with the body such that the external periodic force compensates for the loss of energy in each vibration due to the damping forces.

So if that can be done without resonance being involved, why are sympathetic vibrations referred to with examples of resonance everywhere? Is it that the textbook I'm referring to is entirely wrong or not specific? An intuitive answer would be highly appreciated.

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    $\begingroup$ Related: physics.stackexchange.com/a/684915/226902. Wiki: en.wikipedia.org/wiki/Sympathetic_resonance "Sympathetic resonance is an example of injection locking occurring between coupled oscillators". Injection locking: en.wikipedia.org/wiki/Injection_locking . More links on resonance here: physics.stackexchange.com/a/750711/226902 About " sympathetic vibrations/resonance", see also: physics.stackexchange.com/q/366371/226902 $\endgroup$
    – Quillo
    Commented Feb 20, 2023 at 14:33
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    $\begingroup$ @Quillo thanks for your response. It seems as though all of these links are implying that sympathetic vibrations occur when there’s a harmonic likeliness between a body and an external force i.e. resonance. So is it that the circumstance mentioned in textbook of an external periodic force compensating for loss of energy due to damping while maintaining same amplitude is possible only in an injection locking or resonance related scenario? Or is it that the book is entirely wrong in mentioning all that in reference to sympathetic vibrations? $\endgroup$
    – Kayen Jain
    Commented Feb 20, 2023 at 15:31
  • $\begingroup$ I have the same feeling. The links do not really contain a full answer, but, at least, narrow down the question to exactly what you describe in your comment. I have no clear answer now, but if I'll find something interesting I'll let you know! $\endgroup$
    – Quillo
    Commented Feb 20, 2023 at 16:47

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I think the distinction is that the driving source may be vibrating itself, or it may be simply providing a periodic force. One or the other is essential if the vibration is to be sustained despite damping.

The classic example is pushing a kid on a playground swing. If you push every time at the same point in the swing, small pushes can make the kid swing high. But your pushing is not a vibration, even though it is a periodic force. So I would not call that sympathetic vibration.

However, if you were to place a (very large) loudspeaker behind the kid and play a sine wive through it at the right (very low) frequency, you could conceivably get the kid to swing via sympathetic vibration.

Both of the phenomena are resonance, but only one is sympathetic vibration.

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  • $\begingroup$ Thanks. Got it. Resonance is only sympathetic vibrations when the driving force is coming from a vibrating body, and not just a force periodically acting with a specific magnitude. What exactly qualifies as a periodic force though? Does it have to act with the same direction and magnitude for specific duration once every time period (T) or can it also be of fluctuating magnitude and direction like the force exerted by the sine wave of the loudspeaker on the kid? If yes, is restoring force for a mass attached to a spring in SHM also periodic? (force is proportional to displacement from mean) $\endgroup$
    – Kayen Jain
    Commented Feb 20, 2023 at 20:15
  • $\begingroup$ Anything that repeats on a regular time frame is periodic. But not every periodic motion is a vibration. Incidentally, when you push a kid on a swing, your force doesn't need to have the same magnitude every time, and so it doesn't have to be strictly periodic. It doesn't even have to occur every time. For example, you could push the kid every 2nd or 3rd time and still develop large resonance. $\endgroup$
    – Rich006
    Commented Feb 21, 2023 at 16:15

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