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(The question has its origin because I asked myself in how far frequencies outside our perception can harm our hearing.)

First of all, the energy of a mechanical wave (in this case, the sound wave, which stimulates periodic movements of a gas) is proportional to both amplitude and frequency. Often, I read that it is written that energy depends only on amplitude, but when I have two waves with the same amplitude, and one has a higher frequency than the other, it apparently carries more energy. It has a shorter wavelength and moves the molecules much faster over the same distance. Since $E = F * s$, a wave with a higher frequency must carry a greater force (and therefore energy) to overcome the inertia of the molecules more quickly.

I'm not entirely familiar with how the ear works, but I do know that it has three sections. Waves enter through the outer ear, continue to vibrate the eardrum with their frequency, the middle ear amplifies this, and then somehow it goes into the inner ear, where there's a fluid-filled cochlea with hair cells. Certain frequencies can only reach specific spots due to the cochlea's geometric structure, allowing differentiation of pitch. Specific hair cells vibrate more strongly for certain frequencies, opening ion channels and sending signals to the brain, or something like that.

However, what I'm wondering now is whether it's really about the geometric structure or the fact that the hairs are of different lengths, thus having different natural frequencies, and frequencies that match their natural frequencies simply stimulate the right hairs through resonance? If that's the case, one can consider it similar to a forced oscillation, for example, when ultrasound reaches the hairs, we're far from the resonance case, so the hairs vibrate at ultrasound frequencies, but the displacement is extremely small, is that the reason they don't transmit signals?

How can I understand this? And if ultrasound stimulates the hairs to tiny vibrations, where does the rest of the ultrasound wave energy go? Do the hairs heat up, or do they simply reflect it (since in non resonance case the energy transfere between an exciter and oscillator isn't really good... try to excite a spring with a weight attached to it with a significantly faster frequency than its natural frequency from above, and you'll see that even when you wiggle it with high amplitude, the oscillator hardly moves from its position but only twitches faintly at your specified frequency, so same frequency but lower amplitude means the energy almost stays in the exciter itself)?

TLTR: Are vibrations in the ear stimulated at frequencies beyond our perception, and what is the reason we don't perceive them even when we increase the amplitude (decibels)? (And I'm referring to vibrations that don't originate through bone conduction, I've also read about this possibility.)

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Hyperphysics has a good overview of human hearing from a physics perspective you may want to check out. It should clear up some misunderstanding you have.

First, the idea that hair cells in the cochlea are like strings that vibrate at their resonance frequencies is interesting and I understand how you would think that is what happens, but it is wrong. Each individual hair cell has 50-150 stereocilia linked together. When longest ones move, they pull on the second-to-longest ones, which pull on the third-to-longest ones, and so on until they all move. This collective movement is what triggers the nerves to fire and signal that a sound was heard. (source)

This leaves the question of what causes the longest sterocilia to move. The base of the hair cells are imbedded in the basilar membrane while the sterocilia are connected to the tectorial membrane. Both membranes are between fluid filled canals. This image shows the arrangement very well: Diagram of the Cochlea The basilar membrane had different physical properties along its length. For this reason, when a sound causes vibrations in the canals the basilar membrane only moves relative to the tectorial membrane at certain spots. The relative movement of these membranes causes the sterocilia to bend, which cause the hair cell to trigger the nerves.

Now that all of that is out of the way, we can finally answer the question: why does turning up the amplitude of ultrasonic sounds (i.e. sounds with frequencies above human hearing range) not trigger hair cells eventually? To quote a 47 page report on how humans detect ultrasonic sounds: "The impedance-matching function of the middle ear, necessary to transmit sounds from the outer to the inner ear, is known to be unable to handle ultrasonic frequencies (Pumphrey, 1950)." Literally all that stuff I just talked about is irrelevant. Ultrasounds are unable to be transmitted by the middle ear no matter how large the amplitude is. The only way ultrasounds could ever trigger the nerves is if they bypass the middle ear, such as by vibrating the bones of the skull.

That report's source does not elaborate on why the middle ear is unable to do this, but I found this paper that focuses on the middle ear as a physical system. From figure 3, we see that ultrasounds, which are above 20kHz, have extremely little gain from the middle ear, meaning they are barely transmitted at all:

Gain vs Frequency of the Inner Ear

Sidenote:
"Often, I read that it is written that energy depends only on amplitude" This is only true for electromagnetic waves. In mechanical waves, the energy is proportional to the square of the amplitude and the square of the frequency.

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    $\begingroup$ The "vibrating string" metaphor would apply to the basilar membrane rather than the individual hairs. $\endgroup$
    – Bryan Krause
    Sep 15, 2023 at 20:06

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