Yes, that is precisely how; well, not to overemphasize the point, but entire fields of study take advantage of that!

It may be noteworthy that nonlinear functions are particularly cheap in mechanics -- consider for example the tuning fork. As the tines move back and forth oppositely, they rotate, effectively the forks get very slightly shorter as they deflect. Therefore, the center of mass is moved in and out at
twice the frequency the tines move at, and this is sensible as axial vibration on the handle.
In short, whack a tuning fork and touch near the base of a fork, to a table: you will hear its characteristic frequency. Then, whack it and touch the handle to the table: you will hear twice that frequency (an octave higher).
Linear systems cannot modify frequencies, the frequency is always perfectly equal throughout the system. (The distribution of frequencies might vary -- filtering is definitely a thing. But
a frequency can't be modified.) This is great news for, say, audio amplifiers, radio transmission and optical systems, and some acoustics; but anywhere that assumption is broken, i.e., nonlinearities become significant, you can get weird effects (harmonic generation, mixing, parametric oscillation, chaotic oscillation..).
So, if the audio analysis you're thinking of, can be done through detection of its physical vibration (sensed with some other transducer, say a high-speed camera?), if it behaves linearly, yes, we can perform that analysis.
We might even be able to do it from a single snapshot, if we know how sound moves through the physical system -- for example, if that beeper is hanging off a wire, and we know the velocity of that sound on that wire, we can see its wiggly path from just a single frame -- assuming of course the wire isn't bent like that all the time, of course! This is not a change of frequency, but of wavelength -- the amount of
distance a wave travels in a given amount of time. The time is always the same (the time period being 1/frequency). A visualization:
https://www.seventransistorlabs.com/Images/Refraction.gif See how each part of the wave goes up and down at the same rate, but the distance travelled varies.
Tim