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The propagation of sound waves from their source (e.g. a piano string) are measured at a certain rate of frequency. Most of the physics I have looked into regarding sound frequency treats propagation as constant, yet this doesn't seem to be a satisfactory description.

Let's assume both the source of sound and the receiver are stationary, with the sound source producing the fundamental frequency of the note A4 (440hz). Am I to believe that the 440hz tone propagates perfectly through the air into my ear? That is, the compressions and rarefractions of the air particles are oscillating exactly at the original rate of the sound source.

If there are any variations in frequency, they must be imperceptible. I cannot think of any example where frequency audibly drops over a distance (unlike amplitude). What mechanisms might effect the frequency of a sound wave during propagation? Air pressure variations? Generation of heat energy due friction between air particles? Could phonons and quantum mechanics reveal any inconsistency?

Furthermore, I also very much doubt that sound sources vibrate at a constant frequency rate. Therefore, my second hunch is that there must be some very small variation in frequency over the duration A4 strings on a piano vibrate (even if we round the fundamental vibration to approximately 440hz). Again, what mechanisms might be at work here?


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  • $\begingroup$ I'm afraid this is too many questions at once. Please read e.g. en.wikipedia.org/wiki/Speed_of_sound Sound sources, and musical instruments is a separate topic, yes they might be variations of the frequency generated by musical instruments, for various reasons, depending on the instrument type. $\endgroup$ Commented May 11, 2023 at 18:43
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    $\begingroup$ How is this related to quantum mechanics? $\endgroup$
    – nasu
    Commented May 12, 2023 at 1:25

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Short Answer:

In reality, the received frequency is not the same as the source frequency, but for acoustics it is a very good approximation to assume that they are the same.

Longer Answer:

The concept that the source frequency is the same as the received frequency is associated with the mathematical property known as linearity. The equations of motion for fluids are notoriously nonlinear, and so in general the source frequency would not be the same as the received frequency. However, in a very useful limiting case of very small amplitude waves in a quiet atmosphere (and neglecting all sorts of pathological situations) the equations of motion reduce to a linear set of equations. This means that the received frequency is approximately the same as the source frequency for small waves, and that approximation gets better the smaller the waves.

So, how small are sound waves? The limit is that the velocity of the air pushed by the sound wave, called the particle velocity, is much less than the sound speed. Using the equations of motion we may relate the particle velocity to the sound pressure (what we hear) and we obtain the requirement $$ \frac{p}{\rho c^2} \ll 1, $$ where $p$ is the amplitude of the acoustic pressure wave, $\rho$ is the ambient mass density (1.201 kg/m$^3$ for standard air) and $c$ is the sound speed (343 m/s for standard air). Let us assume that this ratio holds well for $p\le 0.001\rho c^2\approx 141$ Pa. Converting this to decibels gives us about 134 dB, or about the sound of a commercial jet aircraft taking off. A piano is more likely to be around 80 dB, which corresponds to $p/\rho c^2\approx10^{-6}$. So, the linear approximation is extremely good for a piano, and the received frequency should be almost identical to the source frequency.

Now, you also asked what phenomena would affect the frequency through propagation. There are a number of them, and they correspond to different ways to violate the assumptions that led to linear equations. First, is that the sound wave is small amplitude. As the amplitude becomes larger (we are thinking about military aircraft during afterburner, some kinds of biomedical ultrasound, volcanos, and explosions) the speed of sound depends on the amplitude of the wave. In air, the speed of sound increases slightly in the compressions and decreases slightly in the rarefactions. This leads to a distortion of the waveform and the transfer of energy from one frequency to another. Another approach is to consider a receiver that is moving relative to the source, which leads to the doppler effect. You can also have nonlinearities in the geometry, such as the influence of cracks in beams on the vibrations in the beam.

You mentioned molecular/quantum effects. It turns out that these microscopic effects lead to linear macroscopic effects for small amplitudes. The only field I am aware of where I believe that these effects are important is blast physics, and that field is incredibly nonlinear.

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