# Variation of frequency in propagation of sound

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?

• 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. Commented May 11, 2023 at 18:43
• How is this related to quantum mechanics?
– nasu
Commented May 12, 2023 at 1:25

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.