We often seem to only hear the lower beats of music far away, whilst the higher frequency sounds seem to diminish more quickly – remaining unheard. I know that sounds with higher frequency have shorter wavelengths, whilst lower frequencies have longer wavelengths. However, this does not necessarily mean that the sounds travel at different speeds. Under normal pressure and density, sounds are known to travel 331 m/s. How does this information allow me to answer whether the Inverse Square Law is applicable in this real world situation or not? Why can we hear low frequencies from far away, but not high frequencies?
I read very briefly on Stoke's Law just then as I was typing this post. Does it have any relevance to this problem?
Much thanks.
Best Answer
Yes this is related to Stoke's law for sound attenuation, which states that a plane wave decreases amplitude exponentially with a factor $\alpha$ given by:
$$\alpha = \frac{2\eta\omega^2}{3\rho V^3}$$
where you can see that the dependence on the frequency squared $\omega$ of the sound will yield a higher coefficient of attenuation for higher frequency sounds comparatively to lower ones.
So between two plane waves with different frequencies $\omega_H=2\omega_L$, the higher one will attenuate four times faster. That is, if they have at a starting point the same amplitude, after a certain characteristic distance $d=\frac{3\rho V^3}{2\eta\omega_L}$ the wave with higher frequency $\omega_L$ will be $e^3 \approx 20$ times weaker.
For air, using $\rho$=1.225e-3$kg/m^3$, $V$=331$m/s$ and $\eta$=1.8e-5$kg/ms$, and taking the A pitch standard as the high frequency $\omega_L$=440Hz and its lower octave as the lower frequency $\omega_L$=220Hz, the characteristic distance will be 76 732 $km$. And the distance at which their relative amplitudes will be in a one half ratio would be 17 729 $km$.