Question

Cracking Your Knuckles When you "crack" a knuckle, you cause the knuckle cavity to widen rapidly. This, in turn, allows the synovial fluid to expand into a larger volume. If this expansion is sufficiently rapid, it causes a gas bubble to form in the fluid in a process known as cavitation. This is the mechanism responsible for the cracking sound. (Cavitation can also cause pits in rapidly rotating ship's propellers.)

If a "crack" produces a sound with an intensity level of 58 dB at your ear, which is 18 cm from the knuckle, how far from your knuckle can the "crack" be heard? Assume the sound propagates uniformly in all directions, with no reflections or absorption.

127.44m is incorrect.

Answer 1

Threshold of hearing   = 0 dB

dB = 10 log(I/I0)
where I0 = 10^(-12) W/m^2

58 dB has an intensity:

58 = 10 * log(I/10^(-12))
5.8 = log(I/10^(-12))
10^(5.8) = I/10^(-12)
I = 6.31 * 10^-7

this intensity is (6.31 * 10^-7)/(10^-12) = 6.31*10^5 times greater than the threshold intensity

Now distance from knuckle can the "crack" be heard = 18 cm * sqrt(6.31*10^5)
Distance from knuckle can the "crack" be heard = 14298.4 cm
Distance from knuckle can the "crack" be heard = 142.98 m