A siren mounted on the top of towers produces a noise of 120 dB at a distance of 2 meters. Treating the siren as a point particle (ignoring reflection and absorption). What is the sound intensity by an observer 12 m away?
The intensity is 1/36 the original intensity by the 1/r^2 dependence of sound in 3 dimensional space.
The reduction of 120dB by 1/36 becomes 104dB.
Sound operates as a longitudinal wave in 3 dimensions. It propagates outward in bubbles or shells with a surface area of #4pir^2". As you get further away from the source, you receive a smaller proportion of that shell on your ear. For example you get twice as far away, your ear stays the same size, but the wave front "shell" of the sound is now "4pi(2r)^2"... so you receive 1/4 the intensity at twice the distance. In this example, the distance is 6 times the original... so the intensity is 1/(6r)^2 or 1/36 the original (120dB). or 10^12W/m^2
... to convert the ratio of intensity in decibels, use:
Sound intensity = 10log(base10)(I/Io)
Not exactly sure someone double check this but I get 104dB