A weather emergency siren is mounted on a tower. 141 meters above the ground On

Question

A weather emergency siren is mounted on a tower. 141 meters above the ground On one hand, you would like to make the siren very loud so that it will warn as many people as possible On the other hand, safety regulations prohibit you from exceeding an intensity level of 116 dB for workers standing on the ground directly below the siren Assuming that the sound is uniformly emitted, what is the maximum power that the siren can put out? How far away from the base of the tower can a person be and still be able to hear the siren?

Explanation / Answer

Sound intensity level is expressed as,

L = 10log(I/Iref)

Iref = 10^-12 W/m^2 (lowest human hearable sound)

I = Iref*10^(L/10) = 10^-12*10^(11.6) = 0.4 W/m^2

The maximum power can be,

P = I(4*pi)*r^2 = 4**0.4*141^2 = 9.99 x 10^4 W

The farthest distance rmax will be such that,

I = Iref = P/(4*pi*rmax^2)

rmax = (P/(4*pi*Iref)) = (7.95e3/(4*10^-12)) = 2.51 x 10^7 m = 2.51 x 10^4 km

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