a). A weather emergency siren is mounted on a tower, 189 meters above the ground

Question

a). A weather emergency siren is mounted on a tower, 189 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 101 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? (answer in Whatts)

b). How far away from the base of the tower can a person be and still be able to hear the siren? Neglect any absorption of sound energy by the air, though in reality such absorption would be significant at long distances.(answer in meters)

Explanation / Answer

(i) Assuming power levels are measured with respect to the standard threshold of hearing .. P = 10^-12 W/m²
Log P = -12
Let the power at the tower base = P .. .. dB = 10 log(P/P)
dB = 10(log P - log P)
Log P = dB/10 - 12 = 101/10 - 12 .. ..log P = -1.90 .. .. P = 0.0125 W/m²

The original power from the siren has now spread-out uniformly in all directions over a surface with a radius 189m from the siren .. spread over an area A = 4r² (surface area of a sphere)
Siren power .. Ps(W) = 0.0125 W/m² x (4 x 189²)m² .. .. Ps = 5611.04 W

(ii) Assuming at the max distance (R) the power has reduced to the standard threshold value 1.0^-12 W/m²

Ps = 5611.041 W = (1.0^-12 W/m²) x 4R²
R = {5611.041 / (10^-12 x 4)} .. .. R = 2.113x10^7 m

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