Question

A weather emergency siren is mounted on a tower, 153 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 111 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? Neglect any absorption of sound energy by the air, though in reality such absorption would be significant at long distances.

Answer 1

intensity level in dB = 10*log(I / I0)

where I0 is taken to be 1e-12 W/m² (the Threshold of Hearing).

a) 111 = 10*log(I/1e-12) → for I in W/m²

11.1 = log(I/1e-12)

1.26e11 = I / 1e-12

I = 0.126 W/m²

intensity = power / area

and for sound radiated uniformly in all directions, the area is 4πr², so

I = P / 4πr²

P = I * 4πr² = 0126W/m² * 4π * (153m)² = 3.70e4 W

b) At what range does this power reach only the TOH?

3.7e4 W = 1e-12W/m² * 4π * R²

R = 5.43e7 m