Question

The sound intensity 41 m from a wailing tornado siren is 0.09 W/m2.

(a) What is the intensity at 1020 m? µW/m2

(b) The weakest intensity likely to be heard over background noise is 1 µW/m2. Estimate the maximum distance at which the siren can be heard. km

Answer 1

A siren is a point source. The intensity of sound for a point source drops as the square of the distance

Cal the intensity I, then

I=k/r^2 where k is a constant to be determined (see below) and r is the distance.

Determination of k

We know that at r=41m the sound intensity is 0.09 W/m^2
Then 0.09 W/m^2=k/(41^2)
k=0.09*(41^2) W= 151.29 W

1020 m away we get
I(1020m)= 151.29/(1020^2) W/m^2=1.45x10^-4 W/m^2

We need to convert to muW: 1W=10^6 muW, so we multiply the answer above by 10^6 and get

a) I(1200m)=145 µW/m2

To answer part b we need to know at what distance I=1 µW/m2

We already know that I(r)= 151.29/r^2 W/m^2 and we want r such that I(r)=10^-6 W/m^2

Then 10^-6= 151.29/r^2
Solve for r^2
r^2= 151.29 * 10^6 m^2

r=√(151.29 * 10^6) m = 12.30 km