Question

The intensity at distance from a spherically symmetric sound source is 100 W/m2. What is the intensity at five times this distance from the source?

Answer 1

To solve this problem it is necessary to apply the concepts related to intensity as a function of power and area.

Intensity is defined to be the power per unit area carried by a wave. Power is the rate at which energy is transferred by the wave. In equation form, intensity I is

`I = (P)/(A)`

The area of a sphere is given by

`A = 4\pi r^2`

So replacing we have to

`I = (P)/(4\pi r^2)`

Since the question tells us to find the proportion when

`r_1 = 5r_2 \rightarrow (r_2)/(r_1) = (1)/(5)`

So considering the two intensities we have to

`I_1 = (P_1)/(4\pi r_1^2)`

`I_2 = (P_2)/(4\pi r_2^2)`

The ratio between the two intensities would be

`(I_1)/(I_2) = ( (P_1)/(4\pi r_1^2))/((P_2)/(4\pi r_2^2))`

The power does not change therefore it remains constant, which allows summarizing the expression to

`(I_1)/(I_2)=((r_2)/(r_1))^2`

Re-arrange to find
`I_2`

`I_2 = I_1 ((r_1)/(r_2))^2`

`I_2 = 100*((1)/(5))^2`

`I_2 = 4W/m^2`

Therefore the intensity at five times this distance from the source is
`4W/m^2`