Question

The intensity of the sound from a certain source is measured at two points along a line from the source. The points are separated by 11.0 m, the sound level is 66.40 dB at the first point and 55.74 dB at the second point. How far is the source from the first point?

Answer 1

The source is at a distance of 4.56 m from the first point.

Solution:

As per the question:

Separation distance between the points, d = 11.0 m

Sound level at the first point, L = 66.40 dB

Sound level at the second point, L'= 55.74 dB

Now,

`L = 10log_(10)(I)/(I_(o))`

`I = I_(o)10^{(L)/(10)} = I_(o)10^(0.1L) = 10^(- 12)* 10^(0.1* 66.40) = 10^(- 5.36)`

`L' = 10log_(10)(I')/(I_(o))`

`I' = I_(o)10^{(L')/(10)} = 10^(- 12)* 10^(0.1* 55.74) = 10^(- 6.426)`

where

`I_(o) = 10^(- 12) W/m^(2)`

I = Intensity of sound

Now,

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

Similarly,

`I' = (P)/(4\pi (R + 11.0)^(2))`

Now,

`(I)/(I') = ((R + 11.0)^(2))/(R^(2))`

`(10^(- 5.36))/(10^(- 6.426)) = ((R + 11.0)^(2))/(R^(2))`

`R ^(2) + 22R + 121 = 11.64R^(2)}`

`10.64R ^(2) - 22R - 121 = 0`

Solving the above quadratic eqn, we get:

R = 4.56 m