At a distance `r_(0) = 20 m` form an isotropic point source of sound the loudness level is `L_(0) = 30 dB`. Neglicting the damping of the sound wave, find : (a) the loudness level at a distacne `r = 10 m` form the source, (b) the distance form the source at which sound is not heard.

Answer

36.02 dB

Explanation

Intensity of wave at 20 m from point is
`I_(1) = (P)/(4pi(20)^(2))`
Intensity at 10 m from source is
`I_(2) = (P)/(4pi(10^(2)) ) = 4I_(1)`
loudness level of `I_(2)` is
`L_(2) = 10log"((I_(2))/(I_(0)))`
= `10 log"(4I_(1))/(I_(0))`
= =`10log((I_(1))/(I_(0))) + 10 log 4`
= 30 + 6.02
= 36.02 dB
Sound will not be heard when intensity goes below reference level so soddnes becomes 0 dB.
`I_(0) = 10^(-12) W//m^(2)`
given that at 20m distance loudness level is 30 dB
`30 = 10 log" (I_(1))/(I_(0))`
`I_(1) = 10^(-3) w//m^(2)`
Now for point sources we use
`I_(1)x_(1)^(2) = I_(2)x_(2)^(2)`
`10(-9)(20)^(2) = 10^(-12)(x_(2)^(2))`
`x_(2)^(2) = (20)^(2) xx 10^(3)`
`x_(2)^(2) = 400 xx 10^(3)`
`x_(2) = sqrt(4 xx 10^(5) )` = 632.45 m.