Question

At a distance 20 m from a point source of sound the loudness level is 30 dB. Neglecting the damping, find (a) the loundness at 10 m from the source (b) the distance from the source at which sound is not heard.

Answer 1

The loudness at 10m from the source will be 36dB and the distance from the source  at which sound not heard is 632m.

Explanation:

(A)

As we know the intensity related to distance i.e

Intensity I α $\frac{1}{r^{2} }$.... intensity inversely proportional to r²

Then we have

$\frac{I_{1} }{I_{2} } = \frac{r_{2} ^{2} }{r_{1}^{2} } = \frac{10^{2} }{20^{2} } = \frac{100}{400} = \frac{1}{4}$ ,    given $r_{2} = 10m \\r_{1} = 20m$

Then according to loudness formula

Loudness L = 10log₁₀[$\frac{I}{I_{0} }$]

L₁ = 10log₁₀[$\frac{I_{1} }{I_{0} }$]   and  L₂ =  10log₁₀[$\frac{I_{2} }{I_{0} }$]

Then

L₁ - L₂ = 10log₁₀[$\frac{I_{1} }{I_{0} }$] - 10log₁₀[$\frac{I_{2} }{I_{0} }$]

L₁ - L₂ = 10log₁₀[ $\frac{I_{1} }{I_{2} }$]

L₁ - L₂ = 10log₁₀[$\frac{1}{4}$]

given L₁ = 30

then , 30 - L₂ = 10log₁₀[$\frac{1}{4}$]

By solving we get L₂ = 36dB.

(B) By the formula of loudness in term of distance we have

 L₁ = 10log₁₀[$\frac{r^{2}_{3} }{r_{1}^{2} }$] where r be the distance

then 30 = 10log₁₀[$\frac{r^{2}_{3} }{400}$]

by solving this we get r₃ = 632m