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Loudness

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Question

The loudness level at a distance R from a long linear source of sound found to be 40dB. At this point, the amplitude of oscillations of air molecules is 0.01 cm. Then find the loudness level(in dB) at a point located at a distance '10R' from the source.

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Solution

Loudness of the sound $L = 10 l o g_{10} \frac{I}{I_{o}}$ where $I = \frac{I_{o}}{r^{2}}$

Thus $L = 20 l o g_{10} \frac{1}{r}$

$⟹$ $L_{2} - L_{1} = 20 l o g_{10} \frac{r_{1}}{r_{2}}$

Given : $r_{1} = R r_{2} = 10 R L_{1} = 40 d B$

$L_{2} - 40 = 20 l o g_{10} \frac{R}{10 R}$

$L_{2} - 40 = 20 (- 1) ⟹ L_{2} = 20 d B$

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