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Question
The sound level at a point 5.0 m away from a point source is 40 dB. What will be the level at a point 50 m away from the source?
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Solution
Let
\[\beta_A\] be the sound level at a point 5 m (= r1) away from the point source and
∴\[\beta_A\]= 40 dB
Sound level is given by:
\[\beta_A = 10 \log_{10} \left( \frac{I_A}{I_0} \right) . \]
\[ \Rightarrow \frac{I_A}{I_0} = {10}^\left( \frac{\beta_A}{10} \right) . . . . . \left( 1 \right)\]
\[ \beta_B = 10 \log_{10} \left( \frac{I_B}{I_o} \right)\]
\[ \Rightarrow \frac{I_B}{I_0} = {10}^\left( \frac{\beta_B}{10} \right) . . . . . \left( 2 \right)\]
\[\text { From } \left( 1 \right) \text { and } \left( 2 \right), \text { we get: }\]
\[ \frac{I_A}{I_B} = {10}^\left( \frac{\beta_A - \beta_B}{10} \right) . . . . \left( 3 \right)\]
\[\text { Also }, \]
\[ \frac{I_A}{I_B} = \frac{r_B^2}{r_A^2} = \left( \frac{50}{5} \right)^2 = {10}^2 . . . . . \left( 4 \right)\]
\[\text { From } \left( 3 \right) \text { and } \left( 4 \right), \text{ we get: }\]
\[ {10}^2 = {10}^\left( \frac{\beta_A - \beta_B}{10} \right) \]
\[ \Rightarrow \frac{\beta_A - \beta_B}{10} = 2 \]
\[ \Rightarrow \beta_A - \beta_B = 20\]
\[ \Rightarrow \beta_B = 40 - 20 = 20 dB\]
\[\beta_A = 10 \log_{10} \left( \frac{I_A}{I_0} \right) . \]
\[ \Rightarrow \frac{I_A}{I_0} = {10}^\left( \frac{\beta_A}{10} \right) . . . . . \left( 1 \right)\]
\[ \beta_B = 10 \log_{10} \left( \frac{I_B}{I_o} \right)\]
\[ \Rightarrow \frac{I_B}{I_0} = {10}^\left( \frac{\beta_B}{10} \right) . . . . . \left( 2 \right)\]
\[\text { From }\left( 1 \right) \text{ and } \left( 2 \right), \text { we get: }\]
\[ \frac{I_A}{I_B} = {10}^\left( \frac{\beta_A - \beta_B}{10} \right) . . . . \left( 3 \right)\]
\[\text { Also }, \]
\[ \frac{I_A}{I_B} = \frac{r_B^2}{r_A^2} = \left( \frac{50}{5} \right)^2 = {10}^2 . . . . . \left( 4 \right)\]
\[\text { From } \left( 3 \right) \text { and } \left( 4 \right), \text { we get: } \]
\[ {10}^2 = {10}^\left( \frac{\beta_A - \beta_B}{10} \right) \]
\[ \Rightarrow \frac{\beta_A - \beta_B}{10} = 2 \]
\[ \Rightarrow \beta_A - \beta_B = 20\]
\[ \Rightarrow \beta_B = 40 - 20 = 20 dB\]
Thus, the sound level of a point 50 m away from the point source is 20 dB.
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