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Question
If the intensity of sound is doubled, by how many decibels does the sound level increase?
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Solution
Let the intensity of the sound be I and \[\beta_1\] be the sound level. If the intensity of the sound is doubled, then its sound level becomes 2I.
Sound level \[\beta_1\] is given by ,\[\beta_1 = 10 \log_{10} \frac{I}{I_0}\],
where I0 is the constant reference intensity.
When the intensity doubles, the sound level is given by:
\[\beta_2 = 10 \log_{10} \frac{2I}{I_0}\].
According to the question,
\[ \beta_2 - \beta_1 = 10 \log\left( \frac{2I}{I} \right)\]
\[ = 10 \times 0 . 3010 = 3 \text{ dB }\]
The sound level is increased by 3 dB.
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