Question

If the intensity of sound is doubled, by how many decibels does the sound level increase?

Answer 1

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.