Serway & Jewett — Wave Motion: Problem 37

Show that the difference between decibel levels \(\beta_1\) and \(\beta_2\) of a sound is related to the ratio of the distances \(r_1\) and \(r_2\) from the sound source by \[ \beta_2 - \beta_1 = 20 \log \left( \frac{r_1}{r_2} \right) \]