Chapter 14 - Sound - Learning Path Questions and Exercises - Exercises - Page 525: 35

(a) When the distance to a sound source is halved, the sound intensity level will change by a factor of (3) 4. This is because the sound intensity level (SIL) is proportional to the logarithm of the ratio of the sound pressure at a given distance to the sound pressure at a reference distance. When the distance is halved, the ratio of the distances is 1/2, which means the logarithm of the distance ratio is -0.3. Since the sound intensity level is proportional to the negative logarithm of the distance ratio, halving the distance results in a 4-fold increase in the sound intensity level. (b) The change in the sound intensity level depends on the initial sound intensity level and the distance ratio. However, if we assume that the initial sound intensity level is 0 dB (the threshold of human hearing) and the distance ratio is 1/2, then the change in the sound intensity level would be +6 dB. This is because a 4-fold increase in sound intensity level corresponds to a 6 dB increase (since each 10-fold increase corresponds to a 10 dB increase). Therefore, halving the distance to a sound source would result in a 6 dB increase in the sound intensity level.

Sound Intensity Level $SIL = 10 log_{10}\frac {I}{I_o} {}$