Question

A source emits sound uniformly in all directions. There are no reflections of the sound. At a distance r1 from the source the sound is 6.5 dB louder than it is at a distance r2 from the source. What is the ratio r1/r2?

Answer 1

The difference in decibels between the two is defined to be

10 * log (P2/P1)

Where the log function is in base 10, and P2 and P1 are the power of the sources at r2 and r1

6.5 = 10*log(P2/P1)
0.65 = log(P2/P1)

10^0.65 = P2/P1 = 4.46

Power is proportional to the radius squared, so P2/P1 = (r2/r1)^2

r2/r1 = 4.46^(0.5) = 2.11
r1/r2 = 2.11^-1 = 0.47