The intensity of a sound wave is its power/unit area.
I = P/A
In one dimension the intensity is constant as the wave travels. In two or three dimensions, however, the intensity decreases as you get further from the source. In three dimensions, for a source emitting sound uniformly in all directions the intensity drops off as 1/r2, where r is the distance from the source.
To understand the r dependence, surround the source by a sphere of radius r. All the sound, emitted by the source with power P, passes through the sphere. When the sound reaches the sphere its intensity is:
I = P/(4πr2)
That's the surface area of a sphere in the denominator.
Double the distance and the intensity drops by a factor of 4, etc.
How do we perceive the intensity of sound? The human ear is actually an amazing instrument. Not only can it handle sound with frequencies covering three orders of magnitude, it can handle sounds with intensities covering 12 orders of magnitude!
The ear can do this because it responds to sound intensity logarithmically. This is why we use the decibel scale - much like the Richter scale for measuring earthquakes, the decibel scale is logarithmic.
Sounds measured in decibels are given by:
β = (10 dB) * log(I/Io)
where I is the intensity in W/m2 and Io is a reference intensity.
Io = 1 x 10-12 W/m2
This is around the lowest intensity sound we can hear, and it works out to 0 dB. Every 10 dB represents a change of one order of magnitude in intensity. 120 dB, 12 orders of magnitude higher than our reference level, has an intensity of 1 W/m2. Any more intense than this and sounds start getting uncomfortable.
An earthquake measuring 6.0 on the Richter scale is 10 times more powerful as one measuring 5.0, and 1/10th as powerful as one measuring 7.0.
Similarly, a 60 dB sound has ten times the intensity of a 50 dB sound, and 1/10th the intensity of a 70 dB sound.
An increase of x dB means that the sound has increased in intensity by some factor. For instance, any increase by 5 dB represents an increase in intensity by a factor of 3.16.
The decibel equation can also be written in terms of a change. A change in intensity, in dB, is given by:
Δβ = (10 dB) * log( If / Ii )