The intensity of a sound wave is defined as 'the rate of flow of energy per unit area perpendicular to the direction of the wave'.
It may be shown that the intensity of a sound wave is proportional to
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the density of the air,
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the square of the frequency of the sound,
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the amplitude of the wave squared,
We can write therefore
where k is a constant of proportionality.
Usually we cannot vary the density of air, and the frequency of the sound is usually given by the instruments at our disposal. The easiest factor to change is usually the amplitude. We can get a louder sound out of a drum by frankly, banging harder. We cannot very easily change the frequency of vibration of the drum, and the density of air is even more beyond our control.
On the other hand, loudness is a matter of human perception and the varying pressure exerted on the eardrum, which increases with intensity but is not proportional to it. To make matters more complicated, the relationship varies with frequency, so that the human ear is more sensitive to higher than lower frequencies.
On top of this the intensity decreases with increasing distance from the source, but the relationship is not an inverse square one as for light intensity, because some of the energy of the sound is absorbed by the air molecules as energy of random movement – heat energy. The intensity of sound actually decreases faster than would be implied by an inverse square law for this reason.