The amplitude of a sound wave can be quantified in at least three ways:
None of these quantities are used much, however. In fact, the first two are unusually difficult to measure directly. For typical sound waves, the maximum displacement of the molecules in the air is only a hundred or a thousand times larger than the molecules themselves. Any resulting density fluctuations are equally miniscule and very short lived. (The period of sound waves is typically measured in milliseconds.)
Pressure fluctuations caused by sound waves are much easier to measure (animals have been doing it for hundreds of millions of years with ears), but the results of such measurements are rarely ever reported. Instead, amplitude measurements are almost always used as the raw data in some computation. When done by an electronic circuit — like the circuits in a level meter — the resulting value is called the intensity. When done by a neuronal circuit — like the circuits in your brain — the resulting sensation is called the loudness.
Briefly, the intensity of a sound wave is a combination of its rate and density of energy transfer. It is an objective quantity associated with a wave. Loudness is a perceptual response to the physical property of intensity. It is a subjective quality associated with a wave and is a bit complex. As a general rule the larger the amplitude, the greater the intensity, the louder the sound. Sound waves with large amplitudes are said to be "loud". Sound waves with small amplitudes are said to be "quiet" or "soft". (The word "low" is sometimes also used to mean quiet, but this should be avoided. Use the word "low" only to describe sounds that are low in frequency or pitch.) Loudness will be discussed at the end of this section.
By definition, the intensity of any wave is the time-averaged power it transfers per area through some region of space. The traditional way to indicate the time-averaged value of a varying quantity is to enclose it in angle brackets. These look similar to greater and less than symbols but are taller and less pointy. That gives us an equation that looks something like this…
The unit of intensity is the watt per square meter — a unit that has no special name.