The loudness of a sound source is related to how energetically the source is vibrating, consequently making the molecules of the propagation medium vibrate. To measure the loudness of a source, the physical quantity of interest is the intensity. This is defined as the energy emitted per unit of time per unit of area perpendicular to the sound wave's propagation direction. Since the total energy is greater if the source vibrates for a longer duration and over a larger area, dividing the emitted energy by time and area variables accounts for the source's intrinsic loudness.
A sound's source often carries out hundreds or thousands of oscillations per second, but what the human ear perceives is the time average of these oscillations. Therefore, the definition is further modified by replacing power with average power.
It can be shown that the source's intensity is proportional to the square of the oscillation amplitude of the medium's constituents. Since the oscillation amplitude is proportional to the amplitude of pressure creating the oscillation, the intensity is therefore proportional to the square of the pressure amplitude. Since the sound is propagated by the vibration of the medium's particles, it is expected that the intensity increases with the pressure amplitude.
This text is adapted from Openstax, University Physics Volume 1, Section 17.3: Sound Intensity.
