Proximity Effect
In its simplest form, proximity effect is the increase of bass frequencies as a directional microphone comes closer to the sound source. But why? Why do the bass frequencies increase as the microphone gets closer to the sound source?
Pressure Differences
First, we need to understand a bit about how a directional microphone operates. A directional microphone (also called a pressure gradient microphone) gets it's output signal by responding to the difference in pressure from the front of the diaphragm to the rear of the diaphragm. When a sound wave is on axis with the diaphragm, the sound wave must travel a further distance to get to the rear of the diaphragm (figure 1). That means, if we take a snap shot in time, the wave will be slightly delayed at the rear of the microphone.
A typical distance from the front to the rear of the microphone might be 8.5 mm. Thus, the sound wave that hits the back of the diaphragm has to travel an additional 8.5 mm, and hence, is delayed from the front wave. Figures 24 shows that if we have the same 8.5 mm distance at different frequencies (green lines), the change in pressure would also be different. At low frequencies (figure 2), the difference in pressure (red line) between the front and rear is smaller than at mid (figure 3) or high (figure 4) frequencies.

Inverse Square Law
The inverse square law is the rule that helps explain why sound gets softer as it gets further away from the sound source. As the wave disperses from a source, the energy from the original noise keeps expanding over a larger and larger sphere. Since there must be conservation of energy, as the sphere gets larger each point on the sphere actually contains less energy. In figure 7, at 1 meter from the sound source, the blue square contains a certain amount of energy. At 2 meters from the sound source, that same amount of energy must now be spread over the larger green square.
If we place a microphone at the blue square, the distance from the source to the front of the microphone is 1.0000 meters. The distance from the source to the rear of the microphone is 1.0085 meters. As an example, let's say that the source has an output of 1 watt. Calculating the difference in pressure between the front and rear of the microphone at the blue square, yields 0.047 Pa. Calculating the same difference at 2 meters (the green square), we get 0.012 Pa. Thus, we can conclude that as the microphone moves closer to the source, this inverse square component becomes larger. The inverse square component is also equal for every frequency.
Figure 8 shows the inverse square component and the gradient component (from Figure 5) when the microphone is far away from the sound source. At this distance the pressure difference from the gradient component is larger than the pressure difference from the inverse square component. Note that Figure 8 is also with an undamped diaphragm. Once we damp the diaphragm, the graph will tilt, and we get Figure 9.
