What is the source of proximity effect... the boost in lower frequencies as a vocalist gets close to a condenser mic?
Directional mics (eg cardioids) work by allowing sound to reach the rear of the diaphragm as well as the front. Because the sound wave travels further to reach the rear of the diaphragm it is delayed and so is out of phase with the wave at the front of the diaphragm. This phase difference causes a pressure difference because the two waves, one front and one rear, are not at the same amplitude at the same time. The diaphragm moves due to the pressure difference caused by the out of phase waves at it's front and rear.
HOWEVER, because the distance from front to rear of the diaphragm is fixed the phase delay varies with frequency. Low frequencies have long wavelengths, therefore the relatively short distance from the front to the rear of the diaphragm is a small fraction of the wavelength and thus the phase difference is small. Small phase difference = small pressure difference = less output. High frequencies have much shorter wavelengths therefore the distance to the rear of the diaphragm is a much greater fraction of the wavelength and the phase difference is much greater. Bigger phase difference = bigger pressure difference = more output. The sensitivity due to phase difference rises by 6dB per octave as the frequency rises. This is the 'pressure gradient' refered to. To compensate the diaphragm is damped to roll it's frequency response off by 6dB per octave. Result: Flat(ish!) frequency response.
BUT, the sound pressure level falls off as the distance from the source increases. Doubling the distance reduces the sound pressure to approximately 1/4 of its value. Remember the sound has to go an extra distance to the rear of the diaphragm. It loses pressure level simply due to the extra distance travelled. THis effect is not frequency dependant. The closer the mic is to the sound source the more pressure is lost in the extra distance between front and back of the diaphragm because the sound pressure level is greater the closer the mic gets to the source. The mic also reacts to the difference in pressure due to the loss of pressure to the wave at the rear of the diaphragm as a result of the extra distance it has travelled.
So we have two components causing pressure difference at the mic diaphragm. 1) The pressure difference due to phase difference due to the extra distance travelled by the rear wave (pressure gradient component) and 2) the pressure difference due solely to the extra distance travelled by the rear wave (Pressure component). The phase pressure difference is frequency dependant. The distance pressure difference is not, BUT remember, the response of the diaphragm is reduced (damped) by 6dB per octave as the frequency increases to compensate for the frequency dependant effect of the phase difference. This is fine as long as the phase difference component of the sound is dominant. At close proximities the pressure difference due to the extra distance to the rear becomes dominant. Because the diaphragm is damped at 6dB per octave as the frequency rises the higher frequencies are attenuated more the closer the mic is to the sound source. Attenuating the higher frequencies obviously accentuates the lower bass frequencies, therefore the bass response rises as the source gets nearer. In practice the pressure component only really dominates the pressure gradient component at very close distances to the source. Hence, the proximity effect, or 'bass tip up'