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This is something from the text books. Actually you don't have to know this to be a sound engineer - the proof of that is the 90% of sound engineers that don't know it! Yet a little theory never hurt anyone. And if you know how things work in theory, then you can apply them better in practice.
There are two basic types of microphone - omnidirectional and figure-of-eight, which can be made in either dynamic or capacitor forms. Cardioid and hypercardioid microphones are hybrids, combining features of both the omni and figure-of-eight.
The omnidirectional microphone works on the pressure principle. The diaphragm, which picks up sound vibrations in the air, is completely open at one side, but completely closed at the other.
The sound vibration is either pushing the diaphragm against the fixed pressure of the air on the other side, or it is reducing the pressure on the front of the diaphragm allowing the pressure behind to push it out.
One of the features of pressure in a gas is that it pushes equally in all directions. "Equally in all directions"? That makes the mic omnidirectional then.
[For the sake of completeness, it should be mentioned that there is a tiny hole in the capsule to the rear of the diaphragm. This is so the microphone can compensate for long-term variations in air pressure. The hole is too small to affect its sound characteristics.]
The figure-of-eight microphone on the other hand has both sides of the diaphragm fully open to the air. So it doesn't compare the incoming sound pressure with a fixed pressure on the other side of the diaphragm like the omnidirectional microphone. Instead it compares the pressure of the sound wave on one side with the pressure of that same sound wave after it has traveled through to the other side.
Yes, it is a tiny difference in pressure, but strong enough to move the diaphragm.
The difference in pressure between the front and the back of the diaphragm depends on the angle of incidence of the sound wave. In the extreme, if the sound arrives from the side of the diaphragm, then the pressure will be the same at the front and the rear, therefore the diaphragm will not move and there will be no output.
Lastly, the velocity microphone... it doesn't exist! However you will see the term used frequently, particularly in relation to ribbon microphones.
A velocity microphone, if it existed, would respond to the actual velocity of the air molecules striking the diaphragm.
For this to happen though, the diaphragm would have to be so light that it could respond almost instantaneously, and so thin that there was no pressure difference between the two sides of the diaphragm. In practice, the diaphragm is too heavy to acquire the velocity of the air molecules and it has significant thickness.
But a microphone that doesn't make it as a velocity mic, because its diaphragm is too heavy and thick, is still sensitive to pressure gradient. So in practice, these microphones are pressure gradient mics.
So now you know it - the difference between pressure, pressure gradient and velocity microphones.