Ugh, everybody's favorite subject...right next to anthrax!
There's polarity, phase, and phase coherence, all of them tangentially related, all of them quite different, and all of them mistaken for one another quite often by almost everybody (including me) in everyday conversation unless one is very specific about what *exactly* they are talking about.
In short, a polarity switch is a flipping or inversion of the signal around 0 volts DC. It is a change of pls values to minus values and vice versa.
Phase is a value which measures the difference either between two wave forms or between a wave form in it's current state as compared to a previous reference state, and can have a vale anywhere from 0° to 360° depending upon the amount of phase change or difference between the two measurements or waves.
An instantaneous phase change or rotation of 180° has a result identical to a flipping of the wave around it's rest DC value. Assuming a nominal signal with a rest value of 0 volts DC (no DC offset), such an instantaneous "phase flip" of 180° is therefore under those circumstances identical to and indistinguishable from a polarity change.
The problem with most real-life audio situations such as the mixing of signals from two microphones,however, is that the signals are out of phase not because of an instantaneous change in the phase angle of a waveform, but because of a delay or difference in the timing of the signal reaching Mic A and Mic B. because sound travels at a particular speed (and isn't very fast), it can reach two microphones that are different distances from the source at different times. This delay in time is what is causing the two waves from the two microphones to become "out of phase" with each other.
With a pure, simple sine wave, how much they are aout of phase with each other is a simple mathematical calculation, because the wavelength of the wave - the distance from peak to peak in the sine wave, is directly related to the frequency, or number of cycles per second. A sine wave of one kHz means that the wave cycles from peak to peak one thousand times a second.
Therefore a delay of 1/2000th of a second (twice the frequency and therefore half the wavelength of the signal) to the second microphone will cause that signal to become exactly 180° "out of phase" from the first signal and the two will effectively cancel each other out. In such an instance, flipping the polarity will clear up the problem perfectly.
There are two problems in real life though; the first is that the chances of the delay being exactly 1/2000th of a second are extremely small and would be highly coincidental. Lets say the delay were instead 1/4000th of a second. This would be the equivalent of a delay of 1/4th the wavelength, or a phase shift of 90°, exactly halfway between 0 and 180°. In such a case, a polarity flip would have no effect on the phase discrepency, because it would just change the relative phase of the second signal to 270°, which is still exactly halfway between 180° and 0° (a.k.a 360°.) They still would be exactly as much out of phase as they were before the polarity flip.
The second problem, which really muddles things up, is that most of the sound waves we deal with in real life are not simple sine waves of a single fequency, but rather complex waves consisting of a whole witches brew of fundamental frequencies, overtones, and other frequencies, and all at differing amplitudes. One look at the timeline of your DAW software will confirm that.
Therefore a single delay in time between two microphones cannot possibly mean the same degree of phase delay for every frequency in that waveform. It's not a simple matter of hitting the polarity switch to effectively perform a 180° phase shift, because your waveform will be filled with frequencies with phase shifts at practically every point on the scale, many of which the 180° polarity flip will not make that much better, and many it may actually make worse.
The end result is that the polarity flip may make things worse, and it may make things somewhat better, and it may have little effect at all. Frankly it's a roulette spin, with you betting on one color; the odds are slightly worse than even that it'll make things better.
The main reason for the polarity switch is to handle mics that are on the reverse side of a drum skin or other instrument from the first mic, guaranteeing a natural polarity inversion that needs rectifying, or cables or mics that may accidentally have their polarity reversed dur to bad wiring. The invert polarity button was never meant to be an effective tool against time-delay induced phase change except in "today's your lucky day" circumstances.
HTH,
G.
