This goes back to a Seismos column from May 2002 about the various interpretations of the word 'phase'. The meaning at issue here is the apparent time shift associated with a particular phase at a given frequency. Think about a cosine with zero phase. Now let the phase be pi/2 and the cosine becomes a sine, whose peak coincides with the zero crossing of the original cosine. By definition this zero crossing occurs at 1/4 of the period of the cosine. If the phase is pi then the first peak lines up with the first trough of the original cosine, for an apparent time shift of 1/2 of the period. Letting the period of the wave be T=1/f, where f is the frequency in Hz, we can summarize this as follows:
- phase=0 t-shift=0
- phase=pi/2 t-shift=T/4
- phase=pi t-shift=T/2
In other words, if the phase is a linear function of frequency then the apparent time shift will be the same for all frequencies (because the f cancels out). This is in agreement with the numerical tests that showed the summation over all frequencies generated a zero phase, t-shifted waveform.
The figure below is a Mathematica plot that illustrates the connection between phase and apparent time shift for a single frequency. The solid curve is zero phase and the dashed curve is phase shifted. In this case the frequency is 1 Hz, the period is 1 second, and we see that a phase shift of pi/2 moves the peak 0.25 s or one-quarter of the period in accord with our formula.