Consider two waves with equal frequency and amplitude, and .
The difference between these two waves can be computed as follows:
so we may write:
where we have calculated a new angle using
These equations are consistent as a consequence of the division by .
Hence the maximum value of the difference is , which is achieved at
Thus given the maximum distance apart of two out-of-phase but otherwise identical oscillators, we simply say:
Finally, in the case of the maximum being half the amplitude,