Editing Phase (waves) (section)

===For shifted signals===
The phase difference is especially important when comparing a periodic signal <math>F</math> with a shifted and possibly scaled version <math>G</math> of it. That is, suppose that <math>G(t) = \alpha\,F(t + \tau)</math> for some constants <math>\alpha,\tau</math> and all <math>t</math>.  Suppose also that the origin for computing the phase of <math>G</math> has been shifted too.  In that case, the phase difference <math>\varphi</math> is a constant (independent of <math>t</math>), called the 'phase shift' or 'phase offset' of <math>G</math> relative to <math>F</math>.   In the clock analogy, this situation corresponds to the two hands turning at the same speed, so that the angle between them is constant.

In this case, the phase shift is simply the argument shift <math>\tau</math>, expressed as a fraction of the common period <math>T</math> (in terms of the [[modulo operation]]) of the two signals and then scaled to a full turn:
<math display="block">\varphi = 2\pi \left[\!\!\left[ \frac{\tau}{T} \right]\!\!\right].</math>

If <math>F</math> is a "canonical" representative for a class of signals, like <math>\sin(t)</math> is for all sinusoidal signals, then the phase shift <math>\varphi</math> called simply the ''initial phase'' of <math>G</math>.

Therefore, when two periodic signals have the same frequency, they are always in phase, or always out of phase.  Physically, this situation commonly occurs, for many reasons.  For example, the two signals may be a periodic soundwave recorded by two microphones at separate locations. Or, conversely, they may be periodic soundwaves created by two separate speakers from the same electrical signal, and recorded by a single microphone.  They may be a [[radio]] signal that reaches the receiving antenna in a straight line, and a copy of it that was reflected off a large building nearby.

A well-known example of phase difference is the length of shadows seen at different points of Earth.  To a first approximation, if <math>F(t)</math> is the length seen at time <math>t</math> at one spot, and <math>G</math> is the length seen at the same time at a [[longitude]] 30° west of that point, then the phase difference between the two signals will be 30° (assuming that, in each signal, each period starts when the shadow is shortest).