## Phase Shift or Phase Angle?

Murray Bourne at squareCircelZ has taken the time to respond to a comment I made on one of his interactive math pages, so I thought I should make an effort to explain my concern in a bit more detail.

In high school and college precalculus courses, the material on graphing trig functions often includes a definition of "phase shift" which is contrary to the way the term is used by many in applied fields and also by many mathematicians (including me when I have a choice).

The usage demanded by high school examiners corresponds to the horizontal shift of the graph from a purely scaled basic trig function. So, for A*sin(bx+c)+d it would be given by s=-c/b since, with that value for s, we get A*sin(bx+c)+d=A*sin(b(x-s))+d , so the graph of y=A*sin(bx+c)+d comes from y=sin(x) by first scaling to get y=A*sin(bx), and then shifting horizontally by s units along the x-axis (and vertically by d units along the y-axis).

But in fact, the concept of phase arose from a need to identify the part of the cycle being considered (ie rising, peak, falling, mid-point, trough, etc) and is usually identified quantitatively by an angle. So we typically talk of two waves interfering constructively when "in phase" and destructively when "180degrees (or pi radians) out of phase", and we also speak of a process such as reflection or refraction as introducing a "phase shift" of so many degrees or radians in the propagation of the wave. With this usage, the phase shift of A*sin(bx+c) relative to A*sin(bx) is just c (radians) rather than the math teachers' -c/a.

Some authors seek to avoid the conflict by identifying "phase shift" as what the high school teachers insist on and "phase angle" for what the other camp prefers. But I think this is a mistake for several reasons. My main objection is that even if it might be a good idea to implement such a change, it should not be taught to students as fact if it has not in fact yet been established as a convention agreed to universally in the professional mathematics community. There is nothing wrong, and much to value, in admitting to students that not all terms have universally agreed definitions and that when they face such terms it is important to *ask* what convention the user intends rather than to blithely assume something that may be wrong (which is just the sort of thing that leads to expensive space probes crashing into Mars and causes international airliners to run out of gas in the middle of the Atlantic).

But if that particular convention were proposed I would argue against it as I believe it serves no purpose other than to "save face" for the math teachers, and does so at the expense of abusing the language. I say this for three reasons.

First, the word "phase" was introduced to refer to the position in a cycle (which is basically an angle), so to speak of a "phase angle" is redundant.

Second, there is a phase (angle) corresponding to every point on a wave and the term "phase angle" does not properly denote a shift.

Thirdly, to use the term "phase shift" for what in any other graph would be called the "time shift" or "horizontal shift" introduces a completely useless extra bit of language by having a special context-dependent term for something which already has a perfectly good name that works in every other context.

And fourthly (I know you weren't expecting the Spanish Inquisition, but do you know the three kinds of mathematician?) wasting a term where it is not needed makes it unavailable for where it is actually useful.

When two split light waves are brought together again (as in the creation of a hologram) it is not the phase (angle) itself at each point but the angular shift between the two waves that is directly relevant to the outcome rather than the time shift between two signals. We could of course convert the time shift to a phase (angle) shift just by using the known frequency and velocity of propagation, but it would be silly to use both terms to refer to the time shift and leave ourselves without a name for the quantity that is actually most directly relevant.

The convention that makes most sense to me is therefore to use the term "horizontal shift" (or whatever term they'd use for the x-displacement in any other function) for what the math teachers call "phase shift" and keep "phase shift" for its traditional role as what is now being proposed as "phase angle".

February 28th, 2017 at 12:33 pm

I use the term horizontal shift for two reasons.

1. It is universally understood, and it applies to any function.

2. It avoids the confusion around the term phase shift.

For any function f(x), I usually write the new function as

g(x)=af(b(x-h))+k with the argument in factored form rather than simplified form. Thus a positive value of h is a shift to the right.

February 28th, 2017 at 11:45 pm

Thank you Ms H. I wish that everyone did it like you - and especially that textbooks and exam writers didn't feel the need to introduce unnecessary extra terminology (and then use it wrongly to boot!).