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Phase shift or phase angle?

By Murray Bourne, 30 Dec 2006

In the section Graphs of y = a sin (bx + c) in the Interactive Mathematics site, I have a statement

NOTE: Phase angle is not the same as phase shift.

The phase angle for the sine curve y = a sin(bx + c) is the value of c and the phase shift is given by -c/b [...].

Alan Cooper, of alQpr commented:

The use of the term "phase shift" to represent the horizontal shift of a graph is almost universal among high school teachers and text authors at that level, but is, I believe, contrary to the majority usage among university mathematics and physics communities as well as in applied fields. While the proposed distinction between "phase shift" and "phase angle" might be one way of saving face for the teachers, I do not think it is appropriate to require students to adopt linguistic conventions that are not essentially universal, and indeed it is better to let them know that some terms are used differently in different professional communities - and then to refrain from having their grades depend on whether they follow one or other of those conventions.

In my opinion, what high school math teachers do with the notion of phase is worse than having them not mention the topic at all.

He raises an important point here. Are we presenting mathematics for high school students so they can pass some exam (adopting specific linguistic conventions decided on by the assessment writers) or is it what it should be - an insight into how mathematics is used in the "real world"?

Actually, when developing mathematics materials, I have struggled with the desire to keep it simple for newbies, but also to keep it useful so that when students see it again in their physics, chemistry, biology or engineering classes, they will recognise it and will know what to do with it.

Throughout my site, I have tried to use simple variables (I tend to use a, b, c rather than Greek ξ ς χ ψ) in the desire to keep the notation as easy to read as possible. But in a way, this does a disservice to students since they freak out when their other subjects use much more difficult notation. We expect students to be able to transfer their knowledge, but many find it impossible. We leave out too many of the interesting complexities in the desire to make sure they all get it.

So back to the issue of phase shift and phase angle. If I remove the statement defining the difference, my engineering students will miss an important distinction. Thoughts from anyone else on this?

Footnote1 : One of the most confusing things that engineering lecturers do, I feel, is to mix radians and degrees in the same expression. The vast majority of students struggle with radians and I try to keep them separate from degrees where possible.

Footnote 2: When someone challenges you like this, it is good for learning. When you have to justify what you have done, it motivates you to do more reading and thinking. In schools, we don't do enough of Arguing to Learn.

See the 16 Comments below.

16 Comments on “Phase shift or phase angle?”

  1. Alan Cooper says:

    Thanks for following up on this issue. You have correctly understood that my main concern is with how the requirements of testing and grading often appear to be given higher priority than those of the subject itself. I do appreciate the wish to expose students to the concepts without causing unnecessary confusion, but in this context, I believe that it is more important for them to know that there is no universal agreement on a naming convention than it is to suggest one. I also have reservations about your particular proposed terminology, but rather than take up too much of your space here I'll just put the details on a posting in my own blog (at ).

  2. alQpr » Blog Archive » Phase Shift or Phase Angle? says:

    [...] 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. [...]

  3. Murray says:

    Thanks for your considered response, Alan. You have shed some new light on this issue and it has been good to think it through.

    I have modified the page so that the proposed definitions are less dogmatic.

  4. ang says:

    Hi I was just wondering your phase shift when positive I noticed you moved it to the right however I have been thought that a phase shift when positive moves to the left and an negative phase shift moves to the right..which is correct is my teacher wrong. I live in Australia is the teaching different here? Love your interactive site i wish they taught with it. A moving picture is certainly worth a thousand words...
    thanks ang

  5. Murray says:

    Hi Ang. Nope - no difference. Above is say the phase angle is (positive) c, the phase shift is negative c/b (that is, it move to the left).

    The issue in this post is the terminology used. But don't worry - what you are learning in Australia is perfectly correct 🙂

    Thanks for your kind comments on the maths site - glad that you find it useful.

  6. ang says:

    Thank you for clearing that up zac..Whew..I was getting confused..have a great day

  7. Nadine says:

    Phase angle is used in my Electrical Engineering class to differentiate between phase shift, which is displacement. Our professor made the distinction clear especially during the applications in AC circuity calculating Impedence. Phase angle makes more sense because we're calculating the magnitude in relation to an angle as it relates to the other variables.

  8. Tharma Ganesharajah says:

    When the phase angle is given in degrees, we
    have to calculate period by 360/b or 2PI/b.
    I guess it is the second 360/b.
    Radian and degrees are confusing when applied to
    alternating current I=Imax Sin (wt+\Phi).


  9. Murray says:

    Yes, Tharma . We should only use radians to reduce confusion!

  10. teacher and prof says:

    ok so there is a difference between phase angle and phase shift. instead of bitching about math high school teachers you tell me and clarify to me what the phase angle and phase shift is

    i normally tell students that:

    y = a sin ( wt + phi) --> phi is the phase angle

    Rewritten, we can easily see the phase shift:

    y = a sin w [t - (- phi / w) ] --> - phi / w is the phase shift (negative means left direction of shift)

    So, if you think we are so WRONG, then at least you could have "indoctrinated" me on it, instead you went on ranting ABOVE about the difference and not telling me exactly what you know as the DIFFERENCE between the two. Be direct and TELL ME.

    Next question: If the question asks for PHASE ANGLE, is it just the value of phi? (Absolute) or do we care about the sign? (+ anti clockwise, - clockwise)

  11. Alan Cooper says:

    Teacher and prof, what you have called the "phase shift" is what I would call the time shift (if t is time) since it represents a shift in units of t along the t-axis.

    I think most physicists and engineers would say that the phase at any point in time is the full expression for the argument in the trig function (eg wt + phi in your example) and what Murray calls the "phase angle" I would say is just the *initial* phase. Normally phases of the moon are described in terms of position in the cycle rather than in days, and we speak of oscillations being so many degrees, or radians, or cycles, out of phase since that is what determines how they interfere (eg for two waves of different wavelength, the interference is determined by the relative phase in the angular sense much more directly than by the distance or time differences).

  12. katheen says:

    I had an algebra teacher who didnt agree with how the text book taught algebra you had to do it

    his way or it was wrong even though you got the answeer right answeer. He would send you to a tutor and you would be wrong.

  13. Carlos Soto Johnson says:

    There is no problema in Spanish, because all math textbooks only uses the term "Phase Angle".
    En español no hay ningún tipo de problema porque todos los libros de texto únicamente se utilizan el término "Ángulo de Fase"

  14. IanGM says:

    'The phase angle for the sine curve y = a sin(bx + c) is the value of c and the phase shift is given by -b/c [...].'

    The formula for phase shift is -c/b, right? I've also seen it written as c/b. Why the discrepancy?

  15. Murray says:

    @IanGM: Thanks for pointing out the typo, which I've now fixed.

    I guess it could be defined as c/b as long as you indicate clearly the direction (left or right). My earlier comment (#5) is related.

  16. Bill says:

    This thread is way dead, but something about the post rubbed me the wrong way.

    I'd like to point out that the tendency for individuals "downstream" (university?) in the education path to point fingers at individuals "upstream" (high school teachers or perhaps 2-year colleges) as having ulterior motives of just making things easier or "saving face" seems a bit condescending.

    I have a number of differential equations textbooks. After a quick glance through them, I found at least two texts that do not use the terms "phase angle" and "phase shift" as alQpr suggests.

    Cases in point - given the equation y=acos(bx-c):
    On page 199 of Farlow, Hall, McDill, and West's "DEQ and LA," 2nd Ed., "phase shift" is specifically given as c/b while "phase angle" is given as c.

    On page 75 of Gilbert Strang's Diffeq & LA, he states that "the phase shift or lag in this solution will be the angle c."

    So we have examples where (1) the two statements don't agree with alQpr's, (2) the two statements don't even agree with each other, and (3) these aren't high school teachers making these distinctions.

    So perhaps it's not about individuals "saving face." There are also plenty of discussion boards in which this distinction is made and explained, presumably by individuals who are NOT high school teachers.

    In the end, I agree with what I hope is alQpr's main point that we make mountains out of molehills when we force students to treat this linguistic difference as being as important as understanding the conceptual difference. But then again, if all we have at our disposal to accomplish conceptual clarity is a choice of wording, what exactly are we supposed to do if the sources don't necessarily agree on those words? "c" and "c/b" are not the same things, so what, if anything, might we call them, or how (if at all) should we explain their differences?

    While I take every opportunity to engage in conversations with individuals who are much more knowledgeable about these things than I, and I take every opportunity to inform my students that there is a fair amount of "author's choice" in how definitions are expressed, I don't think it's helpful to be condescending about it and make assumptions about others' motives or lack of knowledge or understanding.

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