Phase difference

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jpanknin

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In the book (Mckeague/Turner Trigonometry 8E) there is the following problem that asks for the phase difference.

[MATH]y = sin(2x - pi)[/MATH]
The phase is described in the book as:

1608424958480.png

The image below shows that the line begins at [MATH]pi/2[/MATH] and that the Phase is [MATH]-pi[/MATH]. Using the definition above, it seems to me that the phase (difference) between [MATH]y = sin(x)[/MATH] and the graph here would be [MATH]-pi/2[/MATH] since the graph starts at [MATH]pi/2[/MATH], so why is this phase [MATH]-pi[/MATH]?

1608421477688.png
 
You are confusing a phase shift and a phase. A phase shift is where the graph starts. You graph starts at pi/s so its phase shift is pi/2. Now you comment that The image below shows that the line curve begins at pi/2 and that the Phase is −pi makes no sense at all since you are not comparing the starting points between two sin curves
 
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jpanknin said:
In the book (Mckeague/Turner Trigonometry 8E) there is the following problem that asks for the phase difference.

[MATH]y = sin(2x - pi)[/MATH]
The phase is described in the book as:

View attachment 23879

The image below shows that the line begins at [MATH]pi/2[/MATH] and that the Phase is [MATH]-pi[/MATH]. Using the definition above, it seems to me that the phase (difference) between [MATH]y = sin(x)[/MATH] and the graph here would be [MATH]-pi/2[/MATH] since the graph starts at [MATH]pi/2[/MATH], so why is this phase [MATH]-pi[/MATH]?

View attachment 23877
Click to expand...
Unfortunately, the word "phase" is used in a couple different ways.

One is the actual shift of the graph, which is what you want it to mean here. In the form they are using, that would be -C/B, because the function could be written as sin(B(x + C/B)), and replacing x with x + C/B results in a shift by C/B units to the left (that is, -C/B to the right).

What they are calling phase is the relative angle in radians, which is given by C. Do you see that in this graph, the shift is half a cycle, which means half of 2 pi, namely pi radians? Note the phrase they use, "the fraction of a standard period of 2 pi", which is not quite accurate, but does try to tell you what it means. They don't use the word "shift".

This site calls -C/B the "phase shift", and calls C the "phase angle". I think this is fairly standard at an introductory level, though not in applications such as physics. The terms make sense: the phase shift is the actual movement of the graph itself; the phase angle relates to the sine function itself and its period, and is more important in applications. You just need to recognize which one they are talking about. Your book evidently just uses the one term, "phase", to mean the phase angle, which suggests it is oriented toward actual use.

When I teach this material, I have to check how the textbook uses these terms, and I generally explain these issues to clarify what they are and are not talking about. It's really tricky in tutoring when I don't have your textbook ...
 
You are confusing a phase shift and a phase. A phase shift is where the graph starts. You graph starts at pi/s so its phase shift is pi/2. Now you comment that The image below shows that the line curve begins at pi/2 and that the Phase is −pi makes no sense at all since you are not comparing the starting points between two sin curves
Click to expand...
After reading the passage a third time it does say actually clearly that C (in Bx + C) is the phase. But as I said before, it is not the phase shift. Phase shift and phase have two different meanings, well actually that depends on your book/instructor. But based on what you posted your text clearly has two meanings.
 
Yes, this book uses the term "horizontal shift" (others websites I've explored call it "phase shift") to describe the movements left/right on the graph and "phase" as the lead/lag of a graph as compared to another graph. See the highlighted section in red of the image below that shows a "horizontal shift" of [MATH]pi/2[/MATH] and a "phase" of [MATH]-pi[/MATH].

1608429088823.png

The book's definition of "horizontal shift" is moving C units, in this case [MATH]pi/2[/MATH] to the right from:

1608429383406.png

I'm completely comfortable with the "horizontal/phase shift" aspect and somewhat conceptually with the book's use of "phase." In the chart below, per Jomo's earlier post (that now seems to be deleted), even comparing [MATH]y = sin(Bx)[/MATH] with [MATH]y = sin(Bx + C)[/MATH] it still seems to me that [MATH]y = sin(2x - pi)[/MATH] lags [MATH]y = sin(2x)[/MATH] by only [MATH]-pi/2[/MATH] instead of [MATH]-pi[/MATH].

1608429738357.png
 

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jpanknin said:
Yes, this book uses the term "horizontal shift" (others websites I've explored call it "phase shift") to describe the movements left/right on the graph and "phase" as the lead/lag of a graph as compared to another graph. See the highlighted section in red of the image below that shows a "horizontal shift" of [MATH]pi/2[/MATH] and a "phase" of [MATH]-pi[/MATH].

View attachment 23883

The book's definition of "horizontal shift" is moving C units, in this case [MATH]pi/2[/MATH] to the right from:

View attachment 23886

I'm completely comfortable with the "horizontal/phase shift" aspect and somewhat conceptually with the book's use of "phase." In the chart below, per Jomo's earlier post (that now seems to be deleted), even comparing [MATH]y = sin(Bx)[/MATH] with [MATH]y = sin(Bx + C)[/MATH] it still seems to me that [MATH]y = sin(2x - pi)[/MATH] lags [MATH]y = sin(2x)[/MATH] by only [MATH]-pi/2[/MATH] instead of [MATH]-pi[/MATH].

View attachment 23887
Click to expand...
You say you are (somewhat) comfortable with their use of the word, yet you seem to be rejecting the distinction between the two, and still want the phase to be the horizontal shift.

The phase (angle) is the amount being added to the angle whose sine we are taking. For a given x here, without the phase we would be taking the sine of 2x; we are adding the angle -pi to that. That is half a cycle (180 degrees), and you should clearly see that the shift in the graph is half a cycle (one hump of the sine).

That is what the book means by "the fraction of the period": if you divide the phase by 2pi, the period of the standard sine, you get that fraction 1/2 I have been using to describe how much of a cycle it lags by.

Again, phase is measured relative to a cycle, not as an absolute shift. That's the difference.
 
Hi everyone, I appreciate the responses very much. I think I'm not being clear on which part of this I'm confused about. Again, I understand the shift aspect. What I'm not seeing is the lag/lead aspect. See the passage below. It describes the phase as the "fraction of a standard period of 2pi that a point on the graph of y=sin(Bx+C) lags or leads a corresponding point on the graph of y=sin(Bx).

1608465365858.png

By my understanding of this (marked on the graph below), the shift is certainly [MATH]pi/2[/MATH], but points on the graph of [MATH]y = sin(2x - pi)[/MATH] also lag points on [MATH]y = sin(2x)[/MATH] by [MATH]pi/2[/MATH]. I'm not seeing a lag/lead of [MATH]pi[/MATH] on these graphs. I can see it in the equation, but not in the graphs.

1608466766370.png

Or, per Dr. Peterson's point, since the period of one cycle of these functions is [MATH]2pi / 2 = pi[/MATH], is the phase on the graph relative to the 1/2 of the period of [MATH]pi[/MATH]?
 
For example, in the chart below (with no change to the period), the phase is [MATH]pi/2[/MATH] and the horizontal shift is [MATH]-pi/2[/MATH]. It's easy to see in this one that the lead of [MATH]y = sin(x + pi/2)[/MATH] is [MATH]pi/2[/MATH]:

1608467510737.png

And in this one the phase is [MATH]pi/2[/MATH] while the horizontal shift is [MATH]((pi/2) / (1/2)) = pi[/MATH]. However, it seems that [MATH]y = sin(1/2x + pi/2)[/MATH] leads [MATH]y = sin(1/2x)[/MATH] by [MATH]pi[/MATH] instead of [MATH]2pi[/MATH].

1608467656421.png

I guess my confusion comes when there is a change to the period of these functions and how to visualize the phase on the graph.
 
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jpanknin said:
I think I'm not being clear on which part of this I'm confused about. Again, I understand the shift aspect. What I'm not seeing is the lag/lead aspect. See the passage below. It describes the phase as the "fraction of a standard period of 2pi that a point on the graph of y=sin(Bx+C) lags or leads a corresponding point on the graph of y=sin(Bx).
Click to expand...
I've already said I don't really like their precise wording, "the fraction of a standard period of 2 pi". But the ratio of the phase to 2 pi is the ratio of the shift to the period; it's proportional.

jpanknin said:
Or, per Dr. Peterson's point, since the period of one cycle of these functions is [MATH]2pi / 2 = pi[/MATH], is the phase on the graph relative to the 1/2 of the period of [MATH]pi[/MATH]?
Click to expand...
Yes, that's what I said. You have to think relative, not absolute.

Looking at the (original) graph, you see a shift of pi/2 and a period of pi, so it lags by half a period. In radians, half a (standard) period is 1/2 of 2 pi, which is pi. Looking at the last graph you showed, the shift is pi out of a period of 4 pi, so the phase is 1/4 cycle, which is pi/2.

When you work with sinusoids, you get used to equating pi to half a cycle, pi/2 to a quarter cycle, and so on -- so much so, apparently, that the author thinks of the phase as if it were that fraction!
 
this looks like 2 signals out of phase to the point that they would cancel each other out in the real world. This is how they make noise canceling head phones

1608488127931.png
 
markraz said:
this looks like 2 signals out of phase to the point that they would cancel each other out in the real world. This is how they make noise canceling head phones

View attachment 23909
Click to expand...
Which is an interesting point, because "out of phase" here means "180 degrees out of phase", or pi radians: 1/2 cycle. This is an example of how phase angle is useful in real life: Without knowing other facts like the period, you can immediately know how the two signals interact. "Phase shift" is useful for graphing, which is the typical task in a trig class; "phase angle" is far more useful in actually understanding waves. So this author is wise in using terms that don't let you confuse the two (although the explanation left the reader able to confuse himself).
 
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