Calculating phase shift between two sinusoidal waves

[ngn]

TL;DR Summary

Question about how to calculate phase shift between to sine waves from textbook. Does the book have the correct method?

Hello,
Came across this picture and passage from a textbook. Although the text lays out a method for calculating the phase shift between displacement and acceleration, I am not sure how they are calculating which wave is leading and which is lagging. From their description, it seems like a negative value would suggest that the wave is lagging compared to the reference. However, when I run the same method comparing velocity to displacement (using displacement as the reference), I get:

Displacement = 90+360+180 = 630
Velocity = 180+360+180 = 720

630 - 720 = -90

If negative means lagging, then this would suggest that velocity is lagging displacement by 90 degrees. But is that the case? It seems like the positive peaks for velocity occur earlier in time compared to the positive peaks for displacement? So, shouldn't velocity be leading displacement? Acceleration does look to be lagging displacement, so is there a problem with this method, or am I not considering the waveforms/method correctly?

Thanks!

 

[scottdave]

I agree that the acceleration is lagging displacement, but the way I learned it - lagging would be a positive phase angle.

 

[ngn]

I agree that the acceleration is lagging displacement, but the way I learned it - lagging would be a positive phase angle.

Hi, thank you for the response! So, acceleration is LAGGING, and the text is correct? I am a bit confused then as to what is meant by leading and lagging. I was wondering if you could explain why it is lagging given that by the time the waves reach that point in time, acceleration is further along in its cycling. Wouldn't this suggest that acceleration has "started earlier in time" so to speak and is thus leading? Is there a good way to define what is meant by "leading" and "lagging"? A good definition would allow me to see why one wave is considered to be in the lead.

Also, does that mean that velocity is also lagging?

 

[scottdave]

Hi, thank you for the response! So, acceleration is LAGGING, and the text is correct? I am a bit confused then as to what is meant by leading and lagging. I was wondering if you could explain why it is lagging given that by the time the waves reach that point in time, acceleration is further along in its cycling. Wouldn't this suggest that acceleration has "started earlier in time" so to speak and is thus leading? Is there a good way to define what is meant by "leading" and "lagging"? A good definition would allow me to see why one wave is considered to be in the lead.

Also, does that mean that velocity is also lagging?

It's 180° out of phase, so it could be leading or lagging, and the waveforms would appear the same, in steady state.

Since they depict displacement as a Cosine wave, it starts at the positive peak. Using this as a reference, how far do you need to travel to get to the same place (positive peak) for acceleration. This happens later in time, so it is lagging.

The relationship between Velocity and Displacement is tricky. You have to move forward in time 270° on the velocity curve to get to the positive peak, so it lags by 270°?

 

[Baluncore]

You can think of this as the cause leading the effect or the effect lagging the cause.

Start with the graph of displacement; s. First notice that:
The slope of the displacement graph is velocity; v = ds / dt.
The slope of the velocity graph is acceleration; a = dv / dt.

Given acceleration.
The integral of acceleration, is velocity + initial velocity.
The integral of velocity, is displacement + initial displacement.

You will see that acceleration causes velocity to change,
so acceleration leads velocity, and velocity lags acceleration.

Also, velocity causes displacement to change, so velocity leads displacement, and displacement lags velocity.

 

[ngn]

You can think of this as the cause leading the effect or the effect lagging the cause.

Start with the graph of displacement; s. First notice that:
The slope of the displacement graph is velocity; v = ds / dt.
The slope of the velocity graph is acceleration; a = dv / dt.

Given acceleration.
The integral of acceleration, is velocity + initial velocity.
The integral of velocity, is displacement + initial displacement.

You will see that acceleration causes velocity to change,
so acceleration leads velocity, and velocity lags acceleration.

Also, velocity causes displacement to change, so velocity leads displacement, and displacement lags velocity.

This is a good method when you know the relationships among the phenomena you are studying, but what if you are just given two waves and don't know anything about them or what they plot? Given just two waves, and one is the reference, then which is the best method to calculate the phase shift? Do you take the reference minus the second wave, or the second wave minus the reference, and how do you interpret the positive or negative outcome (i.e., does negative or positive = leading or lagging?). I've gotten different responses about this from different sources so it is a confusing issue.

 

[Baluncore]

Avoid adding or subtracting angles, that will only confuse you.
An integrator or differentiator will restrict the phase shift of a sinusoid to 90°. That makes it easy.

A 180° shift is an inversion, so does not lead or lag.

Plot the waves as phasors, with the reference along the +x axis.
You can then look to see if another leads or lags the reference.

 

[ngn]

Thank you for the replies. They were very helpful. I think there are different conventions for calculating phase lead/lag. Thus, one can either interpret the displacement/velocity shift as either velocity lagging by 270 degrees or leading by 90 degrees. It would depend on the context, how you interpret the phenomena you are measuring, and the method you use to determine the shift (i.e., how you interpret what a lead is and what a lag is).

 

[sophiecentaur]

The phase θ between two sin waves can be measured by (four quadrant = proper) multiplication of the two and then low pass filtering. The trig identity for cos(A). cos(B) gives you
$Cos(ωt + θ) . Cos(ωt) = {\frac {Cos(2ωt + θ) + Cos(θ)} { 2} }$

There are phase meters which can do it this way. You can calculate it using enough samples and then summing the results. The sign of the phase seems to be as you'd expect.