# Phase difference with standing waves

Consider the diagram above. We say the phase difference between two particles vibrating between two consecutive nodes is zero. Hence we say that points B and E are in phase. However we say that points A and D are 180 degrees out of phase.

To me, phase difference for standing waves is unintuitive. For progressive waves we define phase difference to be the difference in phase angle, where phase angle is a quantity representing the fraction of the cycle covered up to time $$t$$. Hence intuitively, phase difference is used to describe how much one wave is in front or behind another wave, represented as an angle. Where 360 degrees is one wavelength ahead / behind.

Now for standing waves, the phase difference is 0 as the waves overlap one another. However, we can also refer to particles on the standing wave itself and desrcibe their phases.

For standing waves, it seems phase difference refers to direction. That is two particles are said to be in phase when they reach the same amplitude at the same time, they travel in the same direction. Hence if this is the case, there should be a non zero phase difference between particles A and B despite virating between the same two consecutive nodes. So how do we find the phase difference of A and B? Do we need to know the distance between the two particles?

What's is the phase difference between A and C? What's the phase difference between A and D?

For progressive waves, phase difference seems to be clearly defined. However for standing waves, it seems unintuitive. What does phase difference really mean in the context of standing waves? Is there a definition of phase difference that applies to both progressive and standing waves?

Now consider phase difference for progressive waves, where two waves have a phase difference of $$90°$$. Could we just as well say that the phase difference is $$360+90=450°$$. Does phase difference have to be between $$0°$$ and $$360°$$?

The concept of phase difference and phase is really quite confusing. Could someone please explain this concept to account for both standing and progressive waves that sufficiently answers my questions stated above?

## 1 Answer

You have a series of misconceptions.

That is two particles are said to be in phase when they reach the same amplitude at the same time, they travel in the same direction.

In general the amplitude does not have to be the same and indeed you do not need to be comparing the oscillations of the same physical property.

Look at the current an potential plot.
I would say that the current and voltage are in phase.
Why?
Because they reach a maximum at the same time go through "zero" at the same time, and reach a maximum in the opposite direction at the same time, and reach one eight of an oscillation from a maximum at the same time, and etc

The picture which you used in your question is really two snapshots of a standing wave taken at different times - they are called wave profile.

Here is a gif which has all these wave profiles shown sequentially to illustrate what a standing wave looks like.

So to answer your specific questions.

So how do we find the phase difference of A and B?

You time the instant each reaches a maximum, $$t_{\rm A}$$ and $$t_{\rm B}$$, and evaluate the difference in time $$\Delta t=t_{\rm A}-t_{\rm B}$$ and evaluate the phase angle $$\dfrac {\Delta t}{T} \times 360^\circ$$.
In this case the time difference is zero and so the phase angle is zero, the motions are in phase.

Do we need to know the distance between the two particles?

You will not that all particles between two adjacent nodes are in phase with one another.

What's is the phase difference between A and C?

More difficult is that particle C is not oscillating and so is at its equilibrium position.

You could say that relative to particles C the phase of particle A varies between $$0^\circ$$ and $$360^\circ$$.

What's the phase difference between A and D?

They differ in time by one half a period so the phase difference is $$\dfrac {(T/2) }{T} \times 360 = 180^\circ$$

Does phase difference have to be between 0° and 360°?
No and the other thing which I have omitted is a statement a to which motion is the reference one.

So you could say that motion X is $$27^\circ$$ ahead of motion Y, or motion Y is $$27^\circ$$ behind motion X

More in answer to [Phase difference in progressive vs standing waves]{https://physics.stackexchange.com/questions/774592/phase-difference-in-progressive-vs-standing-waves?rq=1} which is not quite a duplicate.

• So then what is a consistent definition for phase difference that applies to both types of waves? Aug 8, 2023 at 8:05
• With $\Delta T$ and $T$ defined as above the phase difference is $\dfrac {\Delta t}{T}$ as a fraction of a complete oscillation or the phase difference is $\dfrac {\Delta t}{T} \times 360$ in degrees or $\dfrac {\Delta t}{T} \times 2\,\pi$ in radians or $\ldots$ Aug 8, 2023 at 9:03