# Homework Help: Plane waves

1. Jan 31, 2010

### Niles

1. The problem statement, all variables and given/known data
Hi all

If we look at a harmonic wave with constant amplitude, Ψ(x,t) = Asin(kx-ωt), then a point with constant magnitude (e.g. a crest) moves such that kx-ωt is constant in time.

Now we look at a plane wave Ψ(r,t) = Aexp(i[kr-ωt]). Will a point with constant magnitude (i.e. the whole plane) also move such that i(kr-ωt) is constant in time? If yes, then doesn't this mean that the phase for a plane wave is constant for all times?

2. Jan 31, 2010

### ideasrule

Yes, because if you take the real part of that equation, you'll get an equation almost identical to the one you posted for one-dimensional waves.

It means that whatever phase you choose, you can always find a point in space with that phase at any time. That's logical: the wave spreads, but it's not as if one phase "disappears": it just moves at its phase velocity.

3. Feb 1, 2010

### Niles

Hmm, I don't get that. Say we position the plane wave such that the wavevector lies along the x-axis, i.e. it propagates along the x-axis. It is obvious that (as you said) the real part of the plane wave is just what I wrote in my first example of my OP. Hence all points on that specific plane wave have the same phase, and hence they must maintain that phase as they propagate.

With this explanation I cannot see why I can choose any arbitrary phase; there should only be one?

Thanks.

4. Feb 1, 2010

### vela

Staff Emeritus
What do you mean when you say

5. Feb 1, 2010

### Niles

I mean that our plane wave has the form Ψ(r,t) = Aexp(i[kx-ωt]) (we have aligned it along the x-axis), so each point on the plane wave for some x will have the same phase, i.e. kx-ωt is the same for all points on that plane.

6. Feb 1, 2010

### vela

Staff Emeritus
OK, that's what I thought you meant, but your wording seemed kind of funny, so I wanted to make sure. I'm not sure I understand your question then.
What do you mean about choosing a phase? Choosing it for what?

7. Feb 1, 2010

### Niles

I mean it with respect to this post:

ideasrule's post does not make sense, if there is only one phase that stays constant.

8. Feb 1, 2010

### vela

Staff Emeritus
I think ideasrule just meant if you arbitrarily pick a phase, you can find its corresponding plane, and that plane of constant phase, a wavefront, will propagate at the phase velocity. If you choose a different phase, you're talking about a different wavefront, but it will also propagate with the same phase velocity.

What I found confusing about your initial post was you asked if "the phase for a plane wave is constant for all times." I think you meant "wavefront," not "plane wave." The plane wave fills all of space. The phase at a particular point in space will change with time as the wave propagates, and at an instant in time, different points in space will generally have different phases. A wavefront is a plane of constant phase, and it will propagate with the phase velocity. By definition, its phase won't change over time.

9. Feb 1, 2010

### Niles

I have to go to school now, but when I get home, I will reply.

10. Feb 1, 2010

### Niles

Ok, I agree. My explanations were not that detailed, but I think I get it now. Thanks.