Why is the wave vector direction the direction of propagation?

[stevvvvv]

Can anyone explain why the direction of a wave vector is the direction of wave propagation?

 

[HallsofIvy]

What definition of "wave vector" are you using?

 

[Dale]

The direction of propagation of a wave is given by the change in the location of different points with the same phase, for convenience let's say a phase of 0º. So we have:
cos(wt-k.r) and at t=0 the location of all points with phase of 0º is given by:

k.r=0 (all r locations perpendicular to k)

Then at some time t later we have the position of 0º phase given by:

k.r=wt (all r locations whose normalized projected distance along k is wt)

So the set of points with 0º has moved a certain distance in the k direction.

 

[stevvvvv]

Just a classical 3D wave vector:

$$\psi \left(t , {\mathbf r} \right) = A \cos \left(\varphi + {\mathbf k} \cdot {\mathbf r} + \omega t\right)$$

 

[stevvvvv]

The direction of propagation of a wave is given by the change in the location of different points with the same phase, for convenience let's say a phase of 0º. So we have:
cos(wt-k.r) and at t=0 the location of all points with phase of 0º is given by:

k.r=0 (all r locations perpendicular to k)

Then at some time t later we have the position of 0º phase given by:

k.r=wt (all r locations whose normalized projected distance along k is wt)

So the set of points with 0º has moved a certain distance in the k direction.

Thanks a lot!

 

[Dale]

You are very welcome. It is a nice little convention once you get used to it.

Btw, welcome to PF!