Wave Vector Direction: Explained

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The direction of a wave vector indicates the direction of wave propagation, as it represents the change in location of points with the same phase over time. At time t=0, points at phase 0º are located where k·r=0, which are perpendicular to the wave vector k. As time progresses, these points move to a new position defined by k·r=wt, indicating their displacement along the wave vector direction. This relationship illustrates how the wave's phase front advances in the direction of the wave vector. Understanding this concept is essential for grasping wave behavior in physics.
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Can anyone explain why the direction of a wave vector is the direction of wave propagation?
 
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What definition of "wave vector" are you using?
 
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.
 
Just a classical 3D wave vector:

\psi \left(t , {\mathbf r} \right) = A \cos \left(\varphi + {\mathbf k} \cdot {\mathbf r} + \omega t\right)
 
DaleSpam said:
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!
 
You are very welcome. It is a nice little convention once you get used to it.

Btw, welcome to PF!
 
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