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