Determining direction of wave propagation from the phase?

Suppose you know the phase of a wave is given by

\phi_1 = \vec k \cdot \vec r - \omega t.

How can you determine in which direction this wave is propagating? I guess, more specifically, how does a wave described by this phase differ from a wave described by the phase

\phi_2 = \vec k \cdot \vec r + \omega t

I may not have provided enough detail...please tell me if I haven't! Thanks.
(I was always under the impression that it was the wave vector [tex]\vec k[/tex] that carried the information about which way the wave was propagating, but recent discussions in my E&M class have caused me to question this belief.)


Science Advisor
Gold Member
You first need to define your time progression. If your standard is good ole e^{-i\omega t}, then the phase dependence is given by:

[tex]\mathbf{k}\cdot\mathbf{r}-\omega t[/tex]

The wave propogates in the \hat{k} direction. If you have

[tex]\mathbf{k}\cdot\mathbf{r}-\omega t[/tex]

as your phase progression then you must have changed your time convention to e^{i\omega t} or you are missing a minus sign here where the actual dependence is e^{-\phi_2} in which case we are progressing in the -\hat{k} direction, but this information is not given in what you have.
Take r vector to be in the direction of k vector.
Now, take kx-wt=A (some constant phase) and differentiate it with respect to t . you get dx/dt=w/k which is the velocity of the point whose A(phase) is constant, and hence velocity of wave . It is called phase velocity. Direction of the propagation is the direction of k vector. [Broken]
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