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wave phase velocity vs. group velocity?

Posted: Sun Jan 01, 2012 1:54 am
by hadimotamedi
Dear All
Can you please let me have a clear understanding of the wave phase velocity/group velocity and if phase velocity can exceeds group velocity or vice versa?
Thank you

Re: wave phase velocity vs. group velocity?

Posted: Mon Jan 02, 2012 5:17 am
by King Vitamin
$$v_p = \frac{\omega}{k}$$
$$v_g = \frac{d\omega}{dk}$$

These are the definitions of phase and group velocity respectively. So given different expressions $$\omega(k)$$, we can come up with examples of where one exceeds the other.

$$\omega(k) = c \ k$$ (c is constant)

Here, $$v_g = v_p$$.

$$\omega(k) = k^2$$

Here, $$v_g > v_p$$

$$\omega(k) = \sqrt{k}$$

Here, $$v_g < v_p$$.

It's helpful to know the graphical interpretations of these if you're studying for the GRE, so I'd recommend looking at the graphs of these functions and understanding when either is larger, smaller, or the same.