Verification of of Phase vs. Group Velocity

1. Oct 10, 2004

CollectiveRocker

How do i go about verifying this statement: If the phase velocity is the same for all wavelenghts of a certain wave phenomenon (no dispersion), the group and phase velocities are equal. Do i just put the two formulas equal to each other and solve for something or what?

2. Oct 10, 2004

CollectiveRocker

any ideas on how to even start?

3. Oct 10, 2004

Claude Bile

Phase Velocity = w/k

Group Velocity = dw/dk

If w/k is constant, it really is not that hard to show that the phase and group velocities are equal.

Claude.

4. Oct 10, 2004

CollectiveRocker

5. Oct 11, 2004

Claude Bile

Look at the dispersion relation (w vs k). If w/k is constant, the dispersion relation is just a straight line with gradient w/k.

Since the group velocity is the gradient, dw/dk, if the dispersion relation is a straight line, then the group velocity will be w/k for all w.

Claude.

6. Oct 12, 2004

for3to0won

I think he be suggesting that the local slope of a curve is be equal to the slope of the line joining any two points on it, if the curve is a straight line.

7. Oct 12, 2004

Claude Bile

Spot on. Graphing w vs k may make it less confusing.

Claude.