Verification of of Phase vs. Group Velocity

[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?

 

[CollectiveRocker]

any ideas on how to even start?

 

[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.

 

[CollectiveRocker]

Explain please.

 

[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.

 

[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.

 

[Claude Bile]

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

Claude.