Proving the group velocity equation.

[Lavabug]

Homework Statement

A wave group is generated as a superposition of harmonic waves of average wavelength lambda. Show that
$$V_{g}=V_{p}-\lambda \frac{d V_{p}}{d\lambda }$$

The Attempt at a Solution

All I know is that Vg = dw/dk and Vf = w/k. I am not sure what I need to do.

 

[chrisbaird]

Assume the dispersion relation ω = ω(k) has some well-behaved functional form, expand it in a Taylor series about the average wavelength, differentiate the Taylor series term by term, recognize ω/k as the phase velocity and keep only the first two terms. Note that because we are dropping higher-order terms, the group velocity only really has meaning when the dispersion relation of the material is fairly smooth, i.e. no anomalous dispersion or high dispersion.