Solving Wave Packet Propagation: Analyzing Group and Phase Velocities

[stunner5000pt]

Consider a wave packet represented by

$$\Psi (x,t) = \int_{k-\Delta k}^{k+\Delta k} A \cos\left[k'(x-ct)\right] dk'$$

A constant and ck' is the dispersion relation

SOlve th integral and describe teh propogation properties of this wave packet. Assume this means that the phase and group velocities as well as width of the packet. Also explicitly show if/how the width changes in time.its easy to do this integral...

$$\Psi (x,t) = \frac{2A \Delta k}{\Delta k (x-ct)} cos[k(x-ct)] \sin[\Delta k (x-ct)]$$

Propogation properties...
WEll it seems that the wave is moving right (positive direction) by virtue of the argument of the cosine.

Width of the packet is $\frac{2 \pi}{\Delta k}$Does Delta k change in time??

i cnat think of anything to solve here... do they mean DERIVE the formulas for group and pahse velocities??

 

[StatusX]

You're right, the wave is moving right since the x,t dependence is x-ct (as opposed to x+ct), and the width doesn't change in time, since this is a wave (time only acts to translate the shape in space). The phase and group velocities are the same here since the dispersion relation is linear.