Answer to Question #132822 in Mechanics | Relativity for alfraid

Question

write the relation between phase and group velocity

Answer 1

Phase velocity(wave velocity) is the velocity of monochromatic wave.Velocity propagated when a single wave of definite wave length travel in a medium.

Given by;

$V\scriptscriptstyle p$ =$\dfrac{\omega}{k}$ where k is a constant.

Group velocity: velocity of a number of wave of different wavelength moving with a different velocity in medium.

Given by:

$V\scriptscriptstyle g$ =$\dfrac{d\scriptscriptstyle \omega}{d\scriptscriptstyle k}$

Where wave frequency is given by;

$\omega$ =k$V\scriptscriptstyle p$

Differentiating wave frequency with respect to k;

$V\scriptscriptstyle g$ =$\dfrac{d }{d\scriptscriptstyle k}($ $KV\scriptscriptstyle p$ )

$V\scriptscriptstyle g$ = $V\scriptscriptstyle p$ +$k\dfrac{dV\scriptscriptstyle p}{d\scriptscriptstyle k}$

Taking k =$\dfrac{2\pi }{\lambda}$

$\because$ $\dfrac{d\scriptscriptstyle k}{d\scriptscriptstyle\lambda}$ =$-\dfrac{2\pi }{\lambda^2}$

$d\scriptscriptstyle k$ =$-k\dfrac{d\lambda }{\lambda}$

$V\scriptscriptstyle g$ =$V\scriptscriptstyle p$ $-\lambda\dfrac{dV\scriptscriptstyle p }{d\lambda}$