Question

A wave packet with center frequency $\omega$ is propagating is dispersive medium with phase velocity of $1.5\times {10}^{3}m/s$. When the frequency $\omega$ is increased by $2$%, the phase velocity is found to decrease by $3$%. What is the group velocity of the wave packet?

• A. $0.25\times {10}^{3}m/s$
• B. $1.0\times {10}^{3}m/s$
• C. $0.6\times {10}^{3}m/s$
• D. $0.75\times {10}^{3}m/s$

Answer 1

The correct option is C $0.6\times {10}^{3}m/s$

Phase velolcity, $v_p=\frac{\omega }{k}$
$\frac{d{v}_p}{v_p}=\frac{d\omega }{\omega }-\frac{dt}{k}$
$\therefore \frac{dk}{k}=\frac{d\omega }{\omega }-\frac{d{v}_p}{v_p}=2$% $-(-3$%) $=5$%
Group velocity, $v_g=\frac{d\omega }{dk}$
$\frac{d\omega }{dk}=v_p\left(\frac{d\omega /\omega }{dk/k}\right)=1.5\times {10}^3\left(\frac{2}{5}\right)$
$=0.6\times {10}^{3}m/s$.