Question

A wave packet with center frequency $\omega$ is propagating in dispersive medium with phase velocity of $1.5\times 10^{3}m/s$. When the frequency w 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 × 10³ m/s
• B. 1.0 × 10³ m/s
• C. 0.6 × 10³ m/s
• D. 0.75 × 10³ m/s

Answer 1

Answer: (C)

Phase velocity, $v_p=\frac{\omega }{k}<br/>\frac{d{v}_p}{v_p}=\frac{d\omega }{\omega }-\frac{dk}{k}$
$\therefore \frac{dk}{k}=\frac{d\omega }{\omega }-\frac{d{v}_p}{v_p}=2\%-\left(-3\%\right)=5\%$
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