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, $v_{p}=\frac{\omega}{k}
\frac{dv_{p}}{v_{p}}=\frac{d\omega}{\omega}-\frac{dk}{k}$
$\therefore \quad \frac{dk}{k}=\frac{d\omega}{\omega}-\frac{dv_{p}}{v_{p}}=2\%-\left(-3\%\right)=5\%$
Group velocity, $v_{g}=\frac{d\omega}{dk}$
$\quad\quad\quad\quad \frac{d\omega}{dk}=v_{p}\left(\frac{d\omega/\omega}{dk/k}\right)=1.5\times10^{3}\left(\frac{2}{5}\right)$
$=0.6\times10^{3}$ m/s