A wave packet with center frequency ω is propagating in dispersive medium with phase velocity of 1.5 x 10³ m/s. When the frequency ω is increased by 2%, the phase velocity is found to decrease by 3 %. What is group velocity of the wave packet?

0.25 × 10³ m/s

1.0 × 10³ m/s

0.6 × 10³ m/s

0.75 × 10³ m/s

Solution

0.6 × 10³ m/s

Phase velocity:- It is the ratio of the angular frequency to the wave number.

v_p = $\frac{ω}{k}$

k = wave number

ω = angular frequency

$\frac{{dv}_{p}}{v_{p}} = \frac{dω}{ω} - \frac{dk}{k}$

∴ $\frac{dk}{k} = \frac{dω}{ω} - \frac{{dv}_{p}}{v_{p}}$

= 2 % $- (- 3 %)$

$\frac{dk}{k}$ = 5%

Group velocity:- it is the derivative of the angular frequency with respect to the wave number.

Group velocity is given by

v_g = $\frac{dω}{dk}$

$\frac{dω}{dk} = v_{p} (\frac{\frac{dω}{ω}}{\frac{dk}{k}})$

= 1.5 × 10³ $(\frac{2}{5})$

$\frac{dω}{dk}$ = 0.6 × 10³m/s

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