First slide

Speed of wave

Question

A wave packet with angular frequency $ω_{0}$ is propagating in dispersive medium with phase velocity of $1.5 \times 10^{3}$ m/s. When the frequency is increased by 2%, the phase velocity is found to decrease by 3%. What is the group velocity of the wave packet?

Difficult

A

$0.75 \times 10^{3} m / s$
B

$1.0 \times 10^{3} m / s$
C

$0.25 \times 10^{3} m / s$
D

$0.6 \times 10^{3} m / s$

Solution

Velocity of wave or phase velocity; $v_{p} = \frac{ω}{k}$
$\Rightarrow \frac{{dv}_{p}}{v_{p}} = \frac{dω}{ω} - \frac{dk}{k}$
$\frac{dk}{k} = \frac{dω}{ω} - \frac{{dv}_{p}}{v_{p}} = 2 % - (- 3 %) = 5 %$
Group velocity is given by,
$v_{g} = \frac{dω}{dk} = v_{p} (\frac{\frac{dω}{ω}}{\frac{dk}{k}}) = 1.5 \times 10^{3} (\frac{2}{5})$
= $0.6 \times 10^{3} m / s$

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