In a normal wave, the velocity of wave is given by v = \frac{\omega}{k} where symbols have usual meaning. This is also known as phase velocity. Many a times waves are formed by superposition of a large number of waves with different w and k. In such cases, we see that they form a wave packet which moves with a different velocity known as group velocity. This figure shows a superposition of many such waves. This envelope moves with a velocity known as group velocity. vgroup = \frac{dω}{dk}

Engineering

Physics

Dispersion

Question

In a normal wave, the velocity of wave is given by v = $\frac{ω}{k}$ where symbols have usual meaning. This is also known as phase velocity. Many a times waves are formed by superposition of a large number of waves with different w and k. In such cases, we see that they form a wave packet which moves with a different velocity known as group velocity.

This figure shows a superposition of many such waves. This envelope moves with a velocity known as group velocity. v_group = $\frac{dω}{dk}$

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Linked Question 1

Certain ocean waves travel with phase velocity v = $\sqrt{\frac{gλ}{2 π}}$ . What is the group velocity of these waves in terms of v ?

$\frac{v}{2}$

$\frac{v}{4}$

$\frac{ω}{k} = v = \sqrt{\frac{g}{k}}$

$ω = \sqrt{gk}$

$v_{gr} = \frac{dω}{dk} = \sqrt{g} \times \frac{1}{2 \sqrt{k}} = \frac{1}{2} \sqrt{\frac{g}{k}} = \frac{v}{2}$

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Linked Question 2

When light travels through glass, phase velocity of each wavelength depends on wavelength as $\frac{c}{v} = A + \frac{B}{λ^{2}}$ . (A and B are positive constant, c is velocity of light in vaccum)

Can't be said

The phase velocity for all colors is equal to group velocity.

The phase velocity for all colors is less than group velocity.

The phase velocity for all colors is greater than group velocity

$\frac{ω}{k} = v = \frac{c}{A + \frac{B}{λ^{2}}} λ = \frac{2 π}{k}$

$ω = \frac{kc}{A + \frac{B}{λ^{2}}} = \frac{kc}{A + \frac{B}{4 π^{2}} k^{2}}$

$\frac{dω}{dk} = \frac{c (A + \frac{B}{4 π^{2}} k^{2}) - kc \times (\frac{B}{4 π^{2}} \times 2 k)}{{(A + \frac{B}{4 π^{2}} k^{2})}^{2}}; v_{gr} = \frac{Ac - \frac{{Bk}_{C}^{2}}{4 π^{2}}}{{(A + \frac{{Bk}^{2}}{4 π^{2}})}^{2}}$

$v_{gr} = \frac{c (A - \frac{B}{λ^{2}})}{{(A + \frac{B}{λ^{2}})}^{2}} < \frac{c}{A + \frac{B}{λ^{2}}}$

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