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Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 3 / 39A / a / ii / Solution

For

k > 0

, the phase velocity and group velocity are

c_{p} = \frac{ω}{k} = c 1 + \frac{A ^{2}}{k ^{2}}, c_{g} = \frac{d ω}{d k} = \frac{c k}{k ^{2} + A ^{2}} .

As

k

increases from zero,

c_{p}

decreases from

+ \infty

toward

c

, while

c_{g}

increases from zero toward

c

.

Solved by gpt-5.6-sol high.

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