ϕ = e^{i (k x - ω t)}

- ω^{2} + c^{2} k^{2} + A^{2} c^{2} = 0.

ω (k) = c k^{2} + A^{2} .

ii

Solution

For

k > 0

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}} .

k

increases from zero,

c_{p}

decreases from

+ \infty

toward

c

, while

c_{g}

increases from zero toward

c

Since

c_{p} c_{g} = c^{2},

the phase speed is always larger than the group speed for nonzero finite

k

. Wave crests therefore move faster than the packet envelope.

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