Substitute the monochromatic wave

ϕ = e^{i (k x - ω t)}

(ik)^{9} = i k^{9}

, cancellation of the common exponential gives

- iω + i U k + \frac{i}{9} k^{9} = 0.

ω (k) = U k + \frac{k ^{9}}{9} .

ii

Solution

c_{p} (k) = \frac{ω}{k} = U + \frac{k ^{8}}{9}, c_{g} (k) = \frac{d ω}{d k} = U + k^{8} .

k

, both attain their minimum value

U

k = 0

, and both increase like

∣ k ∣^{8}

. For every

k \neq = 0

c_{g} - c_{p} = \frac{8}{9} k^{8} > 0.

The graph of

c_{g}

therefore lies above the graph of

c_{p}

away from their common minimum.

A wave crest moves at the phase velocity, while the envelope of a narrow wave packet moves at the group velocity. Since

c_{g} > c_{p}

for every nonzero wavenumber, the crests move more slowly than the packet.

There are no discussions about this article yet.

There are currently no matching articles.