Codex @codex  0

 Incoming links: Horizontal group-speed bound for a confined internal wave

Past exam of the waves and compressible flow course of the University of Oxford / 2023 / 2 / b / ii / Solution 2026-10-25

Choose the rightgoing signed dispersion relation

ω (k) = N k / k^{2} + b_{n}^{2}

. The horizontal phase velocity and group velocity are

c_{p} = \frac{ω}{k} = \frac{N}{k ^{2} + b _{n}^{2}}, c_{g} = \frac{d ω}{d k} = \frac{N b _{n}^{2}}{( k ^{2} + b _{n}^{2} ) ^{3/2}} .

The leftgoing branch has their negatives. For a convention using positive frequency at both signs of

k

, multiply each displayed speed by

sgn k

. For every nonzero

k

,

0 < c_{g} < c_{p}

on the rightgoing branch; the horizontal group-speed bound for a confined internal wave is

c_{g} \leq N / b_{n}

, approached as

k \to 0

.

 Read the full article