 Solution

ID: past-exam-of-the-waves-and-compressible-flow-course-of-the-university-of-oxford/2023/2/b/ii/solution

Choose the rightgoing signed dispersion relation

ω (k) = N k / k^{2} + b_{n}^{2}

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}

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

, approached as

k \to 0

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