For a mode $e^{i\mathbf k\cdot\mathbf x+st}$, put $D=s+\nu k^2$. The linear equations become
$$
Du_x-2\Omega u_y=-ik_xP/\rho_0-N^2\theta,\quad
Du_y+\frac12\Omega u_x=0,\quad
Du_z=-ik_zP/\rho_0,
$$
with $s\theta=u_x$ and $k_xu_x+k_zu_z=0$. Eliminating $u_y,u_z,P,\theta$ gives the <dispersion relation>
$$
s(D^2+\varpi^2)+n^2\varpi^2D=0,
\qquad
\varpi^2=\frac{k_z^2}{k^2}\Omega^2,\quad n^2=\frac{N^2}{\Omega^2}.
$$
Expanding,
$$
\boxed{s^3+2k^2\nu s^2+
\left[\varpi^2(1+n^2)+k^4\nu^2\right]s
+n^2k^2\nu\varpi^2=0}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 321 / 3 / c / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/iii/paper-321/3/c/solution