Write the perturbations as $(\sigma,u,v)e^{ik_xx+\lambda t}$, set $k=|k_x|$, and evaluate all unmarked background quantities at $\Sigma_0$. Define
$$
v_s^2=\left.\frac{dP}{d\Sigma}\right|_{\Sigma_0},
\qquad
\beta=\left.\frac{d\ln(\nu\Sigma)}{d\ln\Sigma}\right|_{\Sigma_0}.
$$
The linearized <mass conservation> equation is
$$
\lambda\sigma+ik_x\Sigma_0u=0.
$$
The radial and azimuthal momentum equations are
$$
\left[\lambda+\left(\nu_b+\frac43\nu\right)k^2\right]u-2\Omega v
=-ik_x\left(v_s^2\frac{\sigma}{\Sigma_0}+\phi\right),
$$
$$
(\lambda+\nu k^2)v+(2\Omega-S)u
=-ik_xS\beta\nu\frac{\sigma}{\Sigma_0},
$$
where the <razor-thin disk Poisson kernel> gives $\phi=-2\pi G\sigma/k$. The term proportional to $\beta$ comes from perturbing the density-dependent background shear stress $T_{xy}=-\nu\Sigma S$.

Eliminating $\sigma,u,v$ and using $S=3\Omega/2$, so that the <radial epicyclic frequency> obeys $\kappa_r^2=2\Omega(2\Omega-S)=\Omega^2$, gives the <dispersion relation>
$$
\boxed{
(\lambda+\nu k^2)
\left\{\lambda\left[\lambda+
\left(\nu_b+\frac43\nu\right)k^2\right]
-2\pi G\Sigma_0k+v_s^2k^2\right\}
+\lambda\Omega^2+3\beta\Omega^2\nu k^2=0}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 321 / 1 / c / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/iii/paper-321/1/c/solution