The three terms represent competing physical effects:

* $\Omega^2$ is the stabilizing <epicyclic motion> of a radially displaced fluid element in a <Keplerian disk>.
* $-2\pi G\Sigma|k|$ is the destabilizing self-gravity of a surface-density perturbation, obtained from the <razor-thin disk Poisson kernel>.
* $c_s^2k^2$ is the stabilizing pressure response, strongest at short wavelength.

Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 321 / 2 / a / Solution

In the <razor-thin disk approximation>, the three-dimensional <Poisson equation> is
$$
\nabla^2\Phi_d=4\pi G\Sigma(x,y)\delta(z).
$$
For a horizontal <Fourier mode> with <wavevector> $\mathbf k$ and $k=|\mathbf k|>0$, the <Fourier transform> of this equation is
$$
\left(\frac{d^2}{dz^2}-k^2\right)\widehat\Phi_d
=4\pi G\widehat\Sigma\,\delta(z).
$$
The solution that decays away from the disk is
$$
\widehat\Phi_d(\mathbf k,z)
=-\frac{2\pi G}{k}\widehat\Sigma(\mathbf k)e^{-k|z|}.
$$
Consequently the required <razor-thin disk Poisson kernel> in the midplane is
$$
\boxed{\widehat\Phi_{d,m}(\mathbf k)
=-\frac{2\pi G}{k}\widehat\Sigma(\mathbf k)}.
$$
The spatially uniform $k=0$ background is excluded from this local perturbation formula.

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

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>
$$
\boxed{
(\lambda+\nu k^2)
\left\{\lambda\left[\lambda+
\left(\nu_b+\frac43\nu\right)k^2\right]
-2\pi G\Sigma_0k+v_s^2k^2\right\}
+\lambda\Omega^2+3\beta\Omega^2\nu k^2=0}.
$$

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

For a horizontal Fourier mode of <wavenumber> $k>0$, the midplane gravitational potential of a razor-thin disk is
$$
\widehat\Phi_m(\mathbf k)=-\frac{2\pi G}{k}\widehat\Sigma(\mathbf k).
$$

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