Write the perturbations as $\delta\Sigma=\sigma$, radial and azimuthal velocities $u,v$, and pressure $\delta P=p$, all proportional to $e^{ikx+st}$. Linearization in the <shearing sheet> gives
$$
s\sigma+ik\Sigma_0u=0,
$$
$$
su-2\Omega v=-ik\left(\frac p{\Sigma_0}+\phi\right),
\qquad
sv+\frac12\Omega u=0,
$$
and
$$
sp=-ik\gamma c_0^2\Sigma_0u
-\frac{p-c_0^2\sigma}{\tau},
\qquad
\phi=-\frac{2\pi G}{|k|}\sigma.
$$
The continuity and pressure equations imply
$$
\frac p\sigma=c_0^2\frac{\gamma\tau s+1}{\tau s+1}.
$$
Eliminating $\sigma,u,v,p,\phi$ then yields
$$
\boxed{s^2+\Omega^2-2\pi G\Sigma_0|k|
+c_0^2k^2\left(\frac{\gamma\tau s+1}{\tau s+1}\right)=0}.
$$

When $\tau|s|\ll1$, temperature relaxes during a disturbance and the pressure response is isothermal, $p/\sigma\simeq c_0^2$. When $\tau|s|\gg1$, relaxation is negligible and the response is adiabatic, $p/\sigma\simeq\gamma c_0^2$, so the effective sound speed is $\sqrt\gamma\,c_0$. At finite relaxation time the phase lag between compression and pressure also damps stable waves.

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 321 / 3 / b / ii / Solution

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The shearing sheet is a local Cartesian approximation to a differentially rotating disk. It retains Coriolis force, linearized tidal gravity, and a background linear shear.


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 Incoming links: Shearing sheet