Axisymmetric compressible shearing-sheet waves (${\omega }^4-({\Omega }^2+c^2{k}^2){\omega }^2+{\Omega }^2{c}^2{k}_z^2=0$)

In an unstratified Keplerian shearing sheet with sound speed $c$, the four linear velocity-density equations yield ${\omega }^4-({\Omega }^2+c^2{k}^2){\omega }^2+{\Omega }^2{c}^2{k}_z^2=0$. The lower-frequency branch approaches the inertial wave relation ${\omega }^2={\Omega }^2{k}_z^2/k^2$ for $ck\gg \Omega$; the upper branch is predominantly acoustic.