= Solution
Let $\mathbf u=-(3/2)\Omega x\mathbf e_y+\mathbf v$ and $\rho=\rho_0(1+\delta)$, with $\delta$ and $\mathbf v$ small. Define the <sound speed> by $c^2=(dp/d\rho)_0$, assumed positive. The background <Keplerian shear> balances the radial <Coriolis force> and the <shearing-sheet tidal potential>. For disturbances preserving <axisymmetry> $\partial_y=0$, its advective derivative $\mathbf u_0\cdot\nabla$ vanishes, but $\mathbf v\cdot\nabla\mathbf u_0=-(3/2)\Omega v_x\mathbf e_y$ remains. The linearized <Euler equations> and <continuity equation> are
$$
\boxed{\begin{aligned}
\partial_t v_x-2\Omega v_y&=-c^2\partial_x\delta,\\
\partial_t v_y+\frac\Omega2v_x&=0,\\
\partial_t v_z&=-c^2\partial_z\delta,\\
\partial_t\delta+\partial_xv_x+\partial_zv_z&=0.
\end{aligned}}
$$
The azimuthal coefficient is $(2-3/2)\Omega=\Omega/2$, not $2\Omega$: it combines the background velocity gradient with the <Coriolis force>. The uniform background has no buoyancy or vertical stratification in this stated model.
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