= Barotropic shearing-wave gravitational amplitude system
{title2=$\dot k_x=Sk_y,\quad\widetilde\Phi_{d,m}=-2\pi G\widetilde\Sigma'/k$}
A <shearing-wave ansatz> in a uniform inviscid isothermal <shearing sheet> removes background advection when the radial <wavevector> grows as $k_x(t)=k_x(0)+Sk_yt$. The <razor-thin disk Poisson kernel> reduces the pressure-plus-gravity amplitude to $[c_s^2-2\pi G\Sigma_0/k]\widetilde\Sigma'/\Sigma_0$. Continuity couples it to $ik_xv_x+ik_yv_y$, while the two momentum equations have rotational couplings $-2\Omega v_y$ and $(2\Omega-S)v_x$. The <linearized vortensity> amplitude remains constant, because the barotropic <pressure> and gravitational forces have zero <curl> and the background <vortensity> is uniform.
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