Barotropic shearing-wave gravitational amplitude system (${\dot{k}}_x=S{k}_y,{\tilde{\Phi }}_{d,m}=-2\pi G{\tilde{\Sigma }}^{\prime }/k$)

A shearing-wave ansatz in a uniform inviscid isothermal shearing sheet removes background advection when the radial wavevector grows as $k_x(t)=k_x(0)+S{k}_{y}t$. The razor-thin disk Poisson kernel reduces the pressure-plus-gravity amplitude to $[c_s^2-2\pi G{\Sigma }_0/k]{\tilde{\Sigma }}^{\prime }/{\Sigma }_0$. Continuity couples it to $i{k}_x{v}_x+i{k}_y{v}_y$, 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.