= Comoving perturbations of a homologously collapsing cloud
{title2=$\mathbf k(t)=\mathbf q/R(t)$}
A <Fourier mode> advected with a <homologous spherical flow> has phase $\mathbf q\cdot\mathbf r/R(t)$. Its <material derivative> in the background vanishes because $\dot{\mathbf k}=-(\dot R/R)\mathbf k$. Thus contraction shortens its physical wavelength while preserving its phase on each fluid element. Linearizing the <continuity equation> and pressureless <momentum> equation gives $\dot\delta=-i\mathbf k\cdot\mathbf u_1$ and $\dot{\mathbf u}_1+(\dot R/R)\mathbf u_1=-i\mathbf k\Phi_1$. Combining these with the <Poisson equation for Newtonian gravity> yields $\ddot\delta+2(\dot R/R)\dot\delta-4\pi G\rho\delta=0$ for nonzero compressive <wavenumber>. It is a local bulk calculation, without the moving cloud-boundary conditions.
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