= Solution
After a spatial <Fourier transform>, the preceding equation is
$$
\delta_r''+\frac{k^2}{3}\delta_r=-\frac43k^2\Phi.
$$
The constant particular solution is $\delta_r=-4\Phi$. Hence the <Sachs-Wolfe combination>
$$
S=\frac14\delta_r+\Phi
$$
contains only the homogeneous <harmonic oscillator> modes and satisfies
$$
S''+\frac{k^2}{3}S=0.
$$
For <adiabatic initial conditions> on large scales, $S=\Phi/3=\mathcal R/5$ and $S'=0$ at the start of matter domination. Up to an irrelevant choice of the phase origin,
$$
\boxed{S(k,\tau)=\frac15\mathcal R(k,0)
\cos\left(\frac{k\tau}{\sqrt3}\right)},
\qquad
\boxed{T_S(k,\tau)=\frac15
\cos\left(\frac{k\tau}{\sqrt3}\right)}.
$$
Thus the <radiation acoustic transfer function> is flat at $1/5$ as $k\to0$ and oscillates on smaller scales. Projection onto the sky produces a large-angle Sachs-Wolfe plateau and, after the full photon-baryon physics is included, the acoustic structure of the <Cosmic microwave background power spectrum>.
The <baryon transfer function> should also oscillate soon after <cosmological recombination>: before decoupling, Thomson scattering forced baryons to share the acoustic motion of the <photon-baryon fluid>, and recombination does not instantly erase the resulting density pattern.
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