= Solution
Let $\mathcal H=a'/a$ and $\theta_N=i\mathbf k\cdot\mathbf v_N$. Choose the <synchronous gauge> rest frame of <cold dark matter>, so $\theta_C=0$. Its continuity equation then gives $h'=-2\delta_C'$. Substitution in the trace Einstein equation gives
$$
\boxed{\delta_C''+\mathcal H\delta_C'
-\frac32\mathcal H^2(\Omega_C\delta_C+2\Omega_R\delta_R)=0.}
$$
For <radiation in cosmology> the continuity and Euler equations are $\delta_R'+4\theta_R/3+2h'/3=0$ and $\theta_R'=k^2\delta_R/4$. Differentiate the first, eliminate $\theta_R'$ and $h''$, and obtain
$$
\boxed{\delta_R''+\frac{k^2}3\delta_R-\frac43\delta_C''=0.}
$$
<radiation in cosmology> <pressure> supplies the acoustic restoring term while its <energy density> contributes twice as strongly to the trace gravitational source through $1+3w_R=2$.
Deep in <radiation domination>, $a\propto\tau$, $\mathcal H=1/\tau$, $\Omega_R\simeq1$ and $\Omega_C\simeq0$. At long wavelengths $k\tau\ll1$, the <adiabatic initial conditions> require $\delta_C=3\delta_R/4$. The cold-matter equation reduces to
$$
\delta_C''+\frac1\tau\delta_C'-\frac4{\tau^2}\delta_C=0.
$$
A power $\tau^p$ gives $p^2-4=0$. The regular growing solution therefore has
$$
\boxed{\delta_C=A(\mathbf k)\tau^2,\qquad
\delta_R=\frac43A(\mathbf k)\tau^2,\qquad k\tau\ll1.}
$$
The other adiabatic solution diverges as $\tau^{-2}$ and is discarded for the regular primordial growing mode. The relation between the <density contrasts> fixes their relative entropy to zero, not their subsequent equality inside the acoustic horizon.
One can also solve the leading radiation-era equations explicitly, which proves that the same regular mode develops logarithmic cold-matter growth after horizon entry. Put $x=k\tau/\sqrt3$. The first equation gives $\delta_R=(x^2\delta_{C,xx}+x\delta_{C,x})/3$. Substituting in the second gives
$$
x^2\delta_{C,xxxx}+5x\delta_{C,xxx}+x^2\delta_{C,xx}+x\delta_{C,x}=0.
$$
Define
$$
F(x)=\int_0^x\frac{1-\cos t}{t}dt+
\frac{\sin x}{x}-\frac{1-\cos x}{x^2}-\frac12.
$$
Its useful derivative is $F'=1/x-2\sin x/x^2+2(1-\cos x)/x^3$. Direct differentiation verifies the fourth-order equation and gives the associated <radiation in cosmology> solution:
$$
\delta_C=K(\mathbf k)F(x),\qquad
\delta_R=\frac K3\left[-2\cos x+\frac{4\sin x}x
-\frac{4(1-\cos x)}{x^2}\right].
$$
At zero, $F=x^2/8+O(x^4)$ and $\delta_R=Kx^2/6+O(x^4)$, so this is precisely the regular <adiabatic radiation-era cold-dark-matter transfer solution> with $K=24A/k^2$. The other indicial behaviors of the fourth-order equation are $1,x,x^{-2}$; they are either nonadiabatic at leading order or singular and do not add another regular growing adiabatic mode.
For $x\gg1$, the integral is $\gamma+\ln x-\operatorname{Ci}(x)$, where $\gamma$ is the <Euler's constant> and $\operatorname{Ci}$ the <cosine integral>. Its oscillatory tail cancels the explicit leading $\sin x/x$ term, giving
$$
\delta_C=K\left[\ln x+\gamma-\frac12+O(x^{-2})\right],\qquad
\delta_R=-\frac{2K}3\cos x+O(x^{-1}).
$$
Thus the requested late sound-horizon behavior is
$$
\boxed{\delta_C\simeq B(\mathbf k)\ln(\tau/\tau_*),\qquad
\delta_R\simeq C(\mathbf k)\cos(k\tau/\sqrt3)+D(\mathbf k)\sin(k\tau/\sqrt3).}
$$
An additive constant in $\delta_C$ is absorbed into the reference $\tau_*$; the regular adiabatic solution fixes the acoustic phase, with $D=0$ in the explicit time-origin convention above. A general acoustic mixture has both terms. The physical expansion parameter is $k\tau/\sqrt3$; comparisons with $2\pi/k$ describe the same long- and short-wavelength regimes up to fixed numerical factors.
The contrast grows much more slowly after entry during <radiation domination>: <radiation in cosmology> undergoes pressure-supported acoustic oscillations and cold matter only the logarithmic <Mészáros effect>. Modes entering well before <matter-radiation equality> consequently have suppressed late amplitudes relative to modes that stay outside the horizon until near equality. After <matter domination> begins, pressureless growing modes can grow as $a$. This scale-dependent history produces the turnover and small-scale suppression of the matter transfer function and affects the initial conditions for <large-scale structure of the universe>. The early subhorizon limit requires both $k\tau\gg1$ and $\tau\ll\tau_{\rm eq}$; it is not applicable to a mode that enters only in the <matter domination>.
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