Pure modes. In the absence of sources, the growing mode of the rest frame temperature perturbation matches onto a cosine acoustic oscillation inside the horizon, whereas the decaying mode matches onto a sine oscillation. The oscillator response to an external source is constructed by Greens method from these homogeneous solutions and transformed into the Newtonian frame with Eq.~(9).

Compensation and isocurvature fluctuations. (a) Baryon isocurvature model ($\Phi_s \propto x^{-1}$). Outside the horizon $x \simlt 1$, backreaction from the photons cancels the contribution of the source to the curvature fluctuation. Inside the horizon, pressure prevents significant metric contributions from the photon-baryon fluid and $\Phi \rightarrow \Phi_s$. (b) For source functions $\Phi_s \propto x^{p}$, the ratio of $\RF_a$ to $\RF_b$ amplitudes decreases due to feedback. This leaves the acoustic oscillation mainly in the sine mode at $x \gg 1$.

{Driven oscillations. The self-gravity of the photon-baryon fluid drives a cosine oscillation for adiabatic initial conditions (thin lines) and a sine oscillation (thick lines) for isocurvature initial conditions. The adiabatic model has $\Phi_s(x_i)=0$ and $\Phi_{\gamma b}(x_i)\ne0$. The isocurvature model is the baryonic model of Eq.~(19). The dashed lines show the full potential, the solid lines the effective temperature. In both cases the amplitude in $\Theta_0+\Psi$ increase during the first few oscillations, as described in \S2.5.