= Coherent initial phases are needed for acoustic variance peaks
{title2=$P_\delta=e^{-2k^2/k_D^2}(P_A\cos^2q+P_B\sin^2q+2\operatorname{Re}P_{AB}\sin q\cos q)$}
A damped mode $\delta=e^{-k^2/k_D^2}(A\cos q+B\sin q)$ does not alone imply oscillations in its ensemble <power spectrum>. Averaging its modulus squared gives the displayed formula. Equal independent initial variances, $P_A=P_B$ and $P_{AB}=0$, erase the oscillatory variance through $\cos^2q+\sin^2q=1$. A coherent initial phase, as supplied by a regular growing <adiabatic mode>, selects one common oscillator phase and retains acoustic peaks. Its leading density and velocity oscillations are a quarter-period apart, and angular projection also preserves their correlated contribution.
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