= Solution
Neglecting baryon loading, the <Sachs-Wolfe combination> obeys a harmonic-oscillator equation with sound speed $1/\sqrt3$. <Adiabatic initial conditions> give a nonzero initial displacement and negligible initial velocity, hence
$$
S(k,\tau_*)=S(k,0)\cos(kr_s),
\qquad
r_s=\int_0^{\tau_*}c_s,d\tau.
$$
Projection maps $k$ approximately to $\ell/\chi_*$, and the angular power is quadratic in the transfer function, producing the observed approximate $\cos^2(kr_s)$ sequence of <Cosmic microwave background acoustic peak>[acoustic peaks]. A $\sin^2$ phase would instead indicate vanishing initial displacement and nonzero initial velocity, characteristic of an isocurvature rather than adiabatic primordial mode.
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