= Solution
A blackbody photon density perturbation gives an intrinsic temperature fluctuation $\Theta_0=\delta_\gamma/4$. The baryon/electron bulk velocity produces a <Doppler effect> at last scattering. In the common-potential convention of part (ii),
$$
\theta_b\simeq\theta_\gamma=\frac3{4k^2}\delta_\gamma'.
$$
Ignoring the slowly varying damping derivative at leading acoustic order, differentiating the density cosine produces a velocity sine. Thus \b[density and velocity acoustic sources are approximately a quarter cycle out of phase]. For regular adiabatic initial conditions the density oscillation has approximately a common cosine phase across modes; this phase coherence is what makes a sequence of peaks instead of a smooth incoherent superposition.
Let $D_*=\tau_0-\tau_*$ be the comoving distance to last scattering and $r_s(\tau_*)=\int^{\tau_*}c_s\,d\tau$ its <sound horizon>. In the present limit $r_s\simeq\tau_*/\sqrt3$. Density extrema have $kr_s\simeq m\pi$, whereas velocity extrema have $kr_s\simeq(m+1/2)\pi$. The projection from spatial wavenumber to high angular multipoles is roughly $\ell\simeq kD_*$, giving an acoustic spacing
$$
\boxed{\Delta\ell\simeq\pi\frac{D_*}{r_s}.}
$$
The <Doppler effect> fills some density troughs and modifies the detailed peak shapes; its different projection prevents treating it as an identical second series of narrow peaks.
For clarity, choose the sightline to point outward from the observer and let its angle to $\mathbf k$ have cosine $\mu$. The intrinsic-plus-emitter-Doppler source is then $[\delta_\gamma/4-ik\mu\theta_b]e^{ik\mu D_*}$. Expanding its angular dependence gives the <angular projection of density and Doppler acoustic sources>
$$
\Theta_\ell(k)\simeq\frac{\delta_\gamma(k,\tau_*)}{4}j_\ell(kD_*)
-k\theta_b(k,\tau_*)j_\ell'(kD_*).
$$
The second sign follows from the chosen outward sightline and $v_b^i=ik^i\theta_b$; it changes with the sightline convention together with the velocity projection. The $j_\ell$ are <Spherical Bessel functions>. Both sources are proportional to the primordial curvature. If $\mathcal T_\ell=\Theta_\ell/\zeta_*$, their angular spectrum in the usual continuum Fourier normalization is
$$
C_\ell=\frac2\pi\int_0^\infty k^2dk\,P_\zeta(k)|\mathcal T_\ell(k)|^2.
$$
This formula includes the density-velocity cross term, not just a sum of independent source powers.
Photon diffusion multiplies both leading source amplitudes by $e^{-k^2/k_D^2}$, so their power carries $e^{-2k^2/k_D^2}$. The approximate angular damping scale is
$$
\boxed{\ell_D\simeq k_D(\tau_*)D_*;\qquad\text{high-multipole power is exponentially suppressed}.}
$$
This produces the damping tail in the <Cosmic microwave background power spectrum>. Finite last-scattering thickness adds further small-scale averaging. Gravity, baryon loading and later propagation alter detailed peak locations and heights, but the oscillatory peak sequence, phase-shifted velocity contribution and diffusion damping follow directly from the simplified solution obtained here.
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