Angular projection of density and Doppler acoustic sources (${\Theta }_{\ell }\sim \frac{{\delta }_{\gamma }}{4}{j}_{\ell }(k{D}_{\ast })-k{\theta }_b{j}_{\ell }^{\prime }(k{D}_{\ast })$)

For an outward sightline, Fourier velocity $v_b^i=i{k}^i{\theta }_b$, and sudden last scattering, intrinsic photon temperature projects with a Spherical Bessel function, while the Doppler term projects with its derivative. Density and velocity acoustic sources are approximately a quarter cycle out of phase. Their amplitude is suppressed by photon diffusion, giving a power damping envelope $e^{-2k^2/k_D^2}$. Projection maps the sound scale to peak spacing approximately $\pi D_{\ast }/r_s$ and the damping scale to angular multipoles of order $k_D{D}_{\ast }$.