Acoustic WKB amplitude in an expanding universe (${\delta }_k\propto [a\sqrt{\omega }{]}^{-1},\omega =k{c}_s/a$)

For pressure-dominated linear cosmological density perturbations, ${\ddot{\delta }}_k+2H{\dot{\delta }}_k+{\omega }^2{\delta }_k=0$. Writing ${\delta }_k=\chi /a$ gives $\ddot{\chi }+({\omega }^2-\ddot{a}/a)\chi =0$. In the WKB approximation, the oscillation amplitude is $1/(a\sqrt{\omega })$. For constant sound speed it is proportional to $a^{-1/2}$; the local frozen-frequency amplitude damping time $H^{-1}$ does not imply a universal global envelope $a^{-1}$.