\overset{ˉ}{P}_{r} = \overset{ρ}{ˉ}_{r} /3

and

δ P_{r} = δ ρ_{r} /3

, so both terms proportional to the equation-of-state difference vanish. During matter domination the gravitational potential is constant,

Φ^{'} = 0

θ_{r} = \nabla \cdot v_{r}

, the continuity and Euler equations reduce to

δ_{r}^{'} = - \frac{4}{3} θ_{r}, θ_{r}^{'} = - \frac{1}{4} \nabla^{2} δ_{r} - \nabla^{2} Φ.

Differentiate the first equation and substitute the second:

δ_{r}^{''} = \frac{1}{3} \nabla^{2} δ_{r} + \frac{4}{3} \nabla^{2} Φ.

Thus the radiation perturbation obeys the forced acoustic equation

δ_{r}^{''} - \frac{1}{3} \nabla^{2} δ_{r} = \frac{4}{3} \nabla^{2} Φ .

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