Expand the <density contrast> in <Fourier mode>[Fourier modes], $\delta(\mathbf x,t)=\int\delta_{\mathbf k}(t)e^{i\mathbf k\cdot\mathbf x}\,d^3k$. In the static universe each mode satisfies
$$
\ddot\delta_{\mathbf k}+c_s^2(k^2-k_J^2)\delta_{\mathbf k}=0.
$$
For $k>k_J$ there are <acoustic wave>[sound waves]
$$
\delta_{\mathbf k}=A_{\mathbf k}e^{i\omega_k t}+B_{\mathbf k}e^{-i\omega_k t},
\qquad
\omega_k=c_s\sqrt{k^2-k_J^2}.
$$
For $k<k_J$, <Jeans instability> gives growing and decaying exponentials with rate $c_s\sqrt{k_J^2-k^2}$. At the <Jeans wavenumber> the two solutions are constant and linear in time.

Solution

Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 3 / 14D / i / Solution

{wiki}

An acoustic wave is a propagating pressure and density disturbance in a material medium. Linear sound waves oscillate when pressure restores a displaced fluid element.


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 Incoming links: Acoustic wave