Past exam of the kinetic theory course of the University of Oxford / 2024 / 3 / e / Solution

Represent the perturber by the periodic Dirac delta function at ${\mathbf{x}}_p+{\mathbf{v}}_{p}t$. With spatial Fourier series coefficient normalized by $L^{-3}$, its nonzero density coefficients areThe mean density is subtracted as in d. The Poisson equation for Newtonian gravity givesThe perturbation is zero before switch-on. For $\mathrm{Im}\omega >0$, its Laplace transform isHenceThe switch-on creates a forcing pole at the real frequency $a$, approached causally from the original upper inversion contour.