= Solution
Represent the perturber by the periodic <Dirac delta function> at $\mathbf x_p+\mathbf v_pt$. With spatial <Fourier series> coefficient normalized by $L^{-3}$, its nonzero density coefficients are
$$
\rho_{p,\mathbf k}(t)=\frac{m_p}{L^3}e^{-i\mathbf k\cdot\mathbf x_p}e^{-i\mathbf k\cdot\mathbf v_pt},\qquad t\geq0.
$$
The mean density is subtracted as in <3/d>. The <Poisson equation for Newtonian gravity> gives
$$
\delta\Phi_{\mathbf k}^{\mathrm{ext}}(t)=C_{\mathbf k}e^{-iat},\qquad C_{\mathbf k}=-\frac{4\pi Gm_p}{L^3k^2}e^{-i\mathbf k\cdot\mathbf x_p},\qquad a=\mathbf k\cdot\mathbf v_p.
$$
The perturbation is zero before switch-on. For $\operatorname{Im}\omega>0$, its <Laplace transform> is
$$
\int_0^\infty e^{i(\omega-a)t}\,dt=-\frac1{i(\omega-a)}=\frac1{i(a-\omega)}.
$$
Hence
$$
\boxed{\widetilde{\delta\Phi}_{\mathbf k}^{\mathrm{ext}}(\omega)=-\frac{4\pi Gm_p}{L^3k^2}e^{-i\mathbf k\cdot\mathbf x_p}\frac1{i(\mathbf k\cdot\mathbf v_p-\omega)}.}
$$
The switch-on creates a forcing pole at the real <frequency> $a$, approached causally from the original upper inversion contour.
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