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A pointlike pulse can contain frequencies above the compactification gap even when its travel time and distance greatly exceed the circle size. Its retarded circle-compactified massless propagator can resolve winding images and massive-mode oscillations. Suppressing those modes requires low-frequency or long-wavelength sources/measurements, or suitable smearing; mere coordinate separation is not that restriction. The static Green function across a circle compactification gives a precise long-distance zero-mode limit without this ambiguity.

Define $\Delta t=t-t^{\prime }$, $\mathbf{r}=\mathbf{x}-{\mathbf{x}}^{\prime }$ in the three noncompact spatial directions, and $\Delta w=w-w^{\prime }$. The compact source must be the periodic Dirac deltaFor the wave operator as explicitly defined in the question, a retarded circle-compactified massless propagator isThe poles $\omega =\pm \sqrt{{\mathbf{k}}^2+k_n^2}-i0$ are both below the real axis, making the integral vanish for $\Delta t<0$. Performing just the frequency integral gives an equivalent expressionThe first time derivative of $\Theta (t)\sin (\Omega t)/\Omega$ jumps by one at zero, so applying ${\partial }_t^2+{\Omega }^2$ yields $\delta (t)$. Together with the completeness relations for the Fourier series and Fourier transform this verifies the specified source and normalization. The symbol for the wave operator follows the question's explicit sign convention; it is the negative of $g^{AB}{\nabla }_A{\nabla }_B$ for its mostly-plus metric tensor.