= Low-frequency decomposition of the wave propagator
{title2=$\widehat E(\xi,t)=\sin(ct|\xi|)/(c|\xi|)$}
For zero initial displacement and unit delta initial velocity, the spatial <Fourier transform> of the <wave equation> gives the displayed multiplier. A compact <cutoff function> around zero frequency produces an ordinary smooth function after inversion. The remaining sine splits into two <oscillatory integrals> with phases $x\cdot\xi\pm ct|\xi|$ and symbols proportional to $(1-\chi)/|\xi|$ of order $-1$. Their stationary directions lie on $|x|=c|t|$, proving a light-cone bound for <singular support>. This does not claim vanishing of smooth tails inside the cone.
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