= Rotating acoustic point source
{title2=$\dot\Theta=\Omega/(1-m\cos\Theta)$}
A point source moving at angular speed $\Omega$ on a circle of radius $a$ has observer-direction <Mach number> $m=\Omega a\sin\iota/c_0$. In the far field its retarded phase satisfies $\Theta+\theta-\pi/2=\Omega(t-r/c_0)+m\sin\Theta$. For $m<1$ there is one retarded root, and a constant source strength gives $\rho'=q\dot\Theta/(4\pi c_0^2\Omega r)$. For $m>1$, multiple retarded roots can occur; their contributions require $|1-m\cos\Theta|$ and a sum over roots. Coalescing roots produce a <caustic> in the ideal point-source model.
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