= Weighted acoustic Green-function reciprocity
{title2=$a(x)G(x,y;s)=a(y)G(y,x;s)$}
For an operator $c^{-2}\partial_t^2-a^{-1}\nabla\cdot(a\nabla)$ with positive stationary coefficients and reciprocal boundary conditions, its delta-normalized <Green function> satisfies
$$
a(x)G(x,y;s)=a(y)G(y,x;s).
$$
Multiplying its frequency-domain equation by $a$ and using the bilinear <Green second identity> proves the relation. Both Green functions must use the same outgoing or causal prescription. Exchanging source and receiver preserves time lag, and involves no complex conjugation.
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