Wave reciprocity

In a reciprocal medium, appropriately normalized source and receiver responses can be exchanged: the scalar Green function satisfies $G(\mathbf{r},{\mathbf{r}}^{\prime })=G({\mathbf{r}}^{\prime },\mathbf{r})$. Reciprocity alone does not imply absence of wave absorption.

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 satisfiesMultiplying 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.