= Adjoint reciprocity for a weighted Helmholtz operator
{title2=$L^{\mathrm T}u=\nabla\cdot[f\nabla(u/f)]+k^2u$}
For smooth positive $f$, let $L=f^{-1}\nabla\cdot(f\nabla)+k^2$. Its unweighted bilinear transpose is the displayed operator. If $v$ satisfies $\alpha v+\beta\partial_nv=0$, its transpose boundary condition is $\alpha u+\beta(\partial_nu-u\partial_n\log f)=0$. The resulting Green kernels satisfy $G_L(\mathbf r_2,\mathbf r_1)=G_{L^{\mathrm T}}(\mathbf r_1,\mathbf r_2)$. Equivalently $G_{L^{\mathrm T}}(\mathbf r,\mathbf r_2)=f(\mathbf r)G_L(\mathbf r,\mathbf r_2)/f(\mathbf r_2)$, exhibiting <weighted acoustic Green-function reciprocity>.
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