The directional properties of the source and receiver are often referred to as radiation patterns. To apply the reciprocity principle it is necessary to regard the radiation patterns as attached to the medium, not as being attached to the source and receiver. These general ideas are made more precise in the following derivation. It will be seen that the notion of rays actually turns out to be irrelevant. Reciprocity also works in diffusion and potential problems.
Theoretical treatments are often somewhat hard to read.
They often begin by specifying that the differential operator along with suitable boundary conditions should constitute a self-adjoint problem. This means that when you reexpress the differential equations in difference form you discover that the matrix of coefficients is symmetric. Let us take the example of acoustic waves in one dimension. Newton's equation says that mass density times accelerations equals the negative of the pressure gradient plus the external force F x . Luckily, in this case reciprocity will be exactly true despite the small number of grid points.