## GCPE-M example: reciprocity

According to the acoustical reciprocity principle, the CW acoustic
pressures measured in reciprocal transmissions scheme must be identical in
the absence of flows (see [1] for more rigorois
formulation and generalizations). Flows break reciprocity, but the flow
reversal theorem (FRT) also establishes some symmetry. Namely, in moving
media the fields in reciprocal transmissions will be identical if the
*direction of flow* in reverse propagation is changed to the opposite.
Within the parabolic approximation reciprocity is closely related to
the energy conservation [1, 2].

Reciprocity of the model is crucial if you plan to study impact of
currents on the sound propagation. Oceanic currents are slow and can be
measured acoustically by observing reciprocity breaking effects. To simulate
this process on a computer the model itself must satisfy reciprocity
principle and FRT. Existing PE models are (at best) only approximately
reciprocal. The differences in acoustical pressures predicted for reciprocal
transmissions are often comparable to the effects of currents.

To illustrate reciprocal capabilities of GCPE-M, the figure below shows
acoustic amplitudes and phases predicted for reciprocal transmissions over a
rugged wedge. The final 40 m of the propagation range are shown. The plot
confirms that GCPE-M model comply with the reciprocity principle. In the
example considered the agreement is within the accuracy of round-off errors:
about 10^{-10} dB in amplitude and 10^{-10} degrees in
phase.

[1] Godin, O. A. Reciprocity and energy theorems
for waves in a compressible inhomogeneous moving fluid. *Wave motion*,
**25**, p. 143-167, 1997.

[2] Godin, O. A. Reciprocity and energy
conservation within the parabolic approximation. *Wave motion*,
**29** (2), p. 175-194, 1999