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