The Helmholtz reciprocity principle, which states that a received signal is the same when source and receiver positions are reversed, does not apply to an acoustic system that undergoes the Doppler effect, even in one dimension. The proof advanced here uses a time-domain representation of the Doppler effect, and illustrates the physical cause of the nonreciprocity: Although the fundamental equations are reciprocal, the boundary conditions incur nonreciprocity if they are time-dependent. More generically, the acoustic Doppler effect in the time domain shows that reciprocity failure can arise from the noncommutativity of events along the signal path.
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