Exact result for nonreciprocity in one-dimensional wave transmission

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For sound waves impinging on a one-dimensional medium, we show that nonlinearity can lead to nonreciprocal transmission, without dissipation or broken time reversal invariance. Placing quasi-monochromatic filters at the ends of the nonlinear medium, nonreciprocity can be obtained without the generation of higher harmonics outside the medium. Remarkably, in this configuration the nonreciprocity is found to be proportional to the net energy flow when monochromatic sources of equal strength (at the filter frequency) are simultaneously turned on at both ends. This result is conjectured to be general for one dimensional scattering. It is also shown that although simultaneous monochromatic sources lead to net energy flow, with sources of small but non-zero bandwidth there is no net energy transport, in accordance with the second law of thermodynamics.