  Propagation of sound through a non-uniform medium without scattering is
possible, in principle, if the density and acoustic compressibility assume
complex values, requiring passive and active mechanisms, also known as
Hermitian and non-Hermitian solutions, respectively. Two types of constant
intensity wave conditions are identified: in the first the propagating acoustic
pressure has constant amplitude, while in the second the energy flux remains
constant. The fundamental problem of transmission across an impedance
discontinuity without reflection or energy loss is solved using a combination
of monopole and dipole resonators in parallel. The solution depends on an
arbitrary phase angle which can be chosen to give a unique acoustic
metamaterial with both resonators undamped and passive, requiring purely
Hermitian acoustic elements. For other phase angles one of the two elements
must be active and the other passive, resulting in a gain/loss non-Hermitian
system. These results prove that uni-directional and reciprocal transmission
through a slab separating two half spaces is possible using passive Hermitian
acoustic elements without the need to resort to active gain/loss energetic
mechanisms.
