Constant intensity acoustic propagation in the presence of non-uniform properties and impedance discontinuities: Hermitian and non-Hermitian solutions
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.