$\mathcal{P}\mathcal{T}$-Symmetric Scattering in Flow Duct Acoustics

We show theoretically and experimentally that the propagation of an acoustic wave in an airflow duct going through a pair of diaphragms, with equivalent amounts of mean-flow-induced effective gain and loss, displays all the features of a parity-time ($\mathcal{P}\mathcal{T}$) symmetric system. Using a scattering matrix formalism, we observe, experimentally, the properties which reflect the $\mathcal{P}\mathcal{T}$ symmetry of the scattering acoustical system: the existence of spontaneous symmetry breaking with symmetry-broken pairs of scattering eigenstates showing amplification and reduction, and the existence of points with unidirectional invisibility.

Figure 1

(a) Sketch of the acoustic $\mathcal{P}\mathcal{T}$-symmetric system in an airflow duct. (b) Corresponding 1D model. (c) Numerical simulation with Large Eddies Simulation [47] of the flow in the diaphragm in the presence of an acoustic wave.

Figure 2

Real and imaginary part of the measured impedance parameters $C_1$ and $C_2$. When the imaginary part of $C_2$ is negative the diaphragm gets some gain. At the frequency $f=f_m$, gain and loss are balanced $C_2=C_1^{*}$.

Figure 3

(a) Modulus of the scattering coefficients. (b) Norm of the deviation from acoustic energy conservation property and from the $\mathcal{P}\mathcal{T}$-symmetry property of the scattering matrix. Symbols correspond to experimental measurements and solid lines correspond to the 1D theory.

Figure 4

Spontaneous symmetry breaking of the scattering matrices. (a) Green points: measurements, green line: 1D theory. (b) Blue points: measurements, blue line: 1D theory. In each plot, the red lines correspond to the two SVDs of the scattering matrix. Dashed lines correspond to $|t|=1$, i.e., the phase transition for ${\mathsf{S}}_t$.

Figure 5

(a) Finite periodic case with $N$ cells, green (red) is a scatterer with loss (gain). (b) Transmission coefficient as a function of $kD$ and $kW$ for $N=25$. White regions correspond to band gaps. (c) Transmission for $kW/2\pi =2.1$.