# Doppler-Based Acoustic Gyrator
## 2 Acoustic Phase Shifter  
In this section, we focus our attention on realizing a non-reciprocal acoustic phase shifter that brings about different phase shifts for forward and backward waves. As a starting point, consider the geometry represented in Figure 1b consisting of an acoustic waveguide filled with air on which a velocity of \(\
u=\
u_{0}\hat{y}\) is assumed to be imparted. The overall length of the waveguide is supposed to be \(L=7\lambda=73\) cm, where \(\lambda\) is the wavelength of the operation. When the fluid inside the waveguide is stationary, i.e., when \(\
u_{0}=0\), the signals travelling from the left to the right side of the waveguide acquire an overall phase shift of zero, as the waveguide length is a multiple of the wavelength. Similarly, sound waves injected into the right side of the waveguide are subject to an effective phase shift of zero when reaching the left side, which is consistent with the reciprocity of the device. When assuming a non-zero bias velocity motion (\(\
u_{0}\
eq 0\)), however, the frequency of the signals propagating from the left to the right is Doppler shifted by an amount of \(\Delta f=-\
icefrac{{\
u_{0}}}{{\lambda}}\). Consequently, they obtain an additional phase shift of \(\varphi_{1}=\frac{2\pi\
u_{0}L}{c\lambda}\) (\(c\) is the speed of sound in the air) during their propagation (the red arrow in the figure). Figure 2a (the red line) illustrates the variation of \(\varphi_{1}\) versus the fluid velocity \(\
u_{0}\). The figure reveals that by changing the bias velocity, one can readily tune the overall phase shift that a signal obtains when passing through the waveguide. However, this is not something special by itself, as the same effect could be achieved by changing the length of the waveguide. The salient feature of the proposed structure becomes clear when we look at the propagation of the waves that travel in the opposite direction. In contrast to the previous case, the frequency of these types of signals is upshifted by an amount of \(\Delta f=\
icefrac{{\
u_{0}}}{{\lambda}}\) with respect to the static frequency \(f_{0}=\
icefrac{{\epsilon}}{{\lambda}}\). The corresponding Doppler phase shift is therefore \(\varphi_{2}=-\frac{2\pi\
u_{0}L}{c\lambda}\) in this case, which is negative of \(\varphi_{1}\). Figure 2a (the blue line) plots the dependence of \(\varphi_{2}\) on the fluid motion velocity \(\
u_{0}\). It is evident that the motion of the fluid within the waveguide results in different phase shifts for forward and backward sound waves, making the structure non-reciprocal in phase. Notably, it was not possible to achieve such non-reciprocal behavior by solely changing the length of the waveguide. We further note that since there is no geometrical detour causing reflection or refraction, and since the amount of sound attenuation is quite low at the frequency of operation, the amplitude of the transmitted signals maintains almost the same value, regardless of the propagation direction. One can therefore express the scattering matrix of the waveguide as:  
\[\text{S}=\left( \begin{array}{cc}0&\text{e}^{\text{i}\varphi_{1}}\\ \text{e}^{\text{i}\varphi_{2}}&0\end{array} \right) \tag{1}\]  
Note that the scattering matrix of the waveguide is unitary since we neglected the losses. In order to verify the scattering matrix of the above equation, we perform numerical simulations using the aeroacoustics module of Comsol Multiphysics, which is a full-wave solver working based on finite element method (FEM). In our numerical simulations, we surround the waveguide by sound hard wall boundary conditions, and terminate it at plane wave scattering boundary conditions inducing the excitation. We further assume the material filling the waveguide to be air with a density of \(\rho_{0}=1.2\) kg/m\({}^{3}\) and the bulk modulus of \(K=142\) kPa. Note that the choice of the filling medium is only for the sake of simplicity.