Acoustic asymmetric transmission based on time-dependent dynamical scattering

An acoustic asymmetric transmission device exhibiting unidirectional transmission property for acoustic waves is extremely desirable in many practical scenarios. Such a unique property may be realized in various configurations utilizing acoustic Zeeman effects in moving media as well as frequency-conversion in passive nonlinear acoustic systems and in active acoustic systems. Here we demonstrate a new acoustic frequency conversion process in a time-varying system, consisting of a rotating blade and the surrounding air. The scattered acoustic waves from this time-varying system experience frequency shifts, which are linearly dependent on the blade’s rotating frequency. Such scattering mechanism can be well described theoretically by an acoustic linear time-varying perturbation theory. Combining such time-varying scattering effects with highly efficient acoustic filtering, we successfully develop a tunable acoustic unidirectional device with 20 dB power transmission contrast ratio between two counter propagation directions at audible frequencies.

Measuring the effective parameters of the time-varying medium is significant in interpreting our results. After a more in-depth consideration of this kind of measurement, we discovered several technical difficulties, which prevent us from obtaining the experiment proof.
1) The acoustic field of the effective region might be inhomogeneous, and thus just directly measuring field at several locations in the effective region cannot reflect the whole scenario.
2) With the limits of present measurement techniques available, it is nearly impossible to directly measure the whole sound field of the time-varying region as a function of time.
Neither laser interferometer nor laser Doppler can directly measure the dynamic parameters of this optical reflection-less air medium. Schlieren device has little use here because the acoustic field in our experiment is too weak to be measured. Because the phase distribution is not a stable one, detector array also cannot work in such a dynamical time-varying case for imaging the acoustic field. 3) Transient transmission and reflection method for deducing parameters has its own problem here. In experiment, recording time of signal is finite. Take a sinusoidal signal with a length of T  as an example (Eq. (1)), which resembles the case in our experiment. (1) The amplitude of wave is obtained by Fourier Transform. The frequency component of signal, however, is no longer a single frequency. The frequency spectrum will have a which can also be expressed as If the recording time of signal is too short (to deduce the time-varying parameters, time axis of transient signals should be cut to the small enough parts to match the modulation frequency), we could not get the exact amplitude and phase of pressure in the effective rotation region, which means it is impossible to use this method deducing the effective time-varying parameters.
Taking all the above factors into consideration, we resort to the quasi-static measurement for the effective parameters as shown in Fig.S.3. Since air turbulence in the waveguide caused by the elliptic shape of the blade is minimal, slowly varying approximation with quasi-static effective parameters can still be used in our analysis. Our transmission experimental results in fact well support our theoretical findings.
Since the reviewer still has concerns about our quasi-static method, we feel it is necessary for us to elaborate it in more details and hope this will clarify some of the issues that the reviewer may have in his/her mind. The effective parameters [Phys. Rev. B. 76, 144302 (2007)] of the time-varying region in our work, such as density and acoustic velocity, are deduced from acoustic transmission and reflection fields by utilizing the continuity equation. In theory, as p Z v  g where p is sound pressure, Z is acoustic impedance of medium and v is vibration velocity, both sound pressure and vibration velocity can be used to deduce effective parameters. Nevertheless, the vibration velocity of a medium, e.g., air, subjected to an acoustic field is too difficult to be measured directly. Laser interferometer or laser Doppler vibration meter may be used if there is measurable optical reflection, which is not the case here for air. An alternative way used by many scientists is to measure sound pressure directly with microphones for fluid media like air. Once the phase and amplitude of the pressure fields before and after the time-varying medium (shown in Fig.S.2) are known. The effective parameters of the time-varying medium can be derived. Detailed steps are discussed as follows.

S.2. Schematic of a waveguide section with three regions.
Medium in region 1 and region 3 is air while region 2 contains the time-varying medium. Here, the length D of the effective medium region is 0.02m in both simulation and experiment. The blade is placed in the center of this region.
A three-region waveguide structure under study is shown in Fig.S.2. Medium in region 1 and region 3 is static air while region 2 contains the time-varying medium. We will adopt the continuity equation to deduce the effective mass density of this time-varying medium.
Step One: We need to figure out the expressions for the pressure and velocity in every region. Expressions of region 1: In region 1, sound pressure of incident acoustic waves ( i p ) and air's vibration velocity of incident acoustic waves ( i v ) can be written as: In region 1, sound pressure of reflected acoustic waves ( 1r p ) and air's vibration velocity of reflected acoustic waves ( 1r v ) can be written as: Expressions of region 2: In region 2, sound pressure of transmission acoustic waves ( 2t p ) and air's vibration velocity of transmission acoustic waves ( 2t v ) can be written as: In region 2, sound pressure of reflected acoustic waves ( 2t p ) and air's vibration velocity of reflected acoustic waves ( 2t v ) can be written as: Expressions of region 3: In region 3, sound pressure of transmission acoustic waves ( t p ) and air's vibration velocity of transmission acoustic waves ( t v ) can be written as: Step Two: By applying boundary conditions at boundary 0 x  , we get the following two equations, Similarly at the boundary x D  , we get Acoustic pressure, acoustic impedance of medium and vibration velocity of medium are related according to, where 1 is the acoustic impedance of medium in region 1; where 2 2 2 Z c   is the acoustic impedance of medium in region 2; where 1 is the acoustic impedance of medium in region 3; By substituting Eq. (11), (12) and (13) into Eq. (9) and Eq. (10), we get: Step Three: By substituting Eq. (17) where  is the angular frequency of incident waves.
It is easy to extract the pressures of incident, transmitted and reflected pressure fields to retrieve the effective parameters in simulation (see Fig. S (1 ) (1 ) By repeating the same procedure at the blade's different rotation angle, we can deduce the angle-dependent effective parameters as shown in Fig. S.3. Despite the absolute value for measured and simulated effective modulus and density are slightly different due to the loss in the experiments, harmonically varying effective parameters as a function of the rotation angle is clearly seen in both experiment and simulation. To conclude, the deduced effective parameters for the time-varying medium are consistent with the simulation to great extent, and their periodic change in time as a result of rotation is responsible for the phenomenon of dynamic modulation or dynamic scattering observed in our work. We would love to measure the effective parameters directly. The current available technology, however, prevents us from doing that. In contrast, the quasi-static model we adopted in this work works well, and the simulation results obtained from such model matches with the experimental transmission spectrum to large extent.
In the presence of rotation, the linear acoustic equations can be written as Here equations (20), (21) and (22) where *  is the cofactor of  .
Considering that effective mass density is dependent on the rotation angle with the approximate cosine profile, so 1 ( ) H t can be expressed as: In the above analysis, we have set The coefficient n a must satisfy When single frequency acoustic wave enters the system, the initial condition satisfies The above equation means that the initial system energy level will be split to create two additional sub energy levels, of which the angular frequency are determined according to the following expression,

Details for Experiments
As shown in Figure.S.4, the acoustic waveguide is made of aluminum alloy. Its length is 1.5m and its rectangular cross-section has an outside dimension 25 100 mm mm 

S.5. Acoustic absorption coefficient for various termination conditions.
Since 145-mm-thick sponge has the best absorption performance among these four termination conditions, it was used exclusively in our experimental investigation to seal the waveguide.
S.6. Experimental results for acoustic transmission and energy conversion ratio through the elliptical blade as a function of loudspeaker input voltage. 0