Manipulating Acoustic Wavefront by Inhomogeneous Impedance and Steerable Extraordinary Reflection

We unveil the connection between the acoustic impedance along a flat surface and the reflected acoustic wavefront, in order to empower a wide wariety of novel applications in acoustic community. Our designed flat surface can generate double reflections: the ordinary reflection and the extraordinary one whose wavefront is manipulated by the proposed impedance-governed generalized Snell's law of reflection (IGSL). IGSL is based on Green's function and integral equation, instead of Fermat's principle for optical wavefront manipulation. Remarkably, via the adjustment of the designed specific acoustic impedance, extraordinary reflection can be steered for unprecedented acoustic wavefront while that ordinary reflection can be surprisingly switched on or off. The realization of the complex discontinuity of the impedance surface has been proposed using Helmholtz resonators.

as the design rule of SAI. In addition, we mathematically predict the double reflections and the situation when the ordinary reflection can be switched off.
We assume the time-harmonic factor in this appendix is e −iωt , where ω is the circular frequency, and the coordinate system is that in Fig. 1(a). The incident acoustic pressure can be expressed as: where k 0 = ω/c 0 is the wave number in free space, θ i the incident angle and p i0 (ω) the amplitude.
where β(y, ω) = ρ 0 c 0 /Z n (y, ω) (ρ 0 and c 0 being the given density and sound speed respectively in the upper space) is the normalized acoustical admittance of the locally reacting surface.
We expand β to be β(y, ω) =β(y, ω) + β 0 (ω), where β 0 is a real constant. The ordinary reflection is expressed as: where R is the reflection coefficient and θ ro the angle of p ro . Because p ro observes the usual Snell's law, θ ro = θ i . In order to find the expression of R, we introduce the constant SAI: where n is the normal vector indicated in Fig. 1(a), v i and v ro the acoustic velocities of p i and p ro . Substituting Eq.
(1) and Eq.(3) into Eq.(4) and applying Euler equation ρ 0 ∂ ∂t v = −∇p, we obtain: In Fig. 1(a), the total acoustic field can be written in the integral form: where dl(y 0 , z 0 ) is the infinitesimal segment along the integral contour, n 0 = n(y 0 , z 0 ) and G(y, z, ω; y 0 , z 0 ) is the Green's function corresponding to the following partial differential problem: When the radius of the semicircular contour S approaches ∞, we can regard the contour integral along S is mainly contributed by p i and p ro . Therefore Eq.(6) changes into We can simplify Eq.(8) by substituting Eq.(7) and Eq.
(2) into it. By defining the last part in Eq.(9) as the extraordinary reflection p re (y, z, ω), which is the unique extra component beyond p ro , we obtain The explicit solution of G(y, z, ω; y 0 , z 0 ) in Eq. (7) is where r = (y, z), r 0 = (y, z), and k 0 2 = k y 2 + k z 2 . When r is away from the surface D, k z ≈ k 0 cos θ * holds, where θ * is introduced as a constant. Via this approximation and another definition r † 0 = (y 0 , −z 0 ), it turns out that [3] cos Through Eq.(11), it can be obtained that Applying Eq.(12) into Eq.(10) and using the formula of the cylindrical wave expansion in terms of plane waves, we approach a neat form of the Green's function: where H (1) 0 (·) the Hankel function of the first kind [4]. From the physical insight into Eq.(13), the first part of G is the direct contribution of the point source to the observer through path 2 in Fig. 1(b). The second part is the product of the Green's function excited by the image source and the reflection coefficient R, denoting p ro . According to our interpretation, Fig. 1(b) illustrates path 1 and path 2, visualized as p ro and p re respectively [5]. Due to the expression of R, we figure out that θ * is the effective incident angle regarding to Fig. 1(b). Furthermore, it is reasonable to say that the major contribution of the integral in Eq.(10) is attributed to the vicinity of θ * , in which way R can be regarded as a constant and put outside the integral.

II. DIFFERENCES BETWEEN GSL AND IGSL
Although GSL is not our topic in this paper, the same appearance of IGSL and GSL may cause the false impression that our IGSL is the same as GSL. Actually their mechanisms are totally different.
In terms of phase inhomogeneity, the anomalous reflection p ra actually corresponds to the situation when the ordinary reflection p ro is steered toward a "wrong" direction governed by GSL [6], illustrated in Fig. 2(a). There is only one single direction of reflection all the while. On the contrary in terms of SAI inhomogeneity, it is found that IGSL cannot alter p ro by an SAI interface, but can "turn off" p ro so as to provide insight into the engineering of special wavefronts by SAI interface, illustrated in Fig. 2(b). Moreover, the extraordinary reflection p re governed by IGSL is an additionally unique component in acoustics, which can be "geared" along arbitrary directions, simultaneously with vanishing p ro . Therefore, our proposed IGSL opens up rich effects and unprecedented applications in the community of acoustics. Additionally, GSL can even be phenomenologically considered as one subset of IGSL, when p ro is turned off. In order to stress the irrelevance between IGSL and GSL again, we list the differences: 1. GSL is initiated in electromagnetism with electric properties; IGSL is initiated in acoustics with mechanical properties. 2. GSL is derived Only reflected acoustic pressure is plotted. The propagating path of pre is noted as an arrow with purple crossbars. upon p re . 6. In GSL, the anomalous reflection corresponds to the situation where p ro is tweaked toward a different direction governed by GSL; in acoustics, IGSL cannot alter p ro by SAI interfaces, but is capable of "turning on" or "turning off" p ro .