Acoustic cloning

Cloning refers to producing identical copies of existing objects. Here, we experimentally show how to clone acoustic scattering objects. We acquire a digital twin and bring it back to life—a simple two-step process. First, we use broadband speakers to illuminate the scattering object within a closed receiver aperture. From these recorded reverberative data, we retrieve the object’s scattering Green’s functions using multidimensional deconvolution. In the second step, the acoustic scatterer is holographically reconstructed using the acquired scattering Green’s functions. The hologram scatters any wavefield in real time exactly like the original object would. Low-latency feedback reproduces all orders of interactions between the physical wavefield and the numerically defined hologram. This two-step process is demonstrated by cloning and modifying several rigid scatterers in a two-dimensional acoustic waveguide. Applications range from fully realistic digital scattering models to efficient metamaterial experimentation.

Figure 1

Configuration for scattering Green’s functions retrieval using multidimensional deconvolution. The sound transparent outer surface $S^{\mathrm{O}}$ and inner surface $S^{\mathrm{I}}$ are represented by dotted circles. A recording surface is indicated by blue dots and an emitting surface by red dots. In the physical experiment state (a), a possible ray path for an external excitation (outside of $S^{\mathrm{O}}$) is shown. It undergoes multiple reflections on the rigid surface of the scatterer and waveguide boundary. The desired experiment state (b) has radiation boundary conditions (dashed black circle) with the unknown Green’s functions connecting the arrivals on $S^{\mathrm{O}}$ (c) to their destination on $S^{\mathrm{I}}$. The multidimensional convolution of the incident field on $S^{\mathrm{O}}$ (c) and Green’s function (b) corresponds to the physical experiment (a), such that we can roughly represent Eqs. (1a) graphically as (a) = (b) $\ast$ (c).

Figure 2

Real-time broadband holography configuration. Dotted circles represent outer recording surface $S^{\mathrm{O}}$ (blue) and inner emitting surface $S^{\mathrm{I}}$ (red), respectively. The ingoing field on $S^{\mathrm{O}}$ in (a) is extrapolated using the scattering Green’s functions of the object to be recreated in (b). The holographic field (c) is then any physical field (a) propagating from $S^{\mathrm{O}}$ to $S^{\mathrm{I}}$, where it scatters on the immersive boundary conditions. The IBCs represent the scatterer through Green’s functions (b) as the numerical extrapolator from $S^{\mathrm{O}}$ to $S^{\mathrm{I}}$.

Figure 3

Illustration of wave propagation in the physical domain and simultaneous numerical extrapolation from ${\mathbf{\text{x}}}^o\in S^{\mathrm{O}}$ to ${\mathbf{x}}^i\in S^{\mathrm{I}}$. From two finite-difference pressure recordings (blue triangles), we calculate two finite-difference pressures to apply on the loudspeakers (red stars). Since, in the physical domain, waves can travel any ${\mathbf{\text{x}}}^o\to {\mathbf{\text{x}}}^i$, the same extrapolation has to be done in the numerical domain. Hence there are $N^{\text{rec}}\times N^{\text{emt}}=114\times 18=2052$ extrapolations at each time step.

Figure 4

Visualization of the two-step process for acoustic cloning. Both steps are further subdivided visually in Appendix pp2.

Figure 5

The physical setup is depicted in (a) and sketched in (b). The outermost circle represents a rigid boundary to the 2D waveguide. It is equipped with evenly spaced loudspeakers (red stars). The waveguide contains two sound-transparent surfaces, the outer surface $S^{\mathrm{O}}$ and the inner surface $S^{\mathrm{I}}$ (gray circles). Both surfaces are surrounded by two densely spaced auxiliary circles equipped with microphones (blue dots) or loudspeakers (red dots). For the acquisition of the MDD data, they all contain microphones, and for the holography step, $S^{\mathrm{I}}$ is equipped with loudspeakers (configuration as shown here). In (b) we placed a circular scatterer inside $S^{\mathrm{I}}$ (heterogeneous), whereas (a) is empty (homogeneous). Note that, for illustration purposes, panel (b) is not to scale.

Figure 6

The total pressure field in (a), for the heterogeneous case with circular scatterer, is separated into the incident (b) and outgoing (c) parts. This gives the decomposition $p=p^{\text{in}}+p^{\text{out}}$ or (a) = (b) + (c). The inset of each panel shows a schematic view of the setup, where the black arrows indicate the direction of wavefield propagation. Note that the data shown here represent only the subset for one source position of the illumination data used in multidimensional deconvolution. The source position at ${\varphi }_{\text{src}}=0^{\circ }$ is indicated by the red star in the inset.

Figure 7

Green’s functions retrieved with multidimensional deconvolution from reverberative data in Fig. 6. Only the subset $G({\varphi }^{\mathrm{O}},t\mid {\varphi }^{\mathrm{I}}=0^{\circ })$, i.e., the Green’s functions from the first (top) $S^{\mathrm{I}}$ position to all $S^{\mathrm{O}}$ positions, is shown. The absence of multiple scattering underlines the effectiveness of MDD in retrieving data for radiation boundary conditions. The top panels show $G^{p\mid q}$ for the heterogeneous case $G_{\text{het}}$ in (a), the homogeneous case $G_{\text{hom}}$ in (b), and their difference $G_{\text{scat}}$ in (c) containing the scattering only, i.e., $(\text{a})-(\text{b})=(\text{c})$. The bottom panels show the remaining Green’s functions of this subset, namely $G^{p\mid f}$ in (d), $G^{v\mid q}$ in (e), and $G^{v\mid f}$ in (f). The insets visualize the ray paths as well as source and receiver configurations of the Green’s functions. We use two black dots to denote a monople source or pressure recording, and a black and white dot to denote a dipole source or velocity recording. The color scale for the normal particle velocity fields is scaled by the characteristic impedance $z_0=\rho c$.

Figure 8

Recorded pressure field $p({\varphi }^{\mathrm{O}},t)$ for real-time cloning and reference. In (a),(b) we show the effect of the hologram being off and on, respectively. Note that in both of those cases the waveguide is homogeneous (empty). Any difference is the result of acting with the monopole and dipole sources on $S^{\mathrm{I}}$, i.e., the hologram. For improved visibility, we isolate the real scattering (c) = Fig. 6 - (a) and the holographic scattering $(\mathrm{d})=(\mathrm{b})-(\mathrm{a})$. Comparing the real scattering (c) with the holographic scattering (d) confirms the validity of our approach. The same comparison is provided for the square scatterer in (e),(f) and for the cross scatterer in (g),(h).

Figure 9

Comparison of the scattered angular intensities between the real scatterers and the holographically reproduced scatterers (IBC). The sum of the scattered intensities over time from the results in Fig. 8 are shown. We normalize to the highest angular scattered intensity.

Figure 10

Scattered pressure $p_{\text{scat}}({\varphi }^{\mathrm{O}},t)=p_{\text{het}}({\varphi }^{\mathrm{O}},t)-p_{\text{hom}}({\varphi }^{\mathrm{O}},t)$ for two acoustic cyborgs. The Green’s functions for the circular and square scatterer from Fig. 8 are modified in the numerical domain to exhibit gain and rotation in the clone’s scattering response. The circular scatterer in (a) is cloned and the Green’s functions numerically modified (b) to include a gain in the scattering at $\varphi =0^{\circ }$ (reflection) such that $p_{\text{scat}}^{\text{holo}}(0^{\circ },3.3\text{ms})=1.5\times p_{\text{scat}}^{\text{real}}(0^{\circ },3.3\text{ms})$, and no loss in the scattering at $\varphi ={180}^{\circ }$ (transmission) such that $p_{\text{scat}}^{\text{holo}}({180}^{\circ },3.5\text{ms})=0$ or equivalently $p_{\text{het}}^{\text{holo}}({180}^{\circ },3.5\text{ms})=p_{\text{hom}}({180}^{\circ },3.5\text{ms})$. Formally, we multiply $G_{\text{scat}}({\varphi }^{\mathrm{I}})$ with the gain function $1.5(1+\cos {\varphi }^{\mathrm{I}})/2$. The square scatterer in (c) is physically rotated by ${\varphi }^{\text{rot}}={60}^{\circ }$ as a reference for the same rotation performed in the numerical domain, shown in (d).

Figure 11

Construction of the tangential, local coordinate surface where the origin coincides with the central point of the segment $S^o\subset S^{\mathrm{O}}$.

Figure 12

Wavefield separation method in the frequency-wavenumber domain.

Figure 13

Green’s function retrieval using MDD. Furthermore we have $G\equiv \{G^{p\mid q},G^{p\mid f},G^{v\mid q},G^{v\mid f}\}$, $N^{\text{src}}=52$, and $N^{\text{emt}}=18$.

Figure 14

Real-time broadband holography. With $G\equiv \{G^{p\mid q},G^{p\mid f},G^{v\mid q},G^{v\mid f}\}$, $N^{\text{src}}=52$, and $a+=b$ the addition assignment $a=a+b$.