## 6 Metadiffuser Equivalent Surface Impedance in a Large Space  
For this work, an orchestra pit is chosen for a large scene in which to embed the proposed equivalent surface impedances in a 3D FDTD simulation. The geometry of the orchestra pit of general dimensions [x, y, z](8, 20, 2.5) is idealised as shown in Figure 6a, where a sound source \(S\) located at [x, y, z](2, 10, 1.5) and a receiver \(R\) located at [x, y, z](6, 10, 1.5) are highlighted. Note here that the pit is virtually isolated from the exterior environment of what would be the rest of an opera house. The considered orchestra pit is simulated following two different scattering strategies implemented on the walls. In the first situation, no particular scattering on the boundaries is considered, i.e., the walls are simply assumed perfectly rigid. The FDTD simulation grid resolution was set to 10.5 PPW at 8 kHz, resulting in 4.8 billion computational elements, in order to obtain a numerical dispersion errors less than 1% [41] below such frequency. The simulations involving flat panels (no scatterers) required 30 GB of memory computed in parallel using Nvidia CUDA spread over four Nvidia Titan X GPU cards (Maxwell architecture). Simulations times with flat panels were approximately 55 min for 0.5 s of simulated response. Including the more complex RLC boundary conditions, the FDTD simulation took 65 min and required 3% more memory running on the same GPUs. Thus, the equivalent surface impedance incurs some extra minimal simulation costs (as expected from [29]), and it is also much smaller relative to the simulation costs expected for a full-fledged multiphysics simulation (taking into account QRM details and physics) in this space up to the chosen frequency resolution. The impulse responses captured within this environment are analysed by means of a Spatial Decomposition Method (SDM) [42], which allows the determination of the directions of arrival (DOA) [43; 44] of sound events in a 3D set of spatial impulse responses. The latter are captured through a microphone array and can be windowed over different integration times to show the evolution of the spatial sound field with respect to time. In this case, a \(3\times 3\times 3\) virtual microphone array was used for recording the numerical impulse responses at the receiver location, and is displayed in Figure 6b. For data processing, the SDM Toolbox [45] made available by the Virtual Acoustics Team at Aalto University, Finland, was used. The impulse responses cover a frequency range \(\Delta_{f}=\) [20:8000] Hz  
Figure 6: (**a**) Geometry of an ideal orchestra pit used in a 3D FDTD simulation. (**b**) Sample impulse response with different integration times used for the Spatial Decomposition Method (SDM). Spatio-temporal distributions of sound field energy (\(10\log_{10}|p_{s}|\)) for each coordinate plane in the orchestra pit received at location \(R\) from sound source \(S\) in (**c**–**e**) a pit with flat boundaries, and (**f**–**h**) a pit with sparsely distributed repetitions of equivalent surface impedance (ZEQ) patches. and are integrated over several incrementing time windows so that the cumulative energy of the impulse responses can be observed through time. These span from [0-200] ms to [0-2000] ms to cover most of the recorded information, as shown in Figure 6b. Ultimately, the spatial sound field for each time window can be plotted along the 3 orthonormal polar planes, i.e., lateral, transversal and median planes. Figure 6c displays the spatio-temporal response at \(R\) in the transverse plane (\(xy\)-plane). It can be observed that early acoustic energy coming in the first 20 ms (red area) comes mostly from the front, where the sound source is located, with a significant contribution from the back as well due to the wall reflection.