## 6 Metadiffuser Equivalent Surface Impedance in a Large Space
Later reflections integrated up to 2 s of the impulse responses (orange to blue areas) show an increase of sound energy for many directions of arrival due to a more chaotic state of sound reflections within the environment at those time steps, resulting in a relatively uniform angular late sound field distribution. Still, in most directions, the energy of the late sound field remains 12 dB or more below the initial energy recorded directly in the front and in the back of the receiver. For both early and late spatio-temporal curves, a spatial autocorrelation coefficient for each time window can be estimated in a similar way to the diffusion coefficient in Equation (3). In such manner, the closer the coefficient is to unity, the more uniform the spatial distribution. The ratio of the early to the late spatial autocorrelation coefficients can provide an early-to-late diffuseness coefficient [46],  
\[\mathcal{D}_{e/l}=(1-\delta_{e}/\delta_{l}), \tag{7}\]  
where \(\delta_{e}\) and \(\delta_{l}\) are the early and late spatial autocorrelation coefficients, respectively. This formulation describes the rate of sound field isotropy between early and late integration times. In such case, \(\mathcal{D}_{e/l}\to 0\) implies that the evolution of the early-to-late diffuseness is non-existent, i.e., that both sound fields are identical, whereas \(\mathcal{D}_{e/l}\to 1\) indicates a maximum increase of isotropy between early and late diffuse sound fields, with the early sound field approximating that of a plane wave in free-field conditions and the late sound field approximating a spherical distribution. In the absence of any sound diffuser, in the early [0-20] ms window, this results in an early autocorrelation coefficient \(\delta_{e}=0.0056\), while for the late time window of [0-2] s a value \(\delta_{l}=0.0223\) can be seen. This shows that the early sound field is less uniform than the overall late sound field, which is to be expected in an environment where specular reflections of sound are dominant. In Figure 6c, an early-to-late diffuseness coefficient \(\mathcal{D}_{e/l}=0.75\) is shown, illustrating a great increase in isotropy between the early and late diffuse sound fields. Similar observations can be made for the median (\(xz\)) and lateral (\(yz\)) planes in Figure 6d,e, respectively. In the median plane, an early-to-late diffuseness ratio \(\mathcal{D}_{e/l}=0.56\) is achieved, showing less difference between early and late spatial distributions. This is supported by the open-air nature of the orchestra pit, where little extra reflection directions are enabled. In Figure 6e, a lower early-to-late diffuseness \(\mathcal{D}_{e/l}=0.90\) can be identified for the lateral plane, bringing similar features than those encountered in the transverse plane, i.e., very narrow early spatial distribution significantly widening up in late integration times. In the second pit configuration, an alternative scenario is proposed with clusters of 6 \(\times\) 3 repeated RLC fitted metadiffusers sparsely distributed along the walls. The intention behind this strategy is to distribute more sound energy in the early reflection regime of the pit, which incidentally may also help enhance the acoustic conditions required for musicians to experience a more suitable acoustical comfort while performing in such environment. The use of equivalent surface impedance is again motivated by the limiting constraints of modelling compact and detailed geometrical structures in a large 3D FDTD numerical scenery. The above incentive is illustrated in Figure 6d-f which show the spatio-temporal plots obtained in the second pit configuration for the transverse, median, and lateral planes, respectively. In the transverse plane in Figure 6d, the early time integration area (red) demonstrates the arrival of strong reflections coming from broader directions than in the previous configuration with just rigid walls.