# The Potential of Focusing Acoustic Retroreflectors for Architectural Surface Treatment
## 4 Discussion  
### Discussion of Results  
As has been observed in prior studies [44; 52], the design of non-focusing acoustic retroreflector arrays involves balancing numerosity with size, considering diffraction loss constraints in relation to a target low frequency roll-off (diffraction loss being size and distance dependent). This is evident in the results of the present study, with optimum patch sizes at various non-focusing retroreflector array distances. Putting aside diffraction, a non-focusing retroreflector array returns superimposed images of the reflector patches that are four times the patch area (Figure 3), so large patches project over a large area rather than concentrating only on the source. However, _focusing_ retroreflectors change the array design considerations, because a single patch can concentrate much of the reflected sound at the source. This means that a larger size is not necessarily disadvantageous, as is evident from the results. Multiple focusing retroreflectors yield a more consistent spatial response than a single focusing retroreflector, so there still is an optimum range of sizes for a given source-reflector distance. Even away from the true focus of any patch, a source has a potentially high concentration of sound returned to it from multiple patches, notwithstanding diffraction loss. Also observed in prior studies has been a high frequency bias in reflections from non-focusing acoustic retroreflector arrays, and the present results indicate that this bias is potentially stronger (or extended) when combined with focusing. Prior studies of real acoustic retroreflector arrays in architectural contexts generally see most effects in high octave bands; 2 kHz and above [44; 49; 50; 52]. Fortunately, the spectrum slope from non-focusing retroreflector diffraction loss applies most to small patches, whereas that from  
Figure 16: Spatial distribution of reflected energy level over a 2.4 \(\times\) 2.4 m square for 25 source positions 2 m from reflectors composed of parabolic trihedra (details in the text). Source locations are indicated by the green circles. Values are speech-A-weighted. focusing applies most to large patches in the scenarios tested, so that spectral slopes do not exceed 6 dB/octave. Although not the main theme of the analyses in this paper, the temporal distribution of reflected sound from non-focusing and focusing retroreflector arrays raises issues for consideration. Assuming that retroreflectors are passive devices, their latency is limited by the speed of sound, which introduces a minimum delay proportional to their size. This is one of the counter-indicators against very large focusing retroreflectors, which could become echo-like if a single strong reflection arrives with excessive delay. Retroreflective arrays introduce clusters of arrivals in the impulse response, which might paradoxically be characterized as a focused diffuse reflection. While focusing retroreflector arrays have less contribution from non-adjacent reflectors than their non-focusing counterparts, they do retain this cluster impulse response characteristic (Figure 10). This desirable characteristic of reflections is linked to the reason why focusing retroreflector arrays return less sound than the theoretical maximum coherent pressure focus (Equation (4)), which can only be achieved if all reflected sound arrives at once. The simulation results are similar to theoretical predictions. For simple specular and single-patch focusing cases, ideal simulation results are consistently less than the corresponding theoretical expressions in the Introduction, but all within 1 dB. While the theory is limited by simplification, and ideal case simulation has limitations from discretization and error propagation, results concur sufficiently for the context of architectural acoustics. Furthermore, the parabolic trihedral reflector results are credibly close to those of comparable ideal reflector simulations.