# The Potential of Focusing Acoustic Retroreflectors for Architectural Surface Treatment
## 1 Introduction
### Theoretical Perspectives on Reflected Sound Level  
#### 1.1.1 Reflected Sound Level from a Single Reflection  
The sound returned to a source by a single reflection from an extensive flat hard surface can be simply estimated from Equation (1), where \(r_{\text{min}}\) is the minimum distance from source to the surface (normal incidence), and the result is expressed relative to the emitted sound at 1 m. For example, a surface 2 m from the source yields \(-12\) dB, which is independent of frequency.  
\[L_{\text{retro}}=-20\log(2r_{\text{min}}) \tag{1}\]  
For a finite surface, the reflected sound calculation can also account for diffraction (Equation (2)). Diffraction coefficient \(D\) is frequency-dependent, and is approximately calculated via Equation (3) (limited to the simple case of a square reflector in free space with edge lengths \(l_{x}=l_{y}\) and the source-receiver in line with its centroid), where \(\lambda\) is wavelength, and FresnelC and FresnelS are cosine and sine Fresnel integrals, respectively [42]. For a large surface with a collocated source-receiver in line with the surface's centroid and within a few meters from it, \(D\approx 1\) over the speech frequency range. However, diffraction losses become substantial at low frequencies if small or distant surfaces subtending a small solid angle are considered. Furthermore, diffraction has a ripple response that may boost the reflected sound in parts of the spectrum.  
\[\frac{p_{\text{retro}}^{2}}{p_{\text{1m}}^{2}}=\frac{D}{4r_{\text{min}}^{2}} \tag{2}\]  
\[D=2\Bigg{(}\text{FresnelC}\bigg{(}\frac{l_{x}}{\sqrt{\lambda r_{\text{min}}}} \bigg{)}^{2}+\text{FresnelS}\bigg{(}\frac{l_{x}}{\sqrt{\lambda r_{\text{min}}}} \bigg{)}^{2}\Bigg{)} \tag{3}\]  
The maximum possible sound returned to a source from a single reflection can be calculated by considering a source at the center of a large perfectly reflecting sphere. The reflected pressure at the focus can be expressed via Equation (4) [43]. The squared pressure increases with squared wave number (\(k\)), i.e., at 6 dB/octave. When considering the first reflection only, the size of the sphere does not affect the pressure magnitude, assuming the sphere is sufficiently large. Theta is the angle over which a spherical cap covers (\(0\leq\vartheta\leq\pi\)), with \(\pi\) representing the complete sphere.  
\[\frac{p_{\text{focus}}^{2}}{p_{\text{1m}}^{2}}=p_{\text{1m}}^{2}k^{2}(1-\cos \vartheta)^{2} \tag{4}\]  
Analyses in this paper are evaluated using combined omnidirectional speech-weighting and A-weighting (Section 2.1.3), for which the maximum sound at the focus from Equation (4) is 32.7 dB (\(\vartheta=\pi\)) relative to the emitted sound pressure level at 1 m (calculated over the 125 Hz-8 kHz octave bands). For a hemisphere (\(\vartheta=\pi/2\)), the sound returned to the focus is 26.6 dB.  
Another ideal scenario could be that emitted sound is returned to the source, but incoherently. This ideal can be evaluated by energetically summing the sound that the surface is exposed to, leading to a simple frequency-independent relationship with the surface's solid angle, \(\Omega\) (Equation (5)).  
\[L_{\text{retro}}=10\log(\Omega/4) \tag{5}\]  
If all emitted sound is energetically summed, the result is 5 dB (for \(\Omega=4\pi\) steradians) relative to the emitted sound at 1 m; or 2 dB for an extensive ceiling (\(\Omega=2\pi\)). [PERSON] et al. [44] observed that this simple concept was reasonably good as a predictor of measured retroreflection from building facades comprised of arrays of concave trihedral corners. This equation conservatively points to the potential of retroreflective surface treatments (such as of ceilings), yielding values much greater than a specular surface (when surfaces are extensive), but much less than a coherent focusing surface.