# The Potential of Focusing Acoustic Retroreflectors for Architectural Surface Treatment
## 1 Introduction
### Theoretical Perspectives on Reflected Sound Level
#### 1.1.2 Reflected Sound Level from Room Reverberation  
Classical room acoustics theory provides another important perspective on the potential sound returned to a source, using the assumption of a diffuse field. This approach neglects particular reflections, and instead integrates the whole reverberant sound energy over time, assuming that it is spatially invariant. Relative to the emitted sound at 1 m from an omnidirectional source, the reverberant sound level may be estimated from Equation (6), where \(R\) is room constant, \(S\) is room surface area, and \(\overline{\alpha}\) is average absorption coefficient. \[L_{\text{reverb}}=10\log\bigg{(}\frac{4}{R}\div\frac{1}{4\pi}\bigg{)}=10\log \frac{16\pi\tau}{R}=10\log\frac{16\pi(1-\overline{\alpha})}{S\overline{\alpha}} \tag{6}\]Values for \(L_{\text{reverb}}\) are predictable for rooms of defined volume (\(V\)), and would typically be larger when volume is smaller. A simpler approximation, neglecting the room's surface area, can be made using absorption, \(A=S\overline{\alpha}\approx 24\ln(10)V/cT\), where \(c\) is the speed of sound and \(T\) is reverberation time (Equation (7)). \[L_{\text{reverb}}=10\log\frac{16\pi}{A} \tag{7}\]  
For example, a small room with a volume of 50 m\({}^{3}\) and a reverberation time of 0.5 s would have an approximate \(L_{\text{reverb}}\) value of 4.9 dB; a room with a volume of 1000 m\({}^{3}\) and a reverberation time of 1 s would have an approximate \(L_{\text{reverb}}\) value of \(-5.5\) dB. More generally, a small room can evidently strongly support a source through reverberation, and the example here has a value of \(L_{\text{reverb}}\) that is about the same as the incoherent energy sum from a single reflection returning all energy to the source (Equation (5)). However, a large room dilutes the reverberant energy over its volume, so high values of \(L_{\text{reverb}}\) may not be achievable without excessively long reverberation, and might not be achievable at all, even with excessive reverberation. These values can be related to voice support using the theoretical model proposed by [PERSON] et al. (2015). Voice support (ST\({}_{V}\)) is the reflected energy from a mouth reflected to the ears of a head and torso simulator relative to the direct energy from mouth to ears, and has been used as a physical parameter in influential studies of voice regulation due to room acoustics. Theoretically predicted octave band voice support values (125 Hz-4 kHz bands) are calculated following Equation (8). Here, \(Q^{\star}\) is the directivity factor of speech in the direction of the floor, \(d\) is the transducer height, \(S_{\text{ref}}\) is the reference surface area (approximately 1 m\({}^{2}\), being the ratio of the product of reference power and characteristic acoustic impedance to the squared reference pressure), \(\Delta L_{\text{HRTF}}\) is the diffuse field head-related transfer function of the ears, and \(K\) represents the direct sound level from mouth to ear of a head and torso simulator (see [PERSON] et al. (2015) for details). \[\text{ST}_{V,\text{oct}}=10\log\Biggl{[}\left(\frac{cT_{\text{oct}}}{\ln(10^ {6})V}-\frac{4}{S}+\frac{Q^{\star}_{\text{oct}}}{4\pi(2d)^{2}}\right)S_{ \text{ref}}\Biggr{]}+\Delta L_{\text{HRTF,oct}}-K_{\text{oct}} \tag{8}\]  
Using the octave-band speech-weighting proposed by [PERSON] et al. (2015), a small room with a volume of 50 m\({}^{3}\) and a reverberation time of 0.5 s would have an overall speech-weighted ST\({}_{V}\) value of -6.5 dB; a room with a volume of 1000 m\({}^{3}\) and a reverberation time of 1 s would have an ST\({}_{V}\) value of \(-15.6\) dB (assuming a floor-to-ceiling height of 3 m and a square floor plate). More broadly there is an 11 dB difference between \(L_{\text{reverb}}\) and speech-weighted ST\({}_{V}\) based on evaluation of Equations (6) and (8), with deviations for large rooms and high absorption coefficients (Figure 1). In real rooms, spectral variation in reverberation time, as well as non-diffuse reflections, would be expected to cause further deviations.