# The Potential of Focusing Acoustic Retroreflectors for Architectural Surface Treatment
## 2 Materials and Methods
### Ideal Retroreflection from a Ceiling  
#### 2.1.1 Discrete Surface Transformations for Ideal Reflections  
An ideal acoustic reflector is conceived of as a flat hard surface divided into patches that may undergo transformation between sound incidence and re-emission (reflection). Each patch is divided into numerous elements that are small compared to the shortest wavelength of interest. If no patch transformation occurs (referred to as a simple patch), then the reflection is expected to be specular. Figure 3c illustrates the concept of space-reversal patches, whereby the elements are spatially flipped between incidence and reflection. If patches are sufficiently large, then this should result in ideal retroreflection. In such conditions, each patch's image is projected by the source reflection such that all illuminate the source and the area around it.  
Figure 3: Concepts illustrated by ray analogy in modelling reflections from a flat surface made up of patches, each of which is made up of numerous elements: (**a**) sound is emitted by a source and received by elements; (**b**) if no processing is done, the re-emitted sound from the elements behaves as a specular reflection; (**c**) space-reversal of elements in each patch introduces retroreflection, superimposing the projected images of the patches, with the source at their centres; (**d**) elementwise delays introduce focusing (in this example the source is at the focus of Patch 3); (**e**) the combination of focusing and space-reversal is retroreflective, with a potentially high concentration of reflected rays at and near the source. The ray illustration neglects wave phenomena such as diffraction and interference.  
A further step is to introduce focusing for each of the patches. Focusing can be achieved by introducing a delay that is greatest for central elements in each patch. Thus, the elementwise latency of \(\Delta\tau=2d/c\) is introduced, \(d\) being the depth of an idealized well to create the appropriate delay. Preliminary experimentation showed that focal points on the source plane aligned with each patch's centroid are effective for retroreflection when combined with space-reversal, including when the source-receiver is not at any patch's focus. While the focus may not be geometrically perfect at such off-focus positions, sound concentration is still achieved by focus-like behavior. The required elementwise latency is calculated from the distance of each element to the patch's focus, \(r_{\rm focus}\), by reference to the maximum such distance for the patch (Equation (9)).  
\[\Delta\tau_{i}=\frac{2(r_{\rm focus,max}-r_{\rm focus,i})}{c} \tag{9}\]  
Note that a minimum latency of \(\tau_{\rm min}=2\sqrt{2}a/c\) (using the convention of \(2a\) for reflector patch width) is required for non-focusing space-reversal square patches for the space-reversal mechanism to function at the speed of sound. Focusing space-reversal patches involve additional elementwise latency, so overall latency is the sum \(\tau_{i}=\tau_{\rm min}+\Delta\tau_{i}\).  
Two square reflector sizes were evaluated: a 'large' reflector (7.2 \(\times\) 7.2 m) and a'small' reflector (2.4 \(\times\) 2.4 m). The large case was intended to reasonably represent an extensive reflector, with no source-receivers near the edges of the reflector area. Source-receivers were in a 2.4 \(\times\) 2.4 m square aligned with the reflector's center, so the large reflector's shadow had a 2.4 m margin extending beyond the source-receiver area. The small case represented a local reflector, and had source-receivers over the entire reflector area, up to the edges. Various square patch sizes from 0.3 \(\times\) 0.3 m up to the full reflector size were evaluated.