## 5 Active Coating and its Control Algorithm
### Placement of Microphones
#### 6.5.1 Geometry of the Microphone Grid  
Numerous microphone placement variants are possible. Below we'll describe only one concrete version. In this version the surface \(S_{p}\) is represented by a prominent hexagonal polyhedron with plane six cornered sides (Figure 14) of characteristic linear scale \(D_{P}\) and apexes with ordinal number \(j=\)1,2,3,..., \(N_{w}\). Each group of bearing is spaced near polyhedron's apexes. Title microphone \(\bigstar\) of \(i\) -th group of bearing is spaced in \(j\)-th apex which unites three sides \(S_{j\mu 1}\), \(S_{j\mu 2}\), \(S_{j\mu 3}\). Planes \(\overline{S}_{j\mu 12}\), \(\overline{S}_{j\mu 31}\), \(\overline{S}_{j\mu 32}\) are bisectors of angles between sides \(S_{j\mu 1}\) and \(S_{j\mu 2}\), \(S_{j\mu 3}\) and \(S_{j\mu 1}\), \(S_{j\mu 3}\) and \(S_{j\mu 2}\) respectively. Benchmark microphones \(\bigstar\) are spaced on planes \(\overline{S}_{j\mu 12}\), \(\overline{S}_{j\mu 31}\), \(\overline{S}_{j\mu 32}\). Distances \(D_{j\mu 12}\), \(D_{j\mu 12}\), \(D_{j\mu 13}\), \(D_{j\mu 13}\), \(D_{j\mu 13}\),  
\(D_{j\mu 23},\ D_{j\mu 31}\) do not exceed quarter of corresponding \(S_{p}\)'s edges. Lines \(d_{j\mu 12}\), \(d_{j\mu 23},\ d_{j\mu 31}\) are normal with corresponding edges of \(S_{P}\) and spaced in planes \(\overline{S}_{j\mu 12}\), \(\overline{S}_{j\mu 31}\), \(\overline{S}_{j\mu 32}\) correspondingly. Small (but finite) values \(d_{j\mu 12}\sim D_{j\mu 12}^{2}\) / \(R_{P}<D_{j\mu 12}\), \(d_{j\mu 23}\sim D_{j\mu 23}^{2}\) / \(R_{P}<D_{j\mu 23}\), \(d_{j\mu 31}\sim D_{j\mu 31}^{2}\) / \(R_{P}<D_{j\mu 31}\) are necessary to ensure targeting of incident wave, when its direction is normal to some side \(S_{j\mu 1}\), \(S_{j\mu 2}\), \(S_{j\mu 3}\) of \(S_{P}\). Besides this, the finite distances \(d_{j\mu 12}\), \(d_{j\mu 23}\), \(d_{j\mu 31}>0\) are necessary to ensure sufficient accuracy of the solution of equations (20), (23), (24), (27), (30), (31). \(R_{P}\) denotes the averaged curvature radius of \(S_{P}\) (_i.e._ curvature radius of some smooth surface embracing \(S_{P}\) closely). We choose concrete values \(D_{j\mu 12}\), \(D_{j\mu 23}\), \(D_{j131}\), \(d_{j\mu 12}\), \(d_{j\mu 33}\) to ensure: (a) covering of the full area of directions \(\mathbf{w}_{i}\) of possible arrival of incident waves; (b) simultaneous targeting of several incident waves by several groups of bearing without any conflicts between each other. Total number \(N_{P}\) of microphones, number \(N_{SP}\) of apexes of polyhedron \(S_{P}\), total number \(N_{PG}\) of groups of bearing, total number \(N_{w}\) (expected) of incident waves simultaneously targeted are connected by relation \(N_{P}\) / \(4=N_{PG}=N_{SP}=N_{w}\). The distance \(D_{P}\) between adjacent title microphones in any area of surface \(S_{P}\) with curvature radius \(R_{P}\) one can estimate approximately as \(D_{P}\leq 2R_{P}\theta_{w}\), where \(R_{P}\) refers to some smooth surface embracing all apexes of \(S_{P}\). There \(\theta_{w}\) denotes the expected minimum difference between directions of incident waves, _i.e._\(\theta_{w}=\min\limits_{n\
eq m}[\arccos(\mathbf{w}_{n},\mathbf{w}_{m})]\) (\(n,m=1,2,...,N_{w}\)).