## 3 The Control of Normal Velocities (Displacements) of the Body's Surface  
Here we consider the problem of quick synthesis of a desired distribution of normal oscillatory velocities (displacements) on the outside surface of an active coating during the time, much smaller, than the characteristic temporal scale of the distribution to be formed. In the absence of incident waves the desired DNOD is zero, and in such a case the active coating solves the radiation suppression problem. In the presence of incident waves the active coating (in addition to suppression of radiation) can (see below) suppress the scattering field of the protected body. In the presence of incident waves the active coating suppresses scattered waves jointly with radiation. This is due to the formation on the external surface of the coating of the normal oscillatory velocities distribution, matched for the incident wave, on the base of signals of some group of microphones (bearing group, see Section 6.2.). The active coating is placed immediately on the surface of the protected body. Below we point to several important characteristics and assumptions in the statement of this problem. **1*** Let's assume, that in the absence of physical scattering surface \(S_{V}\) [outside surface of active coating, which protects some body with a characteristic linear dimension \(D_{V}\) and minimum curvature radius \(R_{V}>0\) (Figure 4)] the particle velocities \(\mathbf{V}_{\otimes}\) of a non-viscous fluid have the projection \(\upsilon_{\otimes}=(\lozenge,\mathbf{V}_{\otimes})\) on the vector \(\lozenge=\lozenge\mathbf{(r)}\) (which is normal to the surface \(S_{V}\) and with normalized length \(|\lozenge=1\), \(\mathbf{r}\in S_{V}\)). Now, if we created (by means of the active coating) the DNOV [37]:**  
\[\upsilon=\upsilon_{\otimes} \tag{2}\]  
We would have suppressed the scattering field completely. By the way, the problem of radiation suppression would be solved together with the problem of scattering suppression: without incident waves the normal oscillatory velocity of external surface \(S_{V}\) of the coating must be zero (when nonzero velocity is possible on the internal surface of active coating). The condition (1) must be satisfied in the finite wide operating range of temporal \(\omega\) and spatial frequencies:  
\[\left|\omega\right|\in(\omega_{\min},\omega_{\max})\,,\,\left|\chi\right|\in( \chi_{\min},\chi_{\max})\,, \tag{3}\]  
_i.e._ for a finite area of characteristic spatial and temporal scales:  
\[\lambda\in(\lambda_{\min},\lambda_{\max}),\tau\in(\tau_{\min},\tau_{\max}), \tag{4}\]  
where \(\omega_{\min}>0\), \(\chi_{\min}=\omega_{\min}/c_{w}\), \(\chi_{\max}=\omega_{\max}/c_{w}\), where \(c_{w}\) - sound speed in fluid. Wave dimensions of the protected body with a characteristic space scale \(D_{V}\) can be both  
\[\chi_{\max}D_{V}>>1\text{ and }\chi_{\min}D_{V}<<1. \tag{5}\]  
The minimum frequency \(\omega_{\min}\) must be sufficiently large: this does not permit the displacement magnitude to overcome the border \(A_{\max}\) of the linear characteristics of sensors and actuators [10], _i.e._:  
\[(\omega_{\min}T_{V})^{-1}\,A_{\otimes}<A_{\max}<h\,, \tag{6}\]  
where \(A_{\otimes}\) - desired normal displacement of outside surface of coating, \(h\) - its thickness. At \(\omega_{\min}\)>0 the relation analogous to (2) will be satisfied for normal displacements too. Figure 4: Matching of distribution of normal oscillatory velocities (DNOV) to obtain a “transparent” body (\(\lambda\) - wavelength). **2*** The formation of the current DNOV value \(\upsilon_{\otimes}\) (**r**,_t_) at any moment must be fulfilled by some control actions during the short previous time interval of duration \(T_{V}<<\tau_{\min}\), much shorter than the minimum temporal scale \(\tau_{\min}\) of the wave to be suppressed. **3*** The thickness \(h\) of the active coating (or distance between surfaces \(S_{V}\) and \(\overline{S}_{V}\) (Figure 2b) is much smaller than the length of wave, which must be suppressed, _i.e._\(h<<\lambda\).