Solutions to Problems of Controlling Long Waves with the Help of Micro-Structure Tools

In this chapter we formulate the basic principles of the approach used in this book. In the beginning we present a brief description of known traditional passive and active approaches to the problem of “invisibility” of physical bodies in the wave fields of various physical genesis. All these approaches can be reduced to a combination of linear electric circuits with constant parameters. Due to the physical conditions of far propagation, the wavelengths of incident waves retain of the constant order for many years. We consider the contemporary achievements of electronic actuators and sensors: growth of rapidity and decreasing of geometrical sizes of actuators and sensors together with the growth of speed and accuracy of computer calculations present the stable technological tendency. What benefit we could get from this progress in solution of above mentioned problems? Below we consider several simplest solutions of one-dimensional boundary value problems. The suggested control algorithms use fast parametric and fast logical operations. Therefore, these algorithms can not be reduced to a combination of linear electric circuits with constant parameters (as in traditional approaches above). Control algorithms suggested are characterized by spatial locality and temporal locality of control. This means, for instance, that we need not know the period and wavelength of incident wave for effective control.

The effect of resonant absorption of long waves by the oscillator of little sizes is investigated analytically and numerically. This effect means that absorption cross-section of the oscillator (monopole, dipole...) is defined by wavelength absorbed only, and does not depend on wave geometrical dimensions (much smaller, than the wavelength absorbed) of the oscillator. The expression of optimum amplitudes of excitation of the group of degrees of freedom (or oscillators) in the boundary problem of general type is obtained in the form of generalized velocities and generalized forces. Using linear microstructures (formed by monopoles, which are located on the axis periodically) we investigated the possibility to achieve maximum absorption cross-section of the acoustic waves by these microstructures of small wave dimensions. We consider the examples of linear microstructures, which provide unlimited logarithmic, linear and square growing of the total absorption cross-section, with growing of the quantity of elements (monopoles) in the linear microstructure with wave dimensions remaining small. The examples of cooperative and the individual strategies of absorbing oscillators are also compared.

In this chapter, we use simple models for acoustical waves (Section 3.1) [59], [61], [63], [64], [59], for water surface waves (Section 3.2) [62], [59], [61], [50], and for electromagnetic waves (Section 3.3) [65]. On the basis of these models, we consider the new concept of parametric "black body" with conceptual possibility of designing an active absorbing (nonreflecting) coatings in the form of a thin layer with small-scale stratification and fast temporal modulation of parameters. Algorithms for spatial-temporal modulation of the controlled-layer structure are studied in detail for a one-dimensional boundary-value problem. These algorithms do not require wave-field measurements, which eliminate self-excitation problem, that is the characteristic of traditional active systems. The majority of the considered algorithms of parametric control transforms the low-frequency incident wave to high-frequency waves of the technological band for which the waveguiding medium inside the layer is assumed to be opaque (absorbing). The efficient conditions of use are found for all the algorithms. It is shown that the absorbing layer can be as thin as desired with respect to the minimum spatial scale of the incident wave ensuring efficient absorption in a wide frequency interval (starting from zero frequency) that is bounded from above only by a finite space-time resolution of the parameter-control operations. The structure of a threedimensional parametric "black" coating, whose efficiency is independent of the angle of incidence of an incoming wave is developed on the basis of the studied one-dimensional problems. The general solutions of the problem of diffraction of incident waves from such coatings are obtained. These solutions are analyzed in detail for the case of a disk-shaped element.

An algorithm for the suppression of the radiation and scattering fields created by vibration of the smooth closed surface of a body of arbitrary shape placed in a liquid is designed and analytically explored. The frequency range of the suppression allows for both large and small wave sizes on the protected surface. An active control system is designed that consists of: (a) a subsystem for fast formation of a desired distribution of normal oscillatory velocities or displacements (on the basis of pulsed Huygens' sources, Section 4.6) and (b) a subsystem for catching and targeting of incident waves on the basis of a grid (one layer) of monopole microphones, surrounding the surface to be protected (Section 4.7). The efficiency and stability of the control algorithm are considered. The algorithm forms the control signal during a time much smaller than the minimum time scale of the waves to be damped. The control algorithm includes logical and nonlinear operations, thus excluding interpretation of the control system as a traditional combination of linear electric circuits, where all parameters are constant (in time). This algorithm converts some physical bodies placed in a liquid into one that is transparent to a special class of incident waves. The active control system needs accurate information on its geometry, but does not need either prior or current information about the vibroacoustical characteristics of the protected surface, which in practical cases represent a vast amount of data. Joint suppression of radiating and scattering field by a coating of controlled thickness is considered. The problems of suppressing the sound field generated by a vibrating body in a liquid are considered in another representation too. For solving these problems, an acoustically thin active coating with a real-time thickness control is proposed (Section 4.8). The coating should be placed directly on the surface of the body to be protected. Solutions to the problems of suppressing the radiation and scattering of sound by a body are obtained in the general form based on linear operators, which characterize (i) the sound radiation by a vibrating surface, (ii) the scattering of incident waves by a fixed surface, and (iii) the vibroelastic properties of the body in an acoustic vacuum. Conditions ensuring the stability of the active system are formulated. Forming of directions of zero scattered by minimum tools is considered too.

In monochromatic description (sections 5.1, 5.2), the results of the studies of the physical characteristics of unidirectional acoustic sources used in active sound control systems are presented. A discrete unidirectional source in the form of two phased monopoles (section 5.1) and a planar array of such unidirectional sources is considered (section 5.2.1). One-dimensional boundary-value problems with two (the two-point problem) and three (the three-point problem) controlled parallel planar boundaries between homogeneous media with arbitrary impedances are studied (sections 5.2.2-5.2.6). The boundaries (two or three) are subjected to the action of external forces. The case of the zero sum of external forces applied to the controlled boundaries corresponds to a supportless unidirectional source (SUS). It is shown that a unidirectional source can be created within the twopoint boundary-value problem, whereas a supportless unidirectional source can be created within the three-point problem (sections 5.2.5, 5.2.6). Such parameters such as transparency, small size, absence of support, and broad frequency band can be achieved for a unidirectional source in the form of two piezoelectric layers with the same impedance and velocity of sound as those of the surrounding medium (5.2.7-5.2.9). The aspects of linearity of the transparent SUS and its application to active sound control problems are described (sections 5.2.10, 5.2.11). A spatially one-dimensional model of a plane active double layer between two homogeneous elastic half-spaces is studied analytically in temporal representation (section 5.3). The layer synthesizes a preset smooth trajectory of the controlled boundary between the media without any mechanical support. The outer layer of the coating is piezoelectric, and the inner layer is a polymer that is transparent for low-frequency sound and opaque for highfrequency sound because of dissipation. An algorithm for controlling the piezoelectric elements of the layer on the basis of signals from surface particle velocity sensors is proposed (section 5.3.6), and a method for measuring the particle velocity is developed simultaneously (section 5.3.9). Conditions of stability and efficiency of the synthesis are formulated (section 5.3.7). It is shown that the active layer thickness can be much smaller than the wavelength corresponding to the minimal time scale of the boundary trajectory to be formed. The accuracy of the trajectory synthesis depends on the accuracy of measuring, computing, and actuating elements of the system but does not depend on the vibroacoustic characteristics of the half-spaces separated by the active layer or on the presence of smooth waves in these half-spaces. For the synthesis to be efficient, the operating frequency band and the dynamic range of sensors and actuators should be many times greater than the frequency band and the dynamic range of the trajectory to be formed.

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