On the Alternative Approach to Active Control

Paper is devoted to problem of acoustical visibility reduction and gives brief description of alternative approach to active control. This approach allows satisfy jointly the four conditions: a) effective suppression of acoustical radiation and scattering caused by elastic body immersed in compressible medium (water); b) cloacking at any important temporal frequencies of observant sound waves; c) cloacking at any important spatial frequencies or at any observation directions; (d) thickness of the masking shell is much smaller than the length of suppressed waves. Technological development gives more and more fast and miniature control elements and fast, accurate calculations. On the other hand, the lengths of waves to be damped are constant due to the constant conditions of their far propagation. The approach suggested uses operations of high space-time resolution for long waves controlling. Because the joint fulfillment of the conditions of acoustic support absence and the thinness of the shell and also the causality of control lead to the need to operate at frequencies of much higher than frequencies of waves to be damped. It is assumed that the incident waves are represented by a discrete set of plane waves of finite duration. Also it is assumed only that the characteristic spatial scale of the distribution of normal velocities on the surface of the protected body is limited from below. The boundary value problem with initial conditions is considered.


Introduction
Modern results [1] in the field of cloacking correlate with the famous article [2].
In this paper, an analytical solution is obtained in the case when the incident wave is flat and monochromatic, and the body has a spherical shape.The solution of this problem is tangentially homogeneous radial distribution of medium parameters in the outer spherical layer (that is the time-constant distribution of the passive parameters of the medium in the masking shell).Thus, the distribution of parameters required for the masking inside the protective layer depends on the frequency of the incident wave and does not depend on the direction of arrival of the incident wave.The need to adjust the spatial structure of the masking shell to a nonplanar form and the given direction of the incident wave (or to the non-spherical shape of the body) even in the monochromatic case, makes the practical application of the solution extremely complicated.Moreover the dynamic range of mechanical vibrations of the elements of the masking shell can be significantly exceeded, since all the power of the incident wave through the cross section of the protected body must tangentially pass inside the shell, the thickness of which we seek to reduce (see abstract, (d)).In addition, the above-described shell is not designed to suppress the radiation of the protected body.We will consider below the possibilities of radiation and scattering suppressing for all directions of observation and for all directions of the incident waves in the ranges min

Radiation Suppression
In the most general formulation, we need to create on the active shell surface S a predetermined distribution ( , ) ) of the normal displacements of particles, despite the action of unknown sources of vibrations inside the surface B S .In particular, to suppress radiation, we need to create ( , ) 0 U t , i.e. no incident waves.

Shock Molding of Desired Boundary Form
Let us consider briefly the memory of a compressible or elastic linear medium about impact action, or, in other words, the formulation of a problem maximally different from a monochromatic case [2].For a longitudinal impact to the free end 0 x = of a semi-infinite elastic rod ( 0 x ≤ < ∞ , one dimensional problem, Figure 1(a)), the depth (where ρ -mass density of elas- tic rod, c-speed of longitudinal sound in the rod) of imprint of the rod end remains in time infinitely (imprint life-time # τ = ∞ ) after the switching off the shock force (pressure) # ( ) F t , that acts during the time interval 0 is vacuum).We divide the plane 0 x = into a set of regions in the form of infinite parallel strips: . Suppose that we need to create a δ -like distribution of normal displacements ( , ) U y t that satisfies the condition ) of all the pistons with a time period T.
There is the fact of fundamental importance that (due to the finite lifetime and actual normal displacements of the piston (the center of which is at the point R ) we mean the quantities averaged over the area ˆS σ of the piston.Above we assumed that for the impacts creation we have some unlimited source of mechanical impulse or support (vibrostat).Below we show that it is possible to synthesize a needed distribution ( , ) of the normal displacements of the surface S without mechanical support too.

Transparent Supportless Unidirectional Sources
Let's consider the piezoelectric plane layer 0 h x − < ≤ with the same (for sim- plicity) ρ and c as at x h ≤ − and at h c between pulses.This wavelet is created also without any mechanical support [3].
If the impact duration ( F T τ << ) is negligible, then we can write the wavelet ξ = − , running to right) of single-direction radiation in the follow- ing for m: Summarizing these wave- lets with amplitudes n Y and shifted with respect to each other by time distance

/ T h c =
, we can form a sequence of hooked wave with duration 3 / h c of each: . In this case we ob- tain a sequence of pulses with periodic pauses of duration / h c .Therefore, to create a needed normal displacement ( , ) of the piston at the n-th step, 2[ ( , ) ( , ; , , ,..., )] where n t nT = , 0,1, 2,... n = The above-mentioned spreading of the imprints of the blows (for compensation of which is necessary the impact pumping) is contained in the measured quantity   (between piezoelectric impacts) the current average piston (with the center in the point  ) of the dielectric (piezoelectric) layers:

Measuring
, where 0 ε -dielectric constant of vacuum, ε -relative permittivity of piezoelectric.Now we write down the needed quantities

Scattering Suppression
Suppose that body's radiation is already suppressed by the system described in Section 2. Further suppose that in area of compressible medium (with mass density ρ and sound speed c, identical with outer medium) delineated by surface S we know the particle displacement field ( , ) created by the incident waves.Scattering field does not arise if we create on the outer surface of active shell the distribution ( , ) ( ) ( , ) ( ( ) n r is the outer normal to the surface S in its point ) which coincides with normal component of particle displacement in the incident waves field.

Incident Waves
Further we assume that incident wave field In ξ Θ with leading edges: this means that there is a point n ξ , for which the following condi- tion is satisfied: where min λ is the minimum length of the wave to be damped.

Spacing of Microphones
All microphones are placed in r means the coordinate of any microphone).In addition, all microphones are placed by pairs in points • = r R (farer to S and called "title microphone") and • + = r R (nearer to S and called "reference microphone") on the normal ( ) = n n R to a smooth convex surface S with distance D between them (see

Arrival of 1st Incident Wave
Radiation of internal sources within surface B S is assumed suppressed by the ).Leading edge of some plane incident wave (we will call this wave the 1-st incident wave with direction vector ) achieves some microphone spaced the in the point 1 A • r R at some moment 1 A t t = .This is some space-time node (or event A1): module 1 1 ( , ) A P t R of output signal of microphone spaced in the point crosses at the first time some level q from 1 1 [ , ] We notice a similar event B1 later on some microphone with coordinate crosses at the first time some level q from 1 1 [ , ] And the next similar event C1 in the point crosses at the first time some level q from 1 1 [ , ] Assuming below the ratio I p q σ >> >> (where I p is the characteristic amplitude of the pressure in the incident wave, and σ -the mean square deviation of background noise signal, see Figure 4(b)), we obtain the propagation vector 1 1 ( 1, 1; 1) A B C = w w of the 1-st incident wave from system of equations    in the first incident plane wave in infinite homogeneous compressible medium) and scattering will not arise.More precisely, we will establish a normal average over the area ˆS σ of the piston (with the center at the point R of the surface S) displacement Ξ -characteristic linear dimension of body protected with convex smooth surface B S ) and small thickness 2 h of the active shell (coating between inner and outer surfaces B S and S) of the protected body in comparison with the lengths of quenched waves, i.e.

Figure 1 .
Figure 1.1D impact (a).2D-3D flat impact: structure of a flat impact imprint of one piston (b) and its spreading (space-time conversion) of the flat impact imprint of one piston (c).
Such an ideal plasticity[3] of boundary 0 x = is possible because the region (of thickness F Q cτ = ) of elastic deformation runs to the right with sound speed c.Linearity is guaranteed by the condition Y Q << .Further, instead of the free end of the elastic rod, we consider the free plane boundary of a semi-infinite area ( filled with a compressible medium with the same ρ and c ( 0 x < ) gives us the almost rectangular imprint of depth ε (deformation of boundary 0 x = ).Due to the spreading of the imprint, its lifetime # ~/ L c τ is finite (Figure1(b), Figure1(c)).Therefore, the imprint must be supported by appropriate shock pumping (in time in-

τr
of the print) pumping requires impacts which amplitude is the only a small part / cT L γ ≤ = of the first impact.The total background of pumping impacts is a small part / (1 ) γ γ ≤ − of the magnitude of the first strike.We call a combination Y Q L << << a linear flat blow condition.Now (as in all linear problems), if we can form an almost constant time δ -like distribution ( displacements, then we can also form an arbitrary given distribution with spatial resolution ~L and the scale max ~2 / T π ω >> of temporal variability.Now we give a generalized definition of the piston as an element of an active shell on an arbitrary convex smooth closed surface S. Tangentially (Figure 3(a)) the active shell (spaced between the surfaces B S and S) is lumped into a set of plane pistons with contours of convex polygons.Each piston (of characteristic linear scale ~L) corresponds to some area ˆS σ of the surface B S (or S), as well as the coordinate is piston square) to which all control and measuring signals are addressed.Under the needed ( , ) Pulse of voltage ( ) two normal displacement pulses (of mutually opposite polarity and of the length h) running to the left and to the right (see Figure 2(a)).Piezoelectric forces # F ± and # F  of compression (tension) are mutually balanced and need not mechanical support.Next we consider two piezoelectric layers 2h x h − < ≤ − and 0 h x − < ≤ excited by voltage pulses ( ) t ϕ and ( ) t ϕ (Figure 2(b)) of the same duration but separated in time from each other by the delay / h c and with mutually opposite polarity.It is easy to see (Figure 2(c)) that at 2 x h ≤ − pulses of normal displacement created by voltage pulses (applied to above layers) are mutually compensated.But at 0 x > these pulses of normal displacement form the bipolar wavelet with duration 3 / h c and pause of duration /

Figure 3
Figure 3 presents the structure of active shall.Tangential structure of active shall is presented in Figure 3(a).Transversally (Figure 3(b)) each piston is presented by two piezoelectric layers.It is necessary very quickly and accurately estimate
Distancebetween reference microphone (in the point • + = r R ) and surface S we will de-R presents the vertices of some convex polyhedron.Further, we will not designate microphones with specific numbers to simplify the presentation.At some initial moment, the leading edges of all the incident waves have not yet reached any title microphone.Note that all microphones should be insensitive to pressure fluctuations at a very high frequencies 2 /T π ≥ corresponding to the surface S flat impacts (Section 2).

Figure 4 . 1 w
Figure 4.The geometry of scattering suppression problem (a), space-time event of threshold level crossing by leading edge of incident wave (b).

1 AR
(title microphone) and ) with cardioid directivity pattern having zero in the direction to the surface S (see Figure4(a)).Here / D c τ = -delay, L -linear filter, undistorting signals at frequencies min max where BU t of body surface) is placed immediately under the center R of piston.In the end, we write down the re- incident wave is sufficiantly powerful because one need to detect scattered wave at a large distance from the body.