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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 ob-servant 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 con-stant 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.

Modern results [

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

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 [

shock force (pressure) F # ( t ) , that acts during the time interval 0 < t ≤ τ F ( F # = 0 at t < 0 and t > τ F ). Such an ideal plasticity [

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 ( − ∞ < y , z < + ∞ , x ≥ 0 ) filled with a compressible medium with the same ρ and c ( x < 0 is vacuum). We divide the plane x = 0 into a set of regions in the form of infinite parallel strips: | y − ( L / 2 ) n | < L / 2 , − ∞ < z < + ∞ or “pistons” numbered by n = 0 , ± 1 , ± 2 , ± 3 , ... . Suppose that we need to create a δ ?like distribution of normal displacements U ( y , t ) that satisfies the condition

∫ n L − ( L / 2 ) n L + ( L / 2 ) U ( y , m T ) d y / L = ε ¯ δ ( n ) , where m = 0 , 1 , 2 , 3 , ... , δ = 1 for n = 0 , δ = 0

for n ≠ 0 . Thus at initial condition U ( y , 0 ) = 0 , U t / ( y , 0 ) = 0 we need to apply the first impact of pressure F # = ε ¯ ρ c L / τ F to the strip | y | < L / 2 . This pressure pulse (acting on the interval 0 < t < τ F < < L / c ) gives us the almost rectangular imprint of depth ε ¯ (deformation of boundary x = 0 ). Due to the spreading of the imprint, its lifetime τ # ~ L / c is finite (

Let’s consider the piezoelectric plane layer − h < x ≤ 0 with the same (for simplicity) ρ and c as at x ≤ − h and at x > 0 . Pulse of voltage φ ¯ ( t ) with duration τ F < < h / c creates two normal displacement pulses (of mutually opposite polarity and of the length h) running to the left and to the right (see

If the impact duration ( τ F < < T ) is negligible, then we can write the wavelet Ψ ( ξ ) ( ξ = x − c t , running to right) of single-direction radiation in the following form: Ψ ( ξ ) = { I [ ξ ] − I [ ξ − ( 1 / 2 ) T ] } − { I [ ξ − T ] − I [ ξ − ( 3 / 2 ) T ] } , where T = 2 h / c , I ( ξ ) = 1 at ξ > 0 , I ( ξ ) = 0 at ξ ≤ 0 . Summarizing these wavelets with amplitudes Y n and shifted with respect to each other by time distance T = 2 h / c , we can form a sequence of hooked wave with duration 3 h / c of each: Y 0 Ψ ( t ) + Y 1 Ψ ( t − T ) + Y 2 Ψ ( t − 2 T ) + ⋯ (

the amplitude of the wavelet must be double, in order to provide, on average, the desired value of the normal displacement on the period T = 2 h / c . Thus, we obtain the following expression (control algorithm) of the current amplitude Y n of the wavelet through via amplitude Y n − 1 of the previous and measured displacement U ¯ ⊗ ( R ¯ , t n − 1 ) of the piston, as well as the required displacement U ¯ • ( R ¯ , t n ) value

Y n = Y n − 1 + 2 [ U ¯ • ( R ¯ , t n ) − U ¯ ⊗ ( R ¯ , t n − 1 ; Y 0 , Y 1 , Y 2 , ... , Y n − 1 ) ] , (1)

where t n = n T , n = 0 , 1 , 2 , ... 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 U ¯ ⊗ ( R ¯ , t n − 1 ; Y 0 , Y 1 , Y 2 , ... , Y n − 1 ) . Note that an attempt to synthesize a desired value U ¯ • ( R ¯ , t ) using bipolar wavelets Ψ ( t ) means that the amplitude Y n of the wavelets is proportional to the integral of the quantity U ¯ • ( R ¯ , t n ) − U ¯ ⊗ ( R ¯ , t n − 1 ) . Thus, neither the measured displacement U ¯ ⊗ ( R ¯ , t ) nor the desired value U ¯ • ( R ¯ , t ) should contain time-constant components. To maintain stability, it is necessary to exclude the constant component of the signals U ¯ ⊗ ( R ¯ , t ) and U ¯ • ( R ¯ , t ) , i.e. pass them through a non-distorting differential filter with a time scale τ d > > λ max / c . Now we must note that slow desired trajectory U ¯ • ( R ¯ , t ) of piston in time (with time scale τ max = λ max / 2 c ) requires significant value of wavelet magnitude at some maximum amplitude A max of particle displacement in the waves to be damped. So the condition of linear flat impact becomes the following: ( τ max / T ) A max < < c τ F < < c T < < L .

(between piezoelectric impacts) the current average piston (with the center in the point R ¯ ∈ S ) area r ∈ σ ^ S ⊂ S particle displacement U ¯ ⊗ ( R ¯ , t ) = U B ( R ¯ , t ) + U ˜ ( R ¯ , t ) + U ˜ ˜ ( R ¯ , t ) , where U B ( R ¯ , t ) ?slow displacement of surface S B , where { U ˜ ( R ¯ , t ) , U ˜ ˜ ( R ¯ , t ) } = ( σ S ) − 1 ∫ σ ^ S { d ˜ ( r , t ) , d ˜ ˜ ( r , t ) } d σ ^ ( r ) are instant spatial average in the area r ∈ σ ^ S of the thicknesses d ˜ ( r , t ) , d ˜ ˜ ( r , t ) of the metallized layers of the piezoelectric. The smoothness of the distribution of displacements of the surface S B in space is guaranteed by the condition l min > > L , where l max is maximum spatial scale of displacement distribution on the surface S B ). Therefore, U B ( R ¯ , t ) can be used without spatial averaging over the piston pad. Then we need to measure the instant capacitances C ˜ ( t ) and C ˜ ˜ ( t ) of the flat capacitors (dielectric layers) with varying thicknesses { d ˜ ( r , t ) , d ˜ ˜ ( r , t ) } = h + { d ˜ ~ ( r , t ) , d ˜ ˜ ~ ( r , t ) } (where the variable components are relatively small, i.e. | d ˜ ~ | < < h , | d ˜ ˜ ~ | < < h ) of the dielectric (piezoelectric) layers: { C ˜ ( t ) , C ˜ ˜ ( t ) } = ( ε 0 ε ) − 1 ∫ r ∈ σ ^ S d σ ^ S ( r ) / { d ˜ ( r , t ) , d ˜ ˜ ( r , t ) } , where ε 0 ―dielectric constant of vacuum, ε ―relative permittivity of piezoelectric. Now we write down the needed quantities { U ˜ ( R ¯ , t ) , U ˜ ˜ ( R , t ) } = − h [ { C ˜ ( t ) , C ˜ ˜ ( t ) } − C 0 ] / C 0 , where C 0 = σ S / ( ε 0 ε h ) . These spatial averaging electric operations can be performed almost instantaneously. Inertial accelerometer with output signal U ¨ B ( t ) (i.e. the 2-nd derivative of normal displacement U B ( t ) of body surface) is placed immediately under the center R ¯ of piston. In the end, we write down the remaining component U B ( R ¯ , t ) = ∫ 0 t d ξ ∫ 0 ξ L ¯ ^ U ¨ L B ( η ) d η , where L ¯ ^ means 3-fold processing by differentiating chain with time scale τ d > > λ max / c .

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 U I ( r , t ) created by the incident waves. Scattering field does not arise if we create on the outer surface of active shell the distribution U • ( r , t ) = n ( r ) U I ( r , t ) of normal displacements U • ( r , t ) ( n ( r ) is the outer normal to the surface S in its point r ∈ S , | n ( r ) | = 1 ) which coincides with normal component of particle displacement in the incident waves field.

Further we assume that incident wave field P I ( r , t ) = Σ n = 1 N I P I n ( r , t ) of pressure represents the finite set of N I ≥ 1 planar waves P I n ( r , t ) = Θ I n [ t − ( w n , r ) c − 1 ] , with vectors w n ( | w n | = 1 ) of propagation direction and profiles Θ I n ( ξ ) with leading edges: this means that there is a point ξ n , for which the following condition is satisfied: Θ I n ( ξ ) = 0 at any ξ < ξ n , Θ I n ( ξ ) ≠ 0 at ξ n < ξ < ξ n + ( λ min / 4 ) , where λ min is the minimum length of the wave to be damped.

All the microphones are placed in points r = r • ( 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 n A 1 , n B 1 , n C 1 in

Radiation of internal sources within surface S B is assumed suppressed by the

means described in Section 2. All microphones are waiting for the first incident wave arrival from unknown direction w 1 (output pressure signals of all microphones are denoted by P 1 [ r • , t ] ). 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 r • = R A 1 at some moment t = t A 1 . This is some space-time node (or event A1): module | P 1 ( R A 1 , t ) | of output signal of microphone spaced in the point r • = R A 1 sound pressure P 1 [ R A 1 , t ] crosses at the first time some level q from | P 1 [ R A 1 , t ] | < q to | P 1 [ R A 1 , t ] | > q at the moment t = t A 1 . We notice a similar event B1 later on some microphone with coordinate r • = R B 1 at some moment t = t B 1 ≥ t A 1 : module of pressure P 1 [ R B 1 , t ] crosses at the first time some level q from | P 1 [ R B 1 , t ] | < q to | P 1 [ R B 1 , t ] | > q at the moment t = t B 1 . And the next similar event C1 in the point r • = R C 1 at some moment t = t C 1 ≥ t B 1 : module of pressure P 1 [ R C 1 , t ] crosses at the first time some level q from | P 1 [ R C 1 , t ] | < q to | P 1 [ R C 1 , t ] | > q at the moment t = t C 1 . Assuming below the ratio p I > > q > > σ (where p I is the characteristic amplitude of the pressure in the incident wave, and σ ?the mean square deviation of background noise signal, see ^{st} incident wave) we obtain P I 1 [ R A 1 , t ] = P 1 [ R C 1 , t A 1 ] + τ − 1 [ 1 − ( n A 1 w 1 ) ] − 1 ∫ t C 1 t g 1 ( ξ ) d ξ (note that we have ( n A 1 w 1 ) < 0 for any incident plane wave). Knowing the pressure field P I 1 of the first incident plane wave at a point R A 1 at time t > t A 1 , we can determine the pressure field at any point r (satisfying the condition ( r − R A 1 ) w 1 > 0 ) at time t > t A 1 . In addition, we can determine the normal displacement u S ( r , t ) of the surface S under which the condition u S ( r , t ) = n S ( r ) U I 1 ( r , t ) will be satisfied (the displacement field U I 1 ( r , t ) 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 σ ^ S of the piston (with the center at the point R ¯ of the surface S) displacement

U ¯ • ( R ¯ , t + T ) = U ¯ • ( R ¯ , t ) + δ 1 P I 1 [ R A 1 , t − α 1 ] (2)

(correcting it for a period of duration T, i.e. pressure-velocity-displacement), where δ 1 = ( T / ρ c ) ( w 1 n ( R ¯ ) ) , α 1 = w 1 ( R ¯ − R A 1 ) / c . After inserting (2) into (1), where t + T = t n , the scattering does not occur, then the field of the first incident wave P I 1 [ r , t ] passes without distortion through the region of space occupied by the body and bounded by the surface S. And this means that the pressure field of the first incident wave P I 2 [ r • , t ] (for t > t C 1 ) can be subtracted from the signals of all microphones except for the leading pair (at points R A 1 and R A 1 + ) of the first incident wave. The sound pressure on the microphones at the points r • ≠ R A 1 , R A 1 + will now be denoted as P 2 [ r • , t ] = P 1 [ r • , t ] − P I 1 [ R A 1 , t − β 1 ] , where β 1 = w 1 ( r • − R A 1 ) / c . Thus, we prepared the system for capturing a second plane incident wave, with respect to which we will assume that for the first points of contact of the leading edge with the microphones there will be points r • = R ≠ R A 1 , R A 1 + of placement of the title microphones. The microphones at the points r • ≠ R A 1 , R A 1 + became deaf (insensitive) to the first incident wave. Therefore, one can apply the logical procedure described above to the signals P 2 [ r • , t ] (three events A2 ( t A 2 , R A 2 ), B2 ( t B 2 , R B 2 ), C2 ( t C 2 , R C A 2 )). And so on. Below we give briefly a sequence of next functional steps.

Event A2: t = t A 2 ; r • = R A 2 ; crossing | P 2 | < q ⇒ | P 2 | > q . Event B2: t = t B 2 ; r • = R B 2 ; crossing | P 2 | < q ⇒ | P 2 | > q . Event C2: t = t C 2 ; r • = R C 2 ; crossing | P 2 | < q ⇒ | P 2 | > q .

| ( R B 2 − R A 2 ) w 2 | = c ( t B 2 − t A 2 ) , | ( R B 2 − R A 2 ) w 2 | = c ( t B 2 − t A 2 ) , w 2 = w 2 ( A 2 , B 2 , C 2 ) , g 2 ( t ) = L ^ P 2 [ R A 2 , t ] − L ^ P 2 [ R A 2 + , t − τ ] , δ 2 = ( T / ρ c ) ( w 2 n ( R ¯ ) ) , α 2 = w 2 ( R ¯ − R A 2 ) / c , P I 2 [ R A 2 , t ] = P 1 [ R A 2 , t C 2 ] + τ − 1 [ 1 − ( n A 2 w 2 ) ] − 1 ∫ t C 2 t g 2 ( ξ ) d ξ , β 2 = w 2 ( r • − R A 2 ) / c , U ¯ • ( R ¯ , t + T ) = U ¯ • ( R ¯ , t ) + δ 1 P I 1 [ R A 1 , t − α 1 ] + δ 2 P I 2 [ R A 2 , t − α 2 ] , P 3 [ r • , t ] = P 3 [ r • , t ] − P I 1 [ R A 1 , t − β 1 ] − P I 2 [ R A 2 , t − β 2 ] , for n r • ≠ R A 1 , R A 1 + , R A 2 , R A 2 + .

Event A3: t = t A 3 ; r • = R A 3 ; crossing | P 3 | < q ⇒ | P 3 | > q . Event B3: t = t B 3 ; r • = R B 3 ; crossing | P 3 | < q ⇒ | P 3 | > q . Event C3: t = t C 3 ; r • = R C 3 ; crossing | P 3 | < q ⇒ | P 3 | > q .

| ( R B 3 − R A 3 ) w 3 | = c ( t B 3 − t A 3 ) , | ( R B 3 − R A 3 ) w 3 | = c ( t B 3 − t A 3 ) ⇒ w 3 = w 3 ( A 3 , B 3 , C 3 ) , g 3 ( t ) = L ^ P 3 [ R A 3 , t ] − L ^ P 3 [ R A 3 + , t − τ ] , δ 3 = ( T / ρ c ) ( w 3 n ( R ¯ ) ) , α 3 = w 3 ( R ¯ − R A 3 ) / c , P I 3 [ R A 3 , t ] = P 3 [ R A 3 , t C 3 ] + τ − 1 [ 1 − ( n A 3 w 3 ) ] − 1 ∫ t C 3 t g 3 ( ξ ) d ξ , β 3 = w 3 ( r • − R A 3 ) / c , U ¯ • ( R ¯ , t + T ) = U ¯ • ( R ¯ , t ) + δ 1 P I 1 [ R A 1 , t − α 1 ] + δ 2 P I 2 [ R A 2 , t − α 2 ] + δ 3 P I 3 [ R A 3 , t − α 3 ] , P 4 [ r • , t ] = P 3 [ r • , t ] − P I 1 [ R A 1 , t − β 1 ] − P I 2 [ R A 2 , t − β 2 ] − P I 3 [ R A 3 , t − β 3 ] , for r • ≠ R A 1 , R A 1 + , R A 2 , R A 2 + , R A 3 , R A 3 + .

The main results of this work are the following: 1) transparent supportless unidirectional sources of acoustical wavelets (Section 2), 2) causal sequence of operations to reconcile the normal displacements of the protected surface with the incident waves (Section 3). The presented results are a consequence of the problem formulation with initial conditions and could not be obtained in the widespread stationary monochromatic mathematical model of the problem with complex amplitudes (magnitude A, frequency ω , phase φ , i.e. A exp [ i ( ω t + φ ) ] ) of the fields (like in [

The authors declare no conflicts of interest regarding the publication of this paper.

Arabadzhi, V.V. (2018) On the Alternative Approach to Active Control. Journal of Applied Mathematics and Physics, 6, 2303-2312. https://doi.org/10.4236/jamp.2018.611191