Local Emitters in the Regime of Maximum Absorption - SAGE Journals

LOCAL EMITTERS IN THE REGIME OF MAXIMUM ABSORPTION Vladimir V. ARABADZHI Institute of Applied Physics (RAS), BP 216 603600 Nizhny Novgorod, Russia Ulianov st.46 Received: 3rd October 1998 CONTENTS: I. Abstract. 2. Absorption cross-section. 3. General formulation of regime of maximum absorption: passive and active versions. 4. Examples of the waves of different physical nature. 4.I.Sound waves. 4.2.Surface gravitational waves in liquid. 4.3.Elcctromagnctic waves. 4.4.Bending waves. 5. Linear multipole acoustical antenna, 6. Variational approach to the maximum absorption. 7. Absorbing monopole in the wide-band random sound field. 8. Absorption in the system ofN emitters. 8. I. Algorithm of the maximum total power absorbed by N-emitters. X2. Causality . 8.3. Individual and cooperative strategy. 8.:\1. Infinite lattices. X.3.2. Two pulsing spheres. 8.-l. Competitive behavior. stability lJ. Conclusions. 10.References. I. ABSTRACT. Local emitters of waves (or local actuators of dimensions much less than the wavelength of the wave to be controlled) present a necessary part of most of active control systems. This paper is devoted to some properties of emitters (local emitters at most) in the regime of maximum (resonance-like) absorption. Such an emitter has an absorbing cross-section with dimensions about the wavelength of the wave to be absorbed. This property is referred to the waves of different physical nature, emitters of different constructions, different multipolarity, boundary problems of different geometrical dimension (n=2 and n=3). Especially the local 3-dimensional antennas of simple multipole structure. could ensure the absorbing cross-section unbounded. growing at the increasing of multipolarity order, remaining physically local. Regime of maximum absorption is formulated both in spectral and temporal representations in terms of so called generalized forces and velocities referred to some oscillatory degrees of freedom of boundary problem considered. Some aspects of cooperative and individual strategy of absorbing emitters and problems of their stability and causality of control are considered. Examples when emitters could help or oppose each other are shown.

(b) Fig.l. Two qualitative approaches in active control.

2. ABSORfYf10N CROSS-SECTION The power flux lines of two searchlight antenna-like sources are presented in Fig.l: (+) denotes the source of the wave to be damped and (-j-sourcc of the damping one. 2.3-dimensional wave damping Journal of Low Frequency Noise. I ibrutiou and Actil'c Control li)l. 18 No.3 IC)C)C)

129

LOCAL EMITTERS AND MAXIMUM ABSORPTION

problems have two different representations: (I) cancellation, based on the interference only in the region S to be silenced and without absorption (zero absorbing power W A, Fig. I-a); (2) when (-) is placed in the field of the wave to be damped and this causes nonzero power absorbed W" > 0 (Fig. I-b). This paper is devoted to maximization of power absorbed in a number of versions of this second case. The absorption maximization problem considerated below can find applications in marine wave-energetic (power plant based on the energy of gravitational surface waves by an oscillating float, chamber acoustics (echo reduction), wave umbrella design (maximization of the wave shadow region) etc. Let us consider (I] the local emitter (emitter with small wave dimensions: kD«I, k =wlc is the wave number, D is the characteristic emitter geometrical dimension, w-frequency of the wave to be as a simplest example of space non-one-dimensional boundary problem. For the damped) quantitative description of the efficiency of a single emitter in the absorption regime we introduce the definition of the absorption cross-section rr ~

er: =W;/S; where

A q

W

(2.1.)

is the power absorbed by this emitter, S;

is the power flux density,

n

is the

geometrical dimensionality of the boundary problem. As applied to local emitters, we call the asymptotic expression erA = Lim W AIS w ( 2.2.) q

kDHO

q

q

the absorption cross-section at arbitrary small wave dimensions of the emitter. The cases when the absorption cross-section is finite in spite of the emitter locality (i.e. o ~ > 0 at any kD~O). we call non-local absorption by the local emitter. which becomes in this case the drain channel of the power flux lines. ----

&~b)

..

(a)

(c)

.

•~

~~~= ~F-----.'

.

c

e

\

--,~~ ____

:;l~

Fig.2. Transformation of the power flux picture near the local emitter. Note that absorption and scattering phenomena are closely related with the regime of maximum absorption. The flow of the incident wave power flux lines to the emitter corresponds to absorption and their deformation means scattering of the incident wave. Thus, maximization of the local emitter cross-section leads to the specific relation between the radiation, absorption and scattering characteristics of the emitter. Fig.2 shows the conversion of the pictures of power flux lines of the plane incident wave (Fig.2-a) and monopole emitter (Fig.2-b) into the joint picture (see Fig.2-c) when the emitter is in the absorption regime. There exist the silent region S (see Fig.2-c)without power flow. Any body, installed in this region S, would give a zero part in incident wave scattering for instance. Dimensions of this region S are proportional to the absorption cross-section. Further we will waive the index q (geometrical dimension of the boundary problem) assuming the following: (a) q=2 absorption cross-section presents the one dimensional linear interval on the incident wave front from which absorbing emitter collects the lines of power flux; (b) n=3 absorption cross-section presents the two-dimensional ground on the incident wave front from which the absorbing emitter collects the lines of power flux. Back-scattering cross-section in the sense of far field is proportional to geometrical cross-section a SB _ a g of emitter. So for local emitters one could neglect backscattering. Near field of back-scattering is decreasing with distance as - (l/r 2)and becomes negligibly small at a distance of about the incident wavelength. Absorbing emitter makes the hole in the power flux field (see Fig.2-c). So in the regime of maximum absorption by local emiller one could suppose the forward scattering crosssection equal to absorption cross-section er SF = erA . 130

LOCAL EMITIERS AND MAXIMUM ABSORPTION

3. GENERAL FORMULATION OF REGIME OF MAXIMUM ABSORPTION. As absorption can exist only on a moving boundary (in mechanical terminology) we will consider any boundary problem of absorption of some incident wave as a problem of scattering on a moving boundary. Further. due to conception of linear superposition we will divide the previous boundary problem into a pair of ones: (I)-boundary problem of radiation by the moving boundary in the absence of incident wave and (2)-boundary problem of an incident wave scattering on the fixed boundary. To get the general approach to the problem of absorption of waves of various physical nature by the emitters of various design. we formulate this problem in the terms of the complex amplitudes of generalized forces Fko]. Thus, the emitter is affected by the generalized force F(oo) = E(w) - Z.. V(w) (3.1.) where E(w) is the complex amplitude of the incident wave action on the emitter with the zero generalized velocity V=O, and Zex (CIl) is the impedance of the emitter coupling with space. To generalize the further description note that by the generalized force F we will denote the projection of the medium action on the corresponding oscillation degree of freedom of the emitter (or wave boundary problem). In principle this degree of freedom may be of arbitrary dimensions (not local only). And projection of motions in boundary problem considered on the degrees of freedom.. we call as generalized velocity V. So any boundary problem becomes divided into a discrete set of oscillatory linear independent degrees of freedom. Literal identity of the mentioned forces and our notions takes place only for the monopole emitter. The power

wA absorbed by the

emitter on the average during the period T (T=2woo) equals W A = ReY(w)*F(w)/2 (3.2.)

and achieves its maximum 2

W A = WI~a., =IE(rJ)1 /SReZ.,(r,) at the optimum amplitude of generalized velocity VA «(I)):

(3.3.)

Y(w)= yA(U) = E(w)/2ReZ ex (u) , (3.4.) where Re Z" (00) is the radiation resistance. Expressing the optimum amplitude of velocity yA (00) via measured amplitude F(oo) of medium action on the chosen freedom degree and measured amplitude Vuo ) of velocity. we could get control algorithm yA(w)={F(w)+Z ex (U)Y(w)}/2ReZ ex (00 ) . (3.5) Exact control would mean V = VA. The control (3.5) is local in space, causal, but nonlocal in time, because it operates with the time-spectral estimates of the measured values or complex amplitudes (i.e. LSNT, causal, using the terminology of III ). We should note that this algorithm (3.5) is stable and nonsensitive with respect to the errors of emitter velocity if the value of external impedance Z.. (oo) is known exactly, So any velocity error !'.v leads to the corresponding component of the measured force t.F = -'L"t.V and they are annihilated by each other in (3.5). But sufficiently large error impedance !'.z" may cause selfexcitation. There are two possible versions(see Fig.3-a,b) of realization of optimum velocity (3.4): (I) by switching on an additional load to emitter (passive version at f;=O); (2) applying of additional force s (active version at f; ~). For the incident wave the absorbing emitter presents a load with some internal impedance Z", «(0) satisfying the relation F(oi) = ZI" (to) V (w) .

[=. l;tE 1~11~ (a)

~" I-Z-
=()

In practice it may be hard sometimes to ensure optimum velocity V = V""' by passive ways Re Z,n

(3.12)

= Re Ze-x (i.e. too

soft connection, required by low radiation resistance of emitter) so this lack may be compensated by applying additional outside force = A (similar to electromotive force in electrotechnics): f:A =I{(Z m +Z.,)/2ReZ ex }-11 E

c c

or with corresponding

&A

c= C

S

= [«Z m +Z.,)I2ReZ.,)-11 (F+Z.,V)

(313)

for resonance scattering regime

(3.14) Relations (3.1)-(3.14) refer to emitters of arbitrary wave dimensions. We only should associate them with the oscillating controlled degrees of freeedom of emitters (feeders for instance). gS=2g A

132


4. EXAMPLES OF THE WAVES OF DIFFERENT PHYSICAL NATURE

Note that the absorption non-locality takes place in rather great numbers of boundary problems, waves of various physical nature, geometric dimensionality (two- and three- dimensional) and emitter multipolarity. There are several well known examples of nonlocal absorption for instance: (1) sound resonance absorption by a gas bubble in liquid; (2) light absorption by a dipole; (3) plane sound wave absorption by the infinite equidistant emitter array of low relative pulse duration ("transparent"). Absorption nonlocality phenomenon is studied [I) in the four types of boundary problems. The values a A are given in Tables 1,2,3,4 correspond to the incident wave being in the maximum of the emitter directivity diagram. In the general case the expression for the maximum absorption cross-section should be multiplied by the emitter normalized directivity diagram (with respect to the power) in the point corresponding to the orientation of the emitter. one-side oscillating piston

one-side piston in screen

pulsing sphere

ReZ ex = (npr" / 2c)ro 2 E = 2m 2 p; a A = 'A,2 /21t

012r~

Re Zi,

= (4nr 4p /

C)ro2

2

E = 41tr p; cr A = 'A,2 /41t Re Zex = (npr" / 3C3)ffi 4

oscillating sphere

E = ( 41tr 3k / 3)Pj a A = ').} /3rt

ne z., = (1tpr4d 4 / C 5 )ro 6 E = (41tr 2d (jA

2k 2)P j

= ').} / 2rt

Re Zex = (n 2 pr ~ffil E = (2rtr)Pj 2

pulsing cylinder

aA oscillating cylinder

= 'A /2rt

ReZ ex

= (lt 2 r 2p / 2C3~ffi13

E = (1tr 2k)P j (jA = 'A, / 41t

I

power flux SW = p ,12 /2pc dispersion equation co = ck density for radiation t..:..::::....:..::=='--'

Table 1. ACOUSTICAL EMITTERS. 133

LOCAL EMITIERS AND MAXIMUM ABSORPTION

For more general description we will note below by E and ReZ e• projection of action of incident wave on some freedom degree (in fixed state) of boundary problem and correspondingly radiation resistance ReZ.. = 2W R IIvl 2 connected with this degree of freedom, where V is velocity refered to degree of freedom chosen, W R power radiated in the absence of incident wave. 4.1. SOUND WAVES in fluid in which the particle velocity defined by the wave potential is the Young's modulus, v is Poisson's coefficient, per, t) is the

external pressure, V = [(0 / (}x2) + (0 2 hy») . Table 4 gives the radiation resistance Re Z.. (eo) with respect to the point action; the force and the twisting moment E with which the incident wave affects the source of the force and the moment correspondingly and the corresponding cross2

2

135


sections. A i is the amplitude of normal shift in the incident wave with frequency c.o, the density of the power flux S W and the dispersion equation ro=ro(k) . Note that in the first case E is the force with which the incident wave affects the point ~ fastened by hinges and in the second case E is the moment of forces with which the incident wave affects the point ro freely moving vertically, but with the fixed zero inclination. It is characteristic that in spite of the source "pointness" (not kD~O as before, but D=O) its effect is not localized, because in the point of the application of the force ro (in the absence of the incident wave) either = 0) and power srectral density (PSO) SEem) , and define the following statistical characteristics of the piston with radius r in the rigid screen, operating in the regime of maximum absorption ( 6.3): 2 (d v. (I) Id1 2 ) = 1tr2PE(I) la 2 (7.1) where a 2 = 1tpr' 12c is the "radiating" coefficient (see ((>.3)) of the piston in the screen. (a)the average square of the piston shift from the initial state x=O:

f

< x:(t) >=(ltr 2 la 2)2 SE(w)dw 12ltul" (b) the average piston velosity square:

Then

(7.2)

f

< v: (t) >= (ltf 2 1a 2)2 S E(u) )dw /2ltW'

at the average power flux in the incident wave:

S" =< P~ (t) > Ipc =

fS

E

(w )d(o> 121Cpc,

We see from (7.2), that the piston coordinate is localized near the initial position x=O ( < x:(t) > is finite), if pressure PSD S E(w) satisfies the condition LimSE(w) hl~O

=coWU(co,a. =canst, a. > 5)

(7.3)

In the further estimations we shall use the PSD model of the form: SE(u)=(8Ji 115)t~w6 E.XP(-t~m2) (7.4) (t E is the correlation time-scale of sound pressure PE(I»). Conditionl'l.J] is satisfied by the cutting off the time-constant component of the pressure sensor signal due to several RC-circuits, connected in series. For the chosen PSO model(7.4) we get: 2 < (I) >= (16 115)(C2t~ 1p2r") < P~ >. < v: (r) >= (8 115)(c t; 1p2f') < P~ >. < «. (t) >= (2ltC"t~ ISp) < P~ >.

x:

The average absorption cross-section is: < cr. >=< w. (t) > f.''" = 2ltc" t~ 15 . Note that for the random sound field correlation length fE = cr F. plays the same role as the wavelength A. for the monochromatic field and absorption cross-section does not depend of piston radius r as in previous considerations. Taking into account minimum time-scale of control. t < .one could formulate the condition of neglecting of highest terms (beginning from third term) of piston impedance temporal representation: 140

LOCAL EMIlTERS AND MAXIMUM ABSORPTION

(32r Ie-c.) «1.

(7.5)

APPLICATION OF TWO SOLUTIONS OF ONE-DIMENSIONAL PROBLEM TO TWO-DIMENSIONAL BENDING WAVES IN ELASTIC PLATE.

Differential algorithm (6.4) of autoresonant absorption has the simplest short form v A(t)=(f(t)+zv(t»)/2z in this case of bending waves of elastic plates. This algorithm operates with the point normal shift x(t) (or its velocity v(t) ) of the plate and measuring the force f(t) of final control device interaction with the plate both measured and having known a priori the point impedance z of the plate. But we note that some algorithms designed for space one-dimensional problems may be applied to the bending waves too without knowing z. There are two remarkable properties of bending waves in elastic plate (see Table 4): (1) point (monopole) impedance is the real value only and does not depend on the frequency; and following from this (2) an infinite plate saves the normal shift of coordinate of free plate surface after completion of the force pulse(and during the time interval in which we can take the plate as an infinite one). So this boundary problem in its local sense is very similar to the case of longitudinal waves in an elastic rod. This makes it possible to apply the space-time local algorithms: (a)-algorithm of half-return 141 and (b)-searching algorithm of maximum instant power absorbed 15) , ensuring the time averaged optimum trajectory < v(t) > 'w = V A (t) = Vi (1)/2. which means maximum instant value of absorbing power (v,(t)is normal velocity in the incident wave).

141

LOCAL EMITfERS AND MAXIMUM ABSORPTION

8. ABSORPTION IN THE SYSTEM OF N EMITfERS. Now we will consider the system of N emitters. We must define here the matrices of external and internal impedances. Fig.6 illustrates the definition of matrix lox' lift in the case of system of two acoustical emitters one side oscillating is the matrix of emitters impedances pistons in the pressure field P of incident wave. i. (ell) = IZe'l Jk «

U

I

corresponding to their relation with the space of incident waves and with each other via this space. Zin(ro) = [l;~] is the matrix of the internal impedances of various channels (for example, feeders) escaping the space of incident waves propagation. Similarly to the consideration of item 3 we expand this problem. In principle the components of the vector E(ro)=IE,(ro).E,(ro)•... ,EN(ro)1 of the complex amplitudes of the forces induced by the incident waves on "inhibited" emitters may be of various physical nature which results in various physical dimensions of matrix i.e«(Jl) and Zm «(Jl) •

o

o (a)

~I.-

I-{Z~'-:--..,

Fig.6. On the definition of matrices l.x ,Z m" (a) one-side oscillating pistons: (b) equivalent draft.

8.1. ALGORITHM OF MAXIMUMTOTAL POWER ABSORBED BY N-EMITfERS.

Let us consider (similarly to the item 3 the system of N emitters, not local ones only). The total absorbed W Aand scattered W'(averaged during the period (21t/c'l) of the incident wave) powers are equal to: WA =

Vnez, V*12

W' =

VReZ., V*12

(8.1)

where V(ro) = [V,(w).V,(ro)•....VN(ro)1 is the vector of the complex amplitudes of emitter's velocities. It is simple to obtain that the vector F(w) affecting radiators equals:

=I F, (m ), F, (to), .... F

'l

(w») of the complex amplitudes of forces

F= E- Z~X V.

(8.2)

Note that the vector 'I\ell) describes the projections of the medium's force effects on the corresponding oscillating degrees of freedom of the absorbing system. Taking into account the relation of the vectors

F and V via the matrix of

the internal impedances (analogously to (3.6»

F = Z,,, V

(8.3)

and maximizing the quadratic forms (8.1) over the maximum A W A = Wmax =

EIReZ

under the condition of resonance:

-

Irn Z",(ro)

ex

).I

V• we

obtain that the total absorbed power w A has

E*/ 8

-

= -1m Z.. (el),

(8.4)

-

ReZ,,,(lJl)

-

= + ReZ., (or)

(8.5)

and velocity vector (N-dimensional analog of (3.4)

V == VA == [ReZ., r' E* /2

(8.6)

The control of the emitters system maximizing the total absorbed power has the form

VA =/ReZ

r (F+Z" V)/ 2 J

ex

(8.7)

where VA is optimum (set for the final control element) value of the velocity vector, and F and V are the vectors of the forces affecting emitters and their velocities measured directly on emitters. 142


Exact control would mean V = VA. In this case S

W = W:.ax = 4W:..

WA == W,:... The scattered power has the maximum

under the condition

-

-

ImZ,.(oo)=-lmZ,,(m),

ReZ,. =()

(8.8)

and the vector of the velocities

y=YS=2YA.

(8.9)

When it is impossible to satisfy resonance conditions by passive impedance connections their active version may be used by switching on the additional vector of external forces E (see Fig.6, analog of electromotive force in electric branch)

& = gA = {[lin + l ex Jl2 Re l ex 1'1 -hE or in the form of some algorithm j;A =([lin+ l ex Jl2ReZ ex r' - l }(F + Z ex V) and with optimum vector (';S for maximum scattering j;S =

(8.10)

(lUI)

z&A

(8.12)

The relations (8.1-8.11) (as before (3.1-3.14)) refer to the emitters of arbitrary wave dimension. But in the case of significant wave dimensions of emitters we must understand the absorption and scattering parameters as the ones corresponding to the controlled oscillating degrees of freedom of emitters. 8.2. CAUSALITY. Passive versions of control ensuring the conditions (8.5) may seem more preferable as there one need not know the space spectrum of the incident wave besides its frequency. However there some additional problems are appearing and considered below For a sole local emitter we had local in time (causal) control due to the differential representation ((1.1) where derivatives values of the smooth function E(t) at the moment t are continuous i.e. ljl(n)(t+O) =ljl(n)(t) =(p1n)(t-O) and the causality problem did not appear. So for an isolated emitter. local causal control providing the autoresonance absorption may be also used, while the nonlocal emitter or space developed system of local emitters do not admit the causal control already, because in this case. at the moment teach n-th emitter should "know" about the velocity of each m-th emitter at the moments t ± ~"I c} (e is the sound speed, ~"~,, -are the coordinates of

tlrn - /

n-th and m-th emitters). This condition is caused by the time representation (h(t») of the matrix IRe Z" (00) I-lor by demand that mutual transient function

should be symmetric in time

r:

h~. (t) = h nm (-t) (due to Im[Re Z..(00) = 0) which contradicts the principle of the causality of the adaptive control, when the control action is formed only on the basis of the previous measurements of instantaneous velocities and forces. But among the large class of realized (causal) four-terminal networks it is impossible to choose N elements of the matrix 2:",(Ul) to provide W A = W:.. for a space-developed absorbing system of interacting emitters. Let us define more thoroughly the terms mentioned above: (a) "space-developed system" i.e. kD> I, where k-maximum wave number corresponding to the operating frequency range, D-maximum distance between the emitters; (b)"interacting emitters" i.e. amplitude E of the incident wave pressure on the hindered emitters is less than the pressure .2:" V of the emitters on each other (I~ < where values Y are large and

IZ" vi,

defmed by Ein accordance to (8.2) ).

Fig.7. (a) noncausal active structure. (b) passive structure with causal connections. The collective resonance of the system may be achieved only by the spectral synthesis, i.e. lamination of the static pictures of fields. But the spectral synthesis being of discrete nature, such a 143

LOCAL EMlITERS AND MAXIMUM ABSORPTION

synthesised picture will "collapse" during the time t, - (1 / l!.ro.), where l!.ro. is the discrete frequency of synthesis. In addition, the spectral estimates of fields should be much more long-term, than t,' thus making unsolvable the problem on the causality of control (noncausal only). Thus the optimum velocity distribution (8.2) in the absorbing system exists, but cannot be achieved by the differential causal operations of the successive approximation "measured-tuned". Fig.7-a shows the structure of the system in which the emitter velocities are defined by the optimum values of the electromotive forces of final control elements calculated beforehand in unitary centre under the algorithm (8.10) of maximum total average absorbed power. Fig.7-b shows the structure of the absorbing system with causal connections, which in principle can not provide the collective resonance response in the space developed system of emitters.

Fig.8. (aj-cooperative strategy, (b)-individual strategy ; A-control algorithm. *-microphone. e-emitter.

8.3 COOPERATIVE AND INDIVIDUAL STRATEGY: DEFINITIONS.

Removing the linear circuits between the emitters from our consideration, we would formulate two main strategies for the emitters. (a) Cooperative strategy (Fig.8-a) of n-th emitter: I. Global aim- to maximize total power absorbed by all N emitters. 2. Local aim- to follow the trajectory assigned by unitary control center maximum of total power absorbed by N emitters: 3. Information - wave field Fo(t) on all N (1:5 n:5 N) emitters: 4. Way- tuning of velocities V" (t) of all N (1 ::; 11 ::; N) emitters by unitary control center: 5. All emitters are controlled by one algorithm (8.7) with total information about incident wave and emitters (in unitary centre). (b) Individual startegy (Fig.8-b) of n-th (I :5 n :5 N) emitter: 1. Globalaim-to maximize total power absorbed by all N emitters. 2. Local aim - to maximize power absorbed by n-th emitter: 3. lnfonnation - wave field Fo (t) on n-th emitter: each emitter perceives the total wave field (including the field of adjacent emitters) as the field of incident wave only. following the algorithm (6.4). Each emitter does not know anything about other emitters. their coordinates and velocities besides the total sound pressure on itself. 3. Way - tuning of velocity V o (t) of n-th emitter: 5. each n-th emitter is controlled by algorithm (3.5) or (6.4) -maximum of power absorbed by each emitter. This case presents the minimum dependence of the individual emitter on other ones in the sense of control. But the maximum dependence of instant states of adjacent emitters. Each emitter tries to assign such a velocity on itself which would ensure the maximum work of incident wave field (without distinguishing from fields of adjacent emitters) and absorbing power as it seems.

144


8.3.1. INFINITE LATTICES.

I-

».,

»,

Fig.9. On the same gain of absorption under cooperative strategy. both in two- and three-dimensional cases. The cooperative strategy under the algorithm (8.7) applied to the infinite lot of emitters gives an unexpectedly simple solution in the following version (see Fig.9). Consider all infinite plane screen equipped with piston emitters, spaced equidistantly with period d with normal incident plane wave. It is clear thai maximum of absorption in this case (onedimensional for far-field) is identical to the absence of reflected wave. We compare the absorption cross-sections of only one piston-like emitter on the whole rigid infinite fixed plane screen and one emitter in the infinite equidistant piston lattice on the same screen. It is easy to ensure that piston matching (matching of volume velocity) of this plane lattice with respect to the normal incident wave corresponds to the regime of maximum absorption simultaneously. Maximum possible space period d for this regime is A /2 (halflength of the incident wave), and maximum absorption cross section is (l' :,,,.,,llattlceJ) = "A~ / 4 correspondingly. Absorption cross-section of the sole piston emitter in rigid screen is a;~I(solit;3) =

absorption

i-..1 /2n

(see Table 1). So the under the cooperative strategy.. the emitter has the cross-section of (n /2) times more than under the individual strategy:

a:oop(lanice)

rt

a ,:, (solir.J)

2

a:oop(1anice. 2) . F' --'-'-:."----. Note that we have the same gain two-dimensional case (see Ig.9). a~,(solit;2)

8.3.2. TWO PULSING SPHERES. Above we had considered the case of "forced" individual strategy, when we had a sole emitter for comparison Now we consider two strategies in the monochromatic case of a pair of identical monopolar emitters (pulsing spheres of radius r and distance d from each other) and the matrix of external impedances

[ZliZ" Z" ZI/], where z.,- Z"- =(4nr'krc /d)E..'-':P(ikd), ZI'".--

Z = "

=

=

2

4npcr'k + i(4nr

Jkpc)

, (r«d.

kr'< I ). Lonely pulsing sphere without interaction with another one ensures the absorption cross section (see Table 1) a A (I) = (A.:' /411:). For instance, dipole-like combination of two monopole spheres gives absorption cross section a A (2)

= 10" A (1)Cos 2 S (at S = (). three time more than in monopolar case, where 9 is the angle between the direction of

incident wave and the axis of emitter pair). COOPERATIVE STRATEG'f. Using the matrix of real parts of external impedances -

[+1 Sincf kd)

Re Z,x = Zo "

r'

SinC(kd)] ,. (z [+1 and us inverse one IRe Z e-xj' = . ,'0 , " +1 '. 1-SlI1c-(kd) -SlIIc(kd)

-SinC(kd)] ( I w iere +1

Zo = 4nk 2r 'p C) and equation (8.7) for optimum piston complex amplitudes V1.2 of velocities we get the

asymptotic representations of absorption cross-section for: (I coop(kd« I) = 2U.z / 41t)(3Cos 2S + I) = 2(3Cos 2 & + l)o A (I) = 2{a A (2)+ a A (I)} andacoop(kd» I) = 2(1 A (I) INDIVIDUAL STRATEGY. Following this strategy (see equation (3.5.)) the following are complex amplitudes of piston velocities and total absorbing power: V,.2 = (F;.2 - Z11.22 VI.2 )12(ReZ n), W'od = (1 /2) Re V,'IE, - Z21 V2 - ZII VI J + (1/2) Re V; IE2 - Z12 VI - Z22 V2 ) · ". So in the symmetric case (E, = E2 ) assuming VI = V 2 one could get: (a) in the absence of interaction individual strategy gives trivial value of absorption cross-section (J ind (kd » I) = 2(1 A (I) (b) strong interaction leads to significant decreasing of absorption cross-section a md (kd «

I) -

(I A

(I) 3(kd)2 H O.

And we can note that interaction between the emitters (instead the sole individual emitter) under the individual strategy should lead to more reduction of the absorption cross-section of each emitter, This is the typical case of competition between the local absorbers. 145


There are two aspects of design of absorbing system: (a) spacing of emitters. (b) determination of optimal velocities of emitters already spaced. Taking into account the absorption cross-section of linear multipolar antenna (item 5) and a coop (kd «I) one could guess that the most attractive geometrical configuration of the absorbing system is a linear one (with further optimization of velocities) but perhaps with non-equidistant spacing of emitters. 8.4. COMPETITIVE BEHAVIOR AND STABILITY. Above we used monochromatic representation of cooperative and individual strategies. Now we consider the simplest model of wide-band temporal representation of individual strategy for absorption of normal plane incident wave by a pair of pistons in rigid infinite plane screen (see Fig.lO-a). Cooperative strategy would compensate undesirable influence of mutual impedances. however loosing causality. Really cooperative strategy is impossible (non-causal) in wide-band case. following item 8.1. So we consider only individual strategy on the simple model. where control algorithm (6.4) is used.

(y)

l

U'"mtW,t m k V

V,.2(t)\X)

(b)

(a)

Fig. 10. (aj-two pistons in rigid screen. (b)-dynamic model of servo-drive. Fast oscillations of control system with time-scale t c (see (6.5». caused by the tracking servo-drive. are acting significantly on the adjacent piston as it cannot distinguish this component of pressure field from the field of incident wave, due to differential temporal representation of mutual impedance. In particular. due to this circumstance one could expect the instability of symmetric solution (at Ellt) = E2ll) and symmetric initial conditions). When control is perfectly exact with simplest algebraic relations between assigned and real velocities of pistons, u1. 2 (r) == V1. 2 (t) shown in Fig. to-a , the acoustical-mechanical system is stable. However real connection between U 1.2 (t) and VI•2 (t) is dynamical, one simple model of which we consider below. Point "x" (see Fig.lO-b) has the trajectory in time identical to the trajectory Vu(t) assigned by the control algorithm. Trajectory of the point "y" has dynamic delay and dynamic distortions caused by finite elasticity k, inertial mass m, dissipative factor v. So at infinite elasticity ( k ~ 00 ) one could expect to achieve lll,2 lt) = VI,2 l t) . Dynamics of servo-system presented in Fig.l O-b is described by the system of equations:

)p(t)+(~)u~(t-t).

V/,(t)=( 4C2 pd'

V;/(t)=(

t

-

4C,)f>(r)+(~)u:(r_t). r

pd-'

(8.1)

f2(t) = Uu;/(t) + r-lu~(t) + yu{(t - r) + aP, (t),

fl(t) = uu{i(t) + flu;(t) + yu;(t - t) + af>,(t).

mu{l(t) = -f/(t) - k!u,(t) - V,(t)]- vlu{(t) - V/,(t)]. IllU;/(t) = -f;(t) - k!u 2(r) - V2(t)l- vlu~(t) - vt(t)!, where the following symbols are used: VI (t), V2 (r) -oscillatory velocities assigned by the algorithm (6.4 ) of maximum instant power absorbed under the individual strategy; f l (r), f2 (t) -forces (measured without errors by sensors of sound pressure) applied to the oscillating

pistons from the outside medium; sensors of velocity); u = (p1tr 4 12c); ~

t

= (d 1c) ). u l (r), u 2 ( r) -rcal oscillatory velocities of pistons (measured precisely by

= (8pr) 13) ; y = (1tpr

4

1 ct) ;

('T

= (21tr 2 )

:

r-radius of piston: p-fluid mass density; k -factor of

elasticity of servo-drive of piston controlled: m -inertial factor of servo-drive: d-distance between pistons; v -viscodamping factor of servo-drive. It is easy to show thatib-splittinlof this system (8.1) for parameter k = k(p) = k(jro) • k

_

(P) ...,-P

2ap 4 + 2(m + fJ)pJ + 2'1'2 + (4y I r)p2 exp( -2pr) - (8v 11'2 )exp( -2p1')

(4/2) r exp (2 - p1' ) - P 2

This expression. for instance, always gives poles in right complex half-plane. So wideband temporal representation of individual strategy leads to inevitable instability. Generally speaking. the system under the individual strategy cannot remain within the low-frequency approach (6.1). saving the condition (7.5). due to the following 146

LOCAL EMIlTERS AND MAXIMUM ABSORPTION

chain: quick reply of the one emitter on the field of adjacent emitter require more quick reply of the last and so on... So the absorption regime should be broken quickly at least and selfexcitation is possible. Absorption regime may be saved (in principle) if the space intervals between the emitters would present the low-frequency filters (high-frequency viscodamping mechanism for instance). Cooperative strategy ensure the remaining of all emitters within the low-frequency approach (6.1) and saving of the absorption regime but without wide-band temporal representation and with restrictions on causality. 9. CONCLUSIONS. Any problem of wave absorption maximization may by described by the variables of generalized velocities and generalized forces referred to some controlled degrees of freedom of corresponding boundary problem. Absorption cross-section of local acoustical emitters is proportional (with different coefficients) to the wavelength of the wave to be absorbed independently of their geometry, constructions, multipolarity order. And similar properties for emitters of the waves of other nature: (I) for electromagnetic emitters with specific feature: absorption cross-section for instance does not depend of the number of its current loops of the magnetic dipole and the length of electric dipole; (2) for emitters of gravitational surface waves in liquid' with specific feature: absorption cross-section of pulsing and oscillating spheres under the free surface of liquid does not depend on sphere's geometrical dimensions and the depth of placing: (3) for bending waves in elastic plates.

Absorption cross-section of linear multipole antenna at an unbounded growth of multipolarity order is closing to finite level in two-dimensional boundary problem and increasing linearly (infinitely in principle) in three-dimensional boundary problem. This is accompanied by decreasing of the width of directivity diagram of antenna. The case of acoustical absorbing monopole in the random wideband sound pressure field has been considered. Radius of correlation plays the role of wavelength of the wave to be absorbed in the expression of absorbing cross-section. Control in time differential causal form considered before is impossible for the emitters with external radiation resistance expressed by non-even degree of frequency. This is referred to the two-dimensional emitters of acoustic and electromagnetic waves and for three-dimensional ones of surface gravitational waves. Strict condition of maximum total absorption by emitters in the system of expanded wave dimensions contradictsthe principle of causality in passive version of control. Emitters under the cooperative strategy ensure more total power absorbed than under the individual strategy. And this difference is proportional to the smallness of the wave distance between emitters. Individual strategy applied to emitters in temporal wide-band representation with servo-drives of any finite internal impedance causes instability of damping system. 10. REFERENCES

I V.Arabadzhi "Local emitters in the regime of maximum absorption" , Prcprint N320 of the Institute of Applied Physics Nizhny Novgorod 1992

2. LN.Srctensky "Theory of thc Wave Motions of Fluid", Moscow, "Nauka", 1977, p.512 3. LN.Sretensky 'Theory of the Wave Motions of Fluid", Moscow, "Nauka", 1977, p.516.

4. V.Arabadzhi "On the Space-Time Local Active Control" Journal of Low Frequency Noise, Vibration and Active Control, Vol. 16, No.2, 1997, p.89. 5. Ibid, p.95.

147