Numerical simulation of acoustic wave phase conjugation in active ...

Numerical simulation of acoustic wave phase conjugation in active media S. Ben Khelil and A. Merlena) Laboratoire de Me´canique de Lille ura CNRS 1441 Cite´ scientifique, 59655 Villeneuve d’Ascq, France

V. Preobrazhensky Wave Research Center of General Physics Institute Academy sc., 38 Vavilova Street, 117942 Moscow, Russia

Ph. Pernod Institut d’Electronique et Microelectronique du Nord, DOAE, umr CNRS 8520, 59655 Villeneuve d’Ascq, France

共Received 18 January 2000; revised 18 September 2000; accepted 3 October 2000兲 Godunov-type computation schemes are applied to numerical simulations of wave propagations in time-dependent heterogeneous media 共solids and liquids兲. The parametric phase conjugation of a wide band ultrasound pulse is considered. The supercritical dynamics of the acoustic field is described for one-dimensional systems containing a parametrically active solid. The impulse response function, numerically calculated for a finite active zone in an infinite medium above the threshold of absolute parametric instability, is in a good agreement with the analytical asymptotic theory. The supercritical evolution of the acoustic field spatial distribution is studied in detail for parametric excitations in an active zone of a solid layer, loaded by a semi-infinite liquid on one side and free on the other. © 2001 Acoustical Society of America. 关DOI: 10.1121/1.1328794兴 PACS numbers: 43.25.Dc, 43.25.Lj 关MFH兴 I. INTRODUCTION

The problem of wave propagation in a nonstationary medium when parameters depend on time is of fundamental interest because of its various applications in acoustics and solid state physics. The parametric wave phase conjugation 共WPC兲 in photorefractive media is an example of such applications.1 In acoustics, parametric WPC has been studied for liquids and solids 共water and water with gas bubbles,2,3 piezoelectrics,4,5 magnetics,6 and piezo-semiconductor systems7兲. The modulation of the acoustic parameters of solids is usually carried out by means of rf, microwave, or optical pumping, distributed almost homogeneously in the active zone of the medium. There are no exact analytical solutions to the general problem of parametric WPC. The perturbation theory is applicable for relatively weak parametric interactions under the threshold of absolute parametric instability. Above the threshold 共in a supercritical mode兲, multiscale asymptotic expansion methods 共MSAE兲 can be used to describe narrow band resonance parametric interactions.4,8 Recently the problem of WPC has been discussed extensively in the context of ultrasound time reversal transformation for applications in nondestructive testing and medicine.6,9 Acoustic signals of wide relative frequency band are of practical interest, though the applicability of MSAE methods under such conditions becomes problematical. For this reason, the development of numerical methods adapted to the problem seems to be a productive research direction. On the basis of the propagation properties involved in the phenomenon it is possible to show that the mathematical problem falls within the scope of hyperbolic partial difa兲

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J. Acoust. Soc. Am. 109 (1), January 2001

ferential systems. Therefore the numerical background developed in the last decade within the frame of unsteady aerodynamics can be applied to this problem and, particularly, all the Godunov family schemes10 including the weighted average flux method 共WAF兲 of Toro.11 The present paper represents the first application of this approach to the problem of wave propagation in nonstationary medium. An example of supercritical parametric WPC of a wide band acoustic pulse is considered. In order to compare numerical results with analytical solutions4,8 the model of localized active zone in infinite solid is studied for narrow band pumping. The time evolution of the elastic stress distribution is presented. The shape of the phase conjugate pulse is in satisfactory agreement with the analytical calculation. The problem is also adapted to generic experimental conditions, when the active solid is loaded by a liquid medium on one flat boundary and is free on the opposite one. The parametric amplification of ultrasound reverberations observed earlier in experiments12 is correctly described by the numerical simulation. II. MATHEMATICAL FORMULATION A. Fluid media

The basic idea of the present numerical approach comes from the natural formation of acoustics in fluids. It is well known that in a nondissipative fluid, the linearized Euler equation can be written:

⳵␳ ⫹ ␳ 0 “"V⫽0, ⳵t

⳵V 1 ⫹ ⵜp⫽0, ⳵t ␳0

dp ⫽c 2 , d␳

共1兲

where ␳, p, V, and c are, respectively, the density, pressure, fluid velocity, and sound speed in the medium. 0001-4966/2001/109(1)/75/9/$18.00

© 2001 Acoustical Society of America

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The active medium is defined by a given function C(t) such as c⫽C 共 t 兲 . Hence, since ⳵␳ / ⳵ t⫽(1/c 2 )( ⳵ p/ ⳵ t), system 共1兲 rewritten for p and V reduces to 1 ⳵p ⫹c“"V⫽0, ␳ 0c ⳵ t

冋

册

1 ⳵V ⫹c ⵜ p ⫽0. ⳵t ␳ 0c

共2兲

The acoustic equations usually follow from the change of variable

␪⫽

p⫺p 0 , p 0c

where p 0 is the uniform steady pressure of the medium at rest. Here system 共1兲 becomes

⳵␪ ␪ ⳵c ⫹c“"V⫽⫺ , ⳵t c ⳵t

⳵V ⫹cⵜ ␪ ⫽0. ⳵t

共3兲

If c were a constant, this would be the classical system of acoustics in fluids but, in this case, a source term appears on the right-hand side. In fact, function C(t) is nothing but a small modulation of c 0 , the sound velocity of the medium at rest. In the following, C(t) is assumed, for all practical purposes, to be given by C 共 t 兲 2 ⫽c 20 共 1⫹m cos共 ⍀t⫹⌿ 兲兲 , where m is a small parameter (mⰆ1) referred as the ‘‘modulation depth.’’ It is clear that system 共3兲 can be linearlized and replaced by

⳵␪ ␪ ⫹c 0 “"V⫽m⍀ sin共 ⍀t⫹⌿ 兲 , ⳵t 2 ⍀V ⫹c 0 ⵜ ␪ ⫽0. ⳵t

共4兲

The source term reflects the active effect that becomes sensitive for high frequencies. The left-hand side term is the linear advection operator, which is the basic linear hyperbolic system. All the numerical methods for compressible fluids have been tested on this advection system and, particularly, all the Godunov-type schemes. Owing to flux splitting techniques, these finite volume schemes are essentially a sequence of one-dimensional operators. Therefore, the onedimensional problem is crucial for numerical methods. Moreover, this corresponds to an unavoidable first step in a detailed physical description of the process that can only be provided by numerical methods if these are sufficiently accurate. In many respects, a numerical solution of

⳵␪ ⳵v ⍀␪ ⫹c 0 ⫽m sin共 ⍀t⫹⌿ 兲 , ⳵t ⳵x 2 ⳵v ⳵␪ ⫹c 0 ⫽0, ⳵t ⳵x

共5兲

gives more information on the efficiency of the methods and on the physical behavior of solutions that a direct resolution of a three-dimensional 共3D兲 case with all the unessential features caused by purely geometrical effects or by mesh prob76

J. Acoust. Soc. Am., Vol. 109, No. 1, January 2001

lems. Moreover, the physics involved in the 3D effect is well known: it is essentially the conversion between longitudinal and transverse waves when reflections occur. In this case, the most important goal is the physical understanding of the pumping effects. This means that the source term is more essential than the advection operator. Therefore, a highly refined one-dimensional 共1D兲 simulation is more reliable for a first attempt than a 3D case, which is inevitably less refined because of the limited capacity of the computers. Local refinement techniques are less helpful here than in other applications since the high frequencies of the spatial variations are not localized as in the case of shock waves in fluid mechanics. In such a case, rapid spatial variations concern the whole computational domain. Matters become clearer in 1D, and as a result upgrading to a 3D becomes an engineering problem which is not trivial but generally carried out well by developers. In the frame of flux splitting techniques, this consists in building the 3D operator as a sequence of 1D. Introducing new variables w 1 ⫽ v ⫹ ␪ , w 2 ⫽ v ⫺ ␪ , Eq. 共5兲 can be rewritten:

⳵w1 ⳵w1 ⍀ ⫹c 0 ⫽m 共 w 1 ⫺w 2 兲 sin共 ⍀t⫹⌿ 兲 , ⳵t ⳵x 4 ⳵w2 ⳵w2 ⍀ ⫹c 0 ⫽⫺m 共 w 1 ⫺w 2 兲 sin共 ⍀t⫹⌿ 兲 . ⳵t ⳵x 4

共6兲

The problem finally comes down to two advection equations in opposite directions coupled by linear source terms. Apparently the numerical treatment of such problems is well known but the need to manage high frequencies makes it less trivial than it seems at a first glance. The scheme has to be robust and very weakly dissipative. Before beginning the numerical treatment, it is worth extending the approach to elastic active solid media. In this field, experimental works and analytical theories are much more developed than for fluids.

B. Elastic solid media

For linear elasticity, the formulation in the previous form is less natural. Let it be assumed that the stress tensor is

␴⫽␭ 共 t 兲 “"uI⫹2 ␮ 共 t 兲 ␧. Vector u is the displacement and ␧ the small deformation tensor:

␧⫽ 21 共 ⵜu⫹ 共 ⵜu兲 T 兲 . Here, the Lame coefficients ␭ and ␮ can depend on time but not on the space coordinates. Consequently the classical deviation of the wave equations from the Navier equation for constant ␭ and ␮ still holds. Briefly, by splitting u in ul ⫽ⵜ ␺ and ut ⫽“ÃA 共Helmholtz decomposition兲, the momentum equation ␳ ( ⳵ 2 u)/( ⳵ t 2 )⫽“"␴ gives ⌬␺⫺

1 ⳵ 2␺ ⫽0, c 2l ⳵ t 2

Khelil et al.: Simulation of acoustic phase conjugation

共7兲 76

⌬A⫽⫺

1 ⳵ 2A ⫽0 c 2t ⳵ t 2

共8兲

with c 2l ⫽(␭⫹2 ␮ )/( ␳ ) and c 2t ⫽ ␮ / ␳ . Starting with the compression wave ul , the following change of variables is introduced:

␪ l ⫽⫺

1 ⳵ 2␺ , cl ⳵t2

vl ⫽

⳵ ul ⳵␺ ⫽ⵜ , ⳵t ⳵t

Nevertheless, in one dimension, no mode changes are expected even at the interfaces, and consequently, no shear waves appear unless they exist initially in the medium. For the clarity of the analysis, we now suppose that this condition is fulfilled.

C. Numerical method

Systems 共6兲 and 共13兲 can be written symbolically

and consequently, 1 ⳵ vl ⵜ ␪ l ⫽⫺ , cl ⳵t

⳵U ⳵F共 U 兲 ⫹ ⫽S, ⳵t ⳵x

共9兲

U 共 x,0兲 ⫽U 0 共 x 兲

Moreover the wave equation 共7兲 becomes

␪ l ⫹c l “"ul ⫽0

共10兲

and F共 U 兲⫽

or

⳵␪l ⳵cl ⫹c l “"ul ⫽⫺“"ul , ⳵t ⳵t

共11兲

and

which can be rewritten according to 共10兲:

⳵␪l ␪l ⳵cl ⫹c l “"ul ⫽ . ⳵t cl ⳵t

共12兲

Therefore, Eqs. 共9兲 and 共12兲 once again give system 共3兲 but with a source term of the opposite sign. In one-dimensional 共1D兲 problems and after linearization around the sound velocity c 0 l of the medium at rest, the system becomes

⳵w1 ⳵w1 ⍀ ⫹c 0 l ⫽⫺m 共 w 1 ⫺w 2 兲 sin共 ⍀t⫹⌿ 兲 , ⳵t ⳵x 4 ⳵w2 ⳵w2 ⍀ ⫹c 0 l ⫽m 共 w 1 ⫺w 2 兲 sin共 ⍀t⫹⌿ 兲 . ⳵t ⳵x 4

共13兲

␭⫹2 ␮ ␪ l ⫽⫺ ␪ l ␳ c l . cl

Hence Eq. 共10兲 is just the reduction of the constitutive law for normal stress in direction x. Obviously, solids differ from fluids, even in 1D, by the fact that the stress tensor is not isotropic for solids, the normal stress in the y and z directions being ⫺␭ ␪ l /c l . The case of shear waves can be treated identically but with the variables

␪t ⫽⫺ 77

1 ⳵ ut and Tt ⫽“ut . ct ⳵t


冉

c 0w 1 ⫺c 0 w 2

m

共14兲

᭙x苸R

冊

,

U⫽

冉冊 w1

w1

,

⍀ 共 w 1 ⫺w 2 兲 sin共 ⍀t⫹⌿ 兲 4

⍀ ⫺m 共 w 1 ⫺w 2 兲 sin共 ⍀t⫹⌿ 兲 4

冊

,

depending on the case 共solid or fluid兲. System 共14兲 is solved by an explicit finite volume method. The spatial domain is shared in N cells of length ⌬x and the time step is ⌬t. The numerical solution U ni ⫽U(i⌬x,n⌬t) is obtained at time (n ⫹1)⌬t by ⫽U ni ⫺ U n⫹1 i

The sign of the source term is irrelevant for the amplitude evolution and corresponds only to a phase shift of ␲ between solids and fluids. This difference arises from the fact that, in fluids, the dilatation rate “"V is not directly related to pressure variations by a constitutive law but indirectly, through the mass conservation, by an unsteady thermodynamic process dp⫽c 2 (t)d ␳ . Conversely, for solids, in the onedimensional assumption (u⫽ul , ⳵ / ⳵ y⫽ ⳵ / ⳵ z⫽0), it is easy to verify from expressions of ␴ and c l that the normal stress in direction x is

␴ xx ⫽⫺

S⫽⫾

冉

᭙x苸R,t⭓0,

⌬t 关f 共 U n 兲 ⫺ f i⫺1/2共 U n 兲兴 ⫹⌬tS 共 U ni 兲 , ⌬x i⫹1/2 共15兲

where the numerical fluxes f i⫹1/2 and f i⫺1/2 determine the scheme. This is a first-order accurate method for time integration. Second-order methods are of course available but not of great interest here since the time step is determined more by physical than numerical reasons. It must be emphasized that the pumping is a very quick oscillation of frequency 2⫻107 Hz, leading to characteristic times of order 10⫺8 s. Time steps of order 10⫺9 s are therefore necessary for a good description of the phenomenon. As long as the simulation is 1D the advantage of a second-order time marching technique is not clear since, for the physical reasons mentioned, the time step cannot be one order of magnitude higher and conversely the numerical procedure can be more time consuming. Moreover, for a first simulation of the phenomenon, the first-order time marching method was unavoidable, at least as a reference. The choice of the numerical fluxes f at the cell interface characterizes the scheme and particularly the spatial accuracy, which is much more important that the accuracy of the time procedure because it is linked to the dissipation properties of the scheme. Tests have been performed with the basic first-order Godunov scheme10 and its extension to secondorder using the monotone upstream centered scheme for conversion laws13 approach with limiters. Finally the secondorder WAF11 共weighted average flux兲 and the superbee Khelil et al.: Simulation of acoustic phase conjugation

77

limiter14 appeared to be the most accurate combination for the problem treated here. Hence, the numerical fluxes are n n ,U ni 兲 ⫽ ␣ 1 F 共 U ni 兲 ⫹ ␣ 2 共 U i⫹1 f i⫹1/2共 U i⫹1 兲

共16兲

with ␣ 1 and ␣ 2 obtained by

␣ 1 ⫽ 12 共 1⫹ ␯ 兲 and

␣ 2 ⫽ 12 共 1⫺ ␯ 兲 , where ␯ ⫽c 0 ⌬t/⌬x is the Courant number associated with n ) the wave speed c 0 . The partial fluxes F(U ni ) and F(U i⫹1 represent the upwind and the downwind parts of the total flux. The weights ␣ 1 and ␣ 2 control the contributions of the partial fluxes; ␣ 1 is responsible for stability while ␣ 2 is responsible for accuracy and oscillations. The flux limiter has the role of limiting the contribution of the downwind term. Denoting a flux limiter by B i⫹1/2 the weights ␣ 1 and ␣ 2 are modified in Eq. 共16兲 as follows:

␣ 1 ⫽ ␣ 1 ⫹ 共 1⫺B i⫹1/2兲 ␣ 2 ,

␣ 2 ⫽ ␣ 2 B i⫹1/2 .

共17兲

The flux limiter used here is the so-called superbee. The corresponding limiting function is

B i⫹1/2⫽

冦

共 1⫺2 共 1⫺ 兩 ␯ 兩兲兲 / 兩 ␯ 兩 ,

r i ⭓2

共 1⫺r i 共 1⫺ 兩 ␯ 兩兲兲 / 兩 ␯ 兩 ,

1⭐r i ⭐2

1,

1 2

⭐r i ⭐1

共 1⫺2r i 共 1⫺ 兩 ␯ 兩兲兲 / 兩 ␯ 兩 ,

1/兩 ␯ 兩 ,

共18兲 0⭐r i ⭐ 21

n u ni ⫺u i⫺1 n u i⫹1 ⫺u ni

共19兲

D. The interface problem in 1D

Numerical simulations help to explain phenomena which could not be accounted for by experiments or analytical results. For instance, the instantaneous stress field inside a sample of active medium is not available in experiments, or the effect of wave reflections on the sample boundary requires very complicated theoretical developments in the frame of purely analytical analysis. This latter issue is easily treated in the present approach by solving the classical problem of ‘‘resolution of a discontinuity,’’ which provides the numerical fluxes in the Godunov family scheme under the name of the ‘‘Riemann problem.’’ The interface separates two nonactive media: R 共right兲 and L 共left兲. Equations 共6兲 and 共13兲 are then strictly identical, and, for the 1D case, solid and fluid are hence treated in the same way. In medium R, of sound velocity c R , the initial state is (U R , ␪ R ) and (U L , ␪ L ) in medium L with c L as sound velocity 共Fig. 1兲. In Fig. 1, under characteristic C ⫺ 0 , the lefthand side remains in its original state because no information comes from the interface. The same happens under charac⫺ teristic C ⫹ 0 on the right-hand side. Between C 0 and the interface i, the uniform solution 1 gives the state on the leftJ. Acoust. Soc. Am., Vol. 109, No. 1, January 2001

␪ 2 ⫽ ␶␪ 1 ,

v 2⫽ v 1⫽ v i ,

where ␶ ⫽ ␳ L c L / ␳ R c R is the transmission coefficient. Subscript 1 corresponds to the solution in the medium originally in state L and subscript 2 in the medium originally in state R. For all the C ⫹ characteristics coming from sector L, the invariant relation gives

and for all the C ⫺ coming from sector R:

and ␯⫽0.6.

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hand side where C ⫹ comes from state R and C ⫺ from the interface where the variables are unknown. Solution 2 is the equivalent of 1 for the right-hand side. Nevertheless, characteristics C ⫺ in sector 1 and C ⫹ in sector 2 gives no information. The missing relations for the determination of states 1 and 2 are supplied by the continuity of stresses and normal velocities at the interface, of velocity v i . This gives

v 1⫹ ␪ 1⫽ v L⫹ ␪ L

r i ⭐0

with r i⫽

⫺ FIG. 1. Riemann problem at the interface. Characteristics C ⫹ 0 and C 0 limit the unperturbed zones R and L. Zones 1 and 2 are the uniform states at each side of the interface i.

v 2⫺ ␪ 2⫽ v R⫹ ␪ R .

Since v 1 ⫽ v 2 ⫽ v i and ␪ 2 ⫽ ␶␪ 1 , elimination of ␪ 1 between these four equations leads to v 1⫽ v 2⫽ v i⫽

1 1 关 ␶ v L⫹ v R兴 ⫹ 关 ␶␪ L ⫺ ␪ R 兴 1⫹ ␶ 1⫹ ␶

and consequently,

␪ 1⫽

1 1 关v ⫺ v R 兴 ⫹ 关 ␪ ⫹␪R兴. 1⫹ ␶ L 1⫹ ␶ L

When ␶⫽1 the solution corresponds to the Riemann problem of the linear advection system in an homogeneous medium as given by Godunov. For ␶→0, medium R is infinitely rigid (c R →⬁) and the solution is ␪ 1 ⫽ v L ⫹ ␪ L , v 1 ⫽ v i ⫽ v 2 ⫽0, which corresponds to the ‘‘half Riemann’’ problem often used for reflective boundary conditions for the linear advection system. When ␶→⬁, medium R is a vacuum and ␪ 1 ⫽ ␪ 2 ⫽0, v 1 ⫽ v 2 ⫽ v L ⫹ ␪ L . The present solution provides all the data needed for computing the fluxes at interfaces between nonactive zones in any 1D situation. The boundary between active and passive zones is simulated in a very straightforward way by switching off the source terms in the passive zone. The same is done in an active zone as soon as the pumping is stopped. Khelil et al.: Simulation of acoustic phase conjugation

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W 1 ⫽W 2 ⫽0

FIG. 2. Computational domain: The active zone 共length 2.5 cm兲 is limited by dashed lines, the interface water–solid is at abscissa 0.5 cm, the boundary at 0 cm is nonreflective, and the boundary at 4 cm is stress free 共vacuum兲.

III. RESULTS AND DISCUSSION

The numerical method and the interface problem have been tested on the configuration presented in Fig. 2. From x⫽0 – 0.5 cm the medium is water and the sample of magnetoacoustic ferrite is situated between x⫽0.5 cm and x⫽4 cm. The active zone lies between x⫽1 cm and x⫽3.5 cm. The mesh contains 1000 points in water and 7000 points in the sample. This global refinement is only possible in 1D and underlines the great interest of these simulations in terms of reference for further multidimensional computations. For 2D or symmetry of revolution, a direct extrapolation of the present numerical method is still possible but for 3D the number of points would be too high. Even local refinement– derefinement methods are of little use because the refinement would be needed almost everywhere in the computational domain. A productive research direction is a numerical simulation of the amplitude of the wave without the quick oscillation, the main issue here being the derivation of suitable conditions at the interface. In this case, the boundary condition at x⫽0 cm is nonreflective as if the domain x⬍0 were filled by water too. At x⫽4 cm an absence of stress can be assumed as if a vacuum existed for x⬎4 cm. The initial condition is given in the following form: W 1 ⫽2 sin W 2 ⫽0

冉

2␲ 共 x⫺x L 兲 ␭

冊

for x L ⬍x⬍0.5,

for x L ⬍x⬍0.5,

共20兲

elsewhere.

Abscissa x L is chosen such as (0.5⫺x L )⫽3␭, ␭ being the wavelength: ␭⫽2 ␲ c 0 ␻ with ␻ ⫽2 ␲ 107 s⫺1 and c 0 ⫽1500 m/s. Figure 3 shows this initial condition and the reflection process after 3.75 ␮s is presented in Fig. 3共b兲. The wave in water is traveling back to x⫽0 cm after partial reflection on the interface at x⫽0.5 cm while part of the wave enters the sample traveling to the right-hand side. In Fig. 3共b兲, the normal stress in the x direction is presented after the wave has entered the active zone but before the pumping was switched on. The pumping is applied when the incident wave has covered about one-third of the length of the active zone. Figures 4 and 5 show the next stage of the phenomenon. In Figs. 4 and 5 the beginning of the pumping is taken as the origin of the time. In order to produce the phase conjugation, the pumping frequency is fixed at ⍀⫽2␻.1 The duration of the pumping is T⫽19 ␮s and ⌿ is equal to ␲. Figure 4 shows the normal stress field at different times for m⫽4.1⫻10⫺2 . In Fig. 4共a兲 the pumping has just been initiated and the direct amplified wave begins from the edge of the active zone. In Fig. 4共b兲 the conjugate wave can be clearly observed in the fluid. In the solid, the incident wave is just reflecting at the end of the sample. Figure 4共c兲 illustrates the amplification process and the emission of the conjugate wave in the fluid. The incident wave or its reflection on the edges of the sample is no longer visible due to the high level of amplification of the conjugate and direct waves. In Fig. 4共d兲 the pumping is finished and all the waves are to be evacuated through the interface toward the fluid after many reflections between the end of the sample and the interface. This is a long process, as shown in Figs. 4共e兲 and 共f兲. From the numerical point of view, this test has been very productive. It can be noticed that the scheme succeeds in propagating the waves for a long period without damping their amplitudes. Moreover our boundary conditions work without generation of spurious oscillations, for instance, Figs. 4共a兲 and 共b兲 show no artificial numerical reflections at x⫽0 cm and no oscillations at x⫽4 cm for the reflective boundary. At the interface x⫽0.5 cm, the modified Rieman solver, which respects the continuity of stress and velocity, allows a smooth treatment of the change of amplitude and

FIG. 3. Initial solution. 共a兲 The initial pressure wave in the water zone. 共b兲 The state when the pumping is initiated 共pressure unit: Pa兲.

79



79

FIG. 4. Numerical solution: Spatial pressure and normal stress repartitions in a finite domain water–sample–vacuum at different stages of the process 共stress unit: Pa兲.

sound velocity due to the different impedances on both sides. As to the physical point of view, this test is still too complicated to be compared with analytical results. Another test has thus been performed on an infinite sample with the same active zone and the same pumping conditions except m⫽3.2⫻10⫺2 . The initial wave in the active zone is the 80


same as in previous case. Figures 5共a兲–共f兲 show the evolution of the process. The quasisymmetrical behavior of the direct amplified wave and the conjugate one is not disturbed by the interfaces. At the end of the process, both waves are evacuated in opposite directions. Figures 5共c兲 and 共d兲 present the situation just before and just after the end of the pumpKhelil et al.: Simulation of acoustic phase conjugation

80

FIG. 5. Numerical solution: Normal stress repartitions in an infinite sample at different stages of the process.

ing. The amplification stops immediately and the separation of both waves starts. This process corresponds to the ideal situation that gave rise to an analytical solution for the amplitude of the conjugate wave versus time. This analytical solution still holds in a more realistic situation where the sample is bounded on the left by water and is infinite on the left-hand side. No reflections on the left-hand edge disturb the emission of the con81


jugate wave in the liquid and, hence, the comparison between analytical solution and experiments is relevant for the pressure measurements in water until the first reflection of the right edge reaches the left interface in the experiments. Figure 6 presents the pressure in water for a semi-infinite sample limited on the left by the liquid. It shows total agreement with the results of the supercritical mode theory.4,8 In accordance with the theory of impulse response,8 the first Khelil et al.: Simulation of acoustic phase conjugation

81

amplified waves having the same number of reflections at the interface x⫽0.5 cm. The reflections on x⫽0.5 cm are the only origin of the intensity decay for the waves inside the sample. At the end (x⫽4 cm兲 no energy is lost by the reflected waves. Hence the number of rebounds depends only on ␶. It is clear that in a real situation this number would be dramatically reduced by the 3D effects and by the fact that the direct wave is not as coherent with the conjugate one as in our 1D simulation.

IV. CONCLUSION

FIG. 6. Comparison of the analytical and numerical solutions: Time evolution of the pressure in water due to the conjugate wave at the interface water solid. The sample is semi-infinite, the boundary at 4 cm is nonreflective. Curve ‘‘analytical 1’’ is a partial sinus wave corresponding to the evacuation of the conjugate wave after the end of the pumping. Curve ‘‘analytical 2’’ is the exponential growth.

part of the envelope of the phase conjugate wave corresponds to the supercritical amplification stage due to the presence of electromagnetic pumping. It can be described asymptotically by an exponential function of time with gain ⌫.4 The second part of the signal observed after the end of the pump excitation, can be correlated with the spatial amplitude distribution of the acoustic conjugate wave inside the active zone at the moment the pump is suppressed. In the case of a finite sample, the evolution in time of the acoustic pressure in the fluid side of the interface x⫽0.5 cm is presented in Fig. 7. Numerous reflections at the end of the sample explain the successive rebounds of the signal. The succession of pairs of rebounds having the same intensity is noticeable. This corresponds to conjugate waves and direct

The computation scheme developed in the present paper provides, in principle, direct integration of various problems of wave propagation in nonstationary media. The first results of these numerical simulations of parametric WPC by this scheme clearly show the supercritical dynamics of the acoustic field in the active medium and demonstrate a good agreement with the analytical theory. The method can be easily generalized to the description of the nonlinear stage of parametric WPC 共Ref. 15兲 taking into account pumping depletion and multiple reflections of parametrically coupled waves at the boundaries of the active medium. The problem of wideband pumping is of special interest for applications of the proposed approach because, in contrast with asymptotic methods, it is free of limitations on speed of sound velocity variations. The problem of reverberation noise amplification in parametric WPC under double pumping, recently studied experimentally,12 also does not require considerable modification of the computational technique either. Thus, the computation method proposed in the present paper can find various applications in parametric dynamics of continuous media. Nevertheless, if the extension in 2D is possible through standard numerical techniques like flux splitting, the 3D case cannot be solved without a loss of precision because of the necessary limitation of the number of cells. A fruitful research direction is the derivation of an ‘‘amplitude formulation’’ by a hybrid asymptotic-numerical approach, including a suitable treatment of the boundary conditions which would give only the amplitude evolution.

1

FIG. 7. Time evolution of the pressure in water at the left edge of the finite sample. The boundary at 4 cm is stress free. The conjugate wave 共first burst兲 is followed by the reflection of the direct wave on the right edge 共second burst兲 and next by its own reflection on the left and right edges 共third burst兲, the equivalent double reflection of the direct wave explains burst four. The difference of the amplitude between the first and the second couple of bursts is due to the transmission of energy into water at each reflection on the left interface. The process continues until complete damping of the reflections. 82


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