Elsevier

Wave Motion

Volume 46, Issue 4, June 2009, Pages 255-268
Wave Motion

Duality of the supercritical solutions in magnetoacoustic wave phase conjugation

https://doi.org/10.1016/j.wavemoti.2009.02.003Get rights and content

Abstract

The present study of the paraxial analytical solutions of supercritical wave phase conjugation leads to a new kind of solution showing that the classical ones, obtained hitherto, were of particular type. This dual solution comes over a higher threshold, which explains why it has not been detected before. Transition from one solution to the other strongly depends on the ratio between the length of the active zone and the half wave length. This sensitivity and its practical consequences are numerically illustrated for the multidimensional case.

Introduction

The wave phase conjugation (WPC) in acoustics or ultrasonics is a non-linear coupling between an acoustic wave and an oscillating source of energy inside an active medium. Under the effect of this source of energy, the active medium modulates the elastic properties of the material. When the conditions of parametric resonance are fulfilled, the acoustic wave can pump this energy on a specific frequency ω. The incident wave of frequency ω is then amplified exponentially, although no force is applied to the sample. The momentum conservation leads to the emission of a time-reversed “conjugate” wave with the same amplification and frequency. This phenomenon is called the supercritical mode of WPC because it appears when the pumping intensity exceeds a threshold value [1]. In practice, the magnetoacoustic wave conjugator is generally a cylinder of magnetostrictive material (e.g a Terbium-iron ferrite) which allows an amplitude modulation of about 4% of its bulk modulus by the effect of an oscillatory magnetic field. The field is realized by a coaxial solenoid around the conjugator. This technique that couples giant amplification and time reversal through a physical process is relatively easy to implement and is cheap compared with the alternative time reversal techniques. It opens numerous perspectives in applications such as acoustic imaging [2], [3], [4] or velocimetry [5], [6], [7]. From the theoretical point of view, the paraxial approach is a good first approximation. For the general configuration presented in Fig. 1, such approach describes the major characteristics of the phenomenon.

For the parametric resonance to take place, the incident wave must contain a frequency component ω which is half of the magnetic field one Ω. In this paper, a new type of paraxial solution is obtained which underlines the fact that the classical solutions obtained hitherto [8], [9], [10], [11] are particular cases. It is because these classical solutions generally exhibit the lowest threshold of pumping that they have been observed more easily in the past, both in experiments [12] and in numerical computations [13], [14], [15].

In the paraxial approach we can reduce the elastic properties of the conjugator medium to the speed of the longitudinal elastic waves because the transverse waves are not excited. With the assumption of a linear pumping, the coupling with the magnetic field is introduced through the expression of the sound speed in the active zone: c2=c02[1+mcos(Ωt+φ)], where m1 is referred to as the modulation depth and c0 is the sound speed in the absence of magnetic field. This expression of c2 represents the energy transformation from magnetic field to acoustic field. In the following considerations the arbitrary phase shift φ is set to π without lack of generality. Then, the phenomenon is governed by the equations:θt+cvx=mΩθ2sin(Ωt)vt+cθx=0where θ=σxxρ0c, σxx is the normal elastic stress along the axis in the solid, v is the particle velocity, ρ0 is the density of the conjugator material. For the external media on the left and on the right from the conjugator, the governing equations are the classical wave equations without pumping (m=0). They correspond to the left-hand side of the previous system, where c is replaced by cl or cr, the constant sound velocities respectively in the left and right medium. In a fluid σxx is replaced by -p, the acoustic pressure, in the definition of θ. The interface conditions are determined by τl=ρlclρ0c0 and τr=ρrcrρ0c0 the impedance ratios of the left and right media.

Introducing new variables w1=v+θ for the direct wave and w2=v-θ for the conjugate wave yields:w1t+c0w1x=mΩ4(2w1-w2)sin(Ωt)w2t-c0w2x=-mΩ4(w1-2w2)sin(Ωt)which returns to a linear hyperbolic system with source terms for ensuring the coupling and the creation of the conjugate wave. It is easy to find that the change of variables x-x and w1w2 will not change the problem. The system is solved numerically by the second order Godunov-type explicit finite-volume methods adapted to the heterogeneous media [13], [14], [15], [16], [17] and verified by comparison with spectral methods [18].

In the active zone defined by |x|L2 the phenomenon is governed by Eq. (1). Outside of the active zone the source term is zero and the system comes back to that of 1D linear advection. Although the source term is very small, if the parametric condition of resonance Ω=2ω is fulfilled, it is sufficient to produce an exponential amplification of the component ω in the spectrum of the incident wave by a parametric resonance process. Inside the active zone, the solution of the problem can be written [9] as:w1=-AeΓtsinkα4x+ξ1sin-k2x+Ω2t+φ1+mf1(x,t)andw2=AeΓtsinkα4x+ξ2sink2x+Ω2t+φ2-mf1(-x,t)where ξ1, ξ2 and Γ=Ω8m2-4α2 are the main unknowns to be determined. Function f1 represents the terms of order m in the solution. It can be seen that Eqs. (2), (3) respect the invariance of system (1) for the change of variables x-x, w1w2 under suitable conditions on ξ1, ξ2, φ1 and φ2.

The solution is mainly determined by the boundary conditions at each end of the active zone and by a compatibility equation obtained by substituting Eqs. (2), (3) into system (1) which gives:cos(φ1+φ2)=0andΓsinkαx4+ξ1+Ωα4coskαx4+ξ1=-mΩ8sinkαx4+ξ2sin(φ1+φ2).Eq. (4) yields:sin(φ1+φ2)=1.

Let us introduce the angle γ[0,π/2] bycos(γ)=8ΓmΩandsin(γ)=2αmwhensin(φ1+φ2)=-1or γ[π,3π/2] when sin(φ1+φ2)=1 withcos(γ)=-8ΓmΩandsin(γ)=-2αm.

This leads to a very simple equation:ξ2-ξ1=γ.

As it can be seen here, all the subtle features are therefore hidden in the boundary conditions that are the main subject of the following section which will focus on the slow varying amplitude. After that the problem of the phase of the travelling wave will be solved and compared with numerical experiments. Finally, all new features of the phenomenon will be addressed numerically, including multidimensional effects.

Section snippets

Even and odd solutions

For the resonant component at angular frequency ω, the boundary conditions can be written in the following form [9]:sin-kL2+φ2-φ1=0sin-kαL8+ξ1+cos(-kL/2+φ2-φ1)1-τl1+τlsin-kαL8+ξ2=0at x=-L/2, andsinkL2+φ2-φ1=0sinkαL8+ξ1+cos(kL/2+φ2-φ1)1+τr1-τrsinkαL8+ξ2=0at x=L/2.

Note that, unlike in the previous works [9], the right hand sides are all exactly 0 and not O(m) because it was shown that at order m the solution cannot contain resonant components [19]. Therefore the projection of the boundary

Coupling between amplitude and phases

The previous sections as well as all previous works based on the same analytical approach were focused on the slow varying amplitude. This was in fact the main point since it determined the shape of the signal and the threshold between the subcritical and supercritical regimes. The independent treatment between the phase of the travelling waves and the amplitude of the envelope was postulated in the previous work [9] and is implicitly inferred by the mathematical form of solution (2), (3).

Practical consequences

The previous results have important practical consequences. First of all, they underline the dependency between the phases (φ1, φ2) and the value of kL. Moreover, the analytical solutions hold only for integer values of kL but in practice, even supposing that the pumping is homogeneous, it is impossible to realize a conjugator with an active zone precisely equal to a multiple of λ/4. The question arises: what happens in between the pure odd and even cases? Nevertheless, the main problem is

Conclusion

This paper provides the analytical solution of the wave phase conjugation for a conjugator with an active zone of length equal to a multiple of λ/4 for a given set of physical parameters (pumping frequency Ω, modulation depth m and impedance ratios τl,τr). The phase of the propagating signal is also obtained among four different solutions selected by the system through the establishment process of the stable solution. The solutions are different if kL/π is even or odd with a continuous

Acknowledgement

This work was partially supported by the Russian Foundation for Basic Research, Grant 08-02-92495-NCNIL-a, by the France-Canada Research Foundation and by the French ministry of research.

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