Mechanisms of Interaction between Ultrasound and Sound in Liquids with Bubbles: Singular Focusing

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1 Acoustical Physics, Vol. 47, No., 1, pp Translated from Akusticheskiœ Zhurnal, Vol. 47, No., 1, pp Original Russian Text Copyright 1 by Akhatov, Khismatullin. REVIEWS Mechanisms of Interaction between Ultrasound and Sound in Liquids with Bubbles: Singular Focusing I. Sh. Akhatov and D. B. Khismatullin Institute of Mechanics, Ufa Scientific Center, Russian Academy of Sciences, ul. K. Marksa 1, Ufa, 4 Russia damir-k@anrb.ru Received December 14, 1999 Abstract A two-dimensional interaction between long-wave (sound) and short-wave (ultrasound) pressure perturbations in a rarefied monodisperse mixture of a weakly compressible liquid with gas bubbles is considered. The conditions at which this interaction leads to a singular focusing (an explosive instability) of ultrasound are determined. A numerical study of the defocusing and the singular focusing in a bubbly liquid is carried out. The effect of the long-wave short-wave resonance on the development of two-dimensional disturbances is studied. 1 MAIK Nauka/Interperiodica. A liquid with gas bubbles exhibits pronounced nonlinear acoustic properties due to the nonlinear character of the bubble oscillations and the high compressibility of the bubbles. The theoretical and experimental studies performed in the last few decades revealed many types of nonlinear wave phenomena in bubbly liquids: the self-focusing of ultrasound [1, ], the self-clarification of sound [3], wavefront reversal [4], the acoustic echo [], subharmonic wave generation [6, 7], the focusing and amplification of waves in inhomogeneous bubbly liquids [8, 9], and pattern formation in acoustic cavitation [1 1]. For a bubbly liquid, the dispersion curve representing the frequency dependence of the wave number of the disturbance consists of two branches [13]: the lowfrequency branch and the high-frequency one. Therefore, in this medium, a simultaneous propagation of long (sound) and short (ultrasound) waves is possible. The propagation of these waves in the medium is accompanied by the energy transfer between them through the mechanism of the long-wave short-wave interaction [14]. This interaction is most pronounced when the group velocity of the short waves coincides with the phase velocity of the long wave (the longwave short-wave resonance []). The physical systems in which the long-wave short-wave resonance can occur include, e.g., waves on the water surface [16] and plasma [17]. Our recent studies [18] show that bubbly liquids also belong to such systems. In the one-dimensional case, the nonresonance longwave short wave interaction is described by the nonlinear Schrödinger equation for the short-wave envelope [19] and the resonance interaction is described by the Zakharov system of equations [17]. Both these systems have a solution in the form of the envelope soliton [, 1], which is known to be an example of the one-dimensional focusing of perturbations. In the case of a two-dimensional nonresonance interaction, the Davey Stewartson system of equations [] also describes the focusing process. As a rule, the two-dimensional focusing is singular; i.e., it leads to the development of an explosive instability [3, 4]. Only with specially selected coefficients, does the Davey Stewartson system of equations become integrable and have localized bounded solutions. In our previous paper [], we showed that, in a bubbly liquid, the two-dimensional interaction of long and short waves is described by equations that can be reduced to the Davey Stewartson system and, for some parameters of the liquid, to one of its integrable versions (the Davey Stewartson equations). In this paper, we derive the conditions for the singular focusing of ultrasound in bubbly liquids and perform its numerical study. We analyze the effect of the long-wave short-wave resonance on the development of singular focusing. Let us consider the conditions of a singular focusing. The equations describing the two-dimensional interaction between ultrasound and sound in bubbly liquids were obtained in our previous paper [] by the multiscale method: ( )L ξξ c e L α( S ) ξξ, is τ + βs ξξ + ρ S + γ S S δls. c g c e (1) Here, L is the profile of the long (sound) wave, S is the short (ultrasound) wave envelope, ξ ε(x c g t), εy, and τ ε t (ε is the parameter characterizing the smallness of the short-wave amplitude). The group velocity of the short waves c g ; the equilibrium velocity of the long wave c e ; and the coefficients α, β, ρ, γ, and δ depend on the polytropic index κ, the parameter b /1/47-14$1. 1 MAIK Nauka/Interperiodica

2 MECHANISMS OF INTERACTION BETWEEN ULTRASOUND AND SOUND IN LIQUIDS 141 p ρ 1 l α 1 g C l, the wave number of the short wave k s, and its cyclic frequency ω s []. Applying the substitution L δ 1 ψ/ ξ (δ ) c e and introducing the notation σ ( ), we reduce the system of Eqs. (1) to the Davey Stewartson equations: σψ ξξ + αδc e ( S ) ξ, Ψ () It is known [3, 4] that, in the nonintegrable case, if the coefficients β, ρ, σ, and αδ/ c e are positive and γ > or γ < αδ/ c e, the solution to Eqs. () with the boundary condition tends to infinity for sufficiently large amplitudes within a finite time interval (the effect of singular focusing or an explosive instability). The existence of the singular focusing in nonlinear optics is confirmed by the experimental data [6]. For another choice of coefficients, any initially localized solutions to this system are defocused, i.e., spread in space. If the solution to system () exists and tends to zero when ξ +, the following conservation laws are valid [7]: The first of these integrals can be interpreted as the mass of the short wave, the second and third integrals as the components of the momentum of the short wave, and the fourth integral as the energy of the long-wave short-wave system, although these integrals have nothing to do with the physical conservation laws [3]. c e is τ + βs ξξ + ρ S + γ S S SΨ ξ. P x p y S for ξ + E M S dξd, S* S S S* dξd, ξ ξ S* S S S* dξd, β S + S ξ ρ 1 -- γ S 4 σc e Ψ αδ c e Ψ ξ αδ dξd. c g For the Davey Stewartson equations (), the following relationship is valid [7]: (A and B are the integration constants). In terms of the above representation, the integral I can be considered as the moment of inertia of the short wave. Then, Eq. (3) will be an example of the virial theorem. In the conditions of singular focusing (see above) and for sufficiently large wave amplitudes, the energy E can be negative [3]. Then, according to Eq. (3), the integral I, which is a positive definite quantity, will become zero at some instant of time. Since the mass of the short wave is conserved, its moment of inertia can become zero, if the short-wave perturbation will be concentrated near a single point (a focus). Such a redistribution of the mass of the wave leads to a sharp increase in the wave amplitude at this point and, finally, to an explosive instability. The existence of a negative-energy wave and, as a consequence, the development of the singular focusing is caused by the instability of the solitons of the nonlinear Schrödinger equation envelope to transverse longwave perturbations. This statement is confirmed experimentally for waves on the water surface [8]. Therefore, the theory developed for one-dimensional enved dt ξ S dξd ρ β 8E. This relationship can be easily integrated: I b 1.8 γ 1 > γ 1 < ξ S dξd 8Et + At + B ρ β.79 c e > c g c e < c g γ 1 < γ 1 > k s Fig. 1. Regions of explosive instability (hatched) in the (k s, b) plane. The coefficient is γ l γ + αδ/ c e. (3) ACOUSTICAL PHYSICS Vol. 47 No. 1

3 14 AKHATOV, KHISMATULLIN lope waves does not apply to waves with dimension and higher. For a bubbly liquid, the coefficients β and ρ are always positive and α and δ are of the same sign. Hence, in such a liquid, a singular focusing of an ultra τ. ξ.. τ ξ.. τ ξ.. Fig.. Nonresonance focusing of the short-wave envelope. sonic wave is possible, provided that the following conditions are satisfied: c e > c g, γ > αδ/c e. (4) Figure 1 shows (by hatching) the explosive instability zones determined by conditions (4) in the (k s, b) plane. Figures present the results of the numerical analysis of Eqs. (1). The numerical integration of this system of equations was performed using the Fourier transform method [9] for the first equation and the variable directions scheme [11, 3] for the second equation. We considered the boundary conditions L( ξ, ), S( τ, ξ, ), for ξ + and the initial condition S( τ, ξ, ) S exp{ ( ξ + )}. Here, S is the real amplitude. The coefficients were selected so as to satisfy the condition c e > c g (in this case, the first equation is an elliptic one). We analyzed (a) (b) τ τ τ.9986 k 1.36 b.4 k. b.4. ξ Fig. 3. Effect of the long-wave short-wave resonance on the defocusing process in a bubbly liquid: (a) dependence of the short-wave envelope on the spatial coordinate ξ ( ) at various instants of time; (b) comparison of the dependences of on ξ in the nonresonance (the solid line) and resonance (the dashed line) cases at τ ACOUSTICAL PHYSICS Vol. 47 No. 1

4 MECHANISMS OF INTERACTION BETWEEN ULTRASOUND AND SOUND IN LIQUIDS 143 four cases: (1) k s., b.4; () k s 1.36, b.4; (3) k s., b.; and (4) k s.74, b.. The first case corresponds to a nonresonance stable interaction, because c g c e and the point (.,.4) lies in the τ. ξ.. τ ξ.. τ.931. ξ.. Fig. 4. Nonresonance singular focusing of the short-wave envelope. stability zone (see Fig. 1). The numerical analysis confirms the analytical results: in this case, a defocusing occurs for both long and short waves (Fig. ). In the second case, we also deal with the stability zone, but in this case we approach the resonance curve (c g c e ). From Fig. 3a, one again can see a defocusing of the solution. However, the defocusing process is decelerated and accompanied by a change in the wave form (Fig. 3b). In the third case, an unstable nonresonance interaction takes place. With time, the solution to Eq. (1) shrinks to a point and increases in amplitude (Fig. 4). At some instant of time, this results in the appearance of an explosive instability. A singular focusing is also observed in the fourth (unstable resonance) case. In this case, the focusing is accelerated and accompanied by a strong deformation of the wave profile (Fig. ). It is significant that the minimal amplitude necessary for the development of the singular focusing decreases in the resonance case: S 3 away from the resonance and S at resonance. Therefore, the long-wave short wave resonance increases the probability of the singular focusing. Thus, on the basis of the analysis of the coefficients in the model of a two-dimensional long-wave shortwave interaction, we determined the conditions for a singular focusing (an explosive instability) in bubbly liquids. The existence of a two-dimensional defocusing and a singular focusing in such liquids is confirmed by the numerical integration of the interaction equations. We also analyzed the effect of the long-wave shortwave resonance on the two-dimensional interaction of long and short waves. It was shown that, in the resonance case, the defocusing process is decelerated and accompanied by changes in the wave form. By contrast, the singular focusing is accelerated with the develop- τ τ.1476 τ ξ Fig.. Dependence of the short-wave envelope on the spatial coordinate ξ at various instants of time in the case of the resonance singular focusing. ACOUSTICAL PHYSICS Vol. 47 No. 1

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