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DAYS on DIFFRACTION 2012 1

Theory of selfrefraction effect of intensive fo cused acoustical b eams
V.A. Gusev
Lomonosov's Moscow State University, Physical Faculty, Department of Acoustics, Russia, 119991, Moscow, Leninskie gori; e-mail: vgusev@bk.ru
The theory of selfrefraction of nonlinear acoustical beams is developed based on some exact and approximate analytical equations and solutions. The system of base equations in geometrical acoustics approximation is sequentially derived from Khokhlov­ Zabolotskaya equation for nonlinear fo cused acoustical beams. The generalized metho d of extended characteristics allows to set up the simplified closed equation for ray convergence on the beam axis for the most interesting case of small diffraction, when large amplitudes in the fo cal area are observed. The exact solution is derived in particular case. For the common case of wave parameters there are suggested some analytical approximations and numerical solution. The amplitude dependencies on longitudinal and transversal distances and other wave parameters are obtained. It is shown that at the axis of gaussian beam in the fo cal area the lo cal minimum of amplitude can be formed. Some initial transversal beamforms, such as gaussian, and initial phase mo dulation as parabolic or sinusoidal are analyzed.

2

Model equations and previous results

The first self-consistent method describing the beam propagation with selfrefraction taken into account was suggested in paper [1]. The system of modified nonlinear geometrical acoustics approximation equations was written, where the new term responsible for the selfrefraction was added in the eikonal equation's right hand by analogy with plane waves: µ A + =- , (1) z r 2 r A p p p p = 0. - p- + + + z 2 T r 2 r r (2)

These equations are already written in dimensionless variables, convenient to further calculations. Here = / r -- ray inclination function, -- eikonal, z and r -- dimensionless longitudinal and 1 Introduction transversal coordinates, p -- acoustical pressure, A(z , r) -- dimensionless beam amplitude, T = The problem of discontinuous focused acoustical - /c0 , = t - z /c0 -- retarded time. There are beam propagation and calculation their param- two dimensionless parameters: = F /xs defines eters near focal area is under consideration in the relative contribution of focusing (F -- focal this paper. Discontinuous waves and waves with length) and nonlinearity (xs -- nonlinear length), shock fronts are special ob jects of nonlinear acous F2 FF A0 A0 F 2 tics. They are the general asymptotic solution = µ= 2 2 = 2 =x x c0 r0 xs r0 2pint 2 s d iff at large distances for arbitrary initial wave profile and have some special features. In partic- defines the selfrefraction "strength". Here xdiff = 2 ular the speed of shock front propagation de- r0 /, pint = c2 /2, = r0 /F , -- nonlinearity 0 pends on its peak amplitude. This fact leads to parameter, A0 , r0 -- amplitude and beam radius. However these equations were not derived from such effect as selfrefraction (or nonlinear refraction) of intensive acoustical beams. Let us con- any more exact equations so the question about sider the bounded beam of discontinuous waves for boundaries of their applicability is still open. Beexample gaussian beam. The shock front ampli- sides in paper [1] some suggestions such as paraxial tude and consequently its speed depends on the approximation were used and only numerical sotransversal coordinate so the wave front will be dis- lutions were obtained. The main result of [1] is torted and cause wave defocusing. Moreover the the empirically obtained from numerical calculapropagation speed of all nonsymmetrical acousti- tions expression for the pressure limit in the focal cal pulses differs from the local sound speed and area plim = (1,3 - 1,7)pint 2 , which depends only consequently all these pulses are also influenced by on parameters of medium and beam geometry and very weak on other parameters. selfrefraction.

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2 DAYS on DIFFRACTION 2012 Another approach based on numerical solutions of more exact Khokhlov­Zabolotskaya equation was developed in [2] to calculate the pressure limit. It have been shown that the pressure limit changes weakly for the wide interval of parameters and is in accordance with results of the previous paper [1]. But some questions have not answered yet: 1. Could the simplified equation be derived more correctly and for arbitrary transversal form of beam? 2. Could some analytical estimations be obtained? 3. What is the main factor limiting the pressure in the focus -- diffraction or selfrefraction? 4. Could the improved nonlinear geometrical approximation be used to describe focal area of discontinuous waves instead the Khokhlov­Zabolotskaya equation? This work is an attempt to give answers to these questions. 3 Main equations for selfrefraction effect r = r( , z ). This allows to write the base system of equations describing arbitrary wave beam [4, 5]: µ 1 A r =- , = , z 2 r z A p p 1 p - p- + + z 2 T 2 r r (6) = 0. (7)

The amplitude of initial N-wave with arbitrary transversal form (p(z = 0) = R(r)p0 (T ), p0 (T ) = -T for |T | < 1 and p0 (T ) = 0 for |T | > 1) is defined by expression R( ) A= S 1 , 1 + R( )s S= rr ,
z

s=
0

dz . S

So equations for ray tra jectory r and ray convergence r are µ 1 A 2 r µ 2r =- =- , 2 2 z 2 r z 2 1 A . (8) r

The formal implicit solution for ray convergence at small µ can be written r = 1 + ( )z 0 µ - 2
z


First of all we proceed from the base equation of intensive acoustical beam theory -- Khokhlov­Zabolotskaya equation p p - µp z = p.

d
0

0



A( , ) d r



.

Now one can conclude that selfrefraction 1) limits (3) the peak amplitude in the focal area, 2) moves the maximum amplitude position along beam axis far It can be written for = (p, z ) as the function of from geometrical focus. independent variables p and z [3] 4 Analytical solutions for pressure -1 2 along beam axis. Pressure limit 1 + µp = . + p p z 2 r To obtain analytical solutions we make some sug(4) gestions based on physical sense. It can be shown If we neglect diffraction ( 0) and use the that parameter µ, which describes the selfrefrac"equal area" rule for the shock front area we derive tion "strength", is small for more interesting cases, the following equation for shock front movement including relatively small Mach number, case of 2 strong focusing etc. Besides, large µ corresponds A 1 s s + µ = 0, + (5) to strong defocusing due to diffraction, nonlinear z 2 r 2 absorption and selfrefraction so this situation looks which is coincide in sense with Eq. (1). So Eq. (1) not very useful. Now consider the classical case of gaussian is written in nondiffraction approximation and for beam. Simple equation can be written for field shock front. So there is the principal possibility to calculate diffraction corrections. However we will along the gaussian beam axis in the first order approximation on µ (Q r ( = 0) = r( = 0)): consider further only the nondiffraction model. Now let introduce in Eq. coordinate so called ray ical meaning is the initial of any ray. Thus current
z (1)­(2) new transversal 2 + s µ dz d2 Q = (9) , s= coordinate which physdz 2 2Q2 (1 + s)3/2 Q 0 transversal coordinate transversal coordinate with conditions Q(z = 0) = 1, dQ/dz (z = 0) = -1.

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DAYS on DIFFRACTION 2012 3

Figure 1: Ray convergence (a) and peak amplitude (b) For qualitative estimation of the pressure limit we consider the case of gamma tending to zero when the exact solution can be obtained: z1 = (1 + 2µ)- +
1

Figure 2: Comparison of numerical and asymptotic solutions

1-

Q

2µ ln (1 + 2µ)3/2 Q+

(1 + 2µ)Q - 2µ 1 + 1/ 1 + 2µ Q - 2µ/(1 + 2µ)

(10)

before the turning point and z2 = 2z0 - z1 after the turning point. The turning point z0 is determined by the condition z0 = z1 (Qmin ). This solution is shown on Fig. 1,a for parameters µ = 0,01 (curves 1, 2) and µ = 0,1 (curves 3, 4). There is also shown comparison between first (curves 1, 3) and second (curves 2, 4) order approximations on parameter µ. One can see that corresponding curves are in a good agreement with each other. At the Fig. 1,b the peak amplitude for the same values of parameters are shown. Main results are as follows. We take into account only selfrefraction without any diffraction and obtain the finite pressure limit. Maximum position moves beyond "linear" geometrical focus. Solution (10) allows to find minimum value of ray convergence Qmin = 2µ/(1 + 2µ) and consequently the pressure limit (Amax = Q-1n) in focus: mi Am
ax

Figure 3: Peak amplitude for different shown at Fig. 2 and one can see good agreement between them. At Fig. 3 the peak pressure along beam axis are shown for different . One can conclude that the pressure limit increases with gamma increasing for small gamma and reaches saturation for large gamma. Longitudinal focal area on the contrary decreases with gamma increasing. 5 Transversal structure

=

1 + 2µ 2µ



~ Amax = 1 + 2µ pint 2

(11)

~ (here Amax = A0 Amax -- physical (dimensional) amplitude). This theoretical expression is in a good accordance with empiric estimation from [1, 2]. And we should remember that expression (11) concerns only special case. For nonzero gamma the most effective method to obtain analytical solution is the straight expansion on small µ: Q = Q0 + µQ1 + µ2 Q2 . The solution can be obtained in common case as quadratures. The comparison of this solution with numerical one is

The peak pressure along different rays are shown for small = 0,1 at Fig. 4 and = 1 at Fig. 5. Note that the local minimum of the peak pressure forms near beam axis for small gamma even for initial gaussian beam. It can be obtained that focal area defined by selfrefraction depends weakly on parameter gamma. On the other hand focal area defined by diffraction is / 1 + 2 . So for small gamma diffraction is not significant and for large gamma selfrefraction and diffraction works jointly. The developed model allows to calculate and other spatial-modulated beams, not only gaussian beams. At Fig. 6 the peak amplitude for the beam with periodic modulated wave front 0 ( ) = - sin are shown as example.

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4 DAYS on DIFFRACTION 2012 3. At small gamma diffraction does not influence significantly on the spatial structure of the wave beam and is determined mainly by selfrefraction. At large gamma diffraction can cause some corrections to transversal structure. 4. The developed theory can be used for describing discontinuous waves with arbitrary initial wave front and transversal form. 5. All these results allows to conclude that improved nonlinear geometrical approximation is able to describe propagation of discontinuous waves and waves with shock fronts even in focal area, at least, qualitatively. Acknowledgements The work was financially supported by grants of the RF Presidential Program for Support of Leading Scientific Schools (NSh-2631.2012.2) and the Russian Foundation for Basic Research (pro ject no 1202-01149-a). References [1] Musatov, A.G., O.A., Nonlinear nomena due to p J., 1992. V. 38. Rudenko, O.V., Sapozhnikov, refraction and absorption pheowerful pulses focusing, Acoust. No. 3. P. 502­510.

Figure 4: Peak amplitude along different rays, = 0 ,1

Figure 5: Peak amplitude along different rays, =1

[2] Karzova, M.M., Averiyanov, M.V., Sapozhnikov, O.A., Khokhlova, V.A., Mechanisms for saturation of nonlinear pulsed and periodic signals in focused acoustic beams, Acoustical Physics, 2012. V. 58. No. 1. P. 81­89. [3] Rudenko, O.V., Soluyan, S.I., 1977, Theoretical Foundations of Nonlinear Acoustics, New York: Plenum, Consultans Bureau. [4] Gusev, V.A., Self-refraction of the focused sound beams of sawtooth waves (analytical solutions), Year-book of Russian Acoustical Society. Acoustics of heterogeneous media. Proceedings of Prof. S.A. Rybak's scientific seminar. Moscow, 2007. No 8. P. 103­112.

Figure 6: Peak amplitude for periodicmodulated beam 6 Conclusions

1. The correct derivation of the equation for the [5] Gusev, V.A., Analytical solutions in the theory of the self-refraction of weak shock impulses. shock front propagation is suggested so there is the Proceedings of XIX Session of the Russian principal possibility to take into account diffraction Acoustical Society, Nizhny Novgorod, Septemeffects. ber 24­28, 2007. V. 1. P. 159­162. M.: Geos, 2. Theoretical expressions for the pressure limit are 2007. obtained. The pressure limit does not change significantly for wide interval of parameters.

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