FOCUSED ACOUSTIC BEAM IN NON LINEAR MEDIUM

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FOCUSED ACOUSTIC BEAM IN NON LINEAR MEDIUM

N. Badhvalov, R. Khokhlov, E. Zabolotskaja, I. Zhileikin

To cite this version:

N. Badhvalov, R. Khokhlov, E. Zabolotskaja, I. Zhileikin. FOCUSED ACOUSTIC BEAM IN NON LINEAR MEDIUM. Journal de Physique Colloques, 1979, 40 (C8), pp.C8-56-C8-61.

�10.1051/jphyscol:1979812�. �jpa-00219517�

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JOURNAL DE PHYSIQUE Colloque C8, supplement au n°ll, tome 40, novembre 1979, page c8- 56

FOCUSED ACOUSTIC BEAM IN NON LINEAR MEDIUM

N.S. BADHVALOV - R.V. KHOKHLOV - E.A. ZABOLOTSKAJA - I.M. ZHILEIKIN Moscow State University, USSR

Résumé. - Le problème d'un faisceau acoustique d'amplitude finie se propageant dans un milieu idéal est étudié théoriquement. L'analyse est menée par intégration numérique d'une équation approchée de l'acoustique non linéaire de faisceaux limités par une condition aux limites périodiques. L'amplitude du signal décroit du centre du faisceau vers les bords et la phase dépend de la coordonnée transversale suivant une loi spécifique d'un front d'onde sphérique pour un faisceau paraxial.

Deux types de dépendance de l'amplitude en fonction de la coordonnée transversale sont envi- sagés ici : soit selon un polynôme du quatrième degré, soit suivant la loi de Gauss. On a étudié différents cas correspondant à des rapports variables des propriétés de non linéarité et de diffraction, pour différentes valeurs de la convergence de l'onde. On présente les résultats sous forme de courbes donnant la forme de la perturbation de densité à des distances variables du bord du faisceau. En outre on a tracé les courbes donnant la dépendance de la valeur maximum de la perturbation dans la phase de compression ainsi que de la valeur moyenne par période de l'intensité relativement à la distance de propagation, en différents points de la section transversale du faisceau. On a éga- lement déterminé la distance de formation de l'onde de choc.

Abstract. - The problem of a focused finite-amplitude acoustic beam propagating in can ideal medium is studied theoretically. The analysis is carried out on the basis of numerical integration of an approximated equation of nonlinear acoustic of confined beams with a periodic boundary condition.

The amplitude of the signal decreases from the beam centre to the edge and the phase depends upon the transverse coordinate by the law specifying the sphericity of the wave front for a paraxial beam. Two types of dependence of the amplitude upon the transverse coordinate are discussed here : in the form of a fourth degree polynom and by the Gauss law. Gases have been considered corresponding to the different ratio of nonlinear an diffraction properties at different values of phase convergence of the wave. The results of the calculations are given in the form of graphs illustrating the form of density perturbation in different distances from the border.

Beside that curves have been plotted for the dependence of the peak value of perturbation in the compression phase and of the mean value for the acoustic beam period of intensity upon the propaga- ted distance in different points of the beam transverse section. The shock wave formation coordinate has been determined.

The parametric acoustic array heightened the practical importance of the problem of acoustic beam propagation in nonlinear media /1,2/. It is Known /3,5/ that the distortion of the shape of the signal which is localized in the form of a beam, differs essentially from the change of the profile of a plane or spherical finite-amplitude wave. Focused beams are of special interest.

Nonlinear propagation of a converging beam is determined by nonlinearity, diffraction divergence and phase convergence.

The present work deals with the numerical so- lution of an approximated equation of nonlinear acoustics of confined beams with a periodic boundary condition. The amplitude of the wave decreases from the beam centre to the edge and the phase depends upon the transverse coordinate by the law spedifying the sphericity of the wave front for a paraxial beam. Two types of dependence of the amplitude upon the transverse are discussed here : in the form of a 4-th degree polynom and by the

Gauss law. Cases have been analysed corresponding to the different correlation of nonlinear and diffraction properties at different values of phase convergence of the wave.

Propagation of a paraxial converging acoustic beam in a weak nonlinear medium without dissi- pation of energy is described by the approximated equation / 5 / :

The following symbols are used here : p - perturbation of density referred to the amplitude of a harmonic signal p ; 9 = co(t - £ ) - JH

co dimensionless accompaning time ; Z = x/2£ - cordi- nate along the direction of propagation normed by the doubled diffraction lsngth ; A. - transverse Laplacian ; R - Transverse coordinate referred to the beam width at the border ; coefficient

N = l/£ , where I = to r /c - diffraction length; 2

Article published online by EDP Sciences and available at

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1979812

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JOURNAL DE PHYSIQUE c8- 57

E = l / a

d-

l e n g t o f shock formation i n plane wave, c o e f f i c i e n t a characterizes n o n l i n e a r pro-

p e r t i e s o f t h e medium ct = ( y

+

1)/2po co ; y

-

a d i a b a t i c index i n equation of s t a t e ; po

-

e q u i l i b r i u m d e n s i t y o f medium ; co

-

propa- g a t i o n speed o f weak s i g n a l .

L e t ' s assume t h a t a harmonic converging wave w i t h a dimensionless r a d i u s o f curvature o f t h e

phase f r o n t B w i t h an amplitude depending upon t h e transverse coordinate i s e x c i t e d a t t h e border o f t h e n o n l i n e a r medium.

and t h e dimensionless r a d i u s o f convercence i n boundary c o n d i t i o n s . The parmeter N c h a r a c t e r i z e s t h e r e l a t i ~ z i n t r o d u c t i o n o f n o n l i n e a r and d i f f r a c - t i o n e f f e c t s i n t h e process o f wave form d i s t o r - t i o n , B determines t h e r a d i u s of convergence as r e f e r r e d t o t h e d i f f r a c t i o n length.

The c a l c u l a t i o n s were c a r r i e d o u t f o r t h r e e values o f t h e parameter N : N = 3.25 t h a t corresponds t o t h e r e l a t i v e l y weak n o n l i n e a r i t y b u t suf- f i c i e n t f o r t h e formation o f a shock f r o n t i n t h e

" s e a r c h l i g h t " zone ; N = 5

-

mean nonlineari-ty and N = 10

-

s t r o n g n o n l i n e a r i t y . For each the dimen-

R~ s i o n l e s s r a d i u s convergence adopts the values :

P/,=, =

-

F(R) s i n ( 8 + _{( 2 )} _B ₌_{1 ;}0.5 ; 0.25 ; 0.1. The r e s u l t s are given i n t h e form o f graphs i l l u s t r a t i n g t h e form o f The r a d i u s of curvature o f t h e phase f r o n t d e n s i t y p e r t u r b a t i o n ~ ( e ) i n d i f f e r e n t p o i n t s of i s normed t o t h e d i f f r a c t i o n l e n g t h R - The calcu-

t h e transverse s e c t i o n a t d i f f e r e n t d i s t a n c e s from l a t i o n s were c a r r i e d o u t f o r two amplitude d i s t r i -

the border. For some cases the shape of perturbs- b u t i o n s F(R) :

t i o n i s presented i n t h e form o f the s u r f a c e p(e,R) i n d i f f e r e n t p o i n t s Z. Besides t h a t

( 3 ) graphs were drawn of the dependence o f t h e peak

value o f dimensionless p e r t u r b a t i o n p from t h e coordinates i n d i f f e r e n t p o i n t s of t h e beam t r a n s - 0

and verse section. The mean was c a l c u l a t e d f o r t h e

a c o u s t i c wave i n t e n s i t y p e r i o d and graphs a r e g i - ( 4 ) ven o f the dependence o f t h e mean i n t e n s i t y o f Z There are two dimensionless parameters i n f o r t h e same values

Ri

.

tasks / I / , /2/ : c o e f f i c i e n t N i n t h e equation The b a s i c r e s u l t s a r e presented i n a t a b l e ,

ZS corresponds t o minimum o f t h e f r o n t w i d t h

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c8- 58 N . S . BAKHVALOV, and a1

.

i n c l u d i n g t h e f o l l o w i n g symbols : ZS

-

c o o r d i n a t e of shock wave f o r m a t i o n on t h e beam a x i s ,

P

,

-

maximum o f t h e p e r t u r b a t i o n peak value i n t h e compression phase on t h e a x i s , Z1

-

coordinate o f t h i s maximum, Im

-

maximum value o f wave i n t e n s i t y on t h e a x i s and Z2

-

coordinate corresponding t o

Im

.

The values o f ZS

,

given i n t h e t a b l e , a r e determined from t h e p o i n t o f view o f t h e f o l l o w i n g c r i t e r i o n : the shock p r o f i l e i s considered formed if t h e change o f p e r t u r b a t i o n from t h e n e g a t i v e value t o t h e maximum p o s i t i v e value i s e f f e c t i n A 8 < n/15.

An increase o f t h e peak value o f wave p e r t u r - b a t i o n and i n t e n s i t y on the beam a x i s i s observed s t propagation o f a focused beam. The t a b l e i n d i - cates t h a t the i n t e n s i t y reaches i t s maximum before the peak value o f p e r t u r b a t i o n u n l i k e the l i n e a r case when Z, = 2,

.

The l a c k o f coincidence of

L

t h e i n t e n s i t y maximum w i t h t h e maximum of t h e per- t u r b a t i o n peak value d i s t i n g u i s h e s t h e propagation of t h e converging f i n i te-amp1 i tude a c o u s t i c beam from nonl i n e a r propagation o f a focused 1 in g h t beam. The absence o f d i s p e r s i o n impedes t h e i n t r o - d u c t i o n o f such a c h a r a c t e r i s t i c o f a converging wave as t h e f o c a l l e n g t h because i t i s d i f f e r e n t f o r each harmonic component. The peak values of p e r t u r b a t i o n and i n t e n s i t y increase t o l a r g e values a t a r e d u c t i o n o f t h e convergence r a d i u s .

F i g . 'I.

-

Change o f p e r t u r b a t i o n peak value a t proL pagation on beam a x i s w i t h ampl i tude d i s t r i b u t e d by 4 - t h degree polynom a t d i f f e r e n t convergence.

Curve 1 corresponds t o

B

= 0.1 ; curve 2 t o 8 = 0.25 ; curve 3 t o B = 0.5 ; curve 4 t o B = l . N = 5 .

Graphs a r e g i v e n i n F i g . 1 i l l u s t r a t i n g t h e dependence pp(Z) on t h e a x i s when the amplitude i s d i s t r i b u t e d over t h e beam w i d t h b i t h e 4 - t h degree polynom a t N = 5 f o r d i f f e r e n t convergence r a d i i : curve 1 corresponds t o B = 0.1 ; curve 2 t o 6 = 0.25 ; curve 3 t o B = 0.5 ; curve 4 t o B = 1.

The d i f f r a c t i o n divergence i s e s s e n t i a l l y l e s s i n t h e Gauss beam. This i s i l l u s t r a t e d w e l l when com- p a r i n g i n F i g . 2 which shows t h e change i n t h e wave

F i g . 2.

-

Dependence o f wave i n t e n s i t y on l o n g i t u - d i n a l coordinate i n f o u r p o i n t s o f Gauss beam transverse s e c t i o n : curve 1 corresponds t o R = 0.042 curve 2 t o R = 0.5 ; curve 3 t o

R: = P.9 ; curve 4 t o R: = 1.3. N = 3.25, f? = I.

i n t e n s i t y a t propagation i n d i f f e r e n t p o i n t s o f t h e beam transverse s e c t i o n : on t h e beam a x i s (curve 1) i n t h e zone near t h e a x i s R2 = 0.46 (curve 2), and over t h e p e r i p h e r y R3 = 0.88 (curve 3), R4 = 1.3 (curve 4) w i t h F i g , 3, where t h e same i s shown f o r a beam w i t h d i s t r i b u t i o n o f t h e ampl i t u d e by t h e 4-th degree polynom, curve 1 corresponds t o t h e f u n c t i o n ~ ( 8 ) on t h e beam a x i s , curve 2 t o R2 = 0.4 ; curve 3 t o R3 = 0.7;

curve 4 t o R4 = 1.03.

An increase o f t h e peak value o f wave p e r t u r - b a t i o n and i n t e n s i t y on t h e beam i s observed i n t h e t h e Gauss beam a l r e a d y a t B = 1 (curve 1 i n Fing. 2) w h i l e t h e i n t e n s i t y on t h e beam a x i s i s n o t i n c r e a s i n g i n t h e case o f a beam w i t h amplitude d i s t r i b u t i o n by t h e 4 - t h degree polynom (curve 1 i n F i g . 3 ) . The i n t e n s i t y on t h e a x i s i n

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JOURNAL DE PHYSIQUE C8- 59

F i g . 3.

tude i s polynom curve 2 curve 4

-

D i t t o as i n Fig. 2 f o r beam when ampli- d i s t r i b u t e d over s e c t i o n by 4 - t h degree

.

^Curve¹corresponds t o R = 0.03 ; t o R2 = 0.4 ; curve 3 t o R3 = 0.07 ; t o R4 = 1.03.

a Gauss beam i s accompanied by i t s r e d u c t i o n a t t h e beam edge (curve 4 i n Fig. 2). The beam diverges ges from t h e v e r y b e g i n i n g i n a beam w i t h amplitude d i s t r i b u t i o n by the 4 - t h degree polynom (curve 4 i n F i g . 3 ) . I n t h i s beam a t i3 = 1 t h e phase convergence i s completely suppressed by t h e d i f f r a c - t i o n divergence. I t i s i n t e r e s t i n g t o compare t h e form o f t h e wave i n t h e Gauss beam and i n t h e beam w i t h amplitude d i s t r i b u t i o n by t h e 4 - t h degree polynom. F i g . 4 demonstrates w a v e p r o f i l e s i n a

F i g . 4.

-

Form o f p e r t u r b a t i o n ~ ( 8 ) a t d i s t a n c e Z = 0.12 a t N = 5, B = 0.25 i n f o u r p o i n t s o f beam t r a n s v e r s e s e c t i o n . Amplitude d i s t r i b u t i o n and values R1 a r e t h e same as i n F i g . 3.

focussed beam which i s c h a r a c t e r i z e d by t h e values o f t h e parameters N = 5, B = 0.25, when t h e amplitude i s d i s t r i b u t e d by t h e 4 - t h degree polynom i n t h e same f o u r p o i n t s of t h e beam transverse s e c t i o n as i n F i g . 3 a t a d i s t a n c e Z = 0.12, when t h e form o f p e r t u r b a t i o n on t h e a x i s i s near t o t h e shock shape (curve l ) , w h i l e i n t h e zone near t h e a x i s (curve 2) and over t h e p e r i p h e r y o f t h e beam (curves3 and 4) t h e wave form i s f a r from t h e d i s - c o n t i n u i t y wave. F i g . 5 presents p r o f i l e s a t t h e

F i g . 5.

-

Form o f p e r t u r b a t i o n ~ ( 8 , ) a t d i s t a n c e Z = 0.12 a t N = 5, B = 0.25 i n same f o u r p o i n t s of Gauss beam transverse s e c t i o n as i n F i g . 2.

Curve 4 coincides w i t h a x i s o f abcissae.

same d i s t a n c e and t h e a t same parameters as i n F i g . 4 i n a Gauss beam. The numbers o f t h e curves correspond t o t h e same p o i n t s on t h e beam t r a n s - verse s e c t i o n as i n Fig. 2. Curve 4 coincides w i t h t h e a x i s o f abscissae. When comparing these f i g u - r e s i t i s c l e a r t h a t the p e r t u r b a t i o n on t h e a x i s i n a Gauss beam reaches l a r g e values (curves 1 i n F i g . 4 and 5). Besides t h a t t h e w i d t h o f t h e Gauss beam a t the d i s t a n c e Z = 0.12 i s l e s s than a t t h e border, and t h e r e i s non p e r t u r b a t i o n a t t h e t r a n s - verse s e c t i o n p o i n t s R = 1.3. D i s t o r t i o n o f t h e wave form a t t h e beam edge i s small (curve 4 i n F i g . 4 and curve 3 i n Fig. 5).

An a n a l y s i s o f the t a b l e i n d i c a t e s t h a t i n case o f h i g h n o n l i n e a r i t y N = 10 t h e shock wave forms so f a s t t h a t t h e i n c r e a s i n g convergence i s n o t r e v e a l i n g up t o v e r y small r a d i i o f convergence. The d i s c o n t i n u i t y forms e a r l i e r a t B = 0.1.

I f t h e n o n l i n e a r i t y i s low N = 3.25, t h e increase o f convergence causes a r e d u c t i o n of ZS

,

b u t a t v e r y h i g h convergence B = 0.25 and 0 = 0.1 no shock wave forms i n t h e " s e a r c h l i g h t " zone. This,

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c8- 60 N.S. BAKHVALOV, and a l .

a t f i r s t s i g h t , c o n t r a d i c t o r y f a c t i s e x p l a i n e d by t h e l a c k o f coincidence o f t h e f o c a l zones o f harmonic components t h a t may l e a d t o d i s t o r t i o n of t h e c o r r e l a t i o n between t h e amplitude and phases o f t h e harmonics r e q u i r e d f o r forming a shock wave. Fig. 6 i l l u s t r a t e s t h e form o f a wave

F i g . 6.

-

Form o f wave ~ ( 0 ) on a x i s i n beam w i t h F i g . 7.

-

Form o f wave ~ ( 0 ) on a x i s i n Gauss amp1 i t u d e d i s t r i b u t i o n by 4-th degree polynom a t beam a t d i s t a n c e Z = 0.06

-

curve 1 and a t d i s - d i f f e r e n t distances from border : tance Z = 0.09

-

^{curve 2.} ^N⁼^10,⁸⁼^1.

Z = 0.24

-

curve 1, Z = 0.38

-

^curve^2. N = 3.25, 6 = 0.1.

Z = 0.09

-

curve 2 f o r parameters N = 10, 6 = 1.

on t h e a x i s f o r parameters N = 3.25 and 6 = 0.1.

when t h e amplitude i s d i s t r i b u t e d by t h e 4 - t h degree polynom. Curve 1 corresponds t o t h e coordinate Z = 0.24 when t h e p e r t u r b a t i o n o f d e n s i t y i n t h e phase o f compression i s of t r i a n g u l a r shape ; curve 2 r e f e r s t o Z = 0.38 t h e w i d t h o f t h e f r o n t i s n o t i c e a b l y narrower b u t i t exceeds t h e shock one according t o the adopted c r i t e r i o n . It i s necessary t o s t a t e t h a t t h e character o f d i s t o r t i o n changes a t an increase of n o n l i n e a r i t y . A t values N = 3.25 and N = 5 t h e p o s i t i v e semi- p e r i o d o f p e r t u r b a t i o n becomes a t f i r s t t r i a n g u - l a r w i t h a l a r g e w i d t h o f t h e f r o n t . With propa- g a t i o n t h e apex o f t h e t r i a n g l e moves t o t h e f r o n t and a1 Z = ZS the wave p r o f i l e may be conside- r e d a d i s c o n t i n u i t y one. The process o f f r o n t nar- rowing a t = 0.1 i s h i g h l y prolonged and t h e f r o n t remains s l i n g h t l y w i d e r than t h e shock one (see Fig. 6 ) . I n t h e case o f h i g h n o n l i n e a r i t y d i s c o n t i n u i t y occurs when t h e r e a r f r o n t i s s t i l l rounded (curve 1 i n F i g . 7) and then t h e phase o f combression becomes t r i a n g u l a r (curve 2 i n f Fig.

Fing. 7). This phenomenon i s shown i n F i g . 7 where t h e wave form on t h e axis o f the Gauss beam i s shown a t t h e d i s t a n c e Z = 0.06

-

^curve^1,^{t h e}

shock p r o f i l e has h u s t formed, and the d i s t a n c e

Fig. 8.

-

Form o f d e n s i t y p e r t u r b a t i o n p(0,R) a t distance Z = 0.02 i n beam w i t h amplitude d i s t r i - t i o n by 4-th degree polynom. N = 10, 6 = 0.1.

Figs. 8,9 and 10 i l l u s t r a t e the p e r t u r b a t i o n i n t h e form o f a f u n c t i o n o f two v a r i a b l e s : time and transverse coordinate R a t t h e d i s t a n c e Z = 0.02 a t t h e d i s t a n c e Z = 0.06, i . e . a t a d i s t a n c e when t h e d i s c o n t i n u i t y formed and t h e peak value o f t h e p e r t u r b a t i o n on t h e a x i s reaches t h e maximum, t h e w i d t h o f t h e beam i n t h i s case i s minimum, and a t t h e d i s t a n c e Z = 0.1 when t h e beam s t r o n g l y widened.

The c a l c u l a t i o n i n d i c a t e s t h a t a t propaga-

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JOURNAL DE PHYSIQUE

Fig. 9.

-

D i t t o as i n Fig. 8. Z = 0.06.

Fig. 10.

-

D i t t o as i n F i g . 8. Z = 0.1.

t i o n o f t h e f i n i t e - a m p l i t u d e focused beam t h e po- s i t i v e and n e g a t i v e semi-periods a r e d i s t o r t e d unequally and t h e wave p r o f i l e d i f f e r s from t h e symmetric saw : the compression phase i s o f t r a i n - g u l a r shape w h i l e t h e wave form i n t h e evacuation phase i s rounded and appears l i k e a sinusoid.

The c h a r a c t e r of d i s t o r t i o n changes a t s i m i - l a r convergence w i t h an increase o f n o n l i n e a r i t y . I f i n t h e case o f weak and medium n o n l i n e a r i t y t h e p o s i t i v e semi-period accepts a t f i r s t t h e shape o f w t r a i n g l e and then t h e d i s c o n t i n u i t y forms, then i n t h e case of s t r o n g n o n l i n e a r i t y t h e jump occurs b e f o r e the p o s i t i v e semi-period becomes t r i a n g u l a r .

I n t h e case of weak and medium n o n l i n e a r i t y w i t h an increase o f convergence t h e d i s t a n c e o f

shock wave formation e i t h e r reduces o r remains the same, b u t when t h e cinvergence r a d i u s becomes small i t i s d i f f i c u l t t o speak o f a jump because the f r o n t i s w i d e r than t h e shock one. When N = 10, then t h e d i s t a n c e of jump formation i s i n s e n s i b l e t o an increase o f convergence as the shock f r o n t forms very r a p i d l y . And ZS reduces o n l y a t a very small r a d i u s o f convergence.

A t propagation o f t h e Gauss beam t h e d i f f r a c - t i o n divergence i s s u b s t a n t i a l l y l e s s which i s re- vealed i n a considerable increase o f t h e peak va- l u e o f p e r t u r b a t i o n and i n t e n s i t y on t h e a x i s as w e l l as s t a b i l i z a t i o n o f t h e beam w i d t h a t some d i s t a n c e a f t e r which i t divergences. N o n l i n e a r i t y o f t h e medium and t h e absence o f d i s p e r s i o n cause l a c k of coincidence o f t h e p e r t u r b a t i o n peak value maximum w i t h t h e maximum o f i n t e n s i t y . The i n t e n - s i t y reaches i t s maximum value e a r l i e r than t h e peak value o f p e r t u r b a t i o n i n t h e phase o f compression.

REFERENCES

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