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Optical visualization of non-linear acoustic propagation in cavitating liquids
Optical visualization of non-linear acoustic propagation in cavitating liquids P. CIUTI,
G. IERNETTI
and M.S. SAG00
Non-linearity effects on sound propagation induced by cavitation bubbles are investigated. The convergence of an acoustic wave due to the interaction with the microbubbles produced in the cavitation zone is shown experimentally. In these conditions the theoretical analysis shows that the self-focusing primarily depends on the effective microbubble volume fraction. This fraction turns out to be about 1C6 with a corresponding self-focusing distance of about 9 cm in the Fraunhofer region of a plane circular transducer. Introduction A sound wave of high amplitude appreciably modifies the characteristics of the medium in which the wave is travelling. This results in a non-linearity in the wave propagation which produces a strong self-interaction of the wavele3. This effect is similar to the corresponding one of self-focusing of a laser beam due to the non-linear behaviour of the medium4p5 In this paper we report the results of an experimental investigation of the self-interaction of an ultrasonic wave due to the non-linearity effects caused by microbubbles in the cavitation zone produced by the wave itself. The ultrasonic beam was optically visualized and it was found that, under suitable conditions, the acoustic beam focuses. This phenomenon may be explained as follows. In a cavitation zone, the sound phase-velocity is smaller in the regions where the bubble density is greater, due to the higher compressibility of the gas filling the bubbles with respect to that of the liquid. With the help of the pulsed cavitation technique, we have found that the bubble density depends linearly on the sound-intensity near the threshold of cavitation. On the other hand the intensity of the ultrasonic radiation has in general a non-uniform distribution6. For example, if the intensity distribution has a maximum along the axis of the beam, the phase velocity there is at a minimum. It follows that the central portion of the wavefront travels slower than the outer portion. The wavefront, therefore, becomes concave in the direction of propagation and the ultrasonic beam self-focuses.
these parameters increase the saturation by bubbles of the cavitation zone. Some of these bubbles are visible and owing to their size do not take part in the focusing of the beam and lead to a less regular distribution of the ultrasonic field intensity.7 So, continuous ultrasonic radiation, by a plane transducer, has a poor cavitation efficiency.7 Also continuous ultrasonic radiation produces acoustic streaming in the liquid, which tends to defocus the ultrasonic beam.’ Hence, the ultrasonic pulse technique here employed has proved particularly effective for observing the selffocusing action of the acoustic waves. Apparatus and method The ultrasonic cell used to visualize the self-interaction of acoustic waves is shown in Fig. 1. The cell was filled with distilled water and the cavitation generated by a circular plane transducer emitting an ultrasonic beam parallel to the two glass plates. To absorb the ultrasound an absorption chamber is fixed on the wall facing the transducer. The ultrasonic beam is visualized by the Schlieren’ method. The image of the acoustic beam is photographed on a plate. The onset of the cavitation was checked by a hydrophone, Transparent gloss plate
It should be pointed out that the self-focusing effect was pronounced when the duration of the ultrasonic pulses was in the interval from 100 to 300 ms, with a time of separation of about 0.7 s. The focusing effect diminishes with increasing the pulse duration or the pulse duty-ratio since both P. Ciuti is at the lstituto di Fisica, Universita di Trieste, Trieste, Italy and lstituto Nazionale di Fisica Nucleare, Sezione di Trieste, Trieste, Italy; G. lernetti is at the lstituto di Fisica, Universita di Trieste, Italy; M.S. Sagoo is at the Physics Department, Panjab University, Chandigarh, India. Paper received 15 October 1979. Revised 19 November 1979.
0041-624X/80/0301 ULTRASONICS.
MAY 1980
Transparent glass plate Fig. 1
11-04 $02.00 0
Schematic
view of the ultrasonic
cell
1980 IPC Business Press 111
RF
Pulse generator
Power ompllfler
OSCII lotor
I i
I
X
Trigger
rIJ bYI
0~21 I loscope
u2
Fig. 2
Schematic
view of the electronics
displaying the sound emission of the bubbles on an oscilloscope. A block diagram of the electronics is shown in Fig. 2. The threshold of cavitation was obtained by varying the inverse duty-ratio N (defined as the ratio of the pulsedistance to the pulse-time) of the pulsed radiation and/or the voltage V feeding the transducerrO~“. Theoretical
approach
c = co[1-
wGv/mo )I
It may be emphasized that (5) holds for ultrasonic frequencies well below the bubble resonance frequencyLL-‘*, so that the strains Sb and S, are in phase with the excess pressure due to the acoustic field. In the presence of ultrasound having intensity above the cavitation threshold, the bubble volume-fraction 6 depends on the intensity of the ultrasonic waves. From the diagram shown as Fig. 1 elsewhere16, considering that the inverse pulse-ratio N gives the relative number of the nuclei excitable by the corresponding threshold transducer-voltage V with respect to those excitable at V, (threshold transducer at N = l), the dependence of 6 on the intensity of the ultrasound can be obtained. In fact the variation of the nucleidensity n, with the acoustic intensity I above the threshold intensity It, is given by the linear law l6 % = 4”
WJ
(6)
where 4” is a constant of proportionality. We assume for simplicity the bubble volume-fraction 6 to be: 6 = b (Z-r,)
Let 6 be the mean bubble volume-fraction of the liquid,pe the hydrostatic pressure and p an additional pressure applied to the liquid-bubble mixture. Let us use the suffixes w, b and m to denote quantities relating to liquid, gas and liquidbubble mixture respectively. If S is the volumetric strain, the effective strain S, of the liquid-bubble mixture may be expressed ” in terms of the volume strains S, and Sb in liquid and gas, as follows: S,=S,+6S,
(1)
The bulk modulus K, of the liquid-bubble the equation
mixture satisfies
(7)
with b a new constant of proportionality. The substitution of 6 from (7) into (5) gives the velocity of the ultrasonic waves in the cavitation zone
c=c,
_bKw(Z-4)
1
2yp,
[
6 << 1, it follows that
Km = &/(I + 6 sb/‘%j
(2)
The effective density of the mixture is: Pm ‘Pw
-6 (Pw -Pb)=Pw(l
-6)
(3)
I
As mentioned earlier, the intensity of the ultrasonic radiation is in general non-uniform. Let us now assume the intensity distribution in the paraxial region to be parabolic with respect to the displacement from the point of the extremum intensity16; that is Z(x=O,r)=Zo(l
Under the assumption
(5)
-P//12)
($1
where A is a constant that determines the sharpness of the profile of the intensitydistribution, x the distance alohg the symmetry axis of the beam and r the radial distance from the axis. Under these conditions the propagation of an acoustic wave can be better analysed using the wave equation in cylindrical coordinates’14 . The particle displacement [ therefore satisfies the equation
where p,,, and pb are the densities of the liquid and of the gas content of the bubbles respectively. From (2) and (3), the sound velocity in the presence of bubbles in a liquid can be expressed as Let us now put
c=cO /d/lI1
+@~b/&v)l
(1 -&)I
(41
t = where C,, is the velocity of sound in the liquid without bubbles. Under these conditions, the ratio of the volumetric strains is St,/& = K,/yp, , which is of the order of 104, y being the ratio of the principal specific heats of the bubble content. Therefore, as long as the bubble volume-fraction 6 is less than 10M6,the effective velocity in the bubble-liquid mixture can be written as
c=
CO/
and linearized by series expansion,
112
as
(11)
r=qtt*
b
where w and k are the frequency and the wave vector of the acoustic wave respectively. Substituting 5‘from (11) into (lo), taking into account the expression for I, gives
St’ ar2
d/r1+ @~b/&v>l
Q(w) ev Ii(wf-h)l ,
9
r
‘ar
_2ti
g
ax
=
_k2QbKw YPO
@QQ* -4)
(12)
where a slow variation of Q with the distance of propagation x is assumed. Further, substituting
ULTRASONICS.
MAY 1980
Q = QO(x,9 ew [- iks(x , r>l
(13)
into (12) and comparing real and imaginary components, obtain
we
where 8 = I,,/It. Equation (22) shows that the beam-width parameter f decreases with the distance of propagation of the wave if R2g is greater than unity. The focal distance xf is obtained by putting f = 0. It follows that R ~. xf - Rdg-1
(14)
Finally from (23) and (24) the bubble volume-fraction is given as
(24) 6,,
Results and discussion Assuming an approximate S =;
solution for S as4
B(x) + G(x),
(16)
(15) can be shown to have the solution
Typical photographs of the self-focusing ultrasonic beam in a sample of distilled water are shown in Fig. 3. In both pictures the thick dark line in the acoustic beam near the edge H is the hydrophone used to check the cavitation onset. The division of the ultrasonic beam in two or more shoots can be attributed to the random nature of the cavi-
(17) where F is an appropriate function with f given by the equation
of the argument
r&if),
It can be shown4 that f represents the width of the beam in terms of the initial width, whereas B-’ is the radius of curvature R , of a wavefront. The boundary conditions for f are:
0
f(x=O)=l, g
=;
(1%
l
x=0
From (9) (11) and (17), it follows that Q% (x,
r)
i0 QF
=
[l -
r’/(Aj)‘]
(20) H
ii’
By substituting (20) into (14) and comparing the coefficients of equal powers of r we obtain a differential equation for the dimensionless width parameter f as,
pd;t.g.K$+L=O
(21)
k2A2 The solution of (21) under the boundary is f’ =(l
t&)2
conditions
-gx2
of (19),
(22)
where ,L A2
K,bI, ---..__+TPO
1 k2A2
1
.
By using (7) it can be shown that the expression for g contains the bubble volume fraction 6, as &&e/(6-1) YPO
ULTRASONICS.
MAY 1980
t-
1 k2A2
1
H
(23)
H’
Fig. 3 Pulsed ultrasonic beam without and with focusing. The ultrasonic beam propagates in the direction from H to H’
113
tation process and to more than one maximum in the wavefront emitted by the transducer.
of intensity
The free radius r. of the transducer was 1.25 cm and the wavelength X of the ultrasound was 2.0 mm. Under these conditions, the distance of farthest maxima from the transducer isr7 F ‘%r$/h”- 8 cm, which is the distance of the edge H from the transducer in our experimental set-up. In the present case the corresponding geometry of the acoustic propagation is shown in Fig. 4. The edge marked H in Fig. 3, corresponds to the point P in Fig. 4. This geometry has been chosen to avoid the Fresnel region (near field) with a larger number of small zones of maxima and minima of the acoustic intensity’s, The net effect of the cavitation on the propagation of sound cannot be described easily in the Fresnel region; whereas in the Fraunhofer region (far field) no minima of the acoustic intensity occur, consequently non-linear acoustic propagation due to cavitation is better described by the above-mentioned, simplified model. The divergence of the acoustic wave due to diffraction is overcome by the self-action of the wave. In the example shown in Fig. 3, the ultrasonic beam converges to a focus at a distance xf of about 8.5 cm; that is, the Fresnel zone also contributes to the self-focusing effect. The other parameters of the experiment are as follows. The threshold transducer voltage Vt was 120 V, whereas the transducer voltage, for the self-focusing of the pulse shown in Fig. 3, was 170 V. This gives 0 = lo/It s 2. The ultrasonic frequency used was 690 kHz. The hydrostatic pressure was about 0.1 MPa (1 atmosphere), and the mean depth of the beam below the surface of the water was 7 cm. The value of y is that of a diatomic gas, that is, 1.4. The value ofA, deduced from hydrophone measurements, was roughly equal to rO. Substituting the values of the parameters into (14) the bubble volume-fraction 6, turns out to be about 8 x 10V7. Other authors” have reported that a substantial contribu-
tion to the non-linearity of the liquid is produced by the microbubbles when 6 exceeds about lo-‘. It may be pointed out that the self-focusing of acoustic waves in the presence of cavitation also confirms the observation lg that the cavitation efficiency of a plane transducer is approximately the same as that of a curved transducer in the case of pulsed cavitation. The cavitation efficiency of a curved transducer is greater than that of a plane transducer7 due to the better impedance match with the bubble-containing cavitation zone that develops away from the curved transducer face; however, this also applies for a pulsed plane-beam which becomes convergent due to the self-focusing effect. Acknowledgements The authors are grateful to the Advanced School of Physics at the International Centre for Theoretical Physics in Trieste and also to the International Centre for giving the opportunity to M.S. Sagoo of extending his stay at the Centre. This work was partially supported by a CNR contract and by the convention ENEL-Institute of Physics of Trieste University. References
6 7 8 9 10
“I 11 12 13
c
I-
14
5
Fig. 4 Geometry of the acoustic propagation: d is the transducer diameter, P the point of the farthest maximum on the axis, n, - TT~ the Fresnel region having a large number of maxima and minima, beyond ?i, the Fraunhofer region, and aa the envelope of the cen-
tral lobe
114
15 16 17
Akhmanov. %A.. Sukhorukov, A.P., Khoklov R.V.,Sov Phys Jept 23 (1966) 1025 Akhmauov, S.A., Sukhorukov, A.P., Khoklov, R.V., sol’ &J’S Ups 10 (1961) 609 Mathur,S.S.,Sagoo,M.S.,Can JPhys52 (1974) 1726 So&a, hf.!&, Ghatak, A.K., Tripathi, V.K., Self l’ocus& ot’ kascr Beam, Tata McGraw Hill Pub Co Lrd, New Delhi (1971) Arecchi, F.T., Schultz-Dubois, E.O., Laser Handbook, NorthHolland Pub Co Amsterdam, American Elsevier Pub Co,lnc. New York (1972) Vol. I, Vol. II, part E. Schaafs, W., Molekularakustic, Springer Verlag Kerlin (1963) 64 Rosemberg, L.D., High-Intensity Ultrasonic Fields, Plenum Press, New York (1971) 308 Mathur, S.S., Sagoo,M.S., Can JPhys 52 (1974) 1723 Bergmann, L., Der Ultraschall, S. Ilirzel Vcrlag Zurich ( 1949 1 Kap2,eand Kap5,g Ceschia, M., Iemetti, G., Nabergoj, R., Proc 1973 Symp IinitcAmplitude Wave Effects in I-luids, Ed. L, Bj@rno, IPC Science and Technology Press Ltd Guildford, Surrey, l&$md (1974) 273 Wclsby, V.G. Safar, M.H., Amstica 22 (1969) 177 Flynn, H.G., Physical Acoustics, Vol. IB Mason. W.E., Editor. Academic Press New York (1964) Van Wijngaarden, L., Proc Intern Co11 Drops and Bubbles, California Institute of Technology and Jet Propulsion Laboratory (1974) Vol. II, 405 Krasilnikov, V.A., Kuznetsov, V.P., Sov Phys Acoust 20 (1974) 285 Boguslavskii, Yu. Ya., Sov Phys Acoust 14 ( 1968) 15 1 Iemetti, G., Sagoo, M.S., Acustjca 41 (1978) 32 Rzhevkin, N.S., The theory of sound, I’ergamon Press, Iondon (1963)
18 19
119
Zemanek,J.,J.AcoustSocAm49(1971J Iernctti, G. To be published in Acustjca
181
ULTRASONICS.
MAY 1980
G. IERNETTI
and M.S. SAG00
Non-linearity effects on sound propagation induced by cavitation bubbles are investigated. The convergence of an acoustic wave due to the interaction with the microbubbles produced in the cavitation zone is shown experimentally. In these conditions the theoretical analysis shows that the self-focusing primarily depends on the effective microbubble volume fraction. This fraction turns out to be about 1C6 with a corresponding self-focusing distance of about 9 cm in the Fraunhofer region of a plane circular transducer. Introduction A sound wave of high amplitude appreciably modifies the characteristics of the medium in which the wave is travelling. This results in a non-linearity in the wave propagation which produces a strong self-interaction of the wavele3. This effect is similar to the corresponding one of self-focusing of a laser beam due to the non-linear behaviour of the medium4p5 In this paper we report the results of an experimental investigation of the self-interaction of an ultrasonic wave due to the non-linearity effects caused by microbubbles in the cavitation zone produced by the wave itself. The ultrasonic beam was optically visualized and it was found that, under suitable conditions, the acoustic beam focuses. This phenomenon may be explained as follows. In a cavitation zone, the sound phase-velocity is smaller in the regions where the bubble density is greater, due to the higher compressibility of the gas filling the bubbles with respect to that of the liquid. With the help of the pulsed cavitation technique, we have found that the bubble density depends linearly on the sound-intensity near the threshold of cavitation. On the other hand the intensity of the ultrasonic radiation has in general a non-uniform distribution6. For example, if the intensity distribution has a maximum along the axis of the beam, the phase velocity there is at a minimum. It follows that the central portion of the wavefront travels slower than the outer portion. The wavefront, therefore, becomes concave in the direction of propagation and the ultrasonic beam self-focuses.
these parameters increase the saturation by bubbles of the cavitation zone. Some of these bubbles are visible and owing to their size do not take part in the focusing of the beam and lead to a less regular distribution of the ultrasonic field intensity.7 So, continuous ultrasonic radiation, by a plane transducer, has a poor cavitation efficiency.7 Also continuous ultrasonic radiation produces acoustic streaming in the liquid, which tends to defocus the ultrasonic beam.’ Hence, the ultrasonic pulse technique here employed has proved particularly effective for observing the selffocusing action of the acoustic waves. Apparatus and method The ultrasonic cell used to visualize the self-interaction of acoustic waves is shown in Fig. 1. The cell was filled with distilled water and the cavitation generated by a circular plane transducer emitting an ultrasonic beam parallel to the two glass plates. To absorb the ultrasound an absorption chamber is fixed on the wall facing the transducer. The ultrasonic beam is visualized by the Schlieren’ method. The image of the acoustic beam is photographed on a plate. The onset of the cavitation was checked by a hydrophone, Transparent gloss plate
It should be pointed out that the self-focusing effect was pronounced when the duration of the ultrasonic pulses was in the interval from 100 to 300 ms, with a time of separation of about 0.7 s. The focusing effect diminishes with increasing the pulse duration or the pulse duty-ratio since both P. Ciuti is at the lstituto di Fisica, Universita di Trieste, Trieste, Italy and lstituto Nazionale di Fisica Nucleare, Sezione di Trieste, Trieste, Italy; G. lernetti is at the lstituto di Fisica, Universita di Trieste, Italy; M.S. Sagoo is at the Physics Department, Panjab University, Chandigarh, India. Paper received 15 October 1979. Revised 19 November 1979.
0041-624X/80/0301 ULTRASONICS.
MAY 1980
Transparent glass plate Fig. 1
11-04 $02.00 0
Schematic
view of the ultrasonic
cell
1980 IPC Business Press 111
RF
Pulse generator
Power ompllfler
OSCII lotor
I i
I
X
Trigger
rIJ bYI
0~21 I loscope
u2
Fig. 2
Schematic
view of the electronics
displaying the sound emission of the bubbles on an oscilloscope. A block diagram of the electronics is shown in Fig. 2. The threshold of cavitation was obtained by varying the inverse duty-ratio N (defined as the ratio of the pulsedistance to the pulse-time) of the pulsed radiation and/or the voltage V feeding the transducerrO~“. Theoretical
approach
c = co[1-
wGv/mo )I
It may be emphasized that (5) holds for ultrasonic frequencies well below the bubble resonance frequencyLL-‘*, so that the strains Sb and S, are in phase with the excess pressure due to the acoustic field. In the presence of ultrasound having intensity above the cavitation threshold, the bubble volume-fraction 6 depends on the intensity of the ultrasonic waves. From the diagram shown as Fig. 1 elsewhere16, considering that the inverse pulse-ratio N gives the relative number of the nuclei excitable by the corresponding threshold transducer-voltage V with respect to those excitable at V, (threshold transducer at N = l), the dependence of 6 on the intensity of the ultrasound can be obtained. In fact the variation of the nucleidensity n, with the acoustic intensity I above the threshold intensity It, is given by the linear law l6 % = 4”
WJ
(6)
where 4” is a constant of proportionality. We assume for simplicity the bubble volume-fraction 6 to be: 6 = b (Z-r,)
Let 6 be the mean bubble volume-fraction of the liquid,pe the hydrostatic pressure and p an additional pressure applied to the liquid-bubble mixture. Let us use the suffixes w, b and m to denote quantities relating to liquid, gas and liquidbubble mixture respectively. If S is the volumetric strain, the effective strain S, of the liquid-bubble mixture may be expressed ” in terms of the volume strains S, and Sb in liquid and gas, as follows: S,=S,+6S,
(1)
The bulk modulus K, of the liquid-bubble the equation
mixture satisfies
(7)
with b a new constant of proportionality. The substitution of 6 from (7) into (5) gives the velocity of the ultrasonic waves in the cavitation zone
c=c,
_bKw(Z-4)
1
2yp,
[
6 << 1, it follows that
Km = &/(I + 6 sb/‘%j
(2)
The effective density of the mixture is: Pm ‘Pw
-6 (Pw -Pb)=Pw(l
-6)
(3)
I
As mentioned earlier, the intensity of the ultrasonic radiation is in general non-uniform. Let us now assume the intensity distribution in the paraxial region to be parabolic with respect to the displacement from the point of the extremum intensity16; that is Z(x=O,r)=Zo(l
Under the assumption
(5)
-P//12)
($1
where A is a constant that determines the sharpness of the profile of the intensitydistribution, x the distance alohg the symmetry axis of the beam and r the radial distance from the axis. Under these conditions the propagation of an acoustic wave can be better analysed using the wave equation in cylindrical coordinates’14 . The particle displacement [ therefore satisfies the equation
where p,,, and pb are the densities of the liquid and of the gas content of the bubbles respectively. From (2) and (3), the sound velocity in the presence of bubbles in a liquid can be expressed as Let us now put
c=cO /d/lI1
+@~b/&v)l
(1 -&)I
(41
t = where C,, is the velocity of sound in the liquid without bubbles. Under these conditions, the ratio of the volumetric strains is St,/& = K,/yp, , which is of the order of 104, y being the ratio of the principal specific heats of the bubble content. Therefore, as long as the bubble volume-fraction 6 is less than 10M6,the effective velocity in the bubble-liquid mixture can be written as
c=
CO/
and linearized by series expansion,
112
as
(11)
r=qtt*
b
where w and k are the frequency and the wave vector of the acoustic wave respectively. Substituting 5‘from (11) into (lo), taking into account the expression for I, gives
St’ ar2
d/r1+ @~b/&v>l
Q(w) ev Ii(wf-h)l ,
9
r
‘ar
_2ti
g
ax
=
_k2QbKw YPO
@QQ* -4)
(12)
where a slow variation of Q with the distance of propagation x is assumed. Further, substituting
ULTRASONICS.
MAY 1980
Q = QO(x,9 ew [- iks(x , r>l
(13)
into (12) and comparing real and imaginary components, obtain
we
where 8 = I,,/It. Equation (22) shows that the beam-width parameter f decreases with the distance of propagation of the wave if R2g is greater than unity. The focal distance xf is obtained by putting f = 0. It follows that R ~. xf - Rdg-1
(14)
Finally from (23) and (24) the bubble volume-fraction is given as
(24) 6,,
Results and discussion Assuming an approximate S =;
solution for S as4
B(x) + G(x),
(16)
(15) can be shown to have the solution
Typical photographs of the self-focusing ultrasonic beam in a sample of distilled water are shown in Fig. 3. In both pictures the thick dark line in the acoustic beam near the edge H is the hydrophone used to check the cavitation onset. The division of the ultrasonic beam in two or more shoots can be attributed to the random nature of the cavi-
(17) where F is an appropriate function with f given by the equation
of the argument
r&if),
It can be shown4 that f represents the width of the beam in terms of the initial width, whereas B-’ is the radius of curvature R , of a wavefront. The boundary conditions for f are:
0
f(x=O)=l, g
=;
(1%
l
x=0
From (9) (11) and (17), it follows that Q% (x,
r)
i0 QF
=
[l -
r’/(Aj)‘]
(20) H
ii’
By substituting (20) into (14) and comparing the coefficients of equal powers of r we obtain a differential equation for the dimensionless width parameter f as,
pd;t.g.K$+L=O
(21)
k2A2 The solution of (21) under the boundary is f’ =(l
t&)2
conditions
-gx2
of (19),
(22)
where ,L A2
K,bI, ---..__+TPO
1 k2A2
1
.
By using (7) it can be shown that the expression for g contains the bubble volume fraction 6, as &&e/(6-1) YPO
ULTRASONICS.
MAY 1980
t-
1 k2A2
1
H
(23)
H’
Fig. 3 Pulsed ultrasonic beam without and with focusing. The ultrasonic beam propagates in the direction from H to H’
113
tation process and to more than one maximum in the wavefront emitted by the transducer.
of intensity
The free radius r. of the transducer was 1.25 cm and the wavelength X of the ultrasound was 2.0 mm. Under these conditions, the distance of farthest maxima from the transducer isr7 F ‘%r$/h”- 8 cm, which is the distance of the edge H from the transducer in our experimental set-up. In the present case the corresponding geometry of the acoustic propagation is shown in Fig. 4. The edge marked H in Fig. 3, corresponds to the point P in Fig. 4. This geometry has been chosen to avoid the Fresnel region (near field) with a larger number of small zones of maxima and minima of the acoustic intensity’s, The net effect of the cavitation on the propagation of sound cannot be described easily in the Fresnel region; whereas in the Fraunhofer region (far field) no minima of the acoustic intensity occur, consequently non-linear acoustic propagation due to cavitation is better described by the above-mentioned, simplified model. The divergence of the acoustic wave due to diffraction is overcome by the self-action of the wave. In the example shown in Fig. 3, the ultrasonic beam converges to a focus at a distance xf of about 8.5 cm; that is, the Fresnel zone also contributes to the self-focusing effect. The other parameters of the experiment are as follows. The threshold transducer voltage Vt was 120 V, whereas the transducer voltage, for the self-focusing of the pulse shown in Fig. 3, was 170 V. This gives 0 = lo/It s 2. The ultrasonic frequency used was 690 kHz. The hydrostatic pressure was about 0.1 MPa (1 atmosphere), and the mean depth of the beam below the surface of the water was 7 cm. The value of y is that of a diatomic gas, that is, 1.4. The value ofA, deduced from hydrophone measurements, was roughly equal to rO. Substituting the values of the parameters into (14) the bubble volume-fraction 6, turns out to be about 8 x 10V7. Other authors” have reported that a substantial contribu-
tion to the non-linearity of the liquid is produced by the microbubbles when 6 exceeds about lo-‘. It may be pointed out that the self-focusing of acoustic waves in the presence of cavitation also confirms the observation lg that the cavitation efficiency of a plane transducer is approximately the same as that of a curved transducer in the case of pulsed cavitation. The cavitation efficiency of a curved transducer is greater than that of a plane transducer7 due to the better impedance match with the bubble-containing cavitation zone that develops away from the curved transducer face; however, this also applies for a pulsed plane-beam which becomes convergent due to the self-focusing effect. Acknowledgements The authors are grateful to the Advanced School of Physics at the International Centre for Theoretical Physics in Trieste and also to the International Centre for giving the opportunity to M.S. Sagoo of extending his stay at the Centre. This work was partially supported by a CNR contract and by the convention ENEL-Institute of Physics of Trieste University. References
6 7 8 9 10
“I 11 12 13
c
I-
14
5
Fig. 4 Geometry of the acoustic propagation: d is the transducer diameter, P the point of the farthest maximum on the axis, n, - TT~ the Fresnel region having a large number of maxima and minima, beyond ?i, the Fraunhofer region, and aa the envelope of the cen-
tral lobe
114
15 16 17
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