The shock formation distance in a bounded sound beam of finite amplitude

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The shock formation distance in a bounded sound beamof finite amplitude

Chao Tao, Jian Ma, Zhemin Zhu, and Gonghuan Dua)

State Key Lab of Modern Acoustics & Institute of Acoustics, Nanjing University, Nanjing 210093,People’s Republic of China

Zihong PingHangzhou Applied Acoustics Institute, Hangzhou Fuyang 311400, People’s Republic of China

~Received 10 June 2002; revised 4 April 2003; accepted 14 April 2003!

This paper investigates the shock formation distance in a bounded sound beam of finite amplitudeby solving the Khokhlov-Zabolotskaya-Kuznetsov~KZK ! equation using frequency-domainnumerical method. Simulation results reveal that, besides the nonlinearity and absorption, thediffraction is another important factor that affects the shock formation of a bounded sound beam.More detailed discussions of the shock formation in a bounded sound beam, such as the waveformof sound pressure and the spatial distribution of shock formation, are also presented and comparedfor different parameters. ©2003 Acoustical Society of America.@DOI: 10.1121/1.1579002#

PACS numbers: 43.25.Cb, 43.25.Jh@MFH#

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I. INTRODUCTION

Nonlinear characteristics of a one-dimensional acoucal field, such as plane, spherical, and cylindrical wave, hbeen solved analytically.1 However, most of the boundesound beam, such as the piston field, which is widely usepractical application, cannot be solved analytically, andmerical simulation is necessary. In the last two decades,eral computational algorithms have been developed.2–12 Ad-ditionally, the nonlinear characteristics of an acoustical fiare discussed widely by using these algorithms, a serieproblems has been solved successfully.2–10,13An overview ofthe modern computational techniques can be found in acent monograph on nonlinear acoustics.14 Because of the restriction of huge computational time, one typical and imptant nonlinear characteristic—shock formation—was seldresearched in detail in the early work until some fast alrithms were developed.13,15,16Recently, it has been discussewidely.17–23 In particular, Khokhlovaet al. presented a fasalgorithm to reduce computational time and discussshock formation in the near field of a cw plane piston souby using this algorithm.23

As is known, the shock formation has many particucharacteristics.1,12 It is important to evaluate the shock fomation distance and plot out the shock formation area ifinite amplitude acoustical field for many theoretical apractical problems. Unfortunately, the current nonlinetheory only gives the shock formation distance of ondimensional wave in lossless media.1 Therefore, in order togive insight into the shock formation of a sound beam, in tpaper, we discuss the shock formation distance in the fiamplitude field radiated from an intense piston sourcesolving the KZK equation using Fourier series expansionnumerical method. Several factors, such as diffraction,sorption, and source intensity, are considered to investithe intense nonlinear acoustic field. Study will indicate th

a!Electronic mail: [email protected]

114 J. Acoust. Soc. Am. 114 (1), July 2003 0001-4966/2003

ibution subject to ASA license or copyright; see http://acousticalsociety.org

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the shock formation in the piston field is much more coplex than that in the one-dimensional field. Besides the nlinearity and absorption, the diffraction is another major rothat influences the shock formation in a bounded soubeam. To describe the influence of diffraction, the relatbetween the Rayleigh distance and the shock formationtance at the axis of a piston field is given. Finally, the wavform of pressure and the spatial distribution of shock formtion are also discussed.

This paper is organized as follows: In the second stion, the algorithm is described briefly and the numericmethod of evaluating the shock formation distance is pposed. In the third section, calculation results are preseand discussed. In the last section, we draw the conclusio

II. NUMERICAL MODEL

A. Model equations

Our study is based on the numerical solution of the KZequation:24,25

S 4]2

]t]s2¹'

2 24ar 0

]3

]t3D p521

sD

]2

]t2p2. ~1!

Here t5v(t2z/c0), s5z/r 0 , j5r/a, p5(P2P0)/r0c0u0 , P0 and r0 are ambient pressure and ambient desity, P is the pressure,t is the time,z andr are the coordinatealong the axial and radial direction respectively.¹'

2

5(]2/]j2)1(1/j)(]/]j) is the nondimensional transversLaplace operator with respect toj. Further,v21 is a charac-teristic time anda is a characteristic length transverse to tdirection of propagation. For an axisymmetric piston sinsoidal source,v/2p is frequency anda is source radius,u0

andc0 are the characteristic value of the velocity and isetropic sound speed, respectively.M5u0 /c0 is the Machnumber;r 0 is the Rayleigh distance;a5Dv2/2c0

3 is the ab-sorption coefficient,D being the diffusivity of sound; andsD5 l D /r 0 , wherel D5c0 /bMv is the shock formation dis-

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tance of a plane wave in lossless media. The boundarydition for Eq.~1! is p(s50,j,t)5 f (j,t), whereu0f (j,t) isthe normal velocity distribution in the plane of source.

When the source is periodic in time with period 2p/v,the solution of Eq.~1! can be expanded by using the Fourseries:

p5 (n51

`

~gnsin nt1hncosnt!, ~2!

and its coefficientgn andhn can be obtained by solving thfollowing equations:

]gn

]s52n2Agn1

1

4n¹'

2 hn1Nn

2 S 1

2 (p51

n21

~gpgn2p

2hphn2p!2 (p5n11

`

~gp2ngp1hp2nhp!D ,

~3!]hn

]s52n2Ahn2

1

4n¹'

2 gn1Nn

2 S 1

2 (p51

n21

~hpgn2p

1gphn2p!1 (p5n11

`

~hp2ngp2gp2nhp!D .

Heren is the harmonic number. The nonlinearity paraeterN and the absorption parameterA are defined as

A5ar 0 and N5r 0 / l D51/sD.

Further, we define the nonlinearity absorption ratio as

N/A51/a l D.

The boundary condition forgn andhn at s50 is

gn51

pE2p

p

f sin nt dt and hn51

pE2p

p

f cosnt dt.

~4!

For the case of an axisymmetric piston source which oslates sinusoidally, we have the boundary condition

g151uju<1, g150uju.1

g25g35•••50

h15h25•••50J , when s50.

Equations~3! have been solved numerically.2,3 To reduce thecalculation time, we adopt and improve the calculation tenique presented in Ref. 23. The parameter values of option are chosen as follows: The radial and axial integral stare equal toDj5531023 and Ds52.531025, respec-tively. The initial number of harmonics isn540, and thenumber of harmonics is increased byDn5100 when theabsolute of the last harmonic amplitude exceeds the throld 231024. The size of the spatial window in the radidirectionjmax

(n) is chosen depending on the harmonics numn and the propagation distances,

J. Acoust. Soc. Am., Vol. 114, No. 1, July 2003


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-

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-a-s

h-

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jmax~n! 55

jmax31, 1<n<30,

jmax30.8, 31<n<100,

jmax30.6, 101<n<300,

jmax30.4, 301<n<800,

jmax30.22, 801<n<1000,

here jmax5H 5,0<s<0.3,

7.5,0.3<s<0.5,

10,0.5<s<0.8.

To improve computational speed more, we write the ensource code in C11 language, compile and optimize aiminat Pentium IV 1.6 G CPU by using Intel C11 Compiler5.0.1.

B. Numerical evaluation of shock formation distance

The analytical nonlinear theory evaluates the shock fmation distance of a plane wave in lossless media accorto a multivalueness beginning to appear in the profile ofp1;in other words when the distance approachess5sD , themaximum derivative of sound pressure approaches infini

max0<t<2p

S dp~sD ,t!

dt D5`.

Since the numerical results cannot express an infinite vathis method cannot be used to evaluate the shock formadistance of the piston field. However, it is enlightening ththe maximum derivative of sound pressure can be usedescribe the distortion of the waveform. Further, the chanin the wave amplitude due to diffraction, beam converginetc., may also influence the maximum derivative of soupressure. We therefore use the maximum derivative ofwaveform with normalized peak-to-peak pressure to descthe distortion of the waveform:

Sm5 max0<t<2p

S d

dt S p~s,t!

pppD D , ~5!

whereppp is the peak-to-peak value of pressurep. Becausethe waveformp is normalized in Eq.~5!, this equation cancorrectly describe the distortion of the waveform even whits amplitude is changing and the changes in waveform aplitude will not influence the judgment of the shock formtion distance. The largerSm indicates that the waveform idistorted more seriously. Then, we use a thresholde to judgewhether the shock has formed; i.e.,Sm>e indicates that theshock has formed, otherwise the shock has not formed.next problem is how to choose the thresholde. We noticethat the piston field approaches a plane field whenr 0 ap-proaches infinity. Therefore, the shock formation distancethe piston field evaluated by this threshold value should aapproach the plane wave’s distance whenr 0→`. Accordingto the above discussion, we evaluate the shock formadistance of the piston field as follows:

First, an auxiliary plane wave model is used to detmine the thresholde. For the given sinusoidal wave ampltude at the source, the plane wave propagating in lossy m

115Tao et al.: Shock formation distance of sound beam


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FIG. 1. Determine the thresholde. In ~a! and~b!, the solid line and dashed line are calculated with 3200 harmonics and 5000 harmonics. In~c! and~d!, thesolid line and dashed line are calculated with 500 harmonics and 1000 harmonics.~a! The curve ofSm vs Z with N/A51000.~b! The derivative ofSm withN/A51000.~c! The curve ofSm vs Z with N/A5164. ~d! The derivative ofSm with N/A5164.

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FIG. 2. Steps of evaluating the shock formation distance of a piston fi~a! Compose the waveform of sound pressure withp5(n51
` (gn sinnt1hn cosnt). ~b! Calculate the derivative (d/dt)@p(t)/ppp#. ~c! Draw thecurve ofSm vs s, here in region II, wave is in the shock formation.

116 J. Acoust. Soc. Am., Vol. 114, No. 1, July 2003


is solved by using the following equation with the samvalue ofN/A(N/A@1) as that in the KZK equation:

S 2]

]Z22

A

N

]2

]t2D p5]

]tp2, ~6!

whereZ5z/ l D . Clearly, this equation is obtained by ignoing the diffractive term of Eq.~1! and it is an equivalent ofBurgers equation. Therefore, its solution can be obtainedsolving the coupling partial differential Eqs.~3! with ignoreddiffractive term.

Then Sm @Figs. 1~a! and ~c!# and its spatial-derivativedSm /dZ @Figs. 1~b! and ~d!# over the propagation distancare calculated as a function ofZ. The distanceZ(N/A) isfound where the functiondSm /dZ reaches a maximum. Athis distanceZ(N/A)(z5 l D

(N/A)), the corresponding value oe5Sm(Z(N/A)) is evaluated and used as a threshold for evaating the shock formation in the full diffraction problem. Also, the distancel D

(N/A)5Z(N/A)3 l D is considered theshock formation distance of a plane wave propagating ilossy media (1!N/A,`). For instance, in the lossy mediwith a50.025 33 m21, l D50.24 m andN/A5164, we have

d.

TABLE I. The value ofZ(N/A) with different N/A.

N/A 100 164 395 1000 2000 →`

Z(N/A) 1.158 1.1333 1.091 1.060 1.039 →1.000

Tao et al.: Shock formation distance of sound beam


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FIG. 3. The shock formation distancat the axis of a piston field.

ara

th

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ed

e519.4, Z(164)51.133, andl D(164)50.272 m. Figure 1 illus-

trates the steps determining the thresholde. It also shows thatthe curves calculated with different harmonic numbersalmost identical when the harmonic number is larger thpN/A.1,23

Finally, the bounded sound beam is solved by usingKZK equation. When theSm predicted by the KZK equationexceeds the value of thresholde, it is considered that the

FIG. 4. The waveform of sound pressure in various distances withN51.204 andA50.007 32.



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shock has formed. The shock formation distance is the nest position whereSm.e ~Fig. 2!.

One benefit of the above algorithm is that it is compible with the analytical theory for a strong nonlinear wa(N/A@1). We find the increase ofN/A causes the decreasof Z(N/A) ~Fig. 1 and Table I!. When the value ofN/A ap-proaches̀ , the value ofZ(N/A) will be close to 1 (Z(N/A)

→ l D), which implies the shock formation distance evaluat

FIG. 5. The waveform of sound pressure in various distances withN524.083 andA50.146 41.



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with the above algorithm approaches the analytical solul D in the case of zero absorptiona→0. And whenN/A islarge enough (N/A@1), the difference betweenl D

(N/A) andl D

is small. These meet the requirement of the above discusAnother important reason is that it is of benefit to eva

ate the shock formation distance exactly enough even whehas a slight error, becausee is set as the valueSm(Z(N/A))where the curve of the maximum slope is steepest. Forample, in the lossy media withN/A5164, a 5% error ofeonly brings a 0.7% error to the shock formation distance

III. RESULT AND DISCUSSION

Based on the above model equations and numerevaluation of shock formation distance, we study the shformation distance of a piston field in this section. The pis

FIG. 6. The growing process of the plane-wave-like shock and spherwave-like shock, where shock 1 is the plane-wave-like shock and shockthe spherical-wave-like shock. The parameters areA50.0425,N56.984.



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field with a total of 40 sets of parametersN andA is simu-lated. All the fields have the sameN/A5164. Therefore, weestimate the shock formation distancesl D8 at the axis with thethresholde519.4. Because (N/A)215ar 03( l D /r 0) is iden-tical, we can consider thatl D anda are fixed butr 0 is vari-able. In other words, it can be considered that the sobeams are all radiated from the source with the frequencMHz and the initial pressure amplitude 0.64 MPa, and progate in the same media~in water,r051000 kg/m3, b53.5,c051500 m/s,a50.025 33 m21 at 1 MHz, l D50.24 m!, butthe source radius in differentaP(0.9099 cm, 7.429 cm! cor-respond tor 0P(0.1734 m, 11.56 m!. We therefore get therelation betweenl D8 / l D

(164) and r 0 / l D according to our simu-lation results. The curve ofl D8 / l D

(164) vs r 0 / l D is given in Fig.3. Clearly, the influence ofr 0 on l D8 is very complex and Fig.3 can be divided into three regions. In region III (r 0 / l D

.6.6), l D8 / l D(164) is undulate around 1 and converges at 1

r 0 / l D→`, which meansl D8 → l D(164) . This reasonable resul

can be explained as follows.When r 0@ l D , the shock of a piston field forms in th

nearfield and, according to the theory of a piston beam, innear field of a piston field, its spatial-average characteristisimilar to a plane wave. So, its shock formation distanshould also approach that of its corresponding plane wjust as our simulation results have shown. The agreemalso proves that our numerical simulation and numerievaluation of shock formation distance is effective tosearch the shock formation distance of a piston field.

In region II (0.723,r 0 / l D,6.6), the shock formationdistance increases suddenly at aboutr 0 / l D56.6, and thendecreases with the decrease ofr 0 / l D until r 0 / l D51.204,when 0.723,r 0 / l D,1.204 the shock formation distance increases again. Finally, in region I (r 0/ l 0,0.723), the shockwave cannot happen at all. To describe the relation betwr 0 and l D8 , we fit the curve in region II with the followingformula since the far-field characteristic of a piston fieldsimilar to a spherical field and the shock forms in the ffield:

l D8

l D~164!

5C1r 0 expS C2

l D

r 0D . ~7!

Numerical simulation determines the parametersC150.81and C251.204; the curve ofl D8 / l D

(164) vs r 0 / l D calculatedwith formula~7! is also plotted in Fig. 3 with the dashed linIt is seen that formula~7! can actually approach our calculation results, which implies that the shock formation distanin region II obeys a similar rule to that of a spherical wavAlthough this formula is fitted withN/A5164, it is alsosuitable for the condition thatN/A→` after altering the pa-rametersC1 , C2 slightly.

For an insight into the shock wave in regions II and Iwe give the pressure waveform at various distances frsource with two sets of typical parametersN andA. Figures4 and 5 illustrate the pressure waveform with parameterN51.204, A50.007 32 ~region II!, and N524.083, A50.146 41~region III!. Their shocks form ats50.588 ands50.0489, respectively. Clearly, the waveforms are very dferent. The pressure waveforms in region II are steep

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FIG. 7. The spatial distribution ofmaximum slopeSm . ~a! N524.083,A50.146 41 in region III. ~b!N53.8533,A50.023 425 in region II.

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rical-k’s

vellyeyseeanlike

narrow~Fig. 4! and its shock formation distance is similarthat of a spherical wave; we therefore name this ospherical-wave-like shock. Conversely, the waveform ingion III is more like that of a plane wave~Fig. 5! just as itsshock formation distance approaches that of a plane wwe name it plane-wave-like shock. Actually, we find that ttwo kinds of shock can coexist in the piston field justmentioned in Ref. 23. Figure 6 illustrates the growing pcess of spherical-wave-like shock and plane-wave-like shin piston field with parametersN56.984 andA50.0425. It is



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seen that the plane-wave-like shock forms earlier, andprofile is more like the shock’s profile shown in Fig. 5. Aftethe sound beam propagates a certain distance, sphewave-like shock appears whose profile is just like the shocprofile shown in Fig. 4. Further propagation of the waleads the two shocks to move towards, collide, and finaform a sawtooth wave. Before the two shocks collide, thcoexist in the sound beam within a long distance. We canthat although spherical-wave-like shock forms later, it cfinally grow steeper and narrower than the plane-wave-



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shock since edge wave comes to the axis in a differphase.23 Therefore, in the near field, the maximum slopethe slope of the plane-wave-like shock, but then, withdevelopment of the spherical-wave-like shock, its slope wexceed the plane-wave-like shock’s and the maximum slis that of the spherical-wave-like shock in the area far frsource.

According to the above knowledge of plane-wave-liand spherical-wave-like shocks, we can explain the curvFig. 3 as follows: In region III (r 0.6.6l D) , because theplane-wave-like shock forms early, its slope exceedsthreshold e first; therefore, with the evaluation of oumethod, the shock formation distance is the distance whplane-wave-like shock forms, which agrees with a corsponding plane wave’s shock formation distance. Howewith the decrease ofr 0 , the influence of diffraction will be-come evident and the plane-wave-like shock becomes wand even in region II (0.723l D,r 0,6.6l D), the slope ofplane-wave-like shock cannot exceed the threshold at allthe slope of spherical-wave-like shock can still exceedthreshold. So, the shock formation distance in this regiothe distance where spherical-wave-like shock forms, whicsimilar to a spherical wave’s shock formation distance. Bcause the shock formation distances in regions II andactually represent the shock formation distance of two kiof shock, there is a break point in the curve of shock formtion distance between regions II and III. Finally, in region(r 0,0.723l D), the influence of diffraction is so strong thaboth their slopes cannot exceed the threshold; so in thisgion, the shock formation cannot happen.

Finally, to present the shock formation distance at vaous radii, two-dimensional spatial distributions of the mamum slope are shown in Fig. 7. It can be seen from Fig. 7~a!(N524.083,A50.146 41, in region III! that the shock for-mation distances of various radii are almost identical, jlike the plane wave. However, Fig. 7~b! (N53.8533, A50.023 425, in region II! shows in some regions the maxmum slope at off-axis is larger than that at the axis. It aproximately has a spherical distribution near the axis, justhe sound field is radiated from a spherical source. Thdistributions of maximum slope confirm again that the plawave-like shock is similar to a plane wave shock, butspherical-wave-like shock is similar to a spherical washock.

IV. CONCLUSION

In summary, this paper evaluates the shock formatdistance of a piston field and concentration is primarilythe diffraction. All simulation results show that the diffration plays a major role in the shock formation. When tdiffraction is small~about withinr 0.6.6l D), the shock for-mation distance of a piston field approaches that of a plwave; when the diffraction is large~about within 0.723l D

,r 0,6.6l D), the influence of the Rayleigh distance on tshock formation distance is just like that of the spherisource radius on the shock formation distance of a sphefield. Formula~7! is presented to fit this relation. Furthewhen the diffraction is large enough~about within r 0

,0.723l D), the shock cannot even come into being at



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Therefore, the shock formation distance of a piston field cbe estimated effectively by that of its corresponding plawave only when the influence of the diffraction is mucsmaller than that of the nonlinear intensity.

For giving a reasonable explanation of these phenoena, we study and compare the properties of the shocsound beam from three aspects, the shock formationtance, the pressure waveform, and the spatial distributiothe Sm value. We find that two kinds of shock coexist in thpiston field. One is named plane-wave-like shock, for itsimilar to the shock of a plane wave. Another is namspherical-wave-like shock, for it is more like the shock ofspherical wave. Two shocks compete in the piston fieWhen the plane-wave-like shock is stronger thanspherical-wave-like, the shock formation of the piston fieldjust like a plane wave. Conversely, it is more like a spheriwave. Although our discussion is based on the piston fieldwater, the discussion and conclusion can also be similmade on the other bounded beam with finite amplitude.cause the proposed method of evaluating the shock fortion distance considers fully the nonlinear distortion of twaveform, the changes in the wave amplitude, and so ois also suitable for the other field, focused or unfocused.

ACKNOWLEDGMENT

This work was supported by NSF of China~Grant Nos.10074035 and 19834040!.

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The shock formation distance in a bounded sound beam of finite amplitude

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