IS S N 1063-7710, Acoustical Physics, 2011, Vol. 57, No. 2, pp. 127—135. © Pleiades Publishing, Ltd., 2011.

____________________________________ NONLINEAR_________________________________ACOUSTICS

Finite Amplitude Pressure Field of Elliptical and Rhomboid Transducers in Three Dimensions1

O. A. Kaya", D. Kaleci", and A. Sahin*a Inonu University, Faculty o f Education, Department o f Computer and Education Technology, 44280 Malatya, Turkey

e-mail: oakaya@inonu.edu.trb Inonu University, Science and Art Faculty, Department o f Physics, 44280 Malatya, Turkey

R e c e iv e d N o v e m b e r 2 3 , 2 0 0 9

Abstract—Design of different type of transducers to enhance image quality by forming narrow beams at the principals of nonlinear acoustics is considered in the paper. Thus, the nonlinear pressure fields of elliptical and rhomboid transducers were simulated in three dimensions. The simulation method presented in this study is based on Aanonsen’s model for circular sources, and closely follows the model that recently explored for the nonlinear wave propagation due to square and rectangular sources in three dimensions [Kaya et al. “ Pressure field of rectangular transducers at finite amplitude in three dimensions,” Ultrasound in Med. Biol., vol. 32, no. 2, pp. 271—280, 2006]. It is assumed that elliptical and rhomboid sources are plane sources, and driven at 2.25 M H z fundamental frequency. Typical results of nonlinear acoustical pressure field simulation are presented there in three dimensions for elliptical and rhomboid sources and compared with the results for rectangular source. The similarities and differences between the nonlinear pressure field of rectangular, ellip­tical and rhomboid sources are discussed. The numerical results show that diffraction effects and acoustical beam cross section depend on the source geometry a lot. It is noticeable that the nonlinear pressure field of a rectangular source has a broader beam profile than elliptical and rhomboid source.D O I: 10.1134/S1063771011020217

IN T R O D U C T IO N

N o n lin e a r acoustics has ap p lica tions in m an y areas su ch as in d u stria l an d m ed ica l sectors. P ro p ag a tio n o f h ig h -in ten s ity u ltra so u n d (H IU S ) fro m an acoustic sou rce is a n o n lin e a r process. T h e eq u a tio n som etim es n am ed K hokhlov—Z abolo tskaya (K Z ) [1] an d m ostly n am ed K hokhlov—Z abolo tskaya—K uznetsov (K Z K ) [2] in th e lite ra tu re provides a re liab le descrip tio n o f n o n lin e a r acoustic fie ld , so th e sim u la tio n p resen ted in th is p a p e r was based o n th e 3 D K Z K eq u a tio n in th e frequency d om ain . We cou ld reco m m en d w ith this resp ec t a re c e n t p a p e r by R u denko [3] fo r th e righ t way o f nam in g th a t eq u a tio n an d fo rty years o f h isto rica l progress.

T h e geom etry o f n o n lin e a r acoustic sources o f im p o rtan ce in m ed ica l im aging technology, so th e l in ­ea r an d n o n lin e a r wave p ro p ag a tio n from c ircu la r acoustic sources have b e e n s tud ied in details by A an - o n sen [4], B aker [5], B aker e t al. [6], H am ilto n e t al. [7] fo r c o n tin u o u s wave an d p lan e sources. T h e reason fo r th e extensive use o f c o n tin u o u s wave p ro p ag a tio n in n u m erica l m odels is because it is n o t as d ifficu lt as to m o d e l th e pu lse wave p ro p ag a tio n . T h e p h ase re la ­tions are also n o t as co m p lica ted as pu lsed case. A n o th e r usefu l re sea rch o n m odeling fields o f c ircu la r p lan e sources in frequency do m ain pub lished by V.A. K hokhlova e t al. [8]. M an y o f these stud ies are

1 T h e a r t i c le is p u b l i s h e d in t h e o r ig in a l .

also capab le o f p red ic tin g n o n lin e a r evaluation o f shock fo rm a tio n an d o th e r effects. As th e exact so lu ­tions fo r th e K Z K eq u a tio n have n o t b een kn o w n yet, extensive n u m erica l trea tm en ts have b een used fo r th a t equation .

N o n lin e a r s im u la tions fo r rec tan g u la r sources are n o t as n u m ero u s as c ircu la r sources in th e lite ra tu re . T his is due to th e fac t th a t a c ircu la r sou rce has sym ­m etry w ith resp ec t to th e acoustic axis, u n like a re c t­an g u la r source. T h e first ca lcu la tio n on th e acoustic field o f a rec tan g u la r source was p resen ted by F re e d ­m a n [9, 10]. H e th eo re tica lly exam ined th e am p litu d e an d p h ase o f th e p ressu re field due to a rec tan g u la r p is ­to n v ibrating in an in fin ite rigid baffle fo r ranges dow n to th e o rd e r o f th e p is to n apertu re . F reed m an also co m p ared th e n e a r field o f a c ircu la r p is to n w ith th e n e a r field o f a rec tan g u la r tran sd u ce r an d revealed c e r­ta in in te resting differences betw een th e ir p ressure fields. M o re deta iled com parisons betw een th e fields o f circular, square an d rec tan g u la r transducers w ere m ad e by M arin i an d R ivenez [11] w hose results fo r th e pressure an d in ten sity o f th e u ltraso n ic fields p ro d u ced by rec tan g u la r transducers d em o n stra ted th e genera l charac teristics o f th e p ressu re fields o f various sized rec tan g u la r transducers. However, th ey d id n o t show m u c h o f th e com plex s tru c tu re o f th e pressure field o f rec tan g u la r transducers, especially close to th e source. A m o re gen era l fo rm was p resen ted by Szabo [12] fo r ca lcu la ting th e d iffrac tion field o f a rb itra ry source

127

128 KAYA et al.

ФФ,max

Ax Аф

b.

X max-max

'Act

CT

Fig. 1. The transducer geometry and coordinate system used in the numerical calculation for an elliptical source. The individual field coordinates are indicated by x, Ф, a and the extents of the field calculation in the radial directions (across the acoustic axis), Xmax and <pmax. The extent of the computed field in the axial direction is taken as a max.

fu n c tio n s rad ia ting in iso trop ic an d an iso tro p ic m ed ia . T h is m e th o d is su itab le fo r rap id n u m erica l ca lcu la ­tions fo r th e lin ea r case. S ah in [13] was also m odified th is m e th o d to co m p are th eo re tica l results w ith exper­im en ta l m easu rem en ts. W eyns [14] exam ined th e effect o f sh o rt pulses o n th e n e a r field o f p la n a r c irc u ­lar, square an d a n n u la r transducers. H e also d iscussed th e effect o f tran sd u ce r shapes o n th e b e a m pa tte rn s , co m p arin g th e rad ia tio n fields o f circular, a n n u la r and square rad iato rs.

U ltra so n ic fields from trian g u la r ap ertu res w ere also m o d e led by Jen sen [15] fo r th e lin e a r case using th e im pulse response m eth o d . In th is paper, Jen sen u n d e rlin ed th e im p o rtan ce o f trian g u la r sou rce shape fo r m odeling com plex th ree d im en sio n a l fields. U nfortunate ly , th is w ork is lim ited on ly fo r th e lin ea r case b u t o f im p o rtan ce as th e first tim e it was used fo r trian g u la r sou rce geom etry, in stead o f u su a l c ircu la r an d rec tan g u la r geom etry. B oukuaz an d Jong [16] u n d e rlin ed th e im p o rtan ce o f tran sd u ce r geom etries an d n arro w beam s fo r u ltra so u n d im age quality, an d th e n eed to u n d e rs tan d th e physica l p rocesses involved in th e p ro p ag a tio n o f fin ite am p litu d e so u n d beam s an d issues w h en redesign ing th e transducers. T h ey also m e n tio n e d th e im p o rtan ce o f th e deve lopm en t o f h a r ­m o n ic frequencies due to th e tim e w aveform d is to r­tio n in tim e dom ain . T im e d o m ain s im u la tio n o f n o n ­lin e a r acoustic beam s gen era ted by rec tan g u la r p istons was p resen ted by Yang an d C leveland [17] underly ing th e im p o rtan ce o f rec tan g u la r transducers. Recently, th e n o n lin e a r field g en era ted by rec tan g u la r focused tran sd u ce r driven by pulsed wave was investigated by

K hokhlova e t al. [18]. M o st recen tly th e in flu en ce o f n o n lin e a r an d d iffrac tion effects o n am p lifica tio n fac ­tors o f focused u ltra so u n d system was investigated by B essonova e t al. [19], using a n u m erica l m o d e l fo r th e K Z K eq ua tion . T h e p rob lem s o f n o n lin e a r acoustics w h ich seem to be th e m o st im p o rta n t an d in teresting have b een review ed by O strovsky an d R udenko [20]. T h e sh o ck fo rm a tio n in focused beam s p ro d u ced by spherica l hyd roacoustic tran sd u cers w ith d ifferen t apertu res an d o p era ting frequency o f 3 M H z is s tud ied experim en ta lly by P.N. V ’yug in e t al. [21]. M any o f these papers show th a t th e sou rce geo m etry has im p o r­tan ce in n o n lin e a r m ed ica l im aging an d th is case m otivates th e au th o rs to investigate th e n o n lin e a r field g en era ted by various types o f g eom etrica l sources. T h e re a re lim ited researches in th e lite ra tu re ab o u t e llip tical an d rh o m b o id acoustic sources fo r b o th l in ­ear an d n o n lin e a r cases. T h e reaso n fo r choosing th e e llip tical sou rce is th a t e llip tical source is a s im ila r v er­s ion o f c ircu la r sources, an d rh o m b o id source is a s im ­ilar version o f square an d rec tan g u la r source. T h e re ­fore, it was a im ed to m odify th e n u m erica l m o d e l fo r d ifferen t b o u n d a ry co n d itio n s recen tly p resen ted by K aya e t al. [22] fo r e llip tic an d rh o m b o id sources. T h e n o n lin e a r pressure field s truc tu res o f e llip tical an d rh o m b o id sources in th ree d im ensions an d th e g row th o f th e h a rm o n ics process due to th e p ropag a tio n o f fin ite am p litu d e sou n d beam s w ere also s tud ied in this work.

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F IN IT E A M PLITU D E PRESSU RE FIELD O F ELLIPTICAL 129

Ф

Ax Лф

r b

x max-max

Act

CT

Fig. 2. The transducer geometry and coordinate system used in the numerical calculation for a rhomboid source. The half polygon diagonals are defined as a and b. The individual field coordinates are indicated by x, Ф, a and the extents of the field calculation in the radial directions (across the acoustic axis), xmax and <pmax. The extent of the computed field in the axial direction is taken as a max.

T H E O R E T IC A L M O D E L

T h eo re tica l m o d e l was app lied fo r two d ifferen t sou rce geom etries, e llip tical an d rh o m b o id sources. B o th sources are assum ed to be cen te red a t th e orig in o f th e C artesian co o rd in a tes an d lie in th e a = 0 p lan e as show n a t Figs. 1 an d 2. T h e g eom etrica l pa ram ete rs fo r ellipsoid sou rce are defined as follows. T h e e c c e n ­tric ity o f th e ellipse e, satisfies follow ing co n d itions; 0 > e > 1 ^ 0 > e2 > 1.

As th e ellip tica l source is cen te red a t th e orig in , e

ca n be w ritten as e = V T - ^ V ) w here a is th e m a jo r rad ius o f th e ellipse, defined o n th e ф axis an d b is th e m in o r rad ius o f th e ellipse, defined o n th e x axis. A n a rb itra ry p o in t a t th e ellip tic source p lan e is defined as (x, y , 0) an d r is th e d istance from orig in to th e p o in t (x, y , 0). 0 is th e angle betw een r an d x axis. We c a n

rew rite r as r = b/*Jl - e2c o s( 0 ) .

T h e g eom etrica l p aram eters fo r rh o m b o id sources a re show n in Fig. 2 an d defined as follows. I t is assum ed th a t th e h a lf polygon diagonals a an d b are p e rp en d icu la r an d satisfy th e co n d itio n a 2 + b2 = c2. It is also assum ed th a t th e su rface area (A) an d th e aspect ra tios o f th e ellip tic an d rh o m b o id tran sd u cers are equal to each o ther. In o th e r w ords, b o th transducers p ro p ag a te th e sam e a m o u n t o f acoustic energy in to th e w a te r m ed iu m . S ource p lanes a re tak en as rigid baffles, tran sd u cers are driven a t an g u la r frequency w ( th e fu n ­d am en ta l frequency), rad ia ting a n o n lin e a r b e a m th a t is sym m etrica l ab o u t th e n o rm alized a = z/r 0 axis in to th e h a lf space a > 0, w h ere z is th e d istance from th e

sources o n th e acoustic axis an d r0 is th e R ayleigh d is­tan ce . T h e in itia l b o u n d a ry co n d itio n a t a = 0 is th a t th e p a rtic le velocity h ad c o n s ta n t u n it am p litu d e u0

over th e e llip tical an d rh o m b o id p iston faces an d v an ­ished beyond th e b o u n d a ry o f b o th sources. Similarly, th e in itia l c o n d itio n was u n it am p litu d e an d zero p h ase fo r th e fu n d am en ta l an d th e m agn itudes o f th e all h a rm o n ics re ta in ed in th e ca lcu la tio n w ere set to zero . I t was also co nsidered th a t th e n o n lin e a r p ro p a ­g a tio n occurs in a th e rm o viscous m ed iu m , su ch as w ater. U n d e r these consid era tio n s, th e K Z K eq u a tio n [1, 2] th a t consis ten tly acco u n ts fo r nonlinearity , a tte n u a tio n an d d iffrac tion was used an d it is fo rm is as follows;

d2Wд а д т

а Г0

d V + r - e -д т 3 2 lddx2

( W ) + 1 v 2lw , (1)

w here W = u/ u0 is th e n o rm alized partic le velocity, u is th e fluid velocity in th e d irec tio n o f p ropaga tion , u0 is th e cha rac teris tic velocity am p litu d e , a = z/ r0 is th e n o rm alized axial d istance o n th e acoustic axis, а is th e ab so rp tio n coeffic ien t, т = (wt — kz) is th e re ta rd ed tim e , ld = 1 /p sk is th e acoustic shock d istance , p is th e n o n lin ea rity param eter, s is th e acoustic m a c h n u m -

2 д 2ber, k is th e acoustic wave n u m b e r an d V , = — +д x 2

д 2 is th e transverse L ap lace o p e ra to r defined in C ar-д ytesian coo rd ina tes.

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130 KAYA et al.

Pressure, dB re 1 MPa

Fig. 3. Comparison of the first three harmonics of the axial pressure field of an elliptical and a rhomboid source. The solid line shows the calculated pressure values for elliptical source and the lines with diamond markers show the calculated pressure values for a rhomboid source. Top curves correspond to the first harmonic, the lower ones—to the second, and the lowest—to the third harmonic of initial signal.

T h eo re tica l m odels fo r th e so lu tion o f th e K Z K e q u a tio n can on ly be achieved by nu m erica l m ethods. I t assum es a so lu tio n in th e fo rm o f a F o u rie r series w ith am plitudes an d phases th a t are fu n c tio n s o f th e spatia l co o rd in a tes x , y, an d a :

W(x, y, a, t )

^ (gp - ngp + hp - nhp)p = n + 1

dhn 2 , 1 ( d 2 d 2— = - n a г0 hn + — l — + — J gn4 n l d%2 дф

ад

ад= X { Wn(x,y,a) s in [nt + ^ n (x,y, a ) ] }

n = 1 (2)ад

= X {gn(x, y, a) s in ( nt ) + hn(x, y, a ) c o s (nt )} ,n = 1

w here n is th e h a rm o n ic num ber, Wn is th e velocity am p litu d e in th e F o u rie r expansion o f W, gn an d hn are F o u rie r coeffic ien ts fo r th e n th h arm o n ics an d у n is th e phase o f th e n th harm o n ics . By substitu ting Eq. (2) in to Eq. (1) an d separa ting th e te rm s in cos(nT) and sin(nT), an in fin ite set o f co up led partia l d ifferen tia l eq u a tions is o b ta in ed for th e co m p o n en ts gn an d hn:

dgn

da= - n a rogn + _ L ( _d2 + * -1

Kd% дф 2 J

+ Eo2 1 d

n - 1

2 (gkgn - к — hkhn - к)■ к = 1

(3)

n - 1

2 X (hkhn - к - gkhn - к) (4)к = 1

ад “+ X (hP - ngP + gP - nhP) ,

p = n + 1 -

w here x = x/a an d ф = y/a are n o rm alized spatial co o rd in a tes an d a is th e n o rm alized axial co o rd in a te .

T h e above eq u a tio n s are coup led d ifferen tia l eq u a ­tio n s an d can be in teg ra ted stepw ise by an im p lic it backw ard fin ite d ifference fo rm ula (IB D F ) [23, 24]. T h e a lgo rith m to solve th e above coup led d ifferen tia l eq u a tions follows sim ilar lines to th a t o f K aya e t al. [22]. T h e m a in d ifference is th e ad o p tio n o f e llip tical an d rh o m b o id source pa ram ete rs in to th e equa tions, con seq u en tly in to th e so lu tions.

T h e ap p ro x im ated eq u a tio n s for e llip tical and rh o m b o id sources were ca lcu la ted using M atlab to analyze wave p ro p ag a tio n in th e frequency dom ain . T h e M atlab p ro g ra m th e n c a lc u la te d th e ch an g e in e a c h h a rm o n ic c o m p o n e n t a t every g rid p o in t in

+ n o

2--

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FIN IT E A M PLITU D E PRESSURE FIELD OF ELLIPTICAL 131

3 -D with absorption and nonlinearity simultaneously accounted for each step. The step sizes A x , Лф and A ct were kept as small as possible and varied if necessary, depending on the pressure levels and the degree of har­monic generation. Numerical values obtained for each (A x, Лф, A ct) step sizes at each (i, j , k) space coordi­nate were stored in a matrix form. The results calcu­lated at a field point in the defined volume were stored in a 3 -D matrix. The dimension of matrix was varied, depending on the step size used. Typically, the matrix was of dimensions (100:100:100); these dimensions are enough for many calculations for step sizes A ct = 5 x 10-4, A x = Лф = 0.05. The run time of the program was also varied, depending on the step sizes used, and the maximum harmonic number m retained in the cal­culation. The maximum harmonic number m directly affected the energy transform mechanism from funda­mental to higher harmonics. These effects are dis­cussed in the following section.

R E S U L T S A N D D IS C U S S IO NThe following values were used in all the numerical

calculations: the driven fundamental frequency ro0 = 2.25 M H z continuous wave, the speed of sound in water c0 =1486 m/s, the density of water p0 = 1000 kg/m3, the nonlinear parameter в = 3.5 for water, the attenuation coefficient a = 0.025/2 Np/m and the maximum harmonic number retained in the calcula­tion, m = 10. The typical run parameters of the Matlab program were chosen as A ct = 5 x 10-4, A x = Лф =° .° 5, x max фmax 3.° , and CTmax ° A .

The finite amplitude pressure fields of plane elliptic transducers were calculated for an elliptical source of dimensions as follows; major radius a = 0.0069 m, minor radius b = 0.0046 m, with e = 0.7454. The geo­metrical parameters for the rectangular source were as follows; the vertical size of the rectangle a = 0.0061 m, horizontal size of the rectangle b = 0.0041 m. For the rhomboid source, the half polygon diagonals were cho­sen as a = 0.0086 m and b = 0.0057 m. The aspect ratios were ar = 1.5, the area of the sources A = 1 x 10-4 m2 and it was assumed that the acoustic pressure at the source planes P0 = 0.23 MPa for three kinds of sources.

Figure 3 shows a typical nonlinear acoustic pres­sure distribution of the fundamental, second and third harmonics on the acoustic axis respectively for an elliptical source of the above parameters, which was initially driven at 0.23 MPa. The higher harmonics were also generated but not included in this study as their levels were very low. A t ranges nearer to the trans­ducers, it can be seen from Fig. 3 that there is not much difference between pressure levels. The nonlin­ear pressure field of both sources exhibits some spatial oscillations close to the source, before passing through a final axial maximum (for the fundamental) (—19.92 dB) at about normalized axial range ct = 0.095 for the rhomboid source, and (—19.51 dB) at about ct = 0.21 for the elliptical source then falling off steady

X = x/aFig. 4. Radial pressure fields of (a) the first harmonic, (b) the second, and (c) the third harmonic calculated at a = 0.4 plane. The vertical axes show the normalized pres­sure amplitude and the horizontal axes show the normal­ized across axis coordinate.

at longer ranges. The degree of the variation is almost similar. However, the place of the maxima and minima show a difference. Pressure levels start to separate for all the harmonics beyond the normalized axial range at about ct = 0.04. For example, the first minimum of the fundamental component (—35.59 dB) for the elliptical source is observed at ct = 0.07 axial range, however, the

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132 KAYA et al.

Rectangular Elliptical Rhomboid

00.050.100.150.200.250.300.350.400.45

0.050

0.045

0.040

0.035

0.030

0.025

0.020

0.015

0.010

0.005

0-2 0 2

00.050.100.150.200.250.300.350.400.45

0.040

0.035

0.030

0.025

0.020

0.015

0.010

0.005

0-2 0 2 X

00.050.100.150.200.250.300.350.400.45

0.016

0.014

0.012

0.010

0.008

0.006

0.004

0.002

0-2 0 2

Fig. 5. Harmonic beam patterns of the first three harmonics in terms of contour plots. The left, the middle, and the right vertical column show beam patterns for a rectangular, elliptical, and rhomboid source. The top, the middle, and the bottom horizontal columns show the first, the second, and the third harmonics, respectively.

first m in im u m o f th e fu n d am en ta l co m p o n e n t (—39.05 dB ) o ccu rred a t a = 0.05 axial range fo r th e rh o m b o id so u rce . T h e reaso n is th a t th e edge wave c o n tr ib u tio n to th e d iffracted field is n o t th e sam e as th e so u rce geom etries.

F igure 4 illustrates th e transverse pressure field d is­trib u tio n s ca lcu la ted o n th e a = 0.4 p lan e fo r th e fu n ­d am en ta l (F ig. 4 a ) , th e seco n d h a rm o n ic (Fig. 4 b ), a n d th e th ird h a rm o n ic (F ig . 4c) fo r re c ta n g u la r , e llip tic a l a n d rh o m b o id so u rces respectively . T h is

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F IN IT E A M PLITU D E PRESSU RE FIELD O F ELLIPTICAL 133

a = 0.4 plane is chosen because the first three har­monics have grown enough at this plane. Notice that the rhomboid source has the narrowest beam profile, the elliptical source has broader beam than the rhom­boid source, and finally the rectangular source has the broadest beam profile. All frequency components show a clear central maximum and a number of side lobes developed on each side of the across axis. These fundamental side lobes are well-defined for rectangu­lar case, but for elliptical and rhomboid sources, there are no well defined side lobes on both side of the x axis. The second and third harmonics generated by the nonlinear acoustic beam propagated from the rectan­gular, elliptical and rhomboid sources are also well formed but have no clear side lobes.

Pressure field patterns are shown in Fig. 5 in terms of contour plots for elliptical, rhomboid and rectangu­lar sources. Figure 5 shows that diffraction effect is more dominant in the near field of these three type transducers and diffraction effect is reduced as expected, after the final maximum of fundamental for elliptical, rhomboid and rectangular sources. This sit­uation is the same for the first three harmonics. Notice that the far field region firstly starts to perform in the field of rhomboid source, secondly develops in the field of elliptical source, and finally develops in the field of rectangular source. Figure 5 also provides a comparison for the pressure field of rectangular (Kaya et al. [22]), elliptical and rhomboid transducers, which are of the same source area and aspect ratios, driven at the same pressure and frequency. It is noticeable that the nonlinear pressure field of a rectangular source has a broader beam profile than an elliptical and a rhom­boid source. It was seen once again that a rhomboid source generates the narrowest beam profile. N o t only the parabolic approximation used in the calculations is responsible from the narrow beam profiles, but also the source geometries and diffraction mechanism of each source are also responsible from the propagating beam profiles.

The harmonic content of the beam profiles can also be seen clearly at Figs. 6 and 7 which show a typical 3 -D nonlinear acoustic pressure distribution of the fundamental (top), second (middle) and third (bot­tom) harmonics respectively, for an elliptical source and a rhomboid source of the same aspect ratios ar = 1.5. It is obvious from the Fig. 6 that the rhomboid source not only has the narrowest beam and a main lobe but also have additional, lower level, side lobes. The contour and 3 -D plots also show that harmonic components have not yet developed near the trans­ducer face for elliptical, rhomboid and rectangular transducers, only the pressure field for the fundamen­tal component is effective. The second and third har­monics built up with increasing range, with various maxima and minima mirroring those in fundamental. For example, the second harmonic component starts to build up at about normalized axial range a = 0.025 for the elliptical source, however the same frequency component is observed earlier than this range, at about

Ф

210

-1-2

y/a

First harmonic 1.61.41.21.00.80.60.40.20

Ф

210

-1-2

У/a

■20

Second harmonic 0.180.160.140.120.100.080.060.040.020

Ф

210

-1-2

Third harmonic

0.0250.0200.0150.0100.0050

Fig. 6. Three dimensional pressure field of an elliptical source.

a = 0.015, for the rhomboid source. The similar case applies for the third harmonic components as well. Again, the third harmonic appears earlier for the rhomboid source (at about a = 0.075) than the ellipti-

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134 KAYA et al.

■20

First harmonicФ = У/a

1.61.41.21.00.80.60.40.20

Ф = У/a

Second harmonic 0.100.090.080.070.060.050.040.030.020.010

Ф = У/a

Third harmonic 0.0160.0140.0120.0100.0080.0060.0040.0020

Fig. 7. Three dimensional pressure field of a rhomboid source.

ca l source, nevertheless th e th ird h a rm o n ic g en era tio n starts a t ab o u t a = 0.09 range fo r th e ellip tica l source. T h is is a n expected case as th e d iffrac tion m ech an ism o fv a rio u s g eom etrica l sources an d th e d iffrac tion p ro ­

cess of th e fu n d am en ta l, second an d th ird h a rm o n ics a re d ifferen t. T h is h a rm o n ic g en e ra tio n h isto ry also gives a n id ea ab o u t th e w aveform d is to rtio n in th e tim e d o m ain an d energy tran sfe r m ech an ism from fu n d a ­m en ta l to h arm o n ics . By referring th e h is to ry o f th e h a rm o n ic gen era tio n , it c a n be said th a t th e energy tran sfe r from fu n d am en ta l to h a rm o n ics starts a t sh o rte r ranges fo r rh o m b o id sou rce th a n ellip tical source. N o te th a t th e energy tran sfe r m e c h a n ism from fu n d am en ta l to h a rm o n ics becom es d o m in a n t after th e last m ax im u m o f fu n d am en ta l. T his is tru e fo r b o th e llip tical an d rh o m b o id sources.

T h e fin ite d ifference m o d e l is costly in te rm s o f co m p u te r m em o ry an d ru n tim e. Ideally a n in fin ite n u m b e r o f h a rm o n ics shou ld be in c lu d ed in th e so lu ­tio n ; how ever a t relatively low drive levels on ly a sm all a m o u n t o f energy is transfe rred to h ig h er h a rm o n ics so th a t th e so lu tio n was tru n c a te d in o rd e r to red u ce th e ru n tim e. T h e n u m b e r o f h a rm o n ics in th e ca lcu la ­tions varied w ith in th e p ro g ram m o n ito rin g th e last few harm o n ics; i f th ey exceeded a specified lim it th e n th e to ta l n u m b e r o f h a rm o n ics was increased . T h e p rog ram was ru n w ith 10 h a rm o n ics th a t typ ically re ta in ed in th e n u m erica l ca lcu la tions.

C O N C L U S IO N

Pressure fields o f e llip tica l an d rh o m b o id aco u sti­ca l sources o f th e sam e surface areas an d aspec t ratios, driven by th e sam e frequency, w ere s im u la ted an d co m p ared here . T h e n u m erica l results show th e 2 -D an d 3 -D struc tu res an d typ ica l charac teris tic aspects o f n o n lin e a r pressu re fields o f e llip tical an d rh o m b o id transducers. I t was found th a t th e b eam profiles m u c h d ep en d o n th e sou rce geom etry. T h is w ork c a n be a beg inn ing o f redesign ing d ifferen t geom etrica l tra n s ­ducers fo r e n h an ced im age q u a lity efforts fo r m ed ica l u ltra so u n d system s.

It is ev iden t th a t n u m erica l results revealed the details o f th e n o n lin e a r p ressu re field o f th e ellip tical an d th e rh o m b o id sources b u t needs to be tested experim entally . T h e m a in d isadvantage o f th e n u m e r­ical m o d e l is th a t th e source geom etry has to be ch o sen sym m etrica l a ro u n d th e b o th side o f th e vertical ф axis, o therw ise th e difficulty arises fo r g enera ting th e in itia l m atrix e lem ents fo r in itia l b o u n d a ry cond itio n s.

As a fu tu re w ork, th is study we are go ing to ex tend to th e cases o f focusing sources w ith d ifferen t focal gains an d to m o re com p lica te geom etries su ch as re c t­an g u la r arrays w h ich a re generally u sed in real m ed ica l u ltra so u n d system . T h e au th o rs th o u g h t th a t it is w orth studying o n th e im age q u a lity by using rh o m b o id type sources, as it genera tes lo n g er an d narrow er b eam p ro ­file (see Fig. 5) in co m p ariso n w ith e llip tical an d re c t­an g u la r sources.

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F IN IT E A M PLITU D E PRESSU RE FIELD O F ELLIPTICAL 135

A C K N O W L E D G M E N T S

T h e au th o rs th a n k to T h e U niversity o f In o n u , R esearch P ro jec t U n it (B A P), M alatya, Turkey fo r th e financ ia l su p p o rt during th e course o f th is study.

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