Modelization of Acoustic Waves Radiation from Sources of Complex Geometry Aperture

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1 coustcs 8 ars odelzaton of coustc Waves Radaton from Sources of Complex Geometry perture R. Serhane and T. Boutedjrt Unversté des Scences et de la Technologe Houar Boumedene; Faculté de hysque, B, El-la, DZ-6 lger, lgera raf serhane@hotmal.com 59

2 coustcs 8 ars bstract - The Raylegh ntegral gvng the mpulse response for the acoustc velocty potental cannot always be determned analytcally for all types of transducer apertures. The shape of the transducer surface and the spatal dstrbuton of the exctaton on ts surface can complcate the calculaton. Ths maes usng numercal methods ndspensable. One of these methods conssts n dscretzng the aperture surface n polygonal shape elements. Our wor maes use of both methods of Jensen and Faure. In the former, the transducer surface s subdvded n trangles. The latter one uses addtonal vrtual trangles one sde of whch s that of the physcal element and ts summt s the feld pont projecton on the plane contanng the consdered physcal element. ddtonally, a rotaton of the vrtual trangle around the feld pont projecton s performed. In ths wor, the orentatons of those vrtual trangles are consdered accordng to Jensen method and ther contrbutons to the mpulse response are calculated accordng to the two stuatons descrbed by Faure. The combnaton of the two methods s, then, generalzed to the case of non planar complex surfaces such as concave phased arrays. Introducton The transmttng surface of an ultrasonc transducer can be consdered as a set of pont sources unormly radatng n all drectons. The radated elementary wavelets propagate nto the surroundng medum. t a pont of ths medum, the sound pressure feld results from the superposton of the all these elementary contrbutons. ressure radated by a mono-element planar transducer The determnaton of the acoustc potental Φ (, s based on the calculaton of the Raylegh ntegral, whch s the mathematcal formulaton of the Huyghens-Fresnel prncple [,]; that s : ( t R / c) δ Φ (, = vn (, ds, ( πr S ) wth R =. δ ( s the Drac mpulse, the temporal convoluton product, c propagaton velocty of the ultrasonc wave n the consdered medum and v (, t n ) the partcle vbraton velocty normal to the radatng surface. In the pston mode, ths velocty has a unorm dstrbuton on the surface ( S ). In ths case, the vbraton velocty depends only upon tme ( vn (, = vn (. ): The spatal mpulse response for the acoustc potental Φ (, s defned by: ( t R/ c) () δ Φ (, = ds. () ( s ) π R Ths allows wrtng the acoustc potental at pont, whch s referred by r, by usng a temporal convoluton operaton; that s : Φ (, = v r ( Φ (,. () n s the acoustc pressure p (, and potental are related by : r Φ (, p(, = ρ, (4) t where ρ s the equlbrum densty of the propagatng medum. ethod of dscretzng the source surface The trangles method (Fg. ) has been proposed by two authors. Jensen [4] calculates the response by consderng three geometrc sub-trangles accordng to the feld pont poston, each consttuted by one sde of the consdered physcal trangular element and the opposte summt, whch s the projecton of the feld pont on the plane contanng the element surface (Fg ). To calculate the response of each geometrc sub-trangle, he apples to the latter a rotaton around the projecton of the feld pont so that one of ts rdges s supported by the Ox axs. By usng ths procedure, the response of a sub-element s gven accordng to fve possble stuatons. Faure [5] consders sub-trangles one summt of whch s the projecton of the feld pont. The mpulse response s calculated accordng to two possble stuatons (Fg. ). Our study maes use of both two methods. The orentatons of the sub-trangles are consdered by usng the Jensen method and ther contrbutons to the mpulse responses for the potental are calculated accordng to the two stuatons consdered by Faure (Fg. ). Then the combnaton of the two methods s generalzed to not planar complex surfaces.. Descrpton of the method One should frst dspose of an analytc expresson ( Z = f ( X, Y )), n the (OXYZ) referental, of the geometrc surface of the radatng aperture for whch the mpulse response s calculated. mesh of the (XY ) plane, whch can be regular or not, causes dscrete values at dstance Z. Each set of coordnates ( X, Y, Z ) represents a not of the mesh net, that s pont, whch s the summt of an elementary polygon. In our case, the mesh s trangular, whch corresponds to the smplest polygonal shape. It s consttuted of three neghbourng nots of the mesh (, j, ) = (, B, C) (Fg ). Fg.: Geometry for the dscretzaton method. 594

3 coustcs 8 ars so-called Delaunay trangularzaton functon, whch has been programmed under atlab, furnshes drectly the ndces (, j, ) of the three summts of each trangle lyng on the surface. The spatal dscretzaton nterval s chosen suffcently small so that the trangular element can be consdered as planar. Each trangular element therefore defnes a plane the normal to whch s that of the trangle. s for planar surfaces, the mpulse response for the acoustc potental s proportonal to the angle subtended by the arc of crcle, whch s the ntersecton of the sphere of radus R = ct and whch s centered on the feld pont, wth the consdered element (Fg.). The ntersecton of the sphercal wave front wth the plane supportng ths trangle s a crcle of radus σ ( = c t z and centered n. z s the dstance between the feld pont and ts projecton on the plane of the consdered element. The determnaton of z requres the pror nowledge of the equaton of the plane contanng the consdered element. Ths equaton has the followng form : αx + βy + δ =. (5) Let N(X,Y,Z) an arbtrary pont of the plane contanng the trangular element BC. The coordnates of ponts, B and C very the equaton (6) of the plane. Ths permts relatng the constants α, β, γ and δ to these coordnates by : α = YBZ C Z BYC β = Z B X C X BZC. (6) γ = X BYC YB X C δ = ( αx + βy ) We deduce then the vector n r, whch s normal to the plane of the element; that s: n r ( α, β, γ ). ont ( X, Y, Z ) s the projecton of the feld pont ( X, Y, Z ) on ths plane. and must very r = ξn, where ξ s a proportonalty factor, therefore : X = X ξα Y = Y ξβ. (7) Z = Z ξγ The coordnates of pont very the plane equaton to whch t belongs : αx + βy + δ =. (8) If n addton, we tae account of the equatons system (7), we obtan the proportonalty factor ξ : αx + βy + δ ξ =. (9) α + β + γ By replacng ξ n the precedent system (7), the coordnates of can be deduced. The dstance =z s : z = ( X X ) + ( Y Y ) + ( Z Z ). () Let us consder the actve trangular element BC of ths set (taen alone n Fg. ). To calculate the mpulse response of ths element, we should consder the projecton of the feld pont as the orgn of the new referental (xyz), so that the plane (x,y) be the element plane. Consequently, a change of the orgn and of the axes orentaton s performed. ll the coordnates are then expressed n the new referental (xyz). To determne the mpulse response for the acoustc potental of element BC whch s denoted by m, we should consder the three sub-trangles B, C and BC (Fg.). The response of the element s the sum of three responses, each wth a contrbuton sgn, accordng to the orentaton of ts correspondng sub-trangle; that s : max m φ (, = s h (,. () e = max = and h s the mpulse response of the th trangular sub-element contrbutng to the physcal element response wth the sgn s = ±. For the sgns of the sub-trangles contrbutons, three cases are possble. These are llustrated n Fg.. If the projecton of the feld pont s stuated nsde element BC, then the sgns of the contrbutons are all postve. If not, the observer whch s placed n sees the trangle BC under angle Φ. The nearest sub-trangles formed by the sdes of element BC and contrbute wth a negatve sgn ( B and C (Fg.b) and C (Fg.c)). ll others have postve contrbutons. To mathematcally determne the projecton of the feld pont s stuated on the radatng surface BC or not, t s suffcent to draw a straght lne from that crosses one sde of trangle BC n Q. It should necessarly cross an Fg. : Geometry for the determnaton of the sgn of the sub-elements contrbuton to the mpulse response of the trangular element. 595

4 coustcs 8 ars other sde n R.. The negatve scalar product Q. R sgnes that pont s nsde the element and all other contrbutons are postve (Fg.a). If not, t s outsde the element. In ths second case, ( Q. R ). If n addton Q < R (Fg..b et.c), then the sub-trangle formed by pont and the sde gong through R contrbutes wth a postve sgn. The other sub-trangle, formed by pont and the sde contanng Q, contrbutes wth a negatve sgn. Otherwse ( R < Q), the sgn of the sub-elements contrbutons s nverted.. Sub-element and element responses The determnaton of the mpulse response of a sub-element requres the calculaton of the contrbuton angle θ (,, so that: c h (, = θ (,. () π Let us consder the sub-element B stuated n referental (xyz) whch s relatve to element BC (Fg ). Fg. : Geometry for the determnaton of the sub-element mpulse response. The equaton of the straght lne (B) s gven by [5] : y = m x + y. () The constants m and y are functon of and B coordnates : yb y yb y m = and y = y ( ). x. (4) xb x xb x Let I be the projecton of pont on the straght lne (B). Two cases can be dstngushed: I belongs to segment [B] or not, accordng to the poston of. We defne, then, ρ as the dstance between and the straght lne (B); that s: y ρ =. (5) + m ρ and ρ are the mnmal and the maxmal dstance from or B to respectvely : ρ = mn(, B) = mn( x + y, xb + yb ), (6) ρ = max(, B) = max( x + y, xb + yb ). The dstances from the feld pont to the dferent postons of dscontnutes wth the sub-element sdes are gven by : r = z + ρ, =,..,, wth : ρ =. The correspondng nstants are: τ = r / c. (7) The angles α et α represented n Fg. are gven by : x xb α = tan ( ) tan ( ) (8) y y B and ρ α = cos ( ). (9) ρ th The response of the sub-element ( ) s : cα π c ρ h = ( α.arccos( )) π σ( c ρ ( α α arccos( )) π σ( and c α h = π c ρ ( α + α arccos( )) π σ( τ t τ τ t τ τ t τ I [ B], τ I [ B], τ wth σ ( = c t z. () summaton on the three sub-trangles ( = max ) of the consdered BC element furnshes the mpulse response of that element. Ths mpulse response s represented n Fg. 4.a for dferent postons of the feld pont projecton on the plane of the element. These projectons are llustrated n Fg. 4.b. It should be notced that, every tme the sphercal wave encounter an edge of the trangular element, the spatal mpulse response shows an mportant varaton at the correspondng nstant. The number of these dscontnutes depends on the feld pont poston. The total mpulse response of the transducer s obtaned by summng the mpulse responses of each elements; that s : m φ (, = φ (,, () e m= Fg 4 a) Impulse response for the velocty potental of a trangular element represented n b) for dferent postons of the feld pont on the x axs (y=) and at plane z= mm The effcency of ths method for such geometres depends on ts precson and the calculaton tme of the response. In order to reduce the error rs and to optmze the method, t s frst tested n the case that an analytc expresson of the mpulse response s avalable. e 596

coustcs 8 ars 4 Dscretzaton method appled to a crcular focusng transducer The surface of a crcular focusng transducer s subdvded n a number N of trangular elements (Fg.

Let us consder a crcular focusng transducer of radus a = mm, of focal dstance F = 5 mm and whch s dscretzed n 5584 trangular elements. Fg. 5: esh of a crcular focusng transducer.

5 coustcs 8 ars 4 Dscretzaton method appled to a crcular focusng transducer The surface of a crcular focusng transducer s subdvded n a number N of trangular elements (Fg. 5) so that each element can be consdered as planar and the mesh method prevously descrbed can be appled. Let us consder a crcular focusng transducer of radus a = mm, of focal dstance F = 5 mm and whch s dscretzed n 5584 trangular elements. Fg. 5: esh of a crcular focusng transducer. The spatal mpulse response for the acoustc potental of ths transducer though calculated at the feld pont of coordnates (x=mm, y=, z=mm, relatvely to the focus chosen as orgn) shows an average error of.5 % relatvely to the results gven by the analytc method [6,7] (Fg 6). Ths relatve dference between the mpulse responses obtaned by the two methods s gven by : Δ φ φ (analytc) φ (dscretzed) = φ φ (analytc) Φ /c analytc Φ /c dscretzed N=5584 elements verage relatve error =.5 % Fg.6 : Impulse responses calculated by the two methods for N= 5584 elements at x= mm, z= mm. It can be notced that the relatve error s neglgble n the tme nterval correspondng to the drect wave regme although the transmttng surface s curved. The relatve error s maxmal at the dscontnutes nstants of the mpulse response. 5 Generalzaton of the mesh method to a curved rectangular transducer (cylndrcal) The applcaton of the dscretzaton method to surfaces for whch an analytc expresson of the mpulse response s avalable has permtted testng ts effcency n order to be generalzed to more complex geometry surfaces. curved rectangular transducer of dmensons ( a = 6 mm, b = mm), of curvature radus R = mm (depth e =, 68mm) s represented n Fg. 7. Such transducers are generally used as elements of a lnear Fg. 7 : Curved rectangular transducer of dmensons (a=6mm, b= mm) and curvature radus R= mm, dscretzed n 5 trangular elements. transducers array n order to reduce the focus wdth along (Oy). In the focal plane (z= mm), the response s maxmal for a feld pont the projecton of whch s stuated on the radatng surface ( x mm and y=). These postons correspond to a segment descrbng the focal zone. In the same regon, the pressure feld (Fg 8.a) s composed of a drect wave component ("plane") followed by an edge wave component of lower ampltude. These two contrbutons nterfere because of the exctaton duraton whch s represented n Fg. (8.d). The drect wave component s proportonal to the source acceleraton. Far from the axs (at x = mm) and beng on the segment of the straght lne defnng the focal zone, the edge wave component appears. Its destructve nterference wth the "plane "wave furnshes an ampltude lower than that obtaned on axs ( x = ). t greater dstances from the axs ( Oz ) (x mm), the feld pont s no more n the focal zone and the response for the potental becomes flat. Only the transducer edges contrbute to the pressure feld wth two replcas of nverse polartes. The "plane" wave regme for whch the response stay maxmal and constant s obtaned for ( x < a ). The pressure feld s composed of a "plane" wave whch has the form of the exctaton (Fg 8.b) followed by two edge waves components. In the shadow zone ( x a ), the pressure feld s consttuted of two edge waves components of very low ampltude, whch become more tme separated when x ncreases. c) nearer than the focal plane a) on the focal plane c) farther than the focal plane Fg. 8 : ressure radated by a curved rectangular transducer for dferent postons x of the feld pont : a) n the focal plane (z = mm), b) nearer than focal plane (z=mm), c) farther than the focal plane (z = mm) and d) Exctaton waveform. Farther than the focus, the feld pont receves frst the contrbutons of the nearest transducer edges. If the feld pont has a projecton on the radatng surface ( x 597

6 coustcs 8 ars mm), then, durng a very short tme nterval, a "plane" wave regme appears (Fg 8.c). The "plane" wave component has an nverse polarty relatvely to the exctaton and s preceded by an edge wave of lower ampltude and of the same polarty as the exctaton (Fg.8.c). Out of axs, the response for the acoustc potental has a monotone ncreasng regme then t decreases before t nulles. The greater the radal dstance x, the greater the duraton of the decreasng regme s. In addton, only two edge wave components consttute the pressure feld. 6 Concave lnear array The dscretzaton method prevously descrbed can be generalzed to calculate the feld of a concave array. Fg. 9 : Lnear concave array : N = 6 elements, a=mm, b= mm, d=. mm and R= mm. The transducer represented n Fg. 9 fgure s consttuted of 6 mono-elements of dmensons (a= mm, b= mm) and whch are algned along the (Ox) axs. The dstance between the centers of two successve elements s d=. mm and the curvature radus s R= mm. The mpulse responses for the acoustc potental of the elements for a smultaneous exctaton of the elements are represented n Fg.. feld pont on the axs (x=, y=, z= mm) receves frst the wavelets ssued from the two central elements. The superposton of the responses of two symmetrc elements furnshes a response of double ampltude and the responses of successve elements start at separated tmes. These results are comparable to those gven by [8]. Element poston Fg. : Indvdual mpulse responses for the acoustc potental of a concave array obtaned by the dscretzaton method 7 Effect of the mesh elements number For N 5, the relatve error becomes neglgble and nearly constant (Fg..a). The calculaton tme vares "lnearly" wth N (Fg.b). By lmtng N around 5, the calculaton tme stays reasonable (n the order of 5 s on an Intel Celerone GHz personal computer) and the relatve error s acceptable (n the order of.5%). Fg. : Effect of the number of mesh elements N a) on the average relatve error and, b) on the calculaton tme of the mpulse response by usng the dscretzaton method. 8 Concluson The descrbed method whch uses trangles formed by the edges of the mesh elements can be generalzed to any geometry of the mesh element. It s suffcent that ths geometry s polygonal. In ths case, the summaton of the contrbutons of the trangular sub-elements above defned furnshes the mpulse response of the element consdered (Eq. ), wth max beng the number of sdes consttutng the elementary polygon. The effcency of ths method for such geometres depends upon ts precson and the calculaton tme of the response. To reduce the error rs and optmse the method, the latter has been frst tested n the case that an analytc expresson of the mpulse response s avalable. Ths combnaton has permtted to generalze the mesh method to any type of radatng surface. References []. R. Stepanshen, "Transent radaton from pstons n an nfnte planar baffle", J. coust. Soc. m. 49(5) (97) [] G.R. Harrs, "Transent feld of a baffled planar pston havng an arbtrary vbraton ampltude dstrbuton", J. coust. Soc. m. 7() 86-4 (98) [] J.. Jensen and N. B. Svendsen, "Calculaton of pressure felds from arbtrarly shaped, apodzed and excted ultrasound transducers," IEEE Trans. Ultrason. Ferroelect. Freq. Contr. 9() 6-66 (99) [4] J.. Jensen, "Ultrasound felds from trangular apertures", J. coust. Soc. m. (4) (996) [5]. Faure, D. Cathgnol and J. Y. Chapelon, "Dfracton mpulse response of arbtrary polygonal planar transducers", cta custca (994) [6] H. Djelouah : Contrbuton à l étude du rayonnement des champs ultrasonores mpulsonels dans les lqudes et les soldes. Thèse de Doctorat d Etat, U.S.T.H.B., lger (99). [7] H. T. O Nel, "Theory of focusng radators", J. coust. Soc. m. (5) (949) [8] W. ng, "Spatal mpulse response method for predctng pulse-echo feld from a lnear array wth cylndrcally concave surface", IEEE Trans. Ultrason. Ferroelect. Freq. Contr. 46(5) 8-97 (999). 598

R s s f. m y s. SPH3UW Unit 7.3 Spherical Concave Mirrors Page 1 of 12. Notes

R s s f. m y s. SPH3UW Unit 7.3 Spherical Concave Mirrors Page 1 of 12. Notes SPH3UW Unt 7.3 Sphercal Concave Mrrors Page 1 of 1 Notes Physcs Tool box Concave Mrror If the reflectng surface takes place on the nner surface of the sphercal shape so that the centre of the mrror bulges