SIMPLIFICATIONS AND SOLUTIONS OF DIFFRACTION FUNCTIONS IN NONLINEAR ACOUSTIC PARAMETER MEASUREMENT

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1 ICSV Cain Autalia 9- July 7 SIMPLIFICATIONS AN SOLUTIONS OF IFFRACTION FUNCTIONS IN NONLINEAR ACOUSTIC PARAMETER MEASUREMENT jilali Koutich Launt Alliè Rachid Gula and Mutapha Nadi LIEN Nancy-Univité Faculté d cinc t tchniqu BP Vandœuv Fanc. djilali.outich@lin.uhp-nancy.f Abtact Thi pap pnt an analytical fomulation fo coctin th diffaction aociatd to th cond hamonic of an acoutic wav mo compact than that uually ud. Thi nw fomulation ultin fom an appoximation of th coction applid to fundamntal ma it poibl to obtain impl olution fo th cond hamonic of th ava acoutic pu but ufficintly pci fo mauin th paamt of nonlinaity B/A in a finit amplitud mthod. Compaion with oth xpion quiin numical intation how th olution a pci in th nafild.. INTROUCTION In acoutic paamt maumnt of a mdium it i ncay to ta into account th diffaction ffct of th ultaonic ouc to impov th pciion of maumnt. Th maumnt cll uually ud in tanmiion conit of two cicula tanduc on ud a ouc and th cond a dtcto. In th ituation th dtcto will tanlat into lctic volta th ava acoutic pu on it cption aa. Th analytical olution dcibin thi ava pu can b fomulatd a th um of two tm on copondin to th popaation of a plan wav and th oth includin th ffct of diffaction natd by th omty of th ouc-dtcto unit. Th attnuation and vlocity c can b obtaind in th ca of lina acoutic. iffnt autho [- ] av xact and aymptotic xpion of th ava pu civd by a cicula tanduc. Th xpion pmit to tablih coction function of diffaction in vlocity and attnuation maumnt [5 6]. On th oth hand B/A paamt i maud in th fild of nonlina acoutic. Th fit maumnt of B/A paamt by finit amplitud mthod td on an analytical xpion of th cond hamonic by conidin th popaation of a plan wav [7-9]. Vaiou autho [ ] thn impovd th pciion of th mthod by includin a function to coct diffaction ffct ultin fom th lation tablihd by Innito and William [] fo th ava pu xtd by th cond hamonic. Howv th coction of diffaction obtaind i not vy pactical bcau it can b valuatd only by numical intation. Th objctiv of thi pap i to how that on can obtain a impl and pci fom by implifyin th coction function of diffaction fo th fundamntal. Thn w will iv

2 impl xpion of th ava pu xtd by th cond hamonic includin diffaction and attnuation ffct. W will how that th ult obtaind a quivalnt to tho tablih by Coob and validatd in maumnt ytm []. But bfo tablihin thi ult it i ncay to pnt th vaiou coction of diffaction applicabl to fundamntal fom th acoutic pu.. CORRECTION OF IFFRACTION FOR THE FUNAMENTAL.. Function of diffaction coction fo th fundamntal Fo th nondiipativ ca William [] av th xact xpion of th ava vlocity potntial fiu : φ ju j U π / / j j [ + a co θ ] in θ π Th fit tm pnt th vlocity potntial in th ca of a plan wav thfo th ju j ava vlocity potntial on th aa of cption i φ φ. Th cond pat of quation copond to diffaction ffct on th vlocity potntial with U th ouc amplitud vlocity and th wav numb. Th ava acoutic pu applid on th civ i xpd in th fom: p jρoω φ dθ Tanduc Tanductu ouc émttu Souc o y a Tanduc Tanductu y dtcto détctu P ϕ a x x ufacπa² Aa π a Fiu. Gomtical confiuation of th ouc-dtcto. Th coction diffaction function allow to adapt th thotical plan wav to a al ituation. Conquntly : φ p φ p with p j ' P <p > i th ava pu povidd by th fundamntal in th ca of a plan wav with P ρ c U i th ava acoutic pu th ouc. Thu th modul of th ava pu i ivn in diipativ mdium in th fom:

3 p P 3 Th xact xpion of i obtaind with th William olution : π j j [ + a co θ ] in θ d π θ.. Simplification of th function of coction Fo > a Ba [] av a vy ood appoximation of th olution which can b implifid fo ξ > > l in th fom: π ξ j a π ξ 5 with ξ + a. Thi xpion wa ud by Coob [] to duc and to valuat th ava pu of th cond hamonic. By limitin to th t od th dvlopmnt of [ ] / in th olution Ro t al. [] obtaind a ood appoximation in th fom: a j a + a J jj 6 It i valid fo all th valu of /a if a / >> and th o ta bac by thi implification compad to th xact olution i low than. % fo a fo /a<a / th pcdin condition impli a / > a / >> and on can duc th xpion 6 by uin th aymptotic dvlopmnt of th Bl function : / j π / - 7 π a wh w dfin a th diffaction function latd to th paamt of ouc a and and havin thi popty Lim [ ] plan wav ca. a.3. Compaion of th vaiou xpion of Fiu a and b pnt th modul of th diffnt xpion. W u fo y axi two vaiabl /a and λ/a² π/a². With th vaiabl w can ditinuih th na fild and th fa fild >. Simulation a obtaind with a cm and a 5. Simplification 6 i confud with th xact olution and th aymptotic xpion 7 contitut a ood appoximation in th na fild fi. a. Thy div fom th xact olution fo /a > 6 > 3 fi. b. Th lativ o fi. c confim th an of validity /a < a / < π/a / fo a / >>. Thu th lativ o i low than.7 %. Th low limit bin in any vnt limitd in xpimnt to th appaanc of tandin wav in th mauin cll.

4 .5 Champ Na fild poch Champ lointain Fa fild.95 Plan Ond wav plan a /a b /a ε % + + : 9 : c /a Fiu. Function of diffaction coction. Compaion with th xact olution of William. Fiu c pnt th lativ o btwn xact olution and olution 5 and 7

5 3. CORRECTION OF IFFRACTION OF THE N HARMONIC 3.. Function of diffaction coction fo th cond hamonic Innito and William [] obtain an quation fo th cond hamonic in th ca of monochomatic wav in non diipativ mdium. W can find in [] a ood appoximation of thi olution which can b ud in th diipativ ca. Ava potntial φ i ivn by: j d c φ β φ 8 with : A B + β and : th cond hamonic attnuation p -jρ ο ω φ and B/A i th paamt of non-linaity. Th lation 8 i th fnc analytical olution fo cond hamonic ava vlocity potntial in diipativ mdium. Innito and William [] howd that a ood appoximation conitd in placin <φ ²> by <φ >² in th xpion of <φ >. Thu w can wit: { } [ ] [ ] f φ φ φ 9 with f with th lation and om aanmnt on obtain fo th ava pu of th cond hamonic accodin to : j d KP p with 3 / c A B K ρ ω +. Th function of diffaction of th cond hamonic can b ivn by p p. Thu whil conidin indpndnt of th attnuation which amount to paatin th ffct of th attnuation and diffaction w obtain : d d f 3.. Simplification of and <p > Accodin to th coction i latd to wich can b implifid. Sinc + and [ ] Lim a w can nlct th tm fo la valu of a. Thu th olution 7 and 5 with th followin condition: /a< a / and a / >> bin to thi implifid xpion :

6 /. jπ / 3 π a π ξ j a πξ W can thu ta advanta of th impl xpion 3 to calculat a diffaction function. In thi ca th intal can b valuatd and it iv : C jπ / a C 5 3 π with. 375 Finally with w can tablihd a impl xpion ufficintly pci abl to iv th ava pu povidd by th cond hamonic on a civ with th am dimnion of th ouc: p K P Compaion of olution fo th ava p <p > W imulat th xpion of th lativ ava pu <p > /P in two xtm mdium in tm of attnuation and nonlina ffct: Wat : c 83 m/ ρ /m Npm - H -. B/A5. Glycol : c 99 m/ ρ 6 /m Npm - H -. B/A9. Th condition clo to th Coob xpimnt a: f 3 MH a cm I.5 W/cm² fo wat and I W/cm² fo lycol with I P /ρ c. Th ult a pntd on fiu 3 and on not that ou olution 5- i imila with that obtaind by Coob []. Impotanc of th diffaction coction i viualid by th pntation of th impl ca of a plan wav i.. fo. W alo imulatd th lativ ava pu obtaind with th fnc olution 8 and th Kin intal [3-] fo fundamntal φ. Rlativ o btwn th fnc olution and th olution 5- and - a pntd fiu 3 c-d fo wat and lycol wh imulation a caid out with a tolanc of -6 fo th calculation of th intal with th Mathcad oftwa. Thy how that th olution 6-5 i ovall mo pci than th olution - and und th condition adoptd fo imulation. Moov th computin tim ncay to th olution 6-5 i much wa than that of th fnc olution which includ a tipl intal.

7 P. o o o: Rf. avc 5- Rf olution P /a a.5 P o o o: Rf. avc 5-9 Rf olution P ε % b ε % /a /a /a c d Fiu 3. Simulation of analytic olution of th cond hamonic ava pu fo wat a and lycol b Rlativ vaiation btwn fnc olution and olution fo wat c and lycol d.

8 . CONCLUSION W howd in thi aticl that w can obtain a function of diffaction coction fo th cond hamonic much impl than tho uually ud. Thi nw fomulation i obtaind fom a implification of th coction applid to th fundamntal acoutic pu. W can u thi nw and impl xpion to dcib with a vy ood pciion th cond hamonic pu dtctd by a tanduc. It can b xploitd in maumnt of non-linaity paamt B/A. Anoth intt of th impl analytical olution i th inificant duction of th computin tim whn thy a ud in poc of imulation of ytm woin in th fild of nonlina acoutic. REFERENCES [] A. O. William: Th piton ouc at hih fqunci. J. Acout. Soc. Am [] R. Ba: iffaction ffct in th ultaonic fild of a piton ouc. J. Acout. Soc. Am [3] A. O. William: Intatd inal on cicula piton civ cntd in a piton bam. J. Acout. Soc. Am [] P. H. Ro P. H. Van Bun: An xact xpion fo th Lomml coction intal. J. Acout. Soc. Am [5] A. S. Khuminin: Ultaonic popaation paamt maumnt incopoatin xact diffaction coction. Acutica [6] E. P. Papadai: Th maumnt of ultaonic vlocity and Th maumnt of ultaonic attnuation. Phyical acoutic Vol. XIX Thuton Acadmic p7-55. [7] L. Adl E. A. Hidmann: tmination of th nonlinaity paamt B/A fo wat and m-xyln. J. Acout. Soc. Am [8] W. K. Law L. A. Fill F. unn: Ultaonic dtmination of th nonlinaity paamt B/A fo bioloical mdia. J. Acout. Soc. Am [9] X. F. Gon R. Fn C. Zhu T. Shi: Ultaonic invtiation of th nonlinaity paamt B/A in bioloical mdia. J. Acout. Soc. Am [] W. N. Coob: Finit amplitud mthod fo th dtmination of th acoutic nonlinaity paamt B/A. J. Acout. Soc. Am [] W. K. Law L. A. Fill F. unn: tmination of th nonlinaity paamt B/A of bioloical mdia. Ultaound in Md. & Biol [] F. Innito A. O. William: Calculation of cond-hamonic nation in a piton bam. J. Acout. Soc. Am [3] J. N. Tjφtta S. Tjφtta: An analytical modl fo th nafild of a baffld piton tanduc. J. Acout. Soc. Am []. A. Hutchin H.. Mai P. A. Puhach A. J. Oi: Continuou-wav pu fild of ultaonic tanduc. J. Acout. Soc. Am