BACKSCATTER THE

BROADBAND MEASUREMENTS OF THE ACOUSTIC

BACKSCATTER CROSS SECTION OF SAND PARTICLES IN SUSPENSION

by

Che ng He

A Thesis submittedto the Schoolof Graduate Studiesinpartialfulfillmentof the

requirementsfor the degree of Masterof Science

Departmen tof Physi cs MemorialUniver sityof Newfoundland

October,1991

51 .

.rohn's Newfoundla nd

1!I+1

^Naliooalof Canada^Ubrary

CibiiothCQuco3lionaie

cc caoaoa

Canadian Theses Service Servicedesecscs canaorenoes

The author hasgrantedan Irrevocabl'3non- exclusivelic enceallowing theNatiollalLibrary ofCanada toreproduce ,loan,distributeorsell copiesof his/h erthesisbyanymeansandin anyterm or format,makingthisthesis available 10interested persons.

Theauthor re tains ownershipofthe copyright in his/her thesis.Neuner Ihethesis nor subs tantialextracts homit may be prin te dor otherwisereproduced without his/her per- missi on.

l'auteuraaccoroeunelicenceirrevoca ble at nonexc lusivepermett ent ala Bibliot he que nationaleduCanadade reprodclre,pr~ter.

distribuerau vend redes copiesde sa these decuetouemane reet sous quelqueforme Que cesoil pour mettre des exerroleiresde cettethese aladisposi tiondes ocrsonnes toteressees.

L'auteurconservetaproprich~t1udroitd'auteur Quiprotege sathese.NiIatheseni des exuens substantiels de ceue-ctno doiventfllro lmprlmes au autrementreprcduils sansson autortsauon.

IS8N 0-315-73290-3

Canada

ABSTRACT

calpn'~~lIn 'in1Ill' ncurfield,itrloserl-formt·x l' r\,,.~ i,,"{IJI'1Ill'ilt'Ullsl ,jflllliPId prr"hIH·tlbyIi,"il'l'lIli1rpiston withsilll JlI~'sUI'I",rll·,1hOlmrl"t }'nlllll iti"l1is derived, ('\I"n,lingthe eerllor workof11'lst,!;"w;,col,II.(I!1....:I.[!IS·I)ttliudll,l., till'["~IM'oflIull -un i fo rmvelocityatthepist..nsnr filn','1'111'.Wl'r;l~l·prf':'SlI rt ' 011Ill{'~lJrra("t'oftheTl "CCi \'C'TiI.Safunct ion<tr.listil.lIn'frum Ill"transmit te-ris calrlllalt.JtheoreticallyAndmeasured(·xpt'Tillll·Ilt"lly.'I'll<'IIlt'a."u «,,1 f<'Sulb

lIt;TN:'Illt'lltwirl. thl"th ro n..otk alcalcula tion.i,hh'l1l;;h tht·rl,llllgt· I,r U,C" ,WI'T- fiSt·pressure wilhdistall('f'inthe'ncarfleklis lart.rrfot allfr"fjlWudl'SL1l il ll pr('vio nsl)'thought.IIn wC\·er.itshouldsti11I'('I"l!<sih~'lulJ)(,l\.~llrl· l\1I11irUflIl disLrih uLionof particlessuspen dedinthe- Iwarl id,J.11''I;i/luillgwithI'MksnIL·

tl'ri ng Froma movabk-rigidsphere, a ncarficld,·x jl tl '!l., iullrdatin/;till' ont put

lindtotill'hi, rk~rRlL(·trurmIactor,j"",, (r ),is ohlili ll'·'1.TIll'

r" lIIl

(11I:1." ris<I,.•

Iermiucrl 11)' 1ll"IlMlrillgLll(' bllrkS CilUI'rf',1rl"lillli" u fr"IJlSIl~I'I'lJ'[l ~l lI allJ rl,1

talJ~"(ml /lI~I:\111:1:tlJ:l.l):\lllz...u.lthl'...,1~il•.,.froll1100to.'>OOl'lll<Iiam-

IIrt..[ "<lrli,'~'ril,li'ls Ilml:'Ull r f(' wavcuumhcrillfU·il"'l'!-.Ill.,mensun-d values at.' hltgl,t IhanI't,~l i'·lc..1hythesJlhcriralscetu-rer model.Thisi~similar to

.1111'l"11u' irr".'!,1I11tr1I11<Iw':'«udinhomogenouscompusitioncfl1iL11II;, 1 sall t!

11111,.]1mull'r"ll1plf'xlhlilllill'farfir;d,scatteri ngrro~sscetiou1lk'i1-1IT\'II1("llb n. ll1M'11\;,,1,·illtlu-1l"'lrfit·I.1,withbroadbandpulse-.athighreschnion;11

rn··l lIt·U<:)'.

I

J

ACKNOWLEDG EMENTS

'11\(' nuthurwislu-sI"thanktln-lh-pa r t uu-ut

lIr

I'h)'sit'Sfuriltr"n lill~Ilim till' opp o rt unityofr"llrl'lfl iu~th iswork.l'il rtit'lllilrl}',lilt'1'III"- T\';'ill l!IIflIr.

AlexHayanrl his"lI~I"' li,,"lJI..u ,'st'itin,!!;TI'SI'llrrl,tupicill1111'1,1111,,,r\ti,-I,I ofillter~lar(' Srl ·I111}·IlJlp rl'f'illk,l.

Dr. A.1I,Iy,Dr.l!..1.Cn-illl'/ltd,. Ill'. II.Sill" lt'!'"" n.1111,1 Dr. II.d,- Youngaretohr-''!iPI~j..Uytlli'l1k''i1fur illsightI'ru~'ilh',1ill""UTI'C'"...li,l i ll j.!.

tothiswork.;"h'tllIM·fS..

r

Ill"O'"('illlogTitpl,y<:ru llp111tIll'Ik!' :ilrl I1K'1l1..

r

Physics,in pitrlirlllaTOr.J.Sheng,l\lr.D.Wilson,Mr.t\.l:uIIM ill,e..;"11'. C.lIamm,Ms.A.Crawford,~Ir.R.GamplJl'IlandII.1Ii1lirl,·,h;owills..

provkl cd'"a ll/a h!1"Msi~lillin'at varioussl.l gl'!iurlhi~""..rk.

Iamut"l'pl)'itl'I'It't;lllin'urthr-filladala~~i~la1Wrrmlll:\I"m"riitlltni- vcrsityor;":cw rollll ,lI i1l1.1ill II."rorlltoraUl1iwr~il)'F.'lIuw"ltil',T'·;lrllin.l:

A~i:;talltship,alltlDr,lIay'J;grilnbfromt!J('1)1, 1iIrtllM'ul

fir

Ell,'r,Q',:\liltl'!o.

ann Resources,andtill':\~, tllralSci,'lJel'SamlEII~in"'riligH...-ur..h(''''1Iwil orCanada,

filially,th('plllil'llt"('llUrl lll1.I,·rsla lld illgortile';lllll lur'~wirc'r1l1rill~1111' rcscar ..hperiodisdl"'ply;'Jll'rc~·i;,h'd.

iii

Contents

] INTROD UCTI O N 1.1 Slal"llu'lil of tIll'1'r/JI,I"1Il J.'J. Ilis1.t>rira[Ba('k~rullll, 1 1..1 TIll'!ApproachinThisTIIPs;l'

2 THEORYOFNEARFIEL IJ BEAMPATT ER NS 11

2.1 Background and Ailsi<' Tlif'OTY II

'/" GplwralThocry 17

2.2.1 Ca~l~ofUniformRadiator 2.2.2 SimplySl1llpurh'(\Hadiator 2.:1 Nr-arflr-ld of1\CircularPislul!

2.·' Spil"I'A\'l'tag"I'rl's~lIr,'inthe :'\"Hrlit'ld 1..'"1 ('umpar;slOllwith!\lIlm' rkil[lutcgruti cn.

iv

17

2.

2' :1-1

·HJ

3 l\IEASI;HI~MENTOFSYST I~1\1CONSTANT ANDACO ll S·

TICA LPHESSURE ·IG

:1,1 ~k,'~ llr ill~S.\"~II'I1' 17

I.:! :-'ll't] J"d rll

:).:1ResultsandAnalysis ' •.'1

4 nACI\~CAT'fE R }O~ R O MSUSPENSIOf\;S (;4

1.1 B~"';(~'·ill tt'rt..lPrcs sun-[rom aSill.!!.]"I'"rl i,+ li.'"1 I.'l Hht'k~,il llt'rt',1rm~~lIl"l'hlJ1ll:\IIEIl~t ·t!l ldt,

or

P" r1.id ..s . liS

5 MEASU REMENTOFFO R M FACTORIN TilELAB OIlA - TORY

:i .1 ExpcelmentalDesignAIl,1~lt ·t b (>.1 f)t~<Ti p t ihlJ. .1.2 Backsca t tcrcd Altlpli!.lIllt' sVt'rsll s1'1If1kll'~llIss{'tllll""t11.ril·

tiou

II.:} Estima tio n ofSy S(I'1I1Cons1ant

.j A ~lNI.~lI r{'llH'lltOfTheForm F....-t ur.

6 CONCLUSIONS

!'III

8 APPEN DIX A

9 '

104

10'

List of Fi gures

z.t (;'''''11U·try"f<'irclllilT pistun 'J'J ('lJllrllinalt'~y~1t'1Il

2.:1 V"\",'ilydisl rill1llj'JlI"II11ll'l'is h m~llrf;..-,'fur1111'(ralls,)"...·!

I:!

"

2.4 Acoustirnlpn-ssurc011tlJ{'anlllst ir,al axis ,f=2.0~111~. :\0 :.1.5 :3-d inw lls iongr.lph ufnco ustiealprcss u n-intln-lll'ariiddwlrh

2.6 :J-di lllf' llsiol1grapllofaronstir-alpreSSIITt' inIIIf' m-nrfi-bl with nonuniform~l l l' fil('f'l't·l"dt)'

2.i Avt'rag e acous t ica lpres su re011thr-diJr.' T< '1I1siz,- offl"·,'i\'i ll~

suda n '(r,"lillsil£)producedby lTlllIsl llit 1<'t(r;"lius 111) will.

Ill<"sillll.l.rsupp"rt!',) I'dgt·,,tlf""2.0MHz.

vii

Tl

:U'

T,,1;,1,wolislil'; 01forrr- 1111 Ill"rlilrl'tl'rrlsi/ "ofrco-l viugMlrfan ' ( r",liorsill )lH,,,lllc"c1 I,yInmsrn i11" l'l nrrlill~itI) wil lrIll<'~imply Slll'lh ' 11, ~ll~l~r"iii:!.ll:\1111.

:!.!J A\'<'ri'~"i,nH lsl io'id l'n"' slll'l"mrlilr"f" lllsi~('offt'n·i\·illg plaul' l,rudllc",II,ytrnusmith'rwithuuifonu~llrran'wlrlf'it.l'.ill'.!.O ~1liz :It>

l.WCl!nrdillill l ·.~ysl "111f"r1l11l1wricl11rnlrnlatlon

l.l lTireIotula,·rJll._li,.;,1 f"rn ' ol1thereceivingplau(',for1/10fl1

,Hilif=2 .0Mlb: .

:1.1 Sketrhuf "x l"'l'im"l1litls,'t up

:l.:! Till1!',s,'ril'snmlSlwd,f11111of I he reflectedpul se :1.:\ lJiap,ram il1us1ri,l inglilt'self.n..-iprorilycalibrationme-asure-

II

I,

so

;U Nnrmal ii',{',1umplitudosofthepulserr-fleetedfro m. . .'j(i

;l.!i Thesolid,,1 i1sl ll~l,(fUSSr-utve are measured volta ge out put uf receiverllgi\i t1~ldis!<lIl CI'lIhm gacous t icalaxis Iorff c l) u l' llcy

IA Mllz,2.0 r-.llIzillll!2.SMl!e,respectively, :Ui Attenuationu'[i1l'ollstic\\'ill'('ill·,,'at..,. c11;,ngcswithfrcquC'I1 Q'

1l1llis liI1ll'l'10,(i rl11.

\'iii

.17

S!)

L! Aplan"ware1.1wk~.·al 1t'n~1l>y11ri,!!,i.lsl'l\l'l"l'.

·1.2 (;c'Ollld ry of susj><'ud,"[Si1l1tlIlll'il~,m'lll"ll t

."i. j Skel t:h of('xpl'r in wlltalset"I"SI,,"lili gin,licc,l('s sllsIWtlll,',1 scdiuu-utjet,,ml! sedimentcloudinrnpturr-cone ,j.2 Squured- mcenburksrnttcrti n\'uh 1)as 11 Iunctiouof111'111'11

sandconcent rat io n at I.:l.'i~l lIztol.!)."",Mlla.St ra iglltlilli'S til

i·1

were obt a inedbyleas:SCjll<tH'Sfit10illlmeasured poin t s .llif- f,'rl'ut symbols[('pres"ntclifr"rt'nlfrrCjllrlw i,'s ,11.~shown , . is

·j.3 Squared- mea nhacksca tu-rfluVo1l¹)as11funct ion ofbeach sand concentrat lon1,12.j!)l\lIlz 11l2.if!Mllz. Straightlines

wereobt ainedbyh-ast squaresfitto all1l11"ll.~ur('rlpoint s.Ilif·

Icrentsym bols repre sent dilf" relltfn'Cjll,'ndes, a.s shown.. ill 5.4 Backscat terformfact o rrf",,{r.)Ifor180 -212JIllltljumctr-r

beechsand.Different symbolsr('p n~('n ldifferent ccncentra.

lions,asshown.

.'i..""i Backsca t terfo rmfarlur

I

fox,(J')

I

forIO()·12.'i11111dlnnn-tr-r

bench saud

ix

lUi

s.n

llitl"ks("i,u. 'l'fonufaclor

I

f.,,Jr )

I

fur180-212l'llldiameter

,'1./ lliwks";,II"lformf"..lor

I

j"...(r )

I

fur:IOO-:FJOI'llldiameter

withlll('datil.inFigure5.5,)'j.G,'I.i . :•.!J Harksrnttorformfunction

I

f".,{.r)

I

for beach sandmeasured

fordifferentsize' sand [soc Table5.;i)

87

88

90

9'

List of Tables

:1.1 ,\!l'a sllr('tlsetlsith"i l.y

.'i.I Sil.('r!Iractlcnof1)('",<:"11 sundused furfonufad ,,!,JlIl'aS IIWIIIl'1I1sif>

5.2 Averaged cvcrnllsyst crr.C01I s t lLI1lll,de-tcnuinrdLy),·;,stSCl"ill"'-S to hll(ksl'a L(Cfmeasurements lImll3",<lc'I,(· rt lli ll, ~1fromrd h'("

lionmeasurements.

,1.3 T - Iomperat.urr-, C-acous ticSpt· C't! illwuter,110 .radiuslOr

5..1Thelistof slIU]tl:nd errorsaspero-u ta ge of meanuftlu-siLlllples 89 5.5 T -tem per atur e,C-acoust icspeer!in the\1"<11"1" • !J:!

xi

Chapter 1

INTRODUCTION

1.1 St at em ent of the Problem

Tileim)lOttilul'Cofgrain sizeandconcentration estimationacousti- rail)'hi\~long11('("recognizedin examiningmovement

o r

sediment(Flam- llwr,l!Ili2),coalslurry transport (Davis,1978)andcellsinthebloodstream (Sic'g,,11l1i11111illlll Rf'irl,1!J73). During thc pasL fewdecades, many newcpti- calandurousticalmeasurementmethods and apparatushavebeenproposed iIIulbuilt acrordingto theprinciplethatIt.wavewillbescattered bythe Sll~))('II (hl pa rlid("ll. :\probh-mwiththeopticalme thodinaqu...ms suspen- sinllsisthat111"11'111'1'dC'{'ilrsveryrapidly. AllK1,theopticalwavcl-ugthsused

arc usuallyvery short l'OmpiHl'dtotill"I'<lrticl.,si~,'.Thllsitisdillicllhf..l optira I methods to (!c lo'1:"l~Il S P "Il< h'tlp<lr l irl.", Tt'lIu,I" ly.;," , l lh,'s. -1I1l'1Iu"ls

Theultra so nicpulsc-oboIt ...·lmi,IIU'has],".'11 proventoI..·II:-illll' l,-.nnl efficient metho dfor1l1l"'ilSl1r"I1lI'l1tofJlilr l id l'sliIISIll'I Il !.,.1inwater(1"I;oII11111'r 1962;Grt'('nla w MHl.Iohuson. lnS2jl1olli,lay,IHRi;HayI!llJl;SIIt't1,l\1111'\

lIay, 1991).The("('1111';11prohlr-min ill"lIll.<licalsrillh'ringkrlllli' llwsIi,,,,ill .1"t " rTl1inillgtherr'la tioll~l lil'Iwl \\" '<'1ls" altt' rr'l! inlo'ns ityall,lII...quuutitv ofscattcrlugman-r i al. Thr- ...,:n pl,'x it ,l'uflilt'Ilro ....'ssisdill'1<>t.lu-(111"1 tha tthescatteringcross section is alf,'de. 1Lytill'size,shl lJll',rllll"" ld,rillion andco mpositionofthe scattcnugmaterial,andbythe c1'i1ril.('t .,r is l,lrsIIf till' acoust icalbeam.ForthePT('S Pll tstudy, attcutlo uis foc used UIIdl'lrrl l,i lli ll_~

thebac kscatt ercrossst'diollofnatur alsaudgrains exper iuu-ntully, Two experlmeut almethods,(",111beenappli..d forultrl,sonic dl'lt ' nlli ll;,!i 'm ofsceue rcrsizeIIl1dcnlln~lltr;,liun.Out! ist11<~multiplc-Frcqtu-ncy wchniqtn- usingmanynarrow-bandpulseseachalII.dilTr-rent.Iise rcl,efrc 'lllt!Il'",\'.This met ho d hasbeen use dintheo,"'an forzooplan kton1Il(,ll~lIrC'I1WIII,s(;r""Il' lawlindJohnson ,I!I8:? llcllidny, I!lSi )andfor sllsllenrl.·,1st" li'I"'II !'In'''' ''ll'' '·

mcuts (Shrug,1990,Shengalld11;'.1"199 1).TIll'other 111l'1.llu,1ill\"" I"c',llsill~

InOildl' ;IIIlJI'nJ.~,,:, ,ill II"llid lilsingf' sllor!jlllbf' ofwld«bnudwldthis 11Sf'd1,0 ill' "n'llSI'Llu-lnfonnauo n ronteutperpulseofthe sratt en-dultraso uud . This llwtl m,1 hill'1'1"11USI'd 10study sa udsnlUt' j"ingLysing leclas ticspln-rcs (l.lritAlJIWII(· ('tal.,1974),Broadban d Spl"lllsUjWII tbcpossibilit y ofchar-

;l<'tNj~i n ~witha singletrens.luccr bothlll('arnttcr ersizedistribut ion1111d

"(Ill.,"tr"l,iollI,y anil.llidysis of thearn plitud..andfrequencycontent oftilt' s,';,l!l'j"l'dwuces.Agoo d underst anding or 111('facto rs influencin gthegene r- ntiou.prul,aga ti"n,scat te ring,and reoep t jun ofbroadbandultrasonicpu lses islllTfh l10 obtainquantit ativeinformationonthesizeand conce n trat ion par.nnetc rs. The majorparamet ers in volvedinthepart icle-ultrasoundinter- actionare: theIrcqucucy-dcpcndencc of the to lal anddifferent ialscatterin g cressSI'l'!.i O llS,particleconce nt rati on,and thesceu ercrsize distribution.

1.2 Historical Background

l!ilc!>scaUeringproblemsinvol vingspheric algeomet ries(i.e.solidsp he re, Furnn,1951,sp hericalshells, WerbyandGauuaurd,1987,coated shells,Gau-

IlHU Tl Iand Kalnins,1982,et c.) have1"''1'11 treat edforthe las t 50yea rs.

'I'lu-....1'r(,"I" Ill~orr- umdenontrivia l bytll('statistic ,,1nature ofscattering

from an('ns{'llIl,k'ofsnitIr-n-rs(1\lilTshidl111 ,,1lfiL, d ilid, l.]llr,:\l<IS w,,11i"

by illlllll'nrt'Ssuch 'IS IIIf',l!,r;rin~iZI',sllllJlt"IInrloril 'lIlnliOltl ,illlflllw '1wllily

cauhrn-nwvcdby pr"pc']"signa lilH'rnging, hillIlwr1d,'n lli ni:<tkpr"l.h' lll

u r

ind idcl1l1rlsrnttcrercross sectionnn'nsttrvnu-nfIII' rOll1ll1llnliunisnmdlIc 'li,~

trartnblc.

TIll' Ilrill"ipknudmcthodsforbark sr utu-rlug mcasurr-mcut in~oliclsawl inwnter arr-sililililr,hUItill'{Iellsily oftlll' ."',dl ,'r"t:<insulid:<i"1Ilfl,'IIl"tw 'r tha ninwater. H('sc'a rrhdfortsflittill'llilrksl'lIl1,'rinl!:l,roh ll'lllillsulids havebeen dln-ctcdateharar tvriaingthestnrj,sl irnlrt,la l iutls lrills 1lI'lwc'C'11!.IIC' energy of th,'nlt tasouicwavepropaga tionalliltill'IlI"nl \'ariat.io lluftI,(,srnt- tcrersLl'rit IlS{'ofmanyra ndo mphysicalpar.uuou-rs, suchusgrllinsizt"p;tilill shap e ,;rllinorlcntntiou,qualityofgrai nhOllll<l ill' i,'~,tIl('I'fUl'urti"lIof,-111'111- ical roustitucnts,and multiplesfill l e ring1I1l1mJ,l( gruins .BI'c ~' l la11l(IHf)(i )W:L~

abletodemon st ra te tha t theattenuation orha"h n ' t k ll'c1l'('!l{){'liwit h (lis, tuuc cinmetnlsisrelated to tll{'llV<'rag{' grainSi~A 'ofllll' sp('c,iml:n,Alfl ri,IAc ' (1969) ronfirmcdBcr cha m 'sresul ts ,Iudl'ollf:I'uh'r1 thill,thr- ,list rilllll.iolJuf gr a in scatteringrrossst'rt io ll sranIll'oh t ;,j rll',ltllnmgl lllIult ipl,>lIll'llSflrt'lll<'lll

furllw riml'r<lw',1t1li,~id"it,,, !II'm'ur-..llrilh"ly,1f'l('rmiJll' tIl{'";''1Wr ll'r!ampli-

whb-h-au ;lls"Iof'1l~" . 1for~1I~1 )(,1l,1.,tlparticleuu-as nn-nu-ntill water.

";11prop,'rLi,'.,ofhiuln~i<' ;' ) '~"atll'r,'rsasa comprcs siblctluidsphe re (Johusou, )977, Gro,,'ulaw,1!)S1)or to usr-a semi- em piricalapproach l-asedon laboru, lllry1I\(';lslln '11l" ll l ~ufjypir-nlcross sectio ns(Love,I9ii).

Fo rworkon;"-,,usl·ir s",llte ringbysuspendedsandpar rlcl.-s.bo-ansothe

<Iisl,;l1lr('SIJl'L\\"l '(' uthescatt crc rsusua lly aremuchlargerthaninsolids,and S;lIl1ll' .1fLid,'sarr-netsoirregula r asbiologicalscou crers.themultiplesca t -

tcnngcouldhI'igucrcd. The approachwasthenL(Imodelsandparticlesas

solir!sp lll'flO:;ofdi[fef(' lItmater iels. The sca tte ringand ba ckscattcr iug cros s M'dion s ofsing l('particleorsaudcloudscanbecalcclat ed thcorcticallyand nu-nsurcdinlaborntory, Theoreticalcalcu lat ionsofsca ttering bynonbio, l"Aka lparticulates have1,''1' 11approachedprimaril ythrou gh clastic orrigid materia!{""];lstif "l1Wit1\Sthat shear and compression waves mayprops-

~01t"wil hiulIlt'lIIall' rilll,O1H'!a~rigid"materialin 01l(' illwhich110 sou nd

sui.l ill~h;jIl('S(rrll'r~nlld, 111..I!t:-:O). or~Ii~hl lyIHUrl',·ullIl'liri,1t·,j~l+;,p.·~

surhasfylin. lp rsWilh~Jllll'l'ic;d"II,[.,-;q ."(Xlll11ri dl<'1al ..1!lsI),SI"'II,!',alld

Illll'llll atiollilldilulf'a'I!II'oIIS"ll"IJI'I1~i "J1"IIf "HndwilhIIII'Illt'" ry.IIItIl"

thcoreticalca1c11 1atitll1s.11,,'s,·;,tl t'rer wa sll~slIm,'.ltoIll's1'II1'I,,";ll,nndpi·

pa rt icl e is infinilrl)".1"11"(" )"Pt"rt'~1l11s"J,mn,.tthuttilt'ri.u;i,lllltJ\";,1.1" III." lpI providedthe-!Jt'slht10till',1;1111,hutI.IHlttltl'llll;,1~!,;,tt l'rilill..-r..-,swdi"lI s werestillIargc rthan lheury

Oneilll!1Urlil nlapplieation of~ilIl dgruiullwaSllrl' IIlI'1I1s isI,,,lStilll;l1. ' sedimenttran sp ort Dkl1.in 19·18first SUggl':-tl,(!amethodfClr"hl;,inillg estima tesofsuspended-."lillWli! rcuo-ntretionintheoroanI,y1l1l';,sllrilJ ~ ultrasonic scat ter.Flamme r(l~l6:!)mensurcdtl,ofr('qllt'lll'yc1I'!"'lll l,'IiI'"

or

ultra sonic at tenuationformany dilrl'rl' lI l~illl.1si:wrra l:liv""ill1111'"Ull/ Ill rrc1Illt>ncj'rangeor2.5102;jMHz.J1isrr-suh sw{'fl' rl1collrl, gi.. gnsr"ll;arrl s the pos si bility ofmeasuringtlx-ronccu r rauous oflhe actua lsilud

ir

itssi;f.l' distr ibutionWASknown.,II1 IIM'1j "ta].(IU77,lfJ7!J)devdopl,<J Ill!Ilhr;oslIllir Dopplerscattorcnu-ter,Onsiruil..rpritu-iph-s.various","!Justin ,luu'axurt-.

meutsystems, such;1",\BSS(An>ll~ lit'idHIII"ks '-<lUI'r ill.l;Sy-;!<' lIl, II,.,."~;111.1

Orr.I !J7Q;Orran d II.",If/ii:!;1I.' ~~illlIlB.·,lfunl.1!JS.'i;Lyn chI!lS,,:Lyll d l"t ill..I!J~JJ;Lyu-haml Ap;r,\ w"I, IfHII), :\(':"1(ArllllstifCoucoutrationMoter,

;d., IY!JIJ,IIIlS(lilt r,NJlIi,.. lJlJpph'rSrill1t'TUIIII'I'·r.Sr h ll a f~mil dal..I~ I.' 5 ) . 1111,1Ill<'Ui\STRAN sysklll(Hayct at, Ill:>''':Hay, 1991)wered \'('lnpPfI furnl<'i1~1\ rillgrrmcentrat.ionsand~iz,,,,ofSIl~lH'n d('dpartides. Hay(I!JS:J)

"II1I,luy,'1IIIl!J2kHza(,(JLl stir~ 1i!lll'k!'l'illtel'system totllrllsuresu~p('ll(I ,'dsed- iun-ntina Ilr galivI,lyhuoya ut,mine-tailin g,l i ~fh a rgcplum ein a sldJllHIrine ,'!lil IlIWI.

Allof lht'S(,studiesha w .l"l1lo llslr,,\edthe ut ilityofacoustica lremote

s"Il~illgtechniqueswithrespect toth,' scdlmcnttran sport problem. Oneof

IIu-muainlng difficulties ill thequantita t.iveusc

o r

thesetechniquesisto ,1.·t."I'mil lL'l!11'acoustlebilrk~rat tt~rcross section ofnaturalsand precisely , whichis1H'('(letl loinrertlin'bn('ksraltl'rl~Jsign al toact u alsedimentron - rr-ntratio n.1I11! siz e. The'pLlTlIO~eoftln-workpres en tedhereis to illwsligllte tilt'Ill'!'ofuroadbiltlnsystemsfersuspended flitr ticlc back scat tercross sec- IionUIl'<ISIlI1'IIK'lIt,illorde r(0provl.lohigherresolu tionu-ce suremeuts

o r

the nCJ~s ~1't'1JOIlthnnohtaillC'dpreviously,

1.3 The Approach in Th is The sis

,\II " f1Ill'iun'" ,i~aliol 1 "of~illl. 1I';,rti,'l,·"11H''''' ''!"l'lIl1'ul'Iu'uri"r" ',l "1,,,1'"

1'l"tl<tchillwhichill,toml1>a llll)1111><.·i~ lI~'·o\. Tlli.-I"'l lllil"1l,",ISlIWllIl'IILof

r-t:II. (Ulj.j)iun -"I igall ·,1thefi'la!io nI,d11"' 1'11Ill<'rd k,.h·,j,WilliS!il'idI'tl'."

sureandIrcqurncy fursillg ]('clastir-sp JIl'tl'.,"fuhnuiuum.Iora s s,IIlI t] 11lll gsl! 'l1 t'llr ],j,r" ill\\',,(1' 1"byusing<~!,rwl,ll.a llllh;,u",[n,willtlu-Ir ;1I1Ml l1n'rfllrli..l,1.

tl.ntthepistonmay he cOlls;.lrrf' d<tINlilltsonro-,til!'I,UWN in tl1l' IW1" 11 d",'rril.«'Sasthl'reciprocal of tl1l'Mlll.,t!'oftlu-tlisl; ' lwl'111111 1.11(' distril'lllj,,1t ofP(l\\~'Twithangleand disteureisuun'huum'"lIlllp]i"iI!!"!.'l'hcllWil SIIW·

IllerItsIlres(,1I1,~111('rcar(,mndeintlll'lIl'<lrfj" l<1oftlu-LTalls,IU(l'r.!,,',-allsl' ill tilt'IIrilrfit'1rlthechanges illb-n mr1i;IIIIt'll'rwithrn'qlll'llCy/ln~smnll~lItl thepoweraveragedoverthe beamdiamd t'ris nppruxim atdyiIHlt'J" 'IHlt'1I1

o r

III«nh-rle111 1l1,'r,~1iltl,1 111I'dl;,r;, r\" r isli r~oft111";WUIl~tb-alncarflcld,the HI·i,rli,·l,!;",,,":<li n.lI'rt'SslIr,. ;~"aln lla1<...1 111,....n-t;".,l1yill <'11;1I11, 'r 2"fthis II",sis.'1'1...,,~"inwhirhthevihrnticnatuplitudo "" IIII' surfurcofthe

"XI\ 'II~i, ,",,(Ilf" \';I"ISwork.

lit<"1',lpl"r:l,1\'\'I'rl'SI'II1.measurcnu-nts oftill'v.:nialiml:<lnthcamplitude sl""'lrlllll"ftln-l"dsl' rd l"r! f'<! fro m anair"I\'lIt{'rint.'r filcl'asa function

<Ifclisl.;IlWl'ill tIll'ncarfleld ofthetrnnsmittor. Those Illl'ilsurCllIt'nls are

"Ollll'iln'd wit htlwIlll'o fd ie,,1resultsforrh«III"c rag"nco u-t icalpr cssurc Oil tln-n-ceivvrsur fill'"FromChaptN 2,Theresults areused to obtainestimat es

"fIh,'overallsysh'1Ilsensitivityconstantas afunctionCorfrequency, which an ' 1Il',...lcrlforthedetermlnaticn of absolute SCilttcrlugCfOSS scctious.

In rh'll' h ' r -l,IJI'p,in n ing withbac ksca u cring fro mil sphere, \1'(' obtain an-lnrion1,,' 1weenthe receive routputandtheconre utrnt ion, average size

;lIu lI'arksn ll u-r rross section forsuspe ndedparticleclouds illumina tedbya hrua,lhil1l'larom tir.lllJt'lllll illtlwl1l'ilrfielrl.

Tlw "XI"'riI11l'J1 lillmoasuromontsof till'l>"rksC ilt lt'j"cross ,,{'t'li on

or

sus- pl'llll,...1samli1l"t'l' rl'St'lI l ,'dillChilplt'f5,T1rt'.~"art'newr..~ults,representing

til..fir:-I "",sp'11lIN! I.art id,'I""' k!,<',llh-r,...~!'<"{'li" 11n)l';'~l , ro·n"·l1l...,\·jlll ..

I"w,d 1,;.:_,1~ysl' ·Ill.

('1"'1'1....Ii<"l111,isl,..

r....

ndus i.,lIs.

10

Chapt er 2

THEORY OF NEARFIELD BEAM PAfTERNS

2.1 Background a n d Basic T h eory

IIIonlcrto obtainconsi~tl'lllmeasurements of size andconcentr atio n ursuspondr-dpartid,·sillncarficld,thedifTracliolleffectsrelated tothe fi, lIit,I'npcrt.nro ofIhetransducerandthecharacteristicsof theacousticalfield :llf,,<:ll'.t hy lilt'v{'lority distribntiou ontilt'source surfaceshould heinv, tip-IlIl',1ill IIl'1a;1. TI\l' problem orthr-batllcd piston radiatorhas beenvery wi,I,·lySl l1llil'tlrurwlrioll snpplicnricns. 'I'll,>rornplctcdescriptionofthe

II

z

x

--=* --+--t-- - ^y

SOURCEEI.EMENT

Figure2.1:(lcometryofcirr n!ilrpi~loJl

soundfieldproducedbylUIacous ticpisto ni~1'.·IIt,tillly,lj\'jdedintolim e-parate paris.Dill,'partofthedcscrlptlonislilllilrlitothe"'Sion in til\' nM&hborhoodof thepiston,i.e.intheDt'a rfiddn-gtcn,whi lethe otherI"ut is confinedto the Iaefield, theregionbeyondtilt"I1carfield.Beeeusc oftil..

complex ualureofthebea mpatt ernclose10the pisto n ,especi aUJfor1&1';('ra- tioecfrad iatordiameter towavelen gth.theexpressionforthesound pressure

withinthisareaiscomplexandtht>integra t ionof lobeequationforpressure becomesalmostimpossibletoperform exp licitly.Thereforethisproblem is usuallysolvedbynumericalme tho ds.

There areseve ral('leadCXIJfCSs i(l1Isfor thefil'ldwllk harcusedill dealing withrhisproblem.Theyallrepresentsolutionstothl'l'(lllllt iull[eocrdiuut e

12

(2.1)

thr-l"'II11dtlry. 11,,1'('

'Ii

isLap la ce's opr-rutor .II.p.Itand9(")arenormal ,lil""l i"II, r;lllialcoordinate, pistonradiusandvrloo'il.\"distributionofpisto n Sllrf,W"n-spcrtjvely,~isth('velocitypotential,111111J.:isthewavenumber,

OIWlIf theexp res sions fortIll'IJl'arlielJ isthatdue toKing (1934),who d.·!'i\'t's!Iisn·I,,1.iolifrom<lgeneralized solutionill-yliudricalcoordinates Ololililll'dhyHan-man(1914).Itis giwll by

(2.2)

where<f'is thevelocitypotent ial at the fieldpoilltP,en"=~is thenormal v(']udl)"onthe^SOUTcesurface whichis uniform,andII=(')'2 -P)t,where )' andIIare scpareticnconstants[Harris,1!l81). .Ju(JlP) and)I(,l(l)arc tlu-7l'l"uth,111<1first-order cylindricalBesselfunctions.Onthe baffic outside fl

=

^II,thenormal velocityis zero.Williams(1951)lindGreenspan(1979)

ll~('dthisexpression to compute the ncarficld of II circular piston.

:\1111111<'r expressionisbasedallthl"timc-domninGr.vn's-Iuncticnap- f!roill'h,Thevclorjtypou-ntialdue toIIpiston occupyingaregion.~ora rigid

13

planarhiCm"i~liliwllby

Lockwoodan.lWilka c (I!Ji:I) used this,'x llrrs siulIIu"'>I1 I]' lllt':1·.lim,·n",iulI;'!

accor ,lillg to11Jl>1IIlYS,'lIs'prindpleami lak c'S 1111'fom

(2A)

Thisequation'o\'U110>('(1bySeki(19:).1)andZt,III11lC'k II!J;I)for..akulat iltlt theucarfield.

NUllwrk alint eg rltti ollill;usually requi redto "akllliCl,' Mlulltl I'r''S>iurrill thenearfiekl.However,fo r somecasescspcciolly"'II",rg"r;,tillsof source

dianwtrrtolhl'w",\~'I" ngthand at places very dos,- til!' I,ist ,,"fa.:c'.itis hardlo obtain accurateresults bynumcric alnu-tbo.lsIW'·ltll.~o:tlwilll"gralt~1 Iunctiou dJallg('s\W)'rnpl.lly.Morer-llccfi vemethorls ((-.J;.C'l\"iHlil}!,1I1!IXlI,

;\I/<1!l8i).lISilig I'XPllllSiIJIISorIrllnsforl1laltulls\\"'r('lh,~rd"n'r11·v,·IUllI'r1.

"

Fu r n'i,JsilllilLi/JIIs,the\li.~trilI11liolJofn'/oeit)"011 the SUdilWofrln-pis- ton wit!lIutlwuuifnrUl.Iftil{'rry stid,lispl'lre1l1clltis doruiuatcdbr thefirst 1ll",1,'of vlhratlou[i.r-.l.lH'r('is nonodal cirriI'011thevibrating surface('xn'pt illtln-,>,IS"),Lln-vP!od lYOIlIIll' surface ofthepistonshouldincreaseIromtln- ('(1St'tothe celltl'r,themaximumhcillg al the center. Dekkerl'tal.(197-1 ) 1.111]Gr(,(~lls]l lt ll(1979)r-onsid-rrdI>otll simply-supportedandc]"mjlC'ddisks.

Fc,r11(lisk withsilllJ1I~'supported edges,radialvibratio nof the .lisk is every- 1I'1I('rcJlo~sihlf',hut axialdisplnr-cnn-utsattheedserestingonthesupport aT<'zero.TIle velocity dbtrihlltiollcan be expressedbytIo=Vo[l-(;)Z].For a diskwith clamped edges,radialdisplaceme nts andaxialdiaplacer ncutsat theclampedboundary bot h vanish.ThevelocitydistributioncanLcwritten as1'0::::Vo!1-(;)1)1(for definitionordifferentedge cases,seeTimoshcnko andYoung, 1!l!i!i).Thuc sheu koami Youngdiscussedthe effect oftheveloc- itydis t ribution al thesource surfaceontheultrasonic-beam cha racte ristics alongthe axisofsuchapiston.However,using only axial changes it isdif- ficultto understandhow the acous ticalbehaviourinthe whole nearficldis 1IIfer1l',1by the velocitydist ributional thesource,

(Iuyomarand PowerslJ9S5),lid anotheriuler('stillJ pieceof work.Till.')' rnlculntrdthcac.uuslicalpressure in thenearfleldprodu cedby apistoneru-

15

llt'd(h,d in :(I) (n'r -spa n ' : (2)itrigidloHm!"whidlill~llla h'~llw sound wa\"!' inlrrft'rt'llc(' IwlIl'N'llfront auelbacktrnusrlnrer:lltl(1Cl)iln,~ m"11 1]"11111,,'I'll"

r-ak-ulatiouwns pcrfomu-dill tilt'angular spr-rtrum,lutllaiuilttd Ill<'r<'sults showthatthe'acousth-alpn-ssurr-i.,Sl'riulLslyalfl'l'Il,,111)"till'tYI'('ufI,alll,-, IIpto lIOW,noexactculrula tio nuflln-,ll"(llistir a lprt-ssun-Iortln-,'''1L1in-

\101]S wavecaSt'inthr-w1t1,I,'lll'Mfi,'l,l ,inrluclingnonuniform",·I"dly.listrio buuonon thesource surface.has apponrcd in tilt,litr-rnture. Ila sl,,1OilIIH' work of]lasega ll'a t·ta].(ID8;1,IDlH),whichIl"1'il lc~l llH'tri1lls,lu""rasitri.c,i,1 pistonwithuniformno rm al velocityat itssurfilt·r.this lht'Sispro'S"lIl!'1,11"

,I.-riva l ion ofitfo rmu la for l"ilk nlat illgthe llt'llrfil·ltlsound pft'SS1lft' ofur-lrcu.

lnrpistonwithnonu niformvelocitydistribu tion.'l'hr cc -dimcnsiounl graphs ate used toshow theacoustical pressureilltIll'noatfleldandlo rUIIII';m're-.

suitswit h thos e producedby

, l

transduceroruniformsurfa",'"t·l"dty.TIt!' effec tsofthesurfacevel oci tydistrihutionilltill'nearfk-ldart'c1is(~ltss('(1uurl somephysical explanations arcgivcu. AttheSlllllt'timr-,resultsulJUtilJl'd here are comparedwithtllos"obtaineduslng numerica lmethods .

IG

TRANSMITTING PLANE

P

RECEIVING PLANE Figure 2.2: Coordinatesystem

2.2 Gen er al Theory

2.2.1 Caseof UniformRadiator

Because some resultsin the Hasegawa 1"1at(1983,1984) pape rswillbe usedlater,illthis section we brieflyreview their work .

The coordinatesystemis shownillFig. 2.2. The origin of the cylindrical coordinatesystemis at the center ofthe piston where:

=

0, while the spher- iralcoordinate system originis taken at :: """D 011the axis ofsymmetry.The

17

n·loritypOll'lItiill,0;••nt 11 li.·l,llloinl,t',for ill'lill lO'"i.,I"n~'''11T''~l1rnl1nll l.·.1

whi-rr-l'Uistllf'no rm al\'f·lod ly at.thl'piston~lIrrll"'''Iii~1111'di:-lilllt"I'(rum thefieldpoint totill'~lIfrllrl'dCllll'u lfl'~1011tIlt'pistou, 1In,1i~,!l,i \"t' JlIty

(~.fi )

and

cosF=('OsOcos01

+

sin0sin01r''~(1)-f~.l. (~.7 )

Fo rthe raseof relativesoundprCSSIITt'anduniformvelocity distributionon theso u rce~Ilrfac(',thl' constant~in('qlhl.liIlJl(:!..'i)"1111III'iguorerland llw equationrewriueu as

(2.8)

By applyingtil('relation(Arfke n,1970,]J,7fill) e-ik(r'-2,,"c...r+rll; "'"

(1.2_

2n'lros

r + ,.n! =

-ik

,?;PU +

1)jn(~T)II!:ll(hl )l',, (f"US^{1'), "}<

"I

(:'U1)

18

wln-n-j"isIIII'spllf'r i"1l1H,'.~",('IIuuetlon.It!,~)islIlt'spllt'rirxlHankelfun rt ioll of1.lu·""-",,11,1killd,IllidI:,islhr-Lrl!;o'nrlw pnlj-nmnial,equation(2.8)ran Ill'n-writu-n<is

(2. 11)

Aallff1itl~LIl111('additionIhC'OH'l1l [Arfkcn,19iO,p:;S2)forLl'gr ndrcpoly-

nomials,itfollows tlllll

P,,(w s Or us OI

+

SiIlO,~iIl01cos{6 -<)\)J

=

1',,(m sO)P.,(ro sOil

+2

_{",.. ]}

t (("-

_n+11I.^IIJ));

1',:" (rosn)J~:"(COSOI)cos m( ¢-

4id,

(2.12)

whereP,~'is1heasso clat od Legend re functi onof thefirst kind,11Ill'gra t illg bothsides ofll1l'nbowequa tion withrespecttotjJfrom0to2ll',wehave

19

=

:2;;,/'"( n~Ojf',,(n,,,Ol) [:2,1:1)

";(,,,0)

= ::~ ,~~,('.!/I ⁺

^Jjj"(b'l/',.(r",,Ol!,,(1.o.7.,l. (:.!.l,l)

where

Zo=~TO'Z.

=

b'o,and1'.islilt' IJisla lll"('fromtill'rimof1Ill'Ir illl " l1l illl ' l totheoriginofthesplll'licnl "" "r.lillill!',.1',,11'111(Fig.:2,:2).'1'111'11 " aI','1l,i\"'11 bythe il1tlt'finit"illlt'g raJ

(:2.11i)

andsatis fyLIlt' relations[Hascguwn ctal.,HIS:l)

Pll '

A,,(Z)=

-Z['"(~)II~,~dZl+'~l(- J)"'+J(21!-'1m+l)p"_,,,,,(~)h~,~~~.,,{Z )

^,II::::2

(',!.17) A,,(Z )

+

A..

_~(Z ) := _ Zh!,2!I ( Z)[P,, ( ~)

^-

f!,.-A~)I ,

^(:l.IH)

Here[j]lIle<llls1111'integerpart of~.Fo riLpistonwithallnir" rlll",dol'- itydistribuf.iou,till'ab ovefcnuuleI';L1Iheused(firt'iIlt:lLl atill~tlu-;1'·<JlISI,i...r]

20

l'tf'~SIIr<',Iist.rillllt iollilltIl{' m·i1,'IlI,ld. Ilnwt·\,("t.tilt'boundary conditious il1ll' ''S'',]UI'0 1l tlu-rty.~lalnn- 011\'Ieet orwhi-hwill",Ifl'.-tthevrlocit.yIlislri-

2.2.2 Simply Suppor t e dRadia to r

;,:"wWt'will considerthepruIJI('11Jof nonuniformvelocitydistribut ion

011lit(·radiator.This may,in^{SO: HI'}applica ti ons,1)1'more realislic thana uuih-unvelocitydistr ibut ion.Furlustau- e,apiezoelectriccrystal,asitis

\lSt'<1 illultrasonic transducersto measuresllsp(,llll.'clparticles,fixedatthe ('dW'j;andhackedwithadampingmall-rial,willnot moveas apistonbutwill have11velo cit ydist rib ut ion that isaminimu matthe edge and a maximum illthe center. IIIthecase of ultra sonictransducers,experimenta levidence iudir 'tt,l·slhatthevelocityprofllchas¹¹nonuniform distributionratherthan

ituniformdistribution (Shaw,1956;Shawand Sujir,1960;Kikuchi. 1969) Laura(1966)discussed tll(' Iarficldbcampattern producedbycircular rmliatcrswlthtwoIlilretelllncnna! I'cfodtyprofiles atthesurface. One, ndlt'tltill' simplysupportedcase isgiven bythe expression

(2.20)

21

lind Oiltill'1>"' I1l<lary.

wt.r-rr-till',I";IH'constantsn'l'n '~1'1l1i IlP,till'lI't'il!:hl ~fortlu-,lilfl'rI'1I1vihm-

tionnlmodes.

(2.22) whichs;llislil'l'tlu-llOund;Lryrouditicns

lk-re('"hasthe samemeaning as

n ".

'1'1,1':'1'exprcs sicus n,"slit utl·arrllrllh' approximations forthe'ncnnalvekrcltyof a cirrulnr radi;llor(l.allrn, lllCif;).

Forbot h cases, they represent a reutralmaximumnormnlStLIfan'WllWilynnd zerovelor-ity attheedge.Promtil('shape'S

ur

till'sound pressureIlisl rilml ioll Oiltheacoustic axis (Dekker etal.,197'1andGH~'IISP;1I1 ,1!l7fJ),itrailb(~Sf~'l1 that the,\' arc simila r(or thetwo differentC!ISI' S .I" ort],'rtueasilycalculan-,

tlu-clampedplutocaso can be obtainedillthoSILmt'way.

22

I --.. __...._. .~ • , - - ..._ - . .

p,(cm )

Figllr"2,3:Vt'locit)'distributionontht' pistonsurface forlilt'transducer di;,lIwlNmt'dintht' t'xl>(>rilllenLs

Thefollowinganalysisisbased011the assumption thatthe firstmode of

\'ibra tiondominat esthecrystaldisplacement,Sothevelceity distribution onthedisksurfacecallbeapproximatelyexpressedbythe formshown ill Fig,2,3

{

V,(I -

4)

p,<"

",(p,)= .,

o

PI~til

(2.23)

llcre"0istit!'surfa("t'normalvelocityas brfort'andYoisthesurface velocity attheIr<tllsmiUrrcenter,which is assumedto beunity forlater

23

Byllsing equatlon(1.11).lin-\"(·l",·ity"111110"~lIr r;."""ftlll·l'isloI11,,·

Sllhst ituting1'0(1'1iilllu ['{!'Iali"lls(1.:H).il1Hlll,il1~"'I'li\li"11{:!.~IInIHI"' llI;I' lion('!oj:!)~i\" '~

I ii

c.

;1)-;1 1

';(1",0 )

= J 1.. --//-- '

c-ikll d "J

=

j 1.

^[fl

+ ~)

^-

~II-i~'

^f:{2fl+

I )j,,{kr)"!,~l (hl

)/',.(<'fJ8I')! rI.• ,

" III al "" 0

=j '4{[{l

+~ )

^_

~][-ik

^E('211

⁺

^I)

'0 rl l III ",,0

j" fkr.)},!,11(kl·l l t ' · P,,(co,.I'),IOI]}l"lt11'1

==-2'ifkif(111

+

l)j,,{kr,)p,,(r f,.•O)[(J

+ ~)

,,=0 (II

L :A h~21(hdP,.(~)I'ldl'l

^-

L:· $h!.2)(h-d/·"I~)'frd .

^(1,1fi)

For nonunifo rmvelocitydisirillll liu ll tillthe'surf,ln'of111('l'iSll1l1.lIlt' velocity pou-ufial al thr- pniutPruns b !s

or

Iwu parts.'1'1)(' firstpartissn'lI

From 1111' l..st~('l'lioll10 Ill' Ill!'r-xprr-saicnfortbr-~'ilSf'ofunifor m velocity llislrilHltioli.TIll'1'("'OI I(]piltlis11,('modifyingkrillwhichis causedby th"

v(·I,,,.ilyrhilll~ill/!. Olll lll'pixfunsurfan·,If\I""]l'(

~IJ,,(Z.)-II,,(Z,)

equntiou(2,26)ca u tl u'llben-wriucnas

(2.2;)

(2.18)

-2.ico.. r¹ 1

",(,',(1)=

T

E{2"+1)j" {kl')f',,, (co.~O)({I+;;f )fn(Zo ,Z4)-afk2bn(ZO,Z4)]

(2.29)

Elilliltion (2.28 )issolvedusing intcgratkmbypartsand tilerolarions(Ar- fl;,pll,19iO,/l,52fl,V5<l0)

Zh~,''(z) ~(2" -l)h~,'! " Z)-Zh~,'!,(Z ), (2.30) (11

+

i)J:lf1(Z ) -(2/1

+

I)ZP,,( Z)+IIPn_dZ)⁼0, (:?:H)

25

and

follows(M,(,AppC'udix ):

IIu{Z )

=

^{(_X l}+i2Z

+

2),-' z 1:!.:I:I)

BI(Z )=-7.,,(2

+

iZ)._.Z (:!.:I-I)

B~( Z)

=

Z""1(2 )

+

:IZo" I(Z)-.1..1017.)

+

7.~lto(Z) (:!XI) 83(Z)

=

Z²A3{Z)

+

TZoAA X )-

~:\I(Z) + ~ZUZ"I(Z)

^(:Uli)

8.(Z)"" Z2[:\.(Z ) -

~A2{ZJI-

^{[10A1(Z ) -:!Aopn l}

⁺ ~Au( Z) + ~ZDA3(Z)

^{--rA2(Z )}

⁺ ~ fj2(Z)

^(2.:17)

8,(Z)=Z'! A,(Z)-

~A,(Z))

-2I'A, (Z j- "A,IZ j)

+ ~A,(Z) + ~ZOA4(Z)

^-

~A3(Z) ⁺ ~1JJ(Z}

^(t.:IM)

8, (Z )

=

Z'IA,IZ) -

~A,(Z)I-

^{'19A,(Z I -.IA,( ZI}

+

A,(Z))

+

2[W/h(Zl -

~AD{Z)) +

¥ZlJ/h (ZJ - ]:>A.( Z )

+ ~1J4(7.)

^(t.:m)

Byuse or thes e eq uatio ns,lilt' tl,lalifillhdow r-anhC·lIbt"inf'll.

I'll

II"IX)=

Z'IIA~(Z) -~A"_2{Z)J-:l L ( - I )'~+1 {2,,-4 m +I )A"_2"' (Z}

/I ,,,"'1

+

:l(~IIII-^{I)ZuA. _,(Z)_2(11}-1 ~~2'1^{-3l:t,,_1(Z j}

+

^n:10,,_11/>3 (2.·10) ,\I.U\'(' t'qllHliourail ht,furlhf'T~ill1pJi fi('dbyIt>ttiug

Ii i

!/,,(Z }='~I(-Jr" + '(111- ,1m

+

1)A.' _1",,(Zj

••(Z)~.,(Z)=0

11';?:2 (2.41)

(2.42)

I~ _ ll

H,,_'I(Z )

=

,~(

-

J)"'+1(211 -,1m -:J)A"_ 2'''_2(Z )

Tlu-n,l.11l'followingrecurrence formulas callhe obtained

Il';?:2 (2.'13)

9,,(Z)

+

9"_2(1:)

=

(2u-3)..1"_1(Z ) . II~2 (2.'1,1)

SuJ,~l il llti lLp'9,,(Z)into('(J11ation(2..10)finallygivesthe recurrencefor- 111111afUf1J,,(Z las

27

IJ,,(Z)'" 7,'11:1,,(2)-

~tl,,_~(Zl/ + :!.:.~';;=-!1ZII.·I"

^_dZ)-

:!(II-

1~;211

^-:I)A"_'l(Z)-:!,f/ ,,/Z)+

:I("I~ 1)!I" _~( i:' I+ ~ H,,

^{,li:'I, II:;-:\}

(:lAo'l}

{2.·1"; ),(2.,I.i),ami (2.2f)J . Ikf illl,' l'n"ilfll'r ".xilll'ltjllni~ n~,'.lill1111'

solution,E'l ualion (2..15),'1111I)C'provedhyoIilfl'l'I'llliill1n)! itsrwc si.I" swilli respe ct10s, anddetailedpron·dllwisgi\'l'llillApju-udix11.

2.3 Nea rfield of a Circula r Pist on

In ordertoiu\'('sli ga h> tIll'm-arficldr('p;ioll,ti'l'l\('Oll s lir'i,1 pn ' SSIIf\'which

IIsC'ofth('nhcve cquatjons ,furpo ints !o rall'dOililJ,;ri.l illlilt'r-t:plillll'.'!'I)/' pist o naxis corrospomlstoti,l'::r-oordiuau-(Fig.'1.:lJ. 1'11.1I,lli'"rlIH.,!!,'ltIill

28

uxjs,s"w,'ouly rflnsid,'rS0l1U,1pn'sslireinnil'pIalI(' ofr>0and:>0,

.' w

[,amlll l'tr'rs WI'(hns,' for (f1!(u liltill.e:llw ....ousticel fiC'ldereIIll'sa me as

Tl a'functionsjt<,1t~1). I~,.A",1I1l11B"ill"culculatcd byuseorrecurrence

fOTIlI UIII.WI' dllJSl'til<'c1isl llllfl 'fromtheoriginofthespllC'ril'a1coordinate

S}'sll·tIllu tin'originoftlu- cylilld rir..1 coordluutcsystem tobe I'D=22 em, TIll'(lillJlwll' r of tinl'i~lnllb equal 101.9I'm,theFrequen cyf

=

2,0Mfla audllll'.~PI'l'dof SOlllHlinwa ter b taken tohe 1,IS;)lTlS-I ,Wechose1'0=22 CI1lSIIthat ,tLplan' sbeinglargerthan0.2emfromtransduce rsurfacethey allsati:;fyrdal.io llshipr<1'1illequation ( 2,9).

Fig,2.'1showsthesoun d pressu re ontheacousticalaxisfi.e. theaaxis).

TIll'das lH'dand solid curves.respecti vely, areth e resu lt swith the nonuui- Ienu andunifor m\'t'lodlydist ributi on onthe piston surface.For a circular piston with 111lifun n\'l'!ocilydistribu t ion outhesurface,thelestmaximu m ap[)earing011theacoust icalaxis isdeterminedbythe equa tio n(C layand

(2.46)

whe n' (IIi~[1]('piston radiusand Aisthewa veleng t h .Fo r 0111'case,

f

R::2.lJ

29

NotTJUililCdPressure

_ N

e f e

~ ~ ~ ~

. . , . - , - - - , - - . i_ ..._ 0;;;;;i....=::::!::...

~ ~~~

...

^-

:...-.-

j

~

~

>

~

~ \ \

1

_{~ ;;-}^N

ⁱ

s }

is

g z.

if

Ii

^;;

ⁱ

f

^;:

/

t

,I

.0 '

;;; !

"

^I

l;:

;

!

i<:

if !

~ ^"

.\JHz ;lIul .\::::0.7 nun,tl!is givt'

::,::::12.:!flll . (2.n)

This"1111hseen fromthesolidcnrvo!nHg.2.l Whilebothcurvesin

Fig.:!A .•1,/)\\,llIaxima and minima,itisevidentthat theamplitudeofthe

d;I..IIt~ II·lIrn'is verysmallcomparedwith L1lat ofthe suli,]curve. Aswell,

t.herr-i.~no zr-rnpuirrtforthedashedCUTI'" ,and itslastmaximumpointis 1l(';I lWtil!'piston slirfarcthanthatof thesoli,]curve ( approximately~~).

.\lIlIfl.hc~,'eharactcristieaappea rto be ill good agreementwith theresults

ofDckkernal.(1!l74)ami Greenspan(197ft).

FigIlTl':i2..'iand2.6 arc,3-d il1len~ionalgraphsofacousticalpressurefor uniformandncn-unifcnnVorespectively,Wccansee that theinterference patternrxi~t.~inbothof them,butFig.2.6looks muchmore regular and smooth,ThedifferencesbetweenmaximaandminimaaresmallerinFig.2.6 Ih;11IinFig.2.5.Theseliguresimply that througllcontrollingvelocity dis- trlhntlon, dlllereut spatialacousticalpressure fieldscan be obtained,

31

p

32

·0.665 -'.330

F'illurf'2.6: 3-dil1lt'llsiongrllph ofacoust ical pressureinthe nearfie Jd....it b nonuniformsurfac e \'elocity

2.4 Space Average Pre ssure in t h e N earfl old

nrousticalaxisis ofte nimporta111,\Y•.willnsr-llll'furl11l1ll1Poht.ain..1,. I'on ' rodotenninohull' the acoustk-alpn's,lIrc' "II lin'pliul!'whlrhis])\'rl'!"ldi ,nlar 10the::axisCllil llgCSwitl!axialdistanc e.

Tile'coordinate sy s tem lISpl!inIh,·"IIJ,-lIliltious istln-Silll ll 'as~h(IWIIill Fig.2.:.t Assumethe(,il lrul"l iu llis ran-ip!!»utfurtil('pl alll'::=1"1,im.l tIn' space averagevalueof,)OI"lTthe-fl'n,j \ ';'l,l!,rrys t nlis

(2.4!1)

(1 ....iO)

Subst itut ingequation (2.2!J)iut o!~llI a lion(1..18)gives(lwrl'I'll ''IS!'1'll':'i:'i1ll"<' instea dof velocit ypotential]H'CIUISCthetlifTcwm:cl.CtW(~)1Ith(~lIlisjll:'ilu

31

-I "x.

,.'1

<]1>=r.;;f'.!:iI~'('.!U+ll[fl

+ ;jl1.lZ

o.Zd

_ I/l1p lJ,,(Zo, Zdl i:~I'J L

^{1'i.(kI·)pi'u}

~

^{r1)rd,·dr$}. (t,.';l)}

'I'lp'int''gfilliullIaoo r (:<10 beliinll'tirlC't1as

J ~~J., f· i ..

(L-r)P..(l'u

~

^r,},.drJQ

=

~Rcll;::~" 1

ZlI!,l' (ZjP..(

~ )dZJ

""

~ Rt[J.( Z~,Z:)J ,

^{(2.52 )}

lI"l~....•Z~=(ro-r, )kandZ;=r.2L·wit hr.~h<'iugthe distance (romthe

rim"rtIlt"n'("("i\;ollplanetotheoriginorthe sphericalcoon:lina teIystem. E1lu..ltion(2,51)thenbecomes

<j'

>=

_$'l~

-'~3'l7l'2iE(2"+

_.,,0

I)I(l +~)f,,(Zo. Zl)-+

_{al (l/}

I'1b,,(Zo, ZdIRelf.(Z~.

^Z;)J

(2.53 ) Fig 2.ishowsthevilIIWSolAI'(' ra p;I'aco us ticalpressureonreceivin gplanesor

35

I '2/.1

0.95 -_ _ _ _ _ _ _ _ _ _ _

.•. - _... --'- - -0.9

0.9

. ···-··· · · 0.95

0.8l

·.. ···..····1.0

0. 8

0.75

.•...•...•- -•.•...···-..•...•...·. ··· ·1.1 0.7

0.65

0.6 1.2

10 12

0.llO!----;o---:--~--7---;'n-,~

Figure 2.7: Averag eacousticalpressure on theIlilT~r('ntsize of r('cl..ivillg surfa ce(rad ius a2 ) producedbytra nsrnifter(radiusal)withthe simply suppor tededge,atf=2.r.MHz

36

0.118 o.IJ6

_.,--,--~-~--

a2/.,

.~._•.•.J.4 0.114

~0.112

!

^0.11

~

^0.108

.. ~

^0.106

0.104 0.102

10 12 O.IOL_ _ ~_--c---~----;:----7,;----;';----'

Figure2.8:Total acoustica lforceonthediff("rC'nLsizeof receivingsurface ( radiusa2 )producedbylransmiUt"f(radiu5al) withthe sim plysuppo rted t-'t!gc,a.t2.0MHz

37

-'.

0.8

09: ~ ~:"

~: <:::::: :~ ...

-.-...~

... ^{. ;}

~....~ ~_.-1.0

0.75

. ...

<, .1.1

OO~

^{••••••••• ••• •}• • ••• .. ••••..•• •• 2

10 12

Figure 2.9:Averageeccusticelprcssurcondifferent lti...<'ofreceivingplane produced bytransmit terwithuniformsurfacevelocity....l2.0M lh

38

,!iJr"li ll~~i7.1"Sas aInurtioni,xillldistunrc c.Fromthc«- curves thefollowing

(llAI"l'n,~,'valuesuf pn>:<slll"t' d"'"ll'asl'withinert'asillgradius, andthis Itl"HIlSt.lut lwam imcunit y«mront ratesarou ndthe acousticalaxisz.

(::!)\V1H'11theradiusof111l'receiving arNIisaround 1.0to 1.1times that orIIIl'l,nlllsllIitti ngarea, tIll'1l\",'ragl'pres-nrcalo ng the acoust ical axisis

(:1)\\'11<'11the radius ofthe rcrviving areaisove r1.1times thatof the 1I',IIlsmin ing areathea\'('rilgl'arcusticalpressureincreases alongthe aeous- lit'illllxkThispointismoreeasilysronfromFig .2.8inwhich thecurves Tt'l'rl'SI'III,total acoust icalforce011ther('{'('iv illgpla nea"Jallparam e tersare t['I'"~ill1l<'as for Fig. 2.7.For112=1"1111'the totalaco usticalpressure reaches itsmaximum , an dthenitdecreases again. This isbecauseof the effectsof diffmct tuuamlint erferenc e.Iisuallydivergence of beamwid th isthoughtto Ill'unimp ortant intheucarf kltl,buttheCUT\'CSin Fig.2.8suggestthatit shouldnotbeignoredfersomeapplications.

Fig. :U Ishowsresultswhen['0is uniform.Theparam eters used arc the

S,U1l1',1l'11,0S1'in Fig.2.7. B)'compari ngFigures2.7and 2.9,ilis easily

seen lImttheaverageaco usticalpressure011the receivingareaalongthe 39

acoustir alaxisdHl1Igl'Sless Wh"11theboundaryro.ulitiou furtill'silllpl,'·

~llPllUr\{,tlra-li.uor isapplicd.Till',,11I1lt's<IfI1Il''·Ilr\"('~illFi)!.~.;itT",lIs"

smalh-rtlum thus..inFip;.:.!.!l.

2.5 Conr p a riso n wit h Numer-ical Int egr a ti on

The anJllstkalpressureintlu- IWilTfi" I,1I'xl'n·ss,',1]'Y ('(llliltinn(:1.~4 ) CIlIlbe integrated numerically.Inorder10 dll'rk\Jil Tresult ;1I1l1;llso LliSl'<' whethera numer-ical met hodisrr-ally rlr('ctin'(firtill'ri'S<'of lar,.;"Tilti(Jof sourcesizl'10wa velengt h ncar1.1](~SlIr(an'

ur

tll<' pjsron,wr-IISI'tlw{:1I11SS qlladTahu('mot.hod(Pr ess ctal.HI8G)to inll's,'at"r-quntiuu(2.14).

Thl"coordinatesofthepiston hcing cOllsi, ll'rl"dan-shownillFip;.2.IU, Tilcpist .-,Ilhasradiusa\.amivibr ateswithsimplehunuouic motionnUI'Hut!

toits face. T[n- observationpointwillbe a function ofT,tlw rilllialIli sI. 1UIl'(~

Iromthereuterofthe pisto n,and0,thc eugle hcLwt!"11till'pistcnexisandthe radialvector.The total relative fOTCC(allhewrit te-n as in(~'I\latiol1 (2,24).

Therlis t n nreRis givenby

(:l.M) BCGlllS("ofthe axia lsymmetryof the problem,thetotalfo rceonthis plane

x

y'

Figure2.10: Coordinate systemIornumericalcalculation

is

thepistonandtln-r£'n'i\',' r.!"ro ll] Fig. 2,10W(' muywrit,·

r- r==cosO'

lind

Subst itutingtheabovr-into('q uiltio n(2.55 )antillsingl'tlll:lli,,"(:t.:!:I),l\iv. 'S

1. ( ",,- ' ? 1," 1.

^2• ~"~SiIlOl'l(1^-cl.)

F,(r,)= -iw27r/Jo0 ' II 0 ('- 'II /1("05"'0.'dt/Jlll pl,IO,

(:t."!l)

IIIZrlllau(,k's(197111';'IIl'r,tilt'I,bt u llis,li\i(r,-clinl,,:121f(T )' I·l"IIII·lIls illorde r to obtil inn'!iult~within0.5%error .Apistun{IftI"lillS1I.!j,j"Illallli trau!illlilt illgfr('(lllr nr y

or

2Mltz511011111thus I...·,lh·i,I,...lililuatI"a"t29115 elements. IIIorth-rtngl'l highaccur a cyr(·.lllt,,_19'1poillt

(::"1""

'Iumlril.- turcisusr...1ftlr('Hrh ur111"tlll','t'.1illll'nsiulls

"r

iut' ·J1,rill i•.u.Tlu·!l·r"n·,till' lrans rnitt ingpistonis,Ii"i dcdiu lo31iSfl'l dC·III1·I'Is .

O.ll,.---~--~--

0.109 0.108 0.107

J

^0.106

· 1

^O.IOS

<

0.104 0.103 0.102

1 0 0.10'10~---:,..---~--~--~--~----J'2

Z(cm)

Fi&ure 2.11: TIletotalarullst icalforce onthercceivint; plane,for lJl=

a,

andf=2.0MIh:

.3

IIIFig, 'l.ll ,tlw~()lidrurvr-i~roIJ1;lin"11Inun "'I""ti"ll( :!.;i!l) .;1!L11tl...

,11l~11l~ 1('lIT\"!'isIrvmr-quutiun(:Li:l).:'\1';<1'th,'~'lrf;W"of11\1'l.b l 1.111111' 1I1111H'ri' ;11n-snht'xljihi t~'JS!'illal "r,\'l' I,II;I\'j'"ll 11"1\"'\"1'1', 1"',\,,11111;11",1]1~ nilllu-~t)1idanddashed,·nl'\',·:;a,!!,n~'\'l'r.l·1\",,11. lIll',lilf,'n'lIn':;;,1,-I":;,, rau !!,,'art' !Jt'caust'inll'fr(,rl'lI("(,isIJft1t[ul't',lltlilililyI,yI'ha:;t·dilf"rt'lIn'li.Th,'

tn-an-rau observedpoint is10rlu-sllrfitn 'of1Ilr'],i,",n.til<'11I r~I ' rIII"I.ha", ' tIiJkrt'IW"!ll'twl,t'llsigllilhi1ffi\'ill~rWlllilll.l' 11\'ugil"t'II" I"1Il1'1I1s"rIIIl'Sll ltlT",

lt isdifficult10 gt·thigllil'Tllr<lC)'It"1l11>lI~ill,!!;llUllll'rin\1flwth",[s\\"1ll"lI l'\'al ll ilti ll~rapidlyvaryingflllwti ollS, IIrunI",M"'11(rullls" li,1,'111"\',·ill

l\~. (2.[ I"oscillationsappearat plaer-sdllwluth,'Ir;llIsmit lt'l',~lIrf;It"',W"

havefoundIl la lusing more pointsinlilt'l'IlklJlaliullprod lll"l'S"Illyslight dl;mg('s intheresults.Forlargl'ratios

o r

ra.llator sizl' to \\'II\''' ],'II/o:tllitb

ralcularicn is veryfasL(lI r oll ndt ...otituvsfa ~ l,'rthan1l1111wri"fll lllt'1h",l for our case].

\V('havethusseenthnt,;,ltll('II~111111'[litll!'11i"fitl'''Il.~l i''111pt t'SSlir "ill theucarflcldis complex,b)'comparis ontill'il\'1'l'1I E';I' prr-ssurc untill:n~"'iVt'r

~I]rriln 'lilri,-:;I1lllrl!I('S~illJprl·,"i"I,lywithc1"' lI1',ilJF,dililIHlc,'alongthearcus- tiril1a)li ~ Iti~1"" ';"1~"urlitis that il isf(·,, ~ i hl,·to us,'1..,cks,'" l l('rIromlIlt'

,li_l lil" ll<'rllllliful"lllly"IIaverageacrosslIl('width

u r

the1)\".1111.

45

Chapter 3

MEASUREMENT OF SYSTEM CONSTANT AND ACOUSTICAL PRESSURE

In lilt'prt'\'ioll~dl1'lpll·t.W(',k'f i, '!'<1iI.11JI~,rI1il"illn-sultwlJi.-llSillilY!>

thatthevelocity,lislrilmli" l1011 the~urrart'of a Irilll",I'lo-r pi;,)"!';1lIillll' "r.

IilnlroleillaITt"'l1n,ethopressurefirld.Fu rItrl',\1transduo -r,11...fll"'rilli"u oflrll1Jsolu('{'rsmnyhr>lIrf{'(t N)byothervnrluhlcs,inrlll rli ll~(Iwilt'IIl1st inol

46

Irir,·I"'llt·ll l.Tht'~"fiu·l. m;<illi,IIlIll'IlCl'IIII'Ionuuf IIII'lnlll~d llft'l"out1"11ill t-nn-"rI IIl' 1'11Is,'~hal't'lI11<ll lll's l'a li a l\·iIrial iunSlJf1.lw l' T1·ssIlTl·alll pl ilu d l',

SlId ,asI,ll" t·I,·c lrirn]i'r"lwr li,"IIf 1Ill' puISt'r/rlocl'i \'Crcan playilroh-.

Invh-wof thisv;,ri;,], ilil y. itishnpor tant tod,aracl t'rizl' 1111"lrll rrsd ul'l' rill terms"fIrilUsmill il l~;'U,]1'I'l'('i\'iug('.-pouse Iunrtlon, wit hou t priurkuowl-

,·,I,c,'·

of tIll' ,Il'l /lil,',jiul"rJlIllSII"!f'lnrl' oftil('> Iransdurc rall,1its assncial,· {l

"ll·,.t rullin;, Thisis 1111')ll lrp (lM'ofourexperimentinwh irl ,the- <Jwril llsvs-

d';Illp;illgwithdist1< IlCl'Iroruthctra nsmitt erwillhemeasured.The method tilme-nsurea\'l'ra g.,ptl'SHlfl ';S10 lisethe transducerin thepulse- echo mode, with

nit'

Iralls dllCl'rilsl·lfIlI'illA bo th the transm itter andrec ei ve r .This 1.\'rhll;qlll·rnuyit·J,Jnmplitude.spec tralandspat ialinfor m ati on.

3.1 Measuring System

Agl;oss.sid,·,1tankof 90nn{lo llg) x,10cm(wide) x4 2cm (high)was1JSl~~for Ilu'1I1. 'iISlIfl'I\l,'lll s. ,"HI wnsfillet!withdeionized water,asSllU\\'/1in Fig, 3.1, TIll'tankwasm"ll11h~1('1Iall;"Ijlls l illolj' support,and011the-lopuftill'tank

I.EVELI.INGSCREWS

Figure:U: Sketchof."xjwrilJ)l'lI{i.1se-tup

48

tln-n-wnxI, ,,,, tlll'r IIdj IlS(ill,I,., h·vin' wl,idll·x ll·ll'lt ·,j intowill,',". To lhl'lillll'l' WII:- "t tl' ,.]l1·dau.!/.",nilri" lill.~nJr lllJl{ ·ITi .,ll,n'Il, I I, all ,llril U ~(ll/n·rwllirll WIIS

Wilh' ,"ir illll' rf" n· .TIm renter. uPlwr (- {i,lll ), l'll tll" II"l'r(, 6d D) frequencies of I,ll!'(rlllJs. lll n·rare:!.O!J1\,111:-.,2Ji2\lIIz.iludI.;"W~ll1zresncctively,Thus, 1.1 ...lcuulwjdthisaround6!/.!J% of tIl{'o-utn- frelllll·lICY.Figure 3.:!sho ws IItiIIH'-St'ril'sandspect rumuf tIll' n'Ol'c! I'.1 pulse.TIl('drivingsigna l,a 2/ls

\'lI1!,W'pillsI',isgeneratedhyIIl'i1ra llwl rics50,j.jfl R pn[s\"T/fl' ccivcr.This Iwdiltj-pirn] rist,time of2()nswhenIlri vi nga purdyresistive50

n

load.ThE' sIW IH'(Ifthl' emittvd ultrasonic waveform couldbevariedby adjustingthe 1'llI'rAYunddampingcont ro ls ofthe unit .TIl(' s;,mpling tim e islO'lS,giving i'n.l :-.1111.frequencyresolutionfind 0.75cl11 spatialresolu t ion. Received tiulI"'["l11i1illsigna ls wererecorded hya digit»losr -lllosco pe(LeCroy9400) which\\',' 5rcuucctcd to a personalcomputer.\11<1plotterthrougha GP lB ill!t ·rril:l' .Thel.r e ro}'!J.l00 hastwo indepen dentcha n nels , eachproviding an :'·1Jit sa m plingAIDconverter up to100 rt-H lzsampling rate and 4k length uf nn'11111ry.Two other functjcn mem o r iespermilFast FourierTransform (VF'!').,]r'>l"i1l!:inl!:andother operathmsto[u-performed. The refore,itis p"ssilrll'til1111sonu-datal'TO("l'Ssiugbcforothe dataaretra nsferredtothe

49

I ;

. ..

^~

J

ERFE CTREFLECTOR

Fi~ lIfl ':Cl:l)i;'~I "11lillusrratingtil l'scll-rc-ip rociry faiii,;.,1iOI1 rm-asuremcnt

rullll,utl'r ,

3.2 M ethod

Theprincipleof tileself-reci p rocitymeasu rementisillustrat edin Fig,3.3 [Reid, 197·1).IntIl('self-reciprocit y syst emitmirrorreflectorsu-hasallair-

watr-rinterfacei~usedtofoldthe fieldgeneratedby the transduce rbark upon

the sarnc rransdueer opera ted as a receiver.The measurementcan be made t,jllwrin theIrausdurerncerficldorIarficld.Inthe ncarfieldmeasurement lilt'plane waveapproxima tionismade, andinthe Iarfieldmeasu re ment a Sllht'riralwave is assllrnr d,PllJ~Cllll'th odsarenecessarytoallowtimest' pa- raliollllfllu·trnnsmi!l ,·d ilnd received wave.Wewillusctheself-recipro cit y

51

IlW"~lIn'1Il1'1l 1inIlll'tril1l~, llln'rIwar fir-l.l 1',1Iltilizill,C: sh"rl pub, '"1",'" Io-II I;llo' III<'"\'""llIt,,ovc-rull~y." I " 1I1""n sta nl,,",11o 111"11,111"1'",-I" " ,s al"'I,1!,II",;,,· ,,11~·

tir-alax is,us d"s"l'il""II"'!.>lI' ,IJ,'rilllsl'tlu-111<','SIII'<'II WIII\\'OIS,'" rri,',l '>lllill tlu- trnnsduo-r uenrfir-ld.Ihi'dr"" l

"r

IWill11 wi,lt l, sl'l"l'adiuj';willI".ij.:,IIUIl " 1.

Tilt'on-ax issoundprf':;~\lrt'Tt'1:l'in·d hy

, I

^trnusdnrr-r , which;1I, t ,~1tllij';i lm ll~'

(:1.1)

whore

J'.

and /IJan-thr-Irilll~l1 lit ll', 1nudn-rt-ivi-dal'llilsti,'1'1"1"'';111"1 '"U"",allli o~,an>till' wntcr-alrn-Ilr-cfion("(wHil'i"lItand "ttl'lmatiullin\1',,1,,1'I"l"'l"'''' lively.Let.Vo(t)illltll'i(t)Ill' ,l"l','lwdi v('ly,theoutput;\IId input,\'ulLil~" ortil<' lustrumcnt whichareInurfionsofLinu-I,all(l'~I(l )alIII,~.(I)hi'the-cve-rnll tra ns m itt ing and rrn·idllF;r<'SpUl1M'uf 1111'systemillcllldi lig110tll,·I.. ln1lli,·, andtrans d uce r. TIl<'illSlll1llilll1'UlISjuri'!.'lltpn' sslITI'IJi(/ )and lust illItll11"'llS output voltaget'o(1)of IIll'Il,l"t'ilwarcn'spt~li""l'ly

1"·'1

1'0(1)

=

1l. (I }

* .•

,(l),

.\2

when-1Ill'..stt·ri~krl·PTt~'11 t.S1111'n>ll111lnli" l1npt-ratic u.A"ro nliuAtothe ...1;, ri,,"~llil'1",' wn 'nllll'l1l1 l\lIluliulI"n,llll<'F(l1IripfTfillIsforlll ,WI'haw'for IIII' rn111".n ,'.I'Spl"'"

(:1'<) lInd

SII1,slilllr,illg1':(IIII,t iulI1'(:I,ll,(;I.:J), (:1.<1),(;t.'i)intoequationEqualion (:I,!), tlu-f" lIuwingrt'lalion~hillfo r(';,('hfr''ljllmC')'Gillbe-obtained

I

!

(:1.6)

111our1'lIspl'l1sionnWilsnr('llll' lIlWI'liremoreinterestedillV;(w)•S",(w) ( rilllt'ti overallsystem ronsrant]rath('rthanSM(W).Intheabo ve equatio n It..."nudn...(w)callbecalcula ted,whiletheVoand:aremeasured. Sothe ll\'t'rallsysteruconstantS!/l';(w)is obtained.Tho equat ionforcalculaling rdltorliol\('Ut,mdt' 1l1R_for norma l incidenceis

(3.7)

53

lind nwdilll11 tw« n-sprv-tivelv,IIIourr;'~t '11,t'ttl""i~ilr<11111< 111.!I!I!I1. TIll'

when'TandLure, respectively, tcmpcrntun-"r1I"i11"rrun]rn'([lll'llcy.WlwlI

in'1'111111':J,Iart'ihr-nWilsllrt>t1n-sultsfur0\"('1',,11,'.I"~1 " l1l n'II~ I"1I11/,,,1..1

f(~Hlz) 1.35 1.55 1.75 1.£15 '2. 10 :!.:l5 i,[15'.!.75

B,tIIIV) 550 800 1050 II rlO I:WO It00 81)0 {i.'iO Table:I.I:Mcusunxl~t·n~il.i\'il.y

For the reflectionmeasurements, thedistlllll't'Iromlilt'SUUTCt'I.utilt' rec.ivcrruug.dfroma.5cmlolzcm inincrf>111l'nlsofInn.This('U\"l'rslilt' ent ire- ncarflcldfor our case.To changellwdisI1ll1t·t'(rum l1wlransdlll"t'r tothewatersur face,waterwaswithdrawnIrorn tiletankbysiphon.TIlt' distunccebetweenthesourceandthe waterSUl r,lCl'wasfldt'rmin("(1fly

importantlokee pthesurface orthetransduc erparnlleltolhewatersurra!"!'.

·rI,,·,Ii~jtil l,,~.·i lJ,,~,·.,I'"WIfSIti~J,..n~ll,,\·II....',II··,.'i1'U !,,,Isl 'r/n,,,,'it','rru r

,'\"'Q'trIfIlMuissi,,".an.1 11u-al1ll,li fiMI('di u wasa"(lllito',II,)"till'dig ita lscope atIIri'lo'"r~!)~ll llwil hIII!'n..f!('("h'tll'ulscn'lllrl'!lill 11 tilll<'window10 l' S1" 11,!!;.An FFT

"r

~.;O pt!itll.~with aHnuuwiudcwWi, s[wr f" l'lIlcd,and II...;t\"'ri'~"ur~unn·n·;·",'.lsl",dra\\'.11,clllclllnl.~1leiI'l"wtll rl'IWil\'rritg('d xpcrfrutu.nUill tyII,,·,lill i'II'f' WWilt toCUlIljsulC'1"toIII' st un..1for fur lher l'r"'·'''!'o,ill~. rll"I'\ " 'rlfh ''lll'I','Clr lllllhasfr("(/u'1I1',\'resolu tion

or

0.2 J\lIlz,

l ilt.,

"PIIl'ri1fl1l/o\l'l't!l.'i';t("oll~ d('1lo.'limits MC'1.097 <llId0.90 ·)rC'!ip n:live1y (0 1",,,,,a1ll1 EIlUdlSft ll,19i2),TileHanuwindow(Ul llt'll.1ndEllorhson,19 72) wa.' ("!tUN'llr. lr111C'

Frr

sok-akagC'wouldIX'('Olllilll..

l

tortl'lI Ut'!IC K-l;close 10

3.3 Result s an d Analysis

Iln ', :1.1.unl :1.;"1Shull'IIII'1ll,'i\Sllrcl111'nlsfo rtill'n-lutivcrt'r{'jYl't!voltagein

55

•• •• •• • •• • 11.

10 12

~---_..

_--

^{._--~--_._,-----}

.2

US 1.1 1.05

e

I £

^{Z 0.950.9}

0.85 0.8 0.75 0.70

Z(Cll1)

Figure 3.4:Normalizedamplitudesofthepulsereflected from an air-wate r interfaceas a{unct ionofdistance,al2.0 MHz.ThemC/!,MHcillcn tsare shown bythesolid curve.The dashedand cross curvesarc thecalculatedresult s for anon-uniformlyand uni formly vibratingpiston,rt~pcetivdy.^{The error} barrepresent sthe95% confidencelimits forthe1l1(""~'H(,lII(·lIls.

56

1.2_.- -. -- - - ,.- ---.-.--~ ---~---

LIS 1.1

1.05

~ S ~

11

0.95

1

^0.9

^{- -}

^'\

Z

. ....

0.85

0.8 _-,

_ .- ..

0.75

0.70 10 12

Z(cm)

Figure 3.5:The solid,dashed,cross curvearc measuredvolt ageout putof receiveragainst distanre alongacoustica laxisfor frequency1.4 MHz, 2.U Mllzand2.8MHz,n-speetlvely,

57

"llrwsflor11l,· tlir(',' ,lilf" l't'lltFn-qm-r...j,..."r'"\1'1',1"~imjla r.\l'ltid l1Il1'0I 1iSl1wr IH' ;1I1,wi,l1hinlht'lll·arfi,·),1i~ind"I"'II,I.'1I1"fI) ...fn~I' lt'I"'Y,'\lI"tl"'l'illt,-I'_

""lilt~r"i\I'lrl'islhnt,1'''IIII",rill':'with1111'Ih" Il"li",,1'-111\"<"'('1"'"''"un ""')

lio n surfarr-.Thisis (liM"u"!'t..1illl!" lil ilillIIIl'111''''1""di"lI.'\lllu,,,~IJtill'

tilt'rn"/I .<llrNI"lIl'\'(' s1<n'l1u lJl;iJ;lIili,-it lll.

TIl('Fig.3.6sho \\'sll it'1I11l.l!;ll i l ll lll'{IfII" ,atu-nuatioukrill.,._ 1"..(...)<.rur ,Iilrrn' II1fn' (I'Wlll'ksalIIit'Ilisl ,m "I ':==HI.!.,H I.111',111lit'M'I 'IIIllal"Ut'n·

0.99's 0.99 0.98's

I

^0.98

H

^0.975

0.97 0.965 0.96 0.9's's1

1.2

I.,

1.6 1.8 2.2 2.4 2.6 2.8

F~ency(MHz)

Figure 3.£i:/\Ucllllalionofacoustic...aveinwaterrhangcs ...ithfrequencyat (listitll('(·IO.6ern

59

3.4 The Rel a t ion of Conti n uou s " rav e Fie ld wit h Tr nn sie n t Field

111l'FUllri"r'rOlll~r, 'tl l l"f ,.sj~llill 1 ,ro,1 11,·.~I I I.\·"1" 11,,·,':\, il,~1lr;" I~,hll-" r ,

attIll'"I''''rl";tlj''llf",in,,l ll ritl~"IJIIls. ·,Ill'W,I\"l' ("'111 •Ill'II.·;'...·,.:,nl"",·t,(

lilt' .lil'lIIO'l'·ror,1,.. ' ra u", llln' r1,1,,1tlu-durllliullur ,I...S'U"l,lill"tilll<"...I.!I

hy:1./=I.-I(' -!Jr,'1is0.1I:'",l'1II.U 1.h('sl..~~1of"",11110 1wawillth,'....-ate-r

60

A \~

^<,

a ~~~ -'~~~,;' ~>::~ l

TRANSbUCER

Fit-"l'(":1.7:DinE-TiUllforWUl."path

13.9)

:=OAScm.

So wan'st"('llIingfromtilt,trlln~,luc("redgewillo\'~r1apinthe"rall~~

=

>0.4.')

In III(" over lap region,itiseasytoobta intill'rrJitiollshipbl'twl-"t'llthe jlllbrpro c!utl'dfil~1tlAlI'Itheroutinuouewavefield.Lettingt'O(PI.01.t) and h{R,I)1>('"l')udty.]i"tril'1Jtiuli011thesurfaceortill'lrallslhll:,crandimpulse

61