The actual impedance of non-reflecting boundary conditions : implications for the computation of resonators

O

pen

A

rchive

T

oulouse

A

rchive

O

uverte (

OATAO

)

OATAO is an open access repository that collects the work of Toulouse researchers and

makes it freely available over the web where possible.

This is an author-deposited version published in:

http://oatao.univ-toulouse.fr/

Eprints ID: 7975

To link to this article:DOI: 10.2514/1.1883

http://arc.aiaa.org/doi/abs/10.2514/1.1883

Any correspondence concerning this service should be sent to the repository administrator:

staff-oatao@inp-toulouse.fr

To cite this version:

Selle, Laurent and Nicoud, Franck and Poinsot, Thierry The actual impedance of

non-reflecting boundary conditions : implications for the computation of resonators. (2004)

01234#.101(S&T*+01;?8#PU=V;KCW*+"#$X(6;?2N4#,@*Y&T*+01;K8Z;<=>CD$6P+;K8[&\*+;?CDP ]^_Y`TaBbcdGefghbcijijb)k+l mno?pqrYstmuvwxSyz{Y|}yz{G~€ Qa‚_ƒd\„B…:†‡ˆ‰„DŠG`T‹Œ Ž@ƒ}‘ €’“”~ }•1–—˜ y +•™š €’|}|€’“›œ™œ  Gž Ÿ ¡Tˆ‰b+aBa¢£¤¥@ŠGˆjd\¦’ŠYe—§ œ ˜ r!xSyz{Y|}yz{G~€ kGŽ#˜ p@¨z¨L©zª k xQyL{Y|«yL{G~1€ ¬ €B­®€’¯°±G€’“n² k ªL©z©L³ ´µ¶#·c¸+¹Gº?»6¸ ¼K½z¾‚¿ÀÁO™Ã+ÂÄÅÄÆ1ÇȾLÉ>Ê˽zÌD¾LÍ+ÎBÁOÏÐÅĽz¾LÍDÇÑÆ1Çѽz¾LÒ[ÎBÁÓÂFÂÄÒÒ1ÂV¾‚¿ Æ1ÇÈΒÔDÂÄÔÑÂVÕ#ÂV¾‚Æ1Ò<ÇȾ#ÆÖ×LÂ?ÅĽzÕ-ØDÌLÆÖΒÆ1Çѽz¾#½¢ÙÚÕ[ÎB¾‚ÏqÅĽLÕ@¿ ØDÁOÂÄÒ1ÒÇÛÊLÔÑÂ#Ã+½’Ü?Ò®ÝTÒÖÌLÅÄ×/ÒÇÛÕ#ÌLÔÈΒÆ1Çѽz¾LÒq΂ÁOÂqÞBÂVÁÓÏ/Ò1ÂV¾‚¿ Ò1ÇÑÆ1ÇÑÞBÂßÆ1½Wƙ×L‡ÆÖÁÓÂVΒÆÖÕ#ÂV¾‚Æ/½’Ù5ΒÅĽzÌLÒ1ÆÇàÅßÜ?ήÞBÂÄÒ°Î’Æ ÊD½LÌD¾LÍ+ÎBÁOÇÑÂÄÒVá ¼?½L¾‚¿ÀÁO™Ã+ÂÄÅÄÆ1ÇȾLÉXÅĽz¾LÍDÇÑÆ1Çѽz¾LÒWΒÔÑÔѽ’Ü ΒÅĽzÌLÒÆ1ÇÑÅ>Ü?ήÞBÂÄÒ Æ1½WØDÁO½zØD΢ÉzΒÆ1ÂâÆÖ×DÁO½zÌLÉz×ãÊD½zÌD¾LÍL¿ ÎBÁOÇÑÂÄÒ<Ü?ÇÑÆÖ×-äÄÂVÁO½#½zÁQҙÕ[ΒÔÑÔYÔàÂÄނÂÄÔàÒK½’ÙhÁO™Ã+ÂÄÅÄÆ1Çѽz¾-ÇȾBƽ ÆÖ×LÂ[ÍD½LÕ[ΒÇȾYá°åK½æÜ<ÂÄÞBÂVÁuØËÂVÁ‰ÙçÂÄÅÄÆ1ÔÑχ¾L½z¾‚¿ÀÁO™Ã+ÂÄÅÄÆ1ÇȾLÉ ÅĽz¾LÍDÇÑÆ1Çѽz¾LÒ5Õ#ÌLÒ1Æ#ÊD°ήÞB½BÇÑÍDÂÄÍUÊDÂÄÅVÎBÌLÒÂ5ÆÖ×LÂÄÏfÅVÎB¾ ÔÑÂVΒÍUÆ1½ÐÇÑÔÑÔè¿ÀØ˽BÒ1ÂÄÍ!ØDÁO½zÊLÔÑÂVÕ#ÒtÙé½zÁêÆÖ×L°Õ#ÂVÎB¾ëÃ+½’Üqá ìTÎBÁOÇѽzÌLÒ Õ#ÂÄÆÖ×L½LÍDÒt×DήÞBÂ/ÊDÂÄÂV¾ãØDÁO½zØ˽BÒ1ÂÄÍßÆ1½ÐÅĽz¾‚¿ Ò1ÆÖÁÌLÅÄÆqÊD½zÌD¾LÍ+ÎBÁÓÏíÅĽz¾LÍDÇÑÆ1Çѽz¾LÒ#Ün×LÇÑŔ×fÅVÎB¾íÊËÂ[ÒÖ̂Ù}¿ î ÅÄÇÑÂV¾BÆÔàÏï¾L½z¾‚¿ÀÁÓ™Ã+ÂÄÅÄÆ1ÇȾLÉ:Ùé½zÁëÆÖ×LÂ!ΒÅĽzÌLÒ1Æ1ÇÑÅ î ÂÄÔàÍ Ün×LÇÑÔÑÂuÒ1Æ1ÇÑÔÑԃÕ[΂ðBÇȾLÉ#ÆÖ×LÂnÕ#ÂV΂¾@Ã+½’ÜñØDÁO½zÊLÔÑÂVÕ%Ü<ÂÄÔÑÔè¿ ØD½‚Ò1ÂÄÍGá ò?×LÇÑÒñØD΂ØDÂVÁUÎB¾DΒÔÑÏLÒ1ÂÄÒWÎóÜ?ÇÑÍDÂÄÔàÏIÌLÒ1ÂÄÍ Æ1ÂÄÅÄ×D¾LÇÑôõÌLÂ^Ùç½zÁ<¾L½z¾‚¿ÀÁO™Ã+ÂÄÅÄÆ1ÇȾLÉ[½LÌLÆ1ÔÑÂÄÆ1Òqöç÷?ÌLÍDÏ/ÎB¾LÍ ø ÆÖÁOÇÈð‚Ü<ÂVÁOÍ+ÎDùQú6½BÇȾLÒ1½BÆ#ÎB¾LÍûüÚÂÄÔàÂVýþá>ÿ ÆqÒÖ×L½’Ü?Ò@ÆÖ×DÎ’Æ ÆÖ×LÂ?ÅĽLÁ1ÁOÂÄÅÄÆ1Çѽz¾#ÇȾ‚ÆÖÁO½LÍ+ÌLÅÄÂÄÍ°ÊBÏ#ÆÖ×LÂÄÒÂnÎBÌLÆÖ×L½zÁÓÒQÅVÎB¾ O!Y"1#ç#j$O%}&B"'æ(Ä)*'’+µ,-æ.}/’"1#101!Ú2+3G4®59!G6’7[8Ä9ó,1:Ö$‰'Ä-’$ ;\<

!Y"#ç)è"1=*)è%O7I> !"#$%&'() D"1-®=è"1-’%ç$™7 4®#j,'*+O$

< $,-./',)*=È0 %ç$‰=*=è$012+À$‰#34È,&+À% < 4Ñ# 5.6 '®)è:Ö$‰#j%})è..789u"'V.}&¢$‰=*=*)è$‰#:;3;ç7<9u,™.}/=>6$‰&æ. < -G59!G6?;@A29 B ,0C < -Ú!G!D%E!™7V>™8F#&G$%H9 u"1'V.}&B$‰=*=*)è$‰# !Y$O(V$"I:%þ7’4V#j,'*+O$ <

J 9u$K/LCB$j#G5M;È565 < ;N9u4O)/5)=*=è$À$Q(þ-:PD#j"Q4È$O%ç%ç$j-®#9!Y,F/\)*=*=è$ 6 A 6 B 5>>?!O8F#&#R)D"1-®=è"1-’%ç$Ö7B4®#j,'*+O$ ÔàÂV΢ͰÆ1½#ÔÈÎBÁOÉBÂ#ÁO™Ã+ÂÄÅÄÆ1Çѽz¾wÔàÂÄނÂÄÔàÒuÎB¾LÍоL½z¾‚¿ÀØD×BÏLÒ1ÇÑÅVÎ’Ô ÁOÂÄÒ1½z¾DÎB¾‚ÆqÊDÂV×DήÞLÇѽzÌDÁOÒVá ST ÒÇÛÕ-ØLÔà Ò1ÅVΒÔÑÇȾLÉÐÇÑÒFØDÁO½’¿ ØD½BÒ1ÂÄÍIÆ1½%ÂÄÞæΒÔÈÌDΒÆ1ÂXƙ×L ÁOÂÄÔÈÎUVDΒÆ1Çѽz¾ ÅĽzÂ&W5ÅÄÇÑÂV¾‚Æ ÌLÒ1ÂÄÍ5ÇȾ-ÆÖ×LÂÄÒ1ÂuÕ#ÂÄÆÖ×L½LÍDÒQÙç½zÁTÎ@¾L½z¾‚¿ÀÁO™Ã+ÂÄÅÄÆ1ÇȾLÉ#½zÌLÆO¿ ÔàÂÄÆ®á5ò?×LÂ[ØDÁO½zØ˽BÒ1ÂÄÍíÒÅVΒÔÑÇÛ¾LÉÐÇÑÒqÆÂÄÒ1Æ1ÂÄÍÐÙç½zÁ^Ò1ÇÈÕ[ØLÔÑ ÅVΒÒ1ÂÄ҇ö}Í+ÌLÅÄÆ1ҙý/ÊD½BÆÖפÆÖ×LÂĽzÁOÂÄÆ1ÇÑÅVΒÔÑÔÑÏ ÎB¾LÍ!¾LÌDÕ#ÂVÁOÇ¿ ÅVΒÔÑÔàÏBá

-&'()*+,-.+/0.120.340567)8'9:;'<=<>?+@0.1 AB34CB0.D;E5)*F2=<340.DB+,>?+,340.G HI34) JKFBLB'()?AM3KN,+,= 34)O+@0.=<34PQL.N,'<>?'<N,F J5FRLM'()?AB3KN,+,= LB)*34A.N,'(PSG ./T+UGTE VW'<F LB)834A.N,'(P +@0 PSC.N,>?+@L.NU' X '<N,DBG GYC.=ZJ EWG =<N@EWG?G[+U=(E\N '(0.15+/0.'<'()8+/0.1 :]C.+UD DBFB0BEKPS+,=<G 01234536.3723853923:<^ EW'()*34EW=<3.C.G?>?+,=<G ,;53</=>.3</,/,3</?1 9^ EWG?>9)*34LBJ5F4G?+,=<G /@453</=653</?7 ^_-.+@AB)?EW>?+,340.G`+@0 G?35NU+,DBG /@8 34) '<N,'<=<>Y)*34PaEW140.'<>?+,GYP A/B9Wb cd0e>YJ.' X '<N,DQ3WHB:]C.+,DQPS'<=<JBEK0.+,=<Gf>YJ.'gLB)*'<G?'(0.=<'h3\H EW=<34C.G[>?+,=<Gi=(EK0jE CDk'<=<>h>YJ.'l:;3mno+/0pESD;)[EWG?>?+,=qPaEK056 0.'()a+/0rLB3K>['(0K>?+@EWN,N,FsCB0.G[>YEKA.N,'p=(EWG?'<G /=:.3</=;531B> b`thJ.' 0.CBPS'()*+,=(EWN.G?+@PqC.N@EW>[+U3.0#3\HRGYC.=<Ju:;3WnhGv+/0l=<34PQLB)*'<G[G*6 +@A.N,'Q=<34DB'<Gu+,G#EQPQE EF734) +UG[GYC.'KbiwuCBPS'()*+,=(EWNxAB34CB0.D.6 EK)*FS=<340.DB+,>?+,340.GhPSC.G?>gAB'u>Y)*'(EW>?'<DSnh+,>YJa14)*'(E\>x=(EKC56 >?+,340`>[3yLB)*'<DB+,=<>zAB35>YJ`>YJ.'{E\=<34C.G?>?+,=|nhE}-K'<G_EK0.D >YJ.'~P'(EK0p:;3Wn#b GH€-K'(0p>YJ.34C.1.Jp>?'<=<JB0.+UBC.'<G~JBE}-K' AB'<'(0‚DB'<-5'<NU3.LB'<Dƒ>?3yLB)*'<DB+,=<>|>9J.'{PS'(EK0„:;3Wn…+@0 G?>?'(E\DBFQ=<34PQLB)*'<G[G?+@A.NU'e=<34DB'<GS†‡niJ.'(0_EW=<34C.G[>?+,=<GiEK)*' GYCBLBLB)*'<G[G?'<DzA5Fˆ04CBP'()*+,=(EWN‰3.)hLBJKF.G?+,=(EWNk-.+,G?=<3KG?+,>*FBŠ 34)e>?3j=<340K>9)*3KNg>YJ.'_EW=<34C.G?>?+,=_nhE}-K'<G#+@0‹N,+@0.'(EK)*+,Œ<'<D =<3.DB'<Gs†‡niJ.'()*'>YJ.'yPS'(EK0„:;3Wn…+,G|+@PQLB35G?'<D;Šp0.3 PS'<>YJ.3.D`=(EK0`JBEK0.DBN,'ŽAM3K>YJr>YJ.'|PS'(EK0 :;3Wn EK0.D >YJ.'pEW=<34C.G?>?+,=jn E}-K'<GS+@0yE‹LB'()7H'<=<>nhE(F‘’_AB34CB0.D.6 EK)*Fa=<340.DB+,>?+,340.GhniJ.+,=<J_EWN,NU3Wn ESLM'()“H'<=<>”=<340K>9)*3KN;3\H >YJ.'pPS'(EK0:;3Wn E5)*'_1K'(0.'()?EWN,N,Fs)8'9:;'<=<>?+@0.1ŽEW=<3.C.G*6 >?+,=fnhE}-K'<G‘niJ.+,N,'hLBCB)*'<N,F#0.340567)*'9:;'<=<>?+@0.1&=<340.DB+,>?+,340.G C.G?'<DSHI34)v' !RE5PQL.N,'gH3.)%NU+@0.'(EK)8+UŒ<'<D_G[+/PSC.N@EW>?+,340.G•nh+,N,N EWN,N,3Wn PS'(EK0–:;3mn`-WEWN@C.'<G&>?3SD;)*+H>(b I 0.'h' !BEKPQL.N,'hniJ.'()8'hGYC.=<J#+,G?G9C.'<G—EK)*'g=()*+,>?+,=(EWN!+,G >YJ.'hLB)*'<DB+,=<>?+,340a3WH]=<3.PqABC.G[>?+,340e+/0.G[>YEKA.+,N,+U>[+U'<G”+/0a)*'96 EW=<>?+@0.1S:;3WnhG 1B>5312/,31J1231B4231B6 bKLxN/E5PS'<Gi' !BJ.+@A.+,>iG?>9)*340.1 =<34PSABC.G?>?+,340j+@0.G?>YE5A.+UN,+,>?+,'<GQniJ.'(0˜>YJ.'<FzEK)8'L.N@EW=<'<D +@0ƒE™=<JBEK0B0.'<NIbTthJ.'<G?'‹3KG?=<+,N,N@EW>?+,340.G{EK)8'{D;C.'‹>?3 E‹=<34CBL.N,+@0.1šAB'<>*ng'<'(0y>9J.'ˆ:]E5PS'|EK0.Dš>YJ.'p=<JBEK056 0.'<N‰EW=<3.C.G?>?+,=<G† MLx+U1!b" 67EKŠ<b#$u=<34C.G?>?+,=en E}-K'<GhEK)8'#)*'96 :;'<=<>?'<DšEW>Q+@0.N,'<>?GpEK0.Dy34C.>[NU'<>[G_+/05>?3‹>YJ.'ˆDB3.PQEW+@0 niJ.'()*'S>YJ.'<Fp=(EK0j=<34CBL.N,'Qn”+U>9Jp>YJ.'QJKF.D;)*3.DBFR0BE5Pe6 +,=<GˆEK0.Dy>9J.'zCB0.G?>?'(E\DBFyJ.'(EW>–)8'<NU'(E\G?'Kb $ =<3405-K'96 0.+,'(0K>SPS'<>YJ.3.Dp>?3_=(EK)?EW=<>['()*+,Œ<'q>9J.'<G?'Q+@0.G?>YEKA.+,N,+,>?+,'<G +,Gi>[3QG?>YC.DBF_>YJ.'lH3.)*=<'<Dz)*'<G9LB340.G?'e3WHx>YJ.'&:]EKP'#+@0 >YJ.'g=<JBEK0B0.'<N4nuJ.+UN,' NO[DB+UG[=<340B0.'<=<>?+@0.1 PQ#>9J.'—=<3.CBL.NU+@0.1 nh+,>YJr>YJ.'|EW=<34C.G[>?+,=<G_3WH#>YJ.'p=ZJBE50B0.'<Nb›thJ.+,G_=(EK0 AB'eDB340.'&' !BLB'()*+@PS'(05>YEWN,N,FzA5F~=<JBEK0.1K+@0.1>YJ.'&=<JBEK056

0.'<N—15'<34PS'<>Y)*F|CBL.G?>9)*'(EKP 3.)&DB3Wni0.G?>Y)8'(EKP 3WHh>YJ.' ABCB)?0.'()<bywuCBPS'()*+,=(EWN,N,FK^h>YJ.'_G9EKPS'ˆ)*'<GYC.N,>Q=(E50yAB' EW=ZJ.+,'<-K'<D|A5F~DB+@PS+@0.+,GYJ.+@0B1j>YJ.'qE\=<34C.G?>?+,=&HI'<'<D;ABEW=<V 3WH‰>YJ.'u=ZJBE50B0.'<Nk>?3#>9J.'#ABCB)?0.'()g+Ibœ'KbfA5F~C.G?+@0.1~0.34056 )*'9:;'<=<>?+@0.1šAB34CB0.D;EK)8Fš=<340.DB+,>?+,340.Gp340o+/0.N,'<>?G|EK0.D 34C.>?N,'<>?G–† MLx+U1!b " 67ABŠ<’&>YJ.+,GQEWN,N,3WnhG#34C.>[1K3K+@0.1_nhE(-5'<G >?3SNU'(E}-K'u>YJ.'&DB34PaEW+@0~ABC.>h=(E50.=<'<NUG”>YJ.'#EKPQL.N,+,>YC.DB' 3WHinhE}-K'<GS'(0K>?'()8+/0.1>YJ.'_DB34PQEW+@0 >YJ.'()*'(A5F{DB+@PS+@056 +UG9J.+/0.1‹>YJ.'_LB3KG[G?+@A.NU'j' 2Dk'<=<>?GS3WHi=<34CBL.N,+@0.1ŽP'<=ZJBE}6 0.+UG9PSG(bfc“0=<'()*>YEW+@0=(EWG?'<GuPQEKV5+/0.1S340.N,F_>YJ.'u34C.>?N,'<> 0.340567)*'9:;'<=<>?+@0.1 †MLx+U1!b " 6d=(ŠPQE}FyAB'GYC %~=<+,'(0K>_>[3 G?>YEKA.+,N,+,G?'&>YJ.'”:;3mne^RE\Gf>?'<G?>['<D~+@0aG?'<=<>?+,340 GRBbxthJ.'()*' +UG0.3|1K'(0.'()?EWNu)?C.N,'ˆ+@0.DB+,=(EW>?+@0.1™niJ.'<>YJ.'()QPaEKVK+@0.1 AB34CB0.D;EK)8+U'<Gu0.340567)*'9:;'<=<>?+@0.1Q+@0_Ee1K+,-K'(0a=<340 X 14CB)?E}6 >?+,340|n”+UN,Ng+@0.DB'<'<DsAM'~GYC %=<+U'(05>Q>?3jD;EKPQLp>9J.'~+@056 G?>YEKA.+,N,+,>7FKb

Inlet 2

Inlet 1

Outlet

Entering acoustic waves

Computational Domain

a)

Other parts of the geometry

Outgoing acoustic waves

b)

c)

STUVWXY*Z[\N]G^P_C`5ab`Ccdef`Cg0UhcPW&^[cdijkl`0^P_CUmc[W&ehno`pq2rPstu*v[w ef^PefUmr[cd^Pk$i[rPsj^PUmcxY thJ.'{DB'<=<3.PQLB3KG[+U>[+U3.0š3WHS>YJ.' yH•C.NU'()j'<BCBEW>?+,340.G +/05>?3=ZJBE5)?EW=<>?'()*+,G?>[+U=SnhE}-K'<Gh+,G&nh+,DB'<N,F|C.G?'<Dp>?3~DB'96 )*+,-K'š0.340567)*'9:;'<=<>?+@0.1 AM34CB0.D;EK)*F›=<340.DB+,>?+,340.G(b $ GYLB'<=<+ X =hLB3K+@05>‰nuJ.+U=<J&+,G%=<34PQPS3.0u>?3iEWN,NK=<JBEK)?EW=<>['()“6 +UG[>?+,=—>['<=ZJB0.+,BC.'<Gf+,Gx>?3&LB)*'<G?=()*+@AB'”>YJ.'hEKPQL.N,+,>YC.DB'h3WH >YJ.'g+/0.=<3.PS+@0.1qnhE}-K'<G 4.3</=453</=6537 bxthJ.'i0BEW>YCB)[EWN4=<J.3K+,=<' H34)g0.340567)8'9:;'<=<>?+@0.1~AB34CB0.D;E5)*+,'<Gi+,Gh>?3SG?'<>g>YJ.+,GiE5Pe6 L.NU+,>YC.DB'”>?3iŒ<'()83;bxthJ.+,GfPS'<>YJ.3.Dq+,Gxng'<N,N.H3.CB0.DB'<Dq>[3 =<340K>9)*3KNB>YJ.'iE\=<34C.G?>?+,= X '<NUD|†‡niJ.'(0EW=<34C.G[>?+,=hnhE(-5'<G )*'(EW=<J|>YJ.'_AB3.CB0.D;EK)*F{EW>S0.34)?PQE\Nf+@0.=<+,DB'(0.=<'(Š~ABC.> 0.3K>#>YJ.'_PS'(EK0˜:;3Wn#’eDB+ VDk'()*'(05>QEKC.>YJ.34)8G 01,7231e+@0.DB+6 =(EW>?'x>YJBEW>v>YJ.+,G‰=<J.3K+,=<'—PQE}FuN,'(EWD#>?3”+UN,N67LB35G?'<D~LB)*34A56 NU'(PG(bfcd0.DB'<'<D_G['<>?>?+@0.1eH34)€' !RE5PQL.N,'#E&Œ<'()*3QEKPQL.N,+6

*+"# GIJ&4 K@ !"L0J8MN AO5K<=PQ2#*? C<BIJIJ*B: !.#&RSM*B: G*9TB&4&UV <W34*+2#EX !<?2Y 56&U<?2QV%,Z&4EXE!"%,Z&?N[\]2^ !.#& #$!,_&U<BI%&K:-*+,ZIJ$ !.#&`56&U<?2aV%,Z&4EGEb"%,Z&`0JED075cV%*?EG&4$dFYPe !.#&KEG b<B G&K<B 012 f 2#0J gPh$%*B:C2#EG !,Z&U<Y5i*B(/ !.#&eF%"%,G2#&U,9:C.#&U,Z&j !.#& V%,Z&4EGEb"%,Z&;0JE "#$%k<?2#$W !.#0JE[012Y()*+,G5K<B X0H*#2W0JEL()&4$6F%<l3mT 012? X*n b.#&;$%*+5K<B012c !.%,Z*+"#8#. &! opEG&4 G X072#8W !.#0JE->q<lI7"#& G*ar4&U,_*s0JEQ"#E!"%<BIJIJPt2#*Y u<v8?*#*+$w34.#*?0J34&aF%&43U<?"#EG& !.#&e012Y(x*#,G5K<B G0J*+2h*+2 '#$%i0JEjEG015KV#IJPt0J8+2#*#,Z&4$yF?P !.#&;34*#$%&'IH&U<l$%072#8W G*WV%*?EXEG01F#IH&/$M,_0z() GE{*+2W !.#&C56&U<?2 V%,Z&4EGEb"%,Z&?NL|;.#0JE}V%,Z*+F#IJ&U5 0HE{:-&4IHIMT#2#*B:C2W<?2#$K>B<?,Z0~ *+"#EpEG*?I1"# G0J*+2#E[.%<=>?&-F%&4&U2WV%,Z*+V*?EG&4$€N ' 2#&-*B( b.#&U5 0JE; G*WV%,Z*B>+0J$%&‚<D>q<BI1"#&'()*+, (! &40J !.#&U,-"#EG012#8`<?29<?2Y~ <BIJP# G0J3U<BI€EG*?I1"# G0J*+2ƒ012Q,Z&48?0J*+2 *+,{EG*?IJ>#012#8u<WIJ012#&U<?, V%,Z*+F#IJ&U5kF&4 g:-&4&U2R !.#&[*+"# XIH&4 LEX&434 G0J*+2 „ <?2#$D072 f 2Y~ 0J ZPs@…EG&4&‚,Z&434&U2Y †,_&4>+0J&4:k&4$%0J G&4$‡FYPQ|ˆ*+"%,G,Z&4 G X&‚<?2#$ ‰'<BI1V%&U,G2 ()* A4N

„

"#34.cEG*?I1"# G0J*+2#E[3U<Y2W*+2#IJPKF%&C"#EG&4$9012 34&U,Z !<l072^<l3U<B$%&U5W0J3‚3U<BEX&4E-(x*#,[:'.#0H34.Q<Y2^<?2%<BIJP+ X0H3U<lI EG*?I1"# G0J*+2e5K<=P^F%&D$%&U,Z0J>?&4$aF%&4 Z:-&4&U2^ !.#&D*+"# GIJ&4 C*l( !.#&^34*+5cV%"# !<B G0J*+2%<BI-$%*+5c<B012w<?2#$d !.#& $X072 f 2#0J ZP & 34*+2#$%0J G0J*+2

+*

N$)‡*#,Z&;V%,G<B34 G0J3U<BIŠ@)F%"# {IJ&4EGE}<B343U"%,X<B G&UA EG*?I1"# G0J*+2#Ee.%<=>?&aF%&4&U2yV%,Z*#V%*?EG&4$€N |;.#&eEG015KV#IJ&4EG *+2#&s@)‹Œ072#&U<Y,`-&4I1< !"%<B G0J*+2 *)e&4 !.#*#$€oQ‹€ +)aAW0JE9 G* EG&4 ; b.#&‚<?5KV#IJ0J !"#$%&W*B( ,! <BE;V%,Z*+V*+,Z G0J*+2%<BIˆ G*W !.#& EG !<l G0J3 V%,_&4EGE!"%,Z&;$%0,€&U,Z&U2#34& (-. /(?o-!

"./0

@1#234# % A :C.#&U,Z& 5# 0JE‚ !.#&`V%,_&4$%0H34 X&4$ŽV%,Z&4EXE!"%,Z&`<B / b.#&K*+"# Z~ IJ&4 ‚EG&434 X0H*#2 „ ,#$%0JE‚ !.#&W(‘<?, f &4IJ$wV%,Z&4EGE!"%,Z&9<?2#$

0

.%<BEC b.#&n$%015W&U2#EX0H*#2e*B( <6(‘,Z&4’"#&U2#34P?N $#$†V%V%,_*B~ V%,Z01<B G& 0&^>B<BI1"#&4E‚*B(

0

<?,Z&W&%"%V%&434 G&4$v G*uV%,_*q>#0J$%&`< ’%"%<BEG0L2#*+2Y~g,Z&bSM&434 G012#8‡F*+"%2#$M<?,ZPe34*+2#$%0J G0J*+2“:C.#0JIH& <=>?*?0J$%012#8uV%,_&4EGE!"%,Z&D$M,Z0(x GE=oL0("#k$M,Z0() GE†<=:;<=PW(‘,Z*+5

#

%

(! <l34 GE`<BEK<eE!V%,_072#8“(x*+,_34&^ G*a,Z&4I1< !"6# G*B~ :;<?,Z$%E 7#$% (-1 N

2L>Y&U2 !.#*+"#8#. !.#0JE ,Z&4I1<!"%<B G0J*+2”<YV%V%,Z*+<B34.0JE "#EG&4$K X*+$M<=P‚012K5W"#IJ G01V#IJ&CEX !"#$%0J&4E 3()4?? Z:-*'’"#&4EG X0H*#2#E <?,Z0JEG&?o

8

$/,Z&/ !.#&U,_&/*+V# G015K<BI€34.#*?0J34&4EO(x*+, 09:

8;<

.%<B p0JE[ !.#&-& 5,€&434 L*B(M !.#0JE-,Z&4I1< !"<l G0J*+2D*+2D !.#& 8YIH*#F%<BI‚<B34*+"#EG X0H3•F%&U.%<=>+0J*+"%,ƒ*B(K !.#&aF%*+"%2#$#~ <Y,ZP : ' FY>+0J*+"#EGIJP?#IJ*B:dIJ&4>?&4IJE[*B( 0 5K<=P‚F%&;012#& &'`340J&U2Y ;012 G&U,G56E/*B(;34*+2? b,Z*?IJIJ072#8e !.#&95W&U<?2aV%,Z&4EXE!"%,Z&?N => *+2Y~

L1

Far field

(Region 2)

(Pressure: P)

P

8

(Region 1)

Outlet section (S)

Computational domain

6789:;<$?=>@A3?BC @=AB?CD5AEF DEFE0GHI(G A J%KLD3FH?MKH8!NO8PFH? QRFH? N!S=A GHTUVWJ%8PNO@ KHA#FDE N!S AHI GHX AYQ A ?MNOA I58P?Z: N!S=A#D5FHTUVW@=NOG[\ NO8PFH?MGHJ]KHFZT GZ8P?^<

>?&U,ZEG&4IJP`.#0J8+.9IJ&4>?&4IJE;*B(

0

&U2#E!"%,Z&' !.%<B +#k,Z&U5K<B012#E >?&U,ZPK34IJ*?EG&/ X* J#$%–<Y2#$^ !.#&U2^5K<?TB&O !.#&‚F%*+"%2#$M<Y,ZP V%<?,Z G01<BIJIJP^*+,{(‘"#IJIHPe,Z&bSM&434 G012#8MN;|;.#&U,_&b(x*+,_&‚<K$%01,Z&434 IH012%TR&%"+0JEG GELF%&4 Z:-&4&U2/ b.#&}5K<B8#2#0H b"#$%&[*B(

0

<?2#$‚ !.#& ,Z&bSM&434 G0J*+2934*#& &'`340J&U2? KL—*B(Š !.#&‚F*+"%2#$M<?,ZP?N ($C34 !"Y~ <BIJIHP

0

34*#2? !,_*?IJE62#*Y n*#2#IHP• !.#&^5K<B8+2#0J !"#$%&^*l( ML F%"# 6<lIHEX*Q0J GEKV%.%<BEG&?N N>O.#*+*?EX072#8

0

:;0J !.#*#"# ‚3U<?,Z& 5K<=P/IH&U<l$n X*†2#"%5W&U,Z0J3U<BIM,Z&4Eb"#IH XE[012D:C.#0J3m.W b.#&[*+"# Z~ IH&4 eF%*+"%2#$M<Y,ZPd34*+2#$%0J G0J*+2˜012? b,Z*+$M"#34&4Ej015KV%*+,_ !<?2Y F#07<lEG&4EC012ƒ G&U,G5WE-*B([<B34*+"#EG X0H3WF%&U.%<=>#0H*#"%,4N

|;.#&‚*+F _g&434 G0J>?&4EO*B(™ b.#0HE'V%<?V%&U,'<?,Z&?o

8

G*;$%&U,_0H>Y&}<-EG015KV#IJ&-<?2%<BIJP+ X0H3U<lI+EG*YI7"# X0H*#2‚,Z&4I1<B Z~ 012#8‚ !.#&'5K<B8+2#0J !"#$%&'*B(€ !.#&C,Z&4I1< !"%<B G0J*+2634*+& &'K~ 340J&U2?

0

<?2#$^ !.#&D34*+5KV#IJ& %"Q,Z&bSM&434 G0J*+2ƒ34*+& &'K~ 340J&U2? OL *B(}<9F%*+"%2#$M<?,_PQ34*+2#$%0J G0J*+2a56*+$%&4IJ&4$ :;0J !.^‹€ P)š@

„

&434 G0J*+2 QR?A4

8

G*D>?&U,Z0(xPc !.#0JE ,_&4I7<l G0J*+2^"#EG012#8W34*+,G,Z&4E!V*+2#$%012#8 2#"%5W&U,Z0J3U<BIpEG015n"#I1<B X0H*#2#EC012^:'.#0H34.j !.#&/*+"# GIJ&4 *B(K<aEG015KV#IJ&• !"%F%&v0HE^Eb"%F%5W0J G G&4$y G*d<•.%<?,]~ 5W*+2#0J3WV%,Z*+V%<B8+<l G012#8n:;<=>?&K@„ &434 G0J*+2 ST+A4

8

G*/$%&U5W*#2#EG !,G<B X&; !.#&;& 5,€&434 [*l(ˆ2#*#2Y~g,Z&bSM&434 G012#8 F%*#"%2#$M<?,ZP–34*#2#$%0H X0H*#2›"#EG012#8 ‹€P) *+2œ !.#& 8?IJ*+F%<BI;,Z&4EX*+2%<?2#34&^*B(†<j8?0J>?&U2•$M"#34 =ou.#&U,_&Q< EG015KV#IJ&e3U<BEG&^:;0J !.

f

"#&4$d012#IJ&4 ^>?&4IJ*#340H ZPd<?2#$ 2#*+2Y~g,Z&bSM&434 X072#89‹€ +)”*#"# GIJ&4 ;0JE}EX !"#$%0J&4$QF%*? !. <?2%<BIJP# G0J3U<BIJIJP‡<?2#$Q2#"%5W&U,Z0J3U<BIJIJPŽ@

„

&434 X0H*#2 `?A4

8

f 2%<BIJIJPK G*/V%,Z*+V*?EG&C</5W&4 b.#*+$W G*/EG3U<lIH&O !.#&

!"#$ !% !&'$()*+,&-)./01234 !+2&' (3456 !"789/0:#+;34+2&'"7)#<=>? $! @A +B&C"#) !DE!F

GH+IJK&'5'5LMN$K<;O#"4JP)Q+@O034+R+;O#$KS4345-:#$!<,"TU !"7)*+;1V"D5 +;O#$QWX3D8Y/#&-)#Z[1234+2$Q"4U\+;O#$%8Q"#W0$!<(]^M0:#+_893(`a3T5C<B" !O03D)#ZD$.<2&'Z7)#&b (3D)D+B5C`c+;O#$Y$!&'ZD$()*Sd3T5e:#$!<fFgGh)i"D+;O#$(1 J\"71VW0<(]03.)#"7)."7/#+2&-8Y3T5^ jO#"*&C !$Q"4U 8Y3f`%5'$(34Wk+2" $(121V"#)#$!"7:#<P1V$!<2"#)03D)D+ Ul1V$!m0:#$()# !&'$!<(F

" #

n$%&'on()*+,- !pq"#$%,'rs

tKO#$9Uu"*5C5'"4JK&-)#ZgW0$!SD$!5'"7/08v$()D+2<Y3D1p$%M034<2$!Ww"7)x+;O#$ 8Q$!+;O#"#WyW0$(1V&'SD$!WzM*` {|"D&-)#<2"D+a3D)#Wz}6$!5'$ & (345'5C$!W ~ ? !./"! >H~3fS7&'$(1€A? +B"74$!< !KO03*1234 !+2$(1V&'<2+B&C ,/I"7:0)#W#A 3D1V` !\"7)#W0&'+2&'"7)#<@E!F G€)‚345'5c jO03D1B34 !+2$(1V&'<2+2&' 3D/*A /01V"73T jO#$!<Y+@O#$a8Y34&-)c&'<2<;:#$a&'<Y+@O#$ƒW0$!+2$(128Q&-)034+2&'"7) "4UK+;O#$ƒ3D8Y/#5'&'+;:#W0$!<Y"4UKJK3fSD$!<„$()*+2$(1V&-)#Z[+;O#$Y !"#8…A /0:#+;34+B&C"#)0345„W0"78934&-)6F†Gh)‡+;O#$ˆ~ ? !./"! 8Q$!+;O#"#W +;O#$…W0$!+2$(128Q&-)034+2&'"7)x&C<„M03T<2$!Wa"7)k+;O#$Q34<2<;:089/#+2&'"7) +;O034++;O#$%3*8Y/#5'&C+@:#W0$!<„ (3D)[M0$9"7M#+;34&-)#$!W‰34<=&U\+;O#$ ŠX"4JwJK34<R5-3D8Q&-)03D1!]*"7)#$KW0&-8Q$()#<2&'"7)034563D)#WY&-)*S7&'<2 !&'W >H}

'

(„G€E!F\~K"D+B$ƒ+;O034+9Uu"71 " (‚)#"7)*AhS#&C<B !"7:#<.ŠX"4JK<(] +;O#$=~ ? !0/#!‹8Q$!+@O#"7Wƒ&'<\$!mŒ:#&'S4345'$()D+ )& *+,&-.+2"Y8Y3D)*` "D+;O#$(1IM0"7:0)#WX3*1V`%+@1V$(34+;8Q$()*+2< /0,123/+,1245+,/ 6 <2"…+;O034+I+;O#$ 1V$!<;:#5'+2<\/01V$!<2$()*+2$!W.&e)9<2$! !+2&'"7) $„3D/0/#5'`Y+2"Q345'5X !O03D1€A 34 !+2$(1p&C<B+2&' „+2$! !O0)#&'mŒ:#$!<=:#<2&-)#Z}Ž "%wF

}

'

(„G6$!m0:034+2&'"7)#<P> " +2"78DEŽ5'&-)0=+;O#$IJ 3fSD$r3*8Y/#5'&A +;:#W0$!<=>&12HER3D)#WY+;O#$‘+2$(8Y/0"#123450$!SD"D5-:#+2&'"7)v"4U’/01V&-8…A &'+2&'SD$_~3fS7&'$(1€A ? +B"74$!<KS43D1V&-3DM#5'$!<%> 346]56^]57“]58…]'=E!F|tKO#$ JK3fSD$_3D89/#5C&'+;:#W0$!< (& 29: & &;: & /<: & 5=: & 9 !"#121V$!<;/N"7)#W 1V$!<;/N$! !+2&'SD$!5'`Y+2"=+;O#$‘5C$@UH+r+@123fSD$!5'5C&-)#ZQ34 !"7:#<2+2&' KJK3fSD$ >”<;/N$!$!W >6)*?@!E!]#+;O#$=$()*+;1V"7/*`YJK3fSD$v>u<;/0$!$!W >6“E!]#+;O#$ b

1V<2+=S*"71V+2&' !&'+V`ƒJ 3fSD$ƒ>”<;/N$!$!W A6XE!]“+;O#$Y<B$! !"7)#W[SD"71€A +2&' !&'+V`aJK3fSD$>”<@/0$!$!W B6XE!]^3*)#Wa+;O#$Y1V&'Z7O*+=+;123fSD$!5'5'&-)#Z 34 !"7:#<B+2&' _JK3fSD$%>”<@/0$!$!W B6,CD@ !E!FKtKO#$!&-1$ #$0/01V$!<2<2&'"7) &'<P"#M#+;34&-)#$!Wa+;O01p"7:#Z7O9 jO03D1B34 !+2$(1V&'<2+2&' Q3D)0345'`7<B&C< :&:,& ;4•

E 4 EFG C " @ & H & & C " % >+& 9 C,& 2 EIJ -& > " E E 6 EFG C " % 4K@ >& 9 * & 2 E -& >% E E 7 EFG C & / -& >.$DE 8 EFG C & 5 -& >/#E E ' ELG C " % >& 9 C & 2 E -& >48*E JPO#$(1V$Q+;O#$QJK3(S*$„3D8Y/#5'&'+;:#W0$!< 0& 2<:9M;M9MN: & 9 3D1p$_W0$@A b )#$!W–M*`—• & 2 ->O6 2 *B@!E0P E ' EQR 2 *S4T@ E 6 2 EUR 29V >O!*E & &"-6 2 P @ & E 4 EUR 2 * E ' EUR 2WV >OX*E & / -6 2 E 6 & EUR 2 >OY*E & 5 -6 2 E 6 / EUR 2 >OZ*E & 9 ->O6 2 CD@!E0P E ' EUR 2 C[4K@ E 6 2 EUR 2WV > "'& E <Rm6F " +2" 8=/01V"4S7&'W0$K3K<B&e89/#5C$8Q$!+;O#"#W„+2" jO#"#"D<2$ +;O#$%&-)# !"78v&e)#Z[JK3fSD$%3D89/#5C&'+;:#W0$!<9+2"–MN$%&-8Y/N"D<2$!W 34+3wMN"7:0)#WX3D1V`DF =X"719$ #$Œ3*8Y/#5'$D]KUh1V"78 <Rm6F % ]3 b $7$!W‹SD$!5'"# !&C+V`w&-)#5'$!+ƒ !"7)#W0&'+2&'"7)‹JK&'5'5=1V$!mŒ:#&-1V$[+;O#$ &e)# !"#8Q&-)#Z˜JK3fSD$g3D8Y/#5'&'+;:#W0$ &

9

+B"™M0$g$!m0:0345„+B" +;O#$="7:#+2ZD"*&e)#ZQJK3fSD$ 1&

2

F =R&$#&e)#Z.3Q !"7)#<2+;3*)D+K/01V$!<VA <;:01V$=34+K3D)Y"#:#+25'$!+KJK&'5'5’M0$„34 !O#&'$!SD$!W‰>Uh1V"78 <Rm6F 8*E MD`g<B$!+2+2&-)#Z &

2 -*"&

9

F ( 8v"71V$. (1V&'+2&' (345K<2&'+;:03fA +2&'"7)[3D1V&'<2$!<=+2"ƒ8Q&-8Q&' Y)#"7)*Aš1V$@ŠX$! !+2&-)#Za !"#)#W0&C+B&C"#)#<(• +;O#&'<YJ\"7:#5'W™1V$!mŒ:#&-1V$ƒ+B"–<B$!+_+@O#$ƒ&e)# !"#8Q&-)#ZgJK3(S*$!< +2"‹›!$(1V"XF (P<i&e)#W0&' (34+B$!Wœ$(3D1V5'&'$(1!]9<;:# jOz3 >7/0$(1€A Uu$! !+25'` >„)#"7)*Aš1V$@ŠX$! !+2&-)#Z= !"7)#W0&'+2&'"7)v&C<r)#"*+34W0$!m0:034+2$ M0$! (3D:#<2$w&'+g8Y3f`‹5'$(34W +B"ž3 WX1V&Uu+["4UY+;O#$w8Q$(3D) ŠX"dJ m0:03D)*+2&'+2&'$!<(F t‘O#$‰)#$ #$7+k<2$! !+2&'"7)‹&'5'5-:#<2+;123T+2$!< +;O#$PM0$(O03fS#&'"7:01R"4U’3WX:# !+I"7:#+25'$!+RUH"71|JPO#&' !O%3=)#"7)*A 1V$@ŠX$! !+2&-)#Z% !"#)#W0&C+B&C"#)ƒ&'<P<B"7:#Z7O*+(F \ ]^_ ' Ÿ`'$ab' r |q?! @ A   @c'de   !f' A q @&g nhi jkl ¡ @ # A %_n ŸQm   @ A %B!Vq?! @ A

}6$!+ :#< !"7)#<B&CW0$(1¢+;O#$ /01V"7/034Z73T+2&'"7) "4U "7)#$@A W0&e8v$()#<2&'"7)0345„34 !"7:#<B+2&' –JK3fSD$!<Y&-)‹3i<2$(8Q&Ah&-)

b

)#&C+B$ +;:0M0$k"4U„ !"7)#<2+;3*)D+Y (1p"D<2<.<2$! !+2&'"7) >C=R&CZ“F2$DE!FytKO#$ +;:0M0$=&'<\&-) b )#&C+B$„&-).+;O#$

R n

x=x

L5

L1

B

Outlet

Non reflecting

!"# $ !%& '"# (%)*+,- ./01"23 )*4 5 "6-78 )/3 9:"20*# .*0 92(%5 .*; 9:<=5 9$%>./;%0?@*)*+ &'A.!" )B9:;(%>5 !" # () #$%&"'()*+,-"'()*./012#$/3*454"6#7,8%9:;()*./3*< =%),8%>? /3*@AB*454"6#$%)CDEF,8G2%)1B -/3HI54,8%)12#7"6#7,8%JKLG)M6"<EF*&#NK=O)PQ*@? KL*.%>"6*41RS !T%U(2 >/6KL,8%)#75T:V WX-*;O2/3,8O2 YC8 !"6#$%)CL#$%Z#N%>? 5./3*. !M[#N%)C 12#$/3*454"6#7,8% #QM=#NK=O2,-M6*41RS\]V()*&* "#^O2/_*4M3? M6#7,8%

C

,!`a"@()#QMT54,)KbO)P7* "#&: WX-*c Y"

# () #7Md"' -e!*.% !M.f $ D # $ !"#"* E+,-./012 $3456 g %&% h ]V()*i,8E F3*454"6#7X-*j,!`c"'()*i54,8%)12#7"6#7,8%k !" # 7) #7MT"6,&E2*b%),8%>?l/3*@AB*454"6#$%)CBSTmn%oO2/[ !54"6#754*-pq"6,& .X>,-#71o 1B/3#`r";,!`s"@()*cKL*. -%tO2/_*4M6M'G2/3*-pu"'()*c#$%)54,8Kv#N%)C&:V WX-* -KbO)P7#7"'G)12* %$ C #7MT%),-"0M6*4"V"6,=w4*./3,UE2G)"V"6,uf $ C&#'( g)*+, * -h g %'$ h GxyqS %&% -%)1 %'$ p+"6,-C-*4"'()*./z:V#7"'({| H Icmj/3*4P$ W? "6#7,8%)M $ -%)1 JLP7*. !1&"[,b"'()*TM6H)M6"6*.K ,Y`a*4y2G2 !"6#7,8%)M.f 89:;< : 6 % C K=>?@ g$ !&#"* EA,-./0 2 $3456 , ( g)*+, *.-bh6h # ( :BC : 6 % C K g $ !&#"* EA,-./0 2 $3456 % ( g)*+, *.-bh6h # ( g % /-h ]V()*&M6*454,)%)1J*4y^G2 !"[#Q,)%},!`;M6H)M6"6*.K % /j,8%)P7H}#N%>? X-,-P7X-*4M 0* -%)1i5. -%j*. !M6#7P7HoEF*<M[,-P7X-*41 1K!ST]V()*LM6,-P$G>? "6#7,8%=`r,8/ 1* #QMWf *bg ' h # *.- %DE EF012GHIJ K , !"#"* E ,-. L K , ,M. 012 $3456 g % 28h N )q~.€OPq‚6ƒ@„V‚ *+…†W‡ˆ€‰Š‹Œ.„Ž€ ,-r~Œ@r„nUrT‰Š~Œ‚@‰…ˆU‰Š~Œ„‘‚ ./01 r„n2-3„0.3u’“„QRŒ„4-3‰r„lSPq‚6ƒ'„@”-‰Š~!„sˆW‡€„3‰sƒ@„“‡€ "-n€‰56cŠ€•[Œ‚[‡ TPq[‹.‡€ /-„$U5VWXMYZ[\](^?_B`abcd J –Ge @‰r„—‰Š~!‚'‰x‰Š~!„…†.~Œ‚@r„0€xr„3‰x‰r fl„n’“ ‚@‰ghi[jh5k &‰rVŠ+…†W‡ˆ36+‰Š~Œ„0‚[‡€•'„ /.’“‚ – 7‰R~!‚'˜!0ˆ FRW‹Œ„n2-3„ [™‰Š~!„˜’“„lŠ‹W‡€‰ – K )q~.€d 864r‰r„4+lPš‚@0.„n’r€ƒ@„lj‚'‰0‰Š~!„T6‹!‰Š‡€„l‰ /-6‹WŒW‚6’ 96@” -36!r„mn@‹Œ„n.‰Š‡6@” -346’“›ˆŒ‚@‰r„l‘‚6’“„ op2:4„3L‚[Œ /-@‰Š~qUc‚[Œrs ‚[’“„;3Q‹W2-3‰Š€[Œx <3^‰Š+0„s[.‡6 – EF0 2 HIJ K E E2*4#$%)Cb T54,8%)M6"@ -%-"  #)*41UE>HL#N%)#7"6#$ !Pq54,8%)12#7"6#7,8%)M@h— YP€? : WH)M;X! -%)#7M'()*4Mc:‘#Q"@(i"6#$KL*LM6#$%)54* (vw ( Scm“%o`žG2/n? "'()*./b12*4X>*4PQ,)O2KL*.%-"[M&#Q"&:‘#QP7PcEF*9 YM6M'G2KL*41\"'(2 !"& M6"6*. !12H}M6"' !"[*t(2 !MŸEF*4*.%k/3*. !54()*41  -%)1\"'()#7MU"6*./[K :V#7PQPxE2*+,8KL#7"6"6*41RS

GxyqS % 2DgŠ:‘#Q"@(),8G)"T"'/6 -%)M[#Q*.%>"<O2 >/3"'hT -%)1 LG¡yšS %0$ !P7PQ,!:\"6,c/3*454,8%)M6"@/6G)54"s"'()*0#$%)54,8KL#$%)Cb:V WX-* $

C

-%)1 "'()*054,8KbO)P7* "#b/3*@AB*454"6#7,8%L54,)* =>U54#7*.%-"…,!`š"@()* EF,8G2%)1)? -/3H-SL]V()*U -%2 !P7H)"6#75. !PxXŒ YPNG)*Z,!`0"'()*b/3*@AB*454"6#7,8%i54,!? *=>U54#7*.%-" 03¢#7M.f 3xy < 6 # $ C $ D # , % % , , K 4 L g % J>h ]V()*ŸK= !C8%)#7"'G)12* z{ 3r{= -%)1DO2(2 !M[* q|},!`; t%),8%>? /3*@AB*454"6#$%)Ck,8G)"6P7*4"£KL,812*4P7*41{:V#7"'(¢|R¤ &4 -/3*£12*@? /3#7X-*41i`“/3,8K GxyqS % Jogž]V()*U !M6H2KbO)"6,>"6#75<E2*.(2 WX)#7,8G2/ ,!`}{53r{; >%)1z|j#7MVM'G2KbKb >/3#7w4*41t#$%&"' >E)PQ* % h4f { 3r{ # % ~ % %œg K 4 L h K g % (>h | # ,$6,£ -/_54"' -%Rg $ . ( h g % )>h ¥sG)P7M' !"6#7,8% {53r{ | . # ( % ,$ . # L K C € K ,$6,‚ M . #ƒ„ ( , N ‚ K …O)†% $ '/& ‡ P;' -78Q9:./9:"63 ˆ%>5R(%)*4 "2.*; + . " ) 0?.*0ST +U57 5V3W9:"60/# .*; 9:<=5 98 "292(‰ŠZ‹$Œ $ !;MZ*"#^OF*454"6*41Rp0`r,8/b 9C-#7X-*.%kO2G)PQM@ !"6#7,8% . pi{53r{ C-,8*4M="6, ( :;()*.% ( #7MUM@Kb !P7P;M@(),Œ:‘#N%)CI"'(2 !"b"'()* E2,8G2%)1B -/_H¦54,8%)12#7"6#7,8%§#7Mk#N%)12*4*41¨%),)%>?l/3*@AB*454"6#$%)C g?{ 3r{x ( hd:;()*.% ( #7M+PQ#$KL#7"6*41}"6,&M'K= !P7P©X! !P$G)*4M.S ªV,!:0*4X-*./TP$ -/_C-*&XŒ !P$G)*4MZ,!` ( 12*4M6"'/3,!H£"'()* BX8%),8%>? /3*@AB*454"6#$%)C *Xk5«(2 >/6 !54"6*./z,!`U"'()*JE2,8G2%)1B >/3Hœ54,8%)12#? "6#7,8%qfZ:;()*.% ( C-,8*4ML"6,i#$%  %)#7"3HŽ{ 3r{=C-,8*4M="6, % Kb -e>#N%)C}"'()*£E2,)G2%)1B -/3HJ`“G)P7P7H¬/_*@AB*454"6#$%)CBS !;54"'G>? !P7PQH-pd !MZM'(),!:;%­E>H GxyqS % (I"'()*t54,)%-"'/_,-P;O2 -/6 >K+? *4"6*./+`r,8/ { 3r{=#7M $

./ (

! ! " " >? @-A$%;B*1C " DEFGH%3."<HI3./J&0H%;K"# @/L;K3M5N O5PQR"<*%ST+,$%&0 U( &L3.*V5,WK 1-.-XC)8';<Y,$%;#5%"<34;</BFJ3.-.-Z5%*O&[\,;T8';]^7;<"<&L;<=_FJ3.&0H &0H%;T/] OS`;T/L&]8';#5%aO&0Hb6cdJ3.aVHeCf8g;<Yh$%;#5%"<3.;</TFJ3.-.-Z;# @/'( 3.-.Qi-4;# U>O;I&]H%;R"<*VSj+,$%&0 1&L3.*V5, 1-k=,*VSj 13M5lD #$ !

"

W

FGH%;#8';# @/m>P;#8'Qn-4*1FoC)8';<Yh$%;#5%"<3.;</cFJ3.-.- \p;I/L&08'*%5%aO-.Q 8';]^7;<"<&L;<=qD #$"! # W<6rs)5t+,82 1"<&L3."<;u n"#$%&'()* rC)8';]( Y,$%;#5%"<Q #$

!

/L;#+, O82 1&L;</vFJ U>O;</v&0H, @&vFJ3.-.-h\p; 8g;]^7;<"<&L;<= D%$ $

!

WkC)8'*VS &0H%;`*V5%;</w&0H, 1&wFJ3.-4-x-.;# U>O;`&0H%;B=,*1( ST 13M5yD %$ # $ ! W<6&$ !

34/w=,; : 5%;<=IC)8'*VS "xYb6 # #T\OQbz

$ ! ! ! $ ! " %%$ D # !OW

{JH%3./N=,; : 5%3.&L3.*V5|3MST+,*P/L;</N&0H, @& &$TD%$

! W ! # "'( 6

s)5}&L;#82S`/G*@C~;#5%;#8'aPQN&0H%3./[S`;# O5%/w&0H, 1&w 1&G&]H%;KC)8';]( Y,$%;#5%"<Q )$

!

H, 1-C7*1C7&0H%;Z 1"<*V$%/2&L3."Z;#5%;#8'aOQK3./C€;<=[\, @"‚ 3M5O&2*T&0H%;K"<*VST+,$%&] 1&L3.*V5, 1-v=,*%ST 13M5b6

!x3.a76 %`3./ƒ `+%-.*O&„*1C "xYb6 # "B O5%= # #,6…{JH%;w"#$%&'()* C)8';<Yh$%;#5%"<Q *$

!

3./k8';#+,8';</L;#5P&L;<=R\PQ`&0H%;J>O;#8'&234"# @-,-43M5%;O6 "xYb6 # !K/0$%aOaP;</L&L/Z&]H, 1&ƒ B+,8'*V+,;#8„3M5P&L;#8L+,8';<&0 @&L3.*V5T*@C

"

34/„&L*K>%34;<F +$ ! !'" "

D%%$Wv 1/Z wC)8';<Y,$%;#5%"<QT\,;<-.*1F FGH%3."<HN&]H%;†\p*V$,5%=7 O8'Q}"<*V5%=,3.&L3.*V5nFJ3.-.-Z5%*O&w-.;<&K&0H%; FJ U>O;</k-4;# U>O;K&]H%;[=,*VST @3A5b6

10

-4

10

-3

10

-2

10

-1

10

0

Modulus

10

-1

10

0

10

1

10

2

10

3

10

4

10

5

10

6

Frequency

-1.5

-1.4

-1.3

-1.2

-1.1

-1.0

Phase (x )

cut-off requency f

c

Modulus

Phase

$%&' (,-)* & + ,-./01.-23456), 7)8 4 2 9:+()3;<9#* 9=>?@A&B+-6 >C+ 9./ D > &E9 6F@%G?9 ;H2 .-2+)I;<901?./9?6)> '(J45@ " ! $"$"$"'2 ) K ( %J&w"<*V5%/L&0 P5O& " ‡7&0H%;`+,H, 1/L; *+n*1CZ&]H%;†8g;]^7;<"<&L;<= /L3.aV5, 1-34/w&0H, 1&k*1CZ

:

!%;<=N+,8';</L/0$,8g;[*V$%&L-.;<&kCˆ*V8„-.*1F >? @-A$%;</n*1C 6 $k$%&9 1/I/0H%*1FG5!3M5 !x34a‰6,%7‡wFGH%;#5 &0H%;yC)8';<Y,$%;#5%"<Qq3A5%"#8g;# 1/L;</y&0H%;t\,*V$,5%=7 P8'Q &]8';# 1&'( S`;#5P&n3A5%=7$%"<;</y t=,;<-M UQl&0H, @&9S`*1>O;</}&0H%;Š+,H, @/L; C)8'*VS -,$j&L*.-/0-$

"

6v{JH%3./Z+,H, 1/L;„/0H%3C€&X"# O5‹3A5%=7$%"<;

&0H%;K=,*VST 13M5} 1/J/0H%*1FG5j3M5N/2;<"<&L3.*V5 L#,6

3 4 M%&'( ŽNOLx)*+(,- k)*CM[k‘POQR S R./}‘ 0 (

Ž1(,23( ZX‘POQR S  R4(,5  O6( S ‘

s)5_&0H%3./m/2;<"<&L3.*V5 "xYb6 # #’3./m"<*%ST+, O8';<=y&L*95%$,S`;#8“( 34"# @-Z/L3MS`$%-M 1&L3.*V5%/#6N{ŒH%34/`3./`=,*V5%;R\PQn/L3MST+%-.QŠ8L$,5P( 5%3A5%an jCf$%-.- "Z$%-.;#8B"<*V=,;T*%5N&]H%;T"<*V5 : aV$,8L @&L3.*V5n*1C !x34a‰6 0G O5%=T$%/L3M5%a T"xYb6 #& @&…&0H%;k*V$%&2-4;<&U6~{ŒH%;J"<*V=,; %…” $k• U FJ 1/w$%/L;<=ICˆ*V8Œ&0H%3./G>1 1-.34=7 @&L3.*V5b6 %…” $k•t3./

0 V Cf$%-.-.Qt"<*%ST+,8';</L/L3M\%-.;–—w U>V3.;#8

˜

&L*V‚@;</T/L*O-.>O;#8 $%/L3M5%aR"<H, O8L 1"<&2;#8'3./L&L3."[\,*V$,5%=7 P8'QI"<*V5%=,3.&L3.*V5%/ W 6

{JH%;T*V$%&L-.;<&K"<*V5%=,3.&L3.*V5_3./m5%*V5P(™8g;]^7;<"<&L3M5%aŠDEFJ3.&0H šv›,1uW` 1/[=,;</L"#8g3A\p;<=i3M5u/L;<"<&234*%5 0n O5%=u&]H%;R3M5%-.;<& 34/N y5%*V5P(™8';]^7;<"<&23A5%ar+,$%-./0 1&L;<=r3M5%-.;<& QUIK?6œ{JH%;u5V$P( S`;#8'3."# 1-x/L"<H%;#S`;`3./ 5657 *%8'=,;#8w3M5n/0+, 1"<;T O5%=u$%/L;</ &0c/L&L;#+y›J$,5%aO;]( 89K$%&2&0 NS`;<&]H%*V=lD #05: 7 *V8'=,;#8LWC€*%8 &L3MS`;I3M5O&L;<a%8L 1&L3.*V5b6 2J 1-."#$%-M 1&L3.*V5t+, P8L OS`;<&L;#8g/† P8'; /0$,STST O8g34ž<;<=}3M5I&0 O\%-.; 6 ˜ 3.ž<;`*1Cx&0H%;K=,*VST @3A5 ;<qDˆSjW " 7 # —w$,S`\,;#8Œ*1C~"<;<-.-./ % "$" 1n;# O5c3M5%-4;<&K>P;<-4*%"<3.&'Q–D %= 8 2 ) KLW #'" ˜ *%$,5%=N/0+p;<;<= 9TD%= 8 2 ) K0W 0 % ! !‰*V8'"<3M5%aTC)8';<Y,$%;#5%"<Q >$iDˆdJž#W # "$" !7*V8g"<3A5%ac OST+%-.3.&0$%=,;ND %= 8 2 ) K W " 8 #

.P4%/ (34/*?@X. YTY 4-;0(Z+() 6F./YT9?;=&E>=4-0P2 &EYT./0145@A&E+56Z7[4 D ;\4-Y 9?@A9 ;H2](

!7*V8w&L;<"<H,5%3."# 1-ƒ8g;# 1/L*V5%/#‡b3.&K3./mS†*V8';`"<*V5P>O;#5%3.;#5O& &L*i>? O8gQ

"

1&R&]H%;u*V$%&L-.;<&I&0H, O5!&L*i"<H, O5%aO;n&0H%; C€*V8g"<3A5%aBC)8';<Y,$%;#5%"<Q A$n 1&k&0H%;w3M5%-4;<&U6ƒ{JH%;#8g;]C€*V8g;G&0H%; C€*O-.-.*1FJ3M5%aŠ8g;</0$%-.&L/R+,8';</L;#5P& 5$TD " WT @& : !%;<=B$!3M5P( /L&L;# 1=}*1C 6$TD%$bWJ 1& : !V;<= " 6J{JH%;T8';]^7;<"<&L3.*V5}"<*%;]CŸ( : "<34;#5P&.$q3./w*V\%&0 13M5%;<=n3M5 %…” $k• \PQ &$ ! 7 K " 7 ^ 6 {JH%;J OST+%-.3.&0$%=,;</ 87 K P5%= 7 ^ O8';JS`;# 1/0$,8g;<=† @&…&0H%; *V$%&L-.;<&Z$%/L3M5%a J"xYb6:"K O5%=

#&" O5%=T*V5%;]()/234=,;<=c/0+, 1&23A @-=,;#8'3.>? 1&L3.>O;</U6 C ¡DEFG ( ¢1£)*k¤ ¥1¦0§]¤ +M£2¨ ©#¤'ªˆ¦2«¬­®¤'© «¬¯&+M£2¨,*k¦0ªE«®£2¯V° ¢#ªˆªE¥V°HIH123232 ± -™¤“¨.+M¦/-™² ± +.¨8H0-+M©JH]³b´12KLB¤,3 ± ¢#ª.*„­

0.001

0.01

0.1

1

Modulus

10

1

10

2

10

3

10

4

10

5

10

6

K

-1.5

-1.4

-1.3

-1.2

-1.1

-1.0

Phase (x )

Modulus of R (AVBP)

Modulus of R (Theory)

Phase of R (AVBP)

Phase of R (Theory)

!"# $ !%&"# '%()* +%,-"./'%0 ' !1023 (45 ,-"67-+%89+%02:;<=>?5@'A,B5C<D"E 7-+%8F,G5 "# 5-7 <D"='%0 7@'C5 ! 7 "=5 0H<!+%< +$%D* +%,-<D"=+%868 !I,G5"# 5@7C<=E "=0%#&%J'%3 <D8K5 <L$ !"#$%&'()*+$,$-./$-012345$,2+67$,$-0 0#85.9$-*+:;<-&'=>?@:A.B8#=A&CD 2@:;E)0#?F&10#GH2I"#$J2K"#$,E)*LMNO DMFPQR N1ST:U?F$ !"#2K*+$-./$,=UMV(WE)E#G O=OF:;(XR N1S,R JEW2K"H.9E#GX8#=A8#?Y&10#GZ[5"5&'?\$YE]^X2K"#$!*+$I_X$,<ID 2@:;E)0`<,E#$ #$N<,:;$-012H&]2Y2I"#$VE)8#2@=;$,2a&W*+$V?K85[5$-*L:b.c[5E1?@$,G 2@EZ2I"#$V2K"#$,E)*+$,2@:;<-&'=d<-85*+e1$,?-R ' ( P !"#;QVfRS T)* ghi$%UPjk&' V V4P ! g S T + S+f ( T P T", l)'$*+' Jimf9S T)* nIPJQ T V , g l,- i P T V4S+fRSWP T -./01 2

opq'r5s1t 34uv]twxyz{x 569v1|Jo}r5xyz 789:~Js1€pv1txdz

!"#$‚[5*L$,e):;E)8#?ƒ?@$,<,2@:;E)0#?„"5&Ce1$3?K"#E'6j0…2K"5&]2†0#E)0WD *+$I_X$,<,2@:A0#(N4‡E)850#GX&1*+Mc<,E)0#G5:;2@:;E)0#?j8#?@:A0#(Zˆy‰ $%Š<-&10 "5&Ce1$3:A.Z[5$,GX&10#<,$,?36j"#:;<,"{.Z&1‹]$„2I"#$-. [5&1*+2@:A&'=;=;M *+$I_X$,<,2@:A0#(XR7 !"#:;?!?@$,<,2\:UE#0`?K"#E'6!?72K"#$V:A.Z[#=;:U<-&]2@:;E)0#? E'^H2K"#$,?\$„*+$,?K8#=;2@?`^ŒE)*c2K"#$†<,E#.Z[58#2K&'2@:;E)0 E'^V_XE'6!? 6j"#$-*+$cE)0#$†OE)*V.9E)*L$-SŽ?@$,<,2@:;E)03:;?B./E)G5$,=;$,G%8#?@:A0#( ?K8#<,"†0#E)0WD*L$I_X$,<,2@:A0#(N<,E)0#G5:;2@:;E)0#?CR

&7E)0#?\:UG5$-*&%?@:A.Z[#=;$%GX8#<,2 ‘ =;=U$,G’6“:U2I" &10”"#E'D .9E1(W$-0#$,E)8#?Z()&'?3O:A0%6j"#:;<,"%2K"#$ƒ?K[5$,$,G…E'^a?\E)850#G .“:U?7<,E)0#?\2K&10W2KSJ&'?!*+$-[5*L$,?@$-012\$,GN:A0 IOF:;(XR/05RF !"#$Ž:b0WD =;$,29?K[5$,$,G%:;?Z:A.Z[‡E1?@$,Gy• 1mO 2 ' % 3 4 S ' " X R3 !"#$ ?@2@$-&]G5MZ_XE'6–?@E1=A8#2@:;E)0—:;?ŽE'^J<,E)85*+?\$1• 1mO 2 3 4 S ' " X R !"#:;?`?@M)?\2@$-.Š:U?—&'<,E)8#?\2@:;<-&'=;=UM%G5$ ‘ 0#$,G{4WM‚:;2@?`*+$ID _X$,<,2@:;E)0`<,E#$ #$N<,:;$-012\?a&'272K"#$Ž:A0#=;$,2a&10#G—2I"#$VE)8#2@=;$,2-R &Ž?@?K85.9:A0#(—2K"5&'2“2K"#$HGX8#<,2V:A0#=;$,2V:;?V2K"#$B[5"5&'?\$B*+$I^D

L1

L5

x=0

x=L

Outlet

Non reflecting

Velocity

Imposed

!"# $7;A&<#!'%0 (Y#?3 ,G+%<D"6'A0

$-*+$-0#<,$VE)0#$Ž<-&10`6j*+:;2@$Ž2K"#$H*+$,=A&'2@:;E)0#?Ž45$,2+67$,$-0Z2K"#$ &1.Z[#=;:;2K8#G5$‚E]^H2K"#$‚&'<,E)8#?\2@:;<„6!&Ce1$,?`:A0{2K"#$3GX8#<,2 &10#G—2K"#$B*L$I_X$,<,2@:;E)0—<,E#$ #$N<,:;$-012\?-• = >? ) 5 " ' @AB @ # ) $ % & ' @ #3C '()*+&DE @AB C )*+&DE O 6WS 6j"#:;<˜"™<-&10—45$>6j*+:;2@2@$-0—&'?C• )75 " ),$ % & -5. / D E * ' % O '(% S

&!&'=;<-8#=A&'2@:A0#( 2I"#$ $,:U(W$-0WDŒ^š*+$,P58#$-0#<,:;$,? E'^ 2K"#:;? GX8#<,2Z<,E)0#?@:;?@2@?9:A0}(1:;e#:b0#(2K"#$—e›&'=A8#$,?ZE]^ 0 6Ž"#:U<," &1*+$V?@E1=A8#2@:;E)0#?ŽE'^ MuPœR '(% R -./FG H zJ€QŒžFv1tr5€Q!q'xywFv1txdzJq OXE)*™2K"#:;?Ÿ2@$,?@2- ¡2K"#$‚:A0#=;$,2}<,E)*@*+$,?K[‡E)0#G5?32@E ‘ ")$,G e1$,=;E)<,:;2+M¢<,E)0#G5:;2@:;E)0#?…6j"#:;<˜"£:A.Z[#=;:;$,? )75 " ' R !"#$3E)8#2@=;$,2—:;?†.9E)G5$,=;$,G 8#?\:b0#(¤&‚0#E#0WD*+$I_X$,<,2@:A0#( <,E)0#G5:;2@:;E)0Ÿ6“:U2I"‚ˆy‰ +%¥Rm !"#$-*+$I^ŒE)*+$ MuPœR N—?I"#E›6“? 2K"5&'2Z2K"#$—$ -Zy$,<,2@:;e1$ƒ:A.Z[5$,GX&10#<,$3E'^V2K"#:;?`E)8#2@=;$,2N:;? ),$ % & ' * 1 O * 2 O' 031 , S@S,RF¦0™2K"#:;?j<-&]?@$ MuPœR ' % G5$,(1$-0#$-*@&'2@$,?“2@EX• -5 . / D E 8 9 * 2 ' 0 , : !"# !

"#$%&'()*+,-.%&/0#123 45$6*+4708*249:;0<3=> !?@AB ! # *CD<&EF/1GHIJ K%,L&$%ABM"#$%&NDO&P*+,7K6*CD<4E1+3 ! ! ! !!" ! :L0I48$%&M&/:;QC&PRSJTK6.%,;08*+4U:L1%RV*CR%WX45$%&M:-HM*+QC:-R6*CD<>YK6*CDO4 "# ! ! !!"! Z![:;0\48$%&]W^*CHMK%:-R%Q_ `1GDM*SHIJ K%,L: a FP*+4U:L1%R6!]D9*+49&CBb"#$%&c4U&PHMK61GDU*+,d&/(C1C,-.%49:;1GR%0]1+3 K6D<&/0908.6D<&#*CR%We(C&/,;1GF/:;4<>M*+4f*SRg&/:;QC&PRSJh3iD<&/@6.%&PR%F/>j*SD<& K6D<1GK61%D<49:;1GR6*+,C4U1 " # $%&' " # $%$%(' " % ) ' `:-Rg45$%&7,;:-R%&P*CD D<&/QC:-H[&P!/k

!

<.;'1=$%23>'50(?4.<'501%2@$= A,';B"

!

!

"C$%&'(D*+,'-.(D 01E-F2G*)4.(7-89 A1H2 1=$%23>'50(/4.<'501%2@$I 0,J;BK

1000

900

800

700

600

500

400

300

200

100

0

Eigen-frequency: f

r

(Hz)

10

-1

10

0

10

1

10

2

10

3

10

4

10

5

10

6

10

7

K

|R| -> 1

|R| -> 0

mode 1

mode 2

mode 3

1/4 wave mode

3/4 wave mode

5/4 wave mode

!"# $ ! %"# "&# ' ( )!*+,'!" - ' (./ "0'*123"!45067'#$ +81 5"# 9:3 ;<'=1 > '?+@1*-A1 % ! 3A+25067':;<- / 5 3"!B !"# $ !<$ CL 0MFK "N 01D2O4'5A,G$H,@!P$Q&'(R(7 0M3(S;GTU!VWX(7Y.<'(S;'Z7 0(71R,[! DLYBK ! " \G(SWX1%<'1 % K #]$^\3(SWX_`&' AM'&`\a2@5b<'(71c,@! % $%&'( 1I_'1I$I(S*dWX(71%4.,';'-.1E2@1/ !e$Q&'(f4.WX(71=1%<.WX(`gh(SWi( 6 $'(7-2@$c$Q&'(:,J<'$I50(7$ @E3".23;'-D$Q&'(:(7 AMG(S;GTU!jWX(7Y.<'(S;'Z7 0(71

# $ & k!$% # " l $ ' % k!%! l Z7,JWIWi(71%4.,J;'-O-. bWX(7Z7$I50_O$I,D$%&'( " $ 'mgc2S\G(Rk% & l7" ( $ ']g 2n\3(ek % & " l7"bKQK7K%*o,J-.(71p,@!q2]-F<'Z7$rgc 0$%& 6 b;'50(7$:\3(750,JZ7 0$X_R23;'-s,J<'$=5A(7$:4.WX(71I1Q<.WX(fkUWi(S4.WX(71I(S;G$I(7->G_D&',JWX 0t7,J;G$%2@5850 b;'(71: b; FCL 0MFK%"3l79L$%&'(+>.,J<.;'-F2GWX_H 01 !V<'5A50_sWX(QuF(7Z7$I b;'MFK ) &'(S; %

-.(7ZSWX(S2[1I(71)$%&'(H(7 0M3(S;GTU!VWX(7Y.<'(S;'Z7 A(71H,[! $%&'(`-F<'Z7$E2@501I,s-.(7ZSWX(S2@1=(E-F<'(H$I,m$%&'(D4.&.2@1I(`1%&' !U$ b;'-F<'Z7(7-v>G_H$%&'(?>w,J<.;'-F23WX_m$%WX(S2@$%*o(S;3$c1Q&',agx;H>3_ DLYBK " "wK #c$)\3(SWi_y5A,@g % \@2@5b<'(71H$%&'( "&' 'Rgc2n\3( *+,'-.(:(7\3(S;H-. 01%2G4.4.(S23WX1nK #c$C\3(SWX_)&' 0MJ&`\a2@5b<'(71z,@! % $Q&'(c b*)2@M3 b;.23WX_`4.23WX$ #& c,@!e2@505z(7 0M3(S;GTU!jWi(7Y{<'(S;'Z7 0(71` A1+t7(SWX,|k 0CL 0MFK '3l7"} ~;GT -. 0ZS2@$I b;'M|$Q&.2@$E$Q&'(7_23WX(N;',3$`-F23*)4w(7-8K €]&' A1m 01

(

B‚&'()Sƒ„…nƒ„…a†`‡ %

ˆ

*+*%‰,*iŠ‹ŒI…@ *XŽ/‚=…a -XŒ=…f‘’@‚ &'y“‘’aŒQ“

) "*+ ) ( †%‚7Xp“U‚ )*”•–V‚)'B’@V…,-—†Q‚7Xr“U‚˜ƒ„… +,S…nƒ™“./ ˆ

-500

-400

-300

-200

-100

0

Damping rate: f

i

(Hz)

10

-1

10

0

10

1

10

2

10

3

10

4

10

5

10

6

10

7

K

|R| -> 1

|R| -> 0

mode 1

mode 2

mode 3

!"&#G$./ % *H9:I*#7"0(JIA+ -. KGI<+H5L3 "! 5067' $ +81 5 #29:3 ;<'=1 > ' +8) 1 -A1

%

!*3<+B5067'M;<- / 5 3"!B !"# $ !A$

Z7,J;'1I 01I$I(S;G$zgc 0$%&)$Q&'(z!j2[Z7$^$%&.2@$C$%&'(SWX(x 01 ;',š1I,J<.WXZ7( $I(SWI* b;H$Q&'(/-F<'Z7$:23;'-R>w,J<.;'-F23WX_HZ7,J;'-. 0$I 0,J;'1/2GWX( 2@Z7,J<'1I$I 0ZS2@5050_|Z750,31=(7- kj;',Nuq<'Z7$%<.2[$I 0,J;›,@! +d,JW() 2@505A,@gz(7-Fl7K N >G\J 0,J<'1=5A_œ2@$H1%<'Z7&—5b23WXMG(

%

\@2@5b<'(71 $%&'(H,J<'$I50(7$E>w,J<.;'-F23WX_OZ7,J;'-. 0$I 0,J;O!V2@ 0501E$=,R(7\@2@ZS<GT 2@$I(?2[Z7,J<'1I$I 0Z+gc2S\G(71SK CF,JW]50,@gh(SW]\a2[5~<'(71š,@! % $%&'( *+,J-F<'5b<'1),[! ,/0 1 2e 01+50,agz(SWš$%&.23; " 23;'-O$%&J<'1)2@505 *+,J-.(71x23WX(:-F23*)4w(7-k # k l ! l7K CL AM#K*")23;'- +'+1%&',@gž$%&.2@$]$%&'(:\a2@5b<'(+,@! % *?<'1=$ WX(S*)2@ b;Ÿ1%*)2@505:$I,—4.WX,@\J 0-.(O-F23*)4.(7-œ*+,'-.(71R>.<'$ 2@501I, $%&.2[$v$%&'(¡*+,J-.(71›ge&' 0Z¢& 234.4.(S23W›23WX( ;',G$ 4.&3_'1I 0ZS2@5U9H$%&'(7 bWo!jWX(7Y.<'(S;'Z7 0(71f2GWX(D;',3$+$%&'(H(7 0M3(S;GT !jWX(7Y.<'(S;'Z7 0(71:,@!r$%&'(š-F<'Z7$SK 0 £ ¤q¥L¦,-@O P$% ¤#§F¨8¦z§ %. % / 0 Q]¨ § 1 . ¨2.3-I¦12/¦z§RO@Q P ¥ Q .45 ¥ O6. P §34

#e1H1%&',@ge;› b;—$%&'(O4.WX(7\J 0,J<'1m1I(7Z7$I 0,J;'1m$%&'(s\a2@5b<'( ,@!

%

&.2@1O2—-FWI2@1I$= AZy b;Guq<'(S;'Z7(—,J; $%&'(|WX(71%<'50$I1 ,@!?2v;'<.*+(SWX 0ZS2@5:1= ~*+<'5b2@$I 0,J;BK©€c&' 01D&.2@1f2[5~Wi(S2@-._ >.(7(S;¡4.,3 b;G$I(7-ž,J<'$D>G_—-. &S8(SWX(S;G$R2G<'$%&',JWX1H(7 0$%&'(SW !ª,JW`1=$I(S2@-._ T@U ,JWD<.;'1I$=(S2@-._ T ZS2@50ZS<'5~2[$I 0,J;'1SK«€c&'( *)2@ b;|4.WX,'>'5A(S* !U,JW)1=(7\3(SWI2@5c23<'$%&','WX1) 01H$I,v¬';',@g &',ag—$I,­Z¢&',',31I(

%

~;m4.WI2@Z7$I

0ZS2@5}ZS2@1I(71SKL®c<'-._D23;'-¯

$%WX b¬3gz(SWX-F2 T@U 1%<'M3M3(71I$=(7-)$%&'(z!U,35050,ag] ~;'M+1IZS2[5A b;'M+,@!

%

! !

+ ""# , +! #-,

./0'1)231 $# "#(4 50'1 $6789:0;&'<=>?@=1)2A$8BC 50'1D>?1)7-&EFC$8.GH I 50'1;(!$'<=&'JK(5LM1N1NJO7-&'J ! 50'1GJ=$%>P78"Q&R(!"#SN1-T

UI<=>?1)23"#9)78VW(!"Q>X<'VQ78 Y"Z$'&'( !"#$ #$ %& (50'$8.[1NJ\ ]0=78 ^7-& $%L' !"Q>?<=> '"R"#(\231)789N0'1NJ_B`$%2 ! %&!"!'( ) # ( ! " " (#$a./0'"#VZ1 50'1b 50'1N$%23c;$8BdeE<'J=cX7-&'J f 523"Qg-.D1)23JC7 !" (5<'h-h-1N(Y !1NJ 7-&R$%L' Y"i>?<=> j878VQ<'1?$8B ! *&$"!'( +,-! " " ! %MT

k^0'1lL=231N(Y1)&- 6.D$%2Yglhm"Zjm1N(n7o&'1N. "Q&m !1)2!L=231N ]7pq !"#$%& $8B ./0'"#9:0 "Z(\&'$8. V#"Q&=g81NJo !$6 ]0'1K9)<' 3qr$ ) Br231Nst<'1)&'9Ncu$8B; 50'16@=$%<=&'JC7-2vcl@-c *wsdT $=Tok^0'<'( +-h-$'$'J ,+Kjx7yVi<'1N(P$8B 7-231z 50'$m(!1z./0'"#9N0_78V#VZ$8. 78V#V JC<'9N {789N$'<'(! !"#9K>?$'J=1N(z !$KV#1)7pj-1\ 50'1\J=$'>z78"Q&|Tn}~ .^78(I(50'$8./&a"i&(!1N9N !"#$%& %#R 50=7y IB~231Ns=<'1)&'9N"#1N(?VZ$8.D1)2 50=7-& " % ! & +& '€,/7m231{231]FC1N9N !1NJ€Hb7-&'J6Br231Ns=<'1)&mq 9N"#1N([0'"#h%0'1)2w 50=7-& #" % V#1)7pj-1D 50'1D9N$%>zL=<' 57y !"#$%&=78V'J=$8q >z78"Q&|T}` "#( 50'1)&‚231NV#1Njx7-&m  !$I9:0'$'$-(!1 (Y$/ 50=7y "

%

"#(/V#$8.D1)2D 50=7-&a78V#V4789N$%<'(Y !"#9;Br231Nst<'1)&'9N"#1N(;1 #$=L=1N9N !1NJ "Q&\ 50'1;9N$'>zL=<' 578 !"#$%&=7yV€J=$%>z78"Q&|T

%I(X7-&a1 #$=7->zL'V#1z 50'"#(G(Y 52!78 !1Nhmc\"Z(?&'$8. 7-L=L'V#"#1NJ !$\ 50'1?JC<'9N ‚$8Bƒ(!1N9N !"#$%& (MT

f

"Q&'9N1Rj-1NV#$%9N"# 3c„"Z(‚"Q>‚q L=$m(!1NJ…78 K ]0'1n"Q&'V#1N †7m&'J L=231N(Y(5<=231l78 a 50'1_$%<' 3q V#1N )HW 50'1zV#$8.D1N(! X789N$'<'(! !"#9zBr231Ns=<'1)&'9Nc6"#(G 50=7y G$8B/7 s=<=7-23 !1)2‡qr.^7pj-1G>?$'J=1-T %E ˆ7Ph-"#j-1)& '$6789:0a&%<=>?@M1)2 # 50'1;s=<=7-23 !1)2‰qr. 7pj-1;>?$'J=1IB~2v1Nst<'1)&'9Nc\"#()* " -(! + ""# , & + ! &%,

.d2v$%> *s|T $o7-&'J ! &o 50'1_0'"Zh'0'1N(! „jx78VQ<'1l$yB B`$%2; 50'"#(?9)78V#9)<'VQ78 !"#$%&Š./0'"#9N06.E"ZV#VD(!"#h%&'"‹ 9)7-&m !V#c JC7->zLŒ 50'1 ! &u. 7pj-1Š>?$%J=1 +~78(K.D1NV#Vz78(„$- 50'1)2 0'"#h%0'1)2E$%23J=1)2^>?$'J=1N(5,D"#()* (./01 ! 'ƒ+ ""# , +! (-, k^0'"#(;(!9)78V#"Q&'hR$8B "Z(Ij-1)23cR(!"Q>?"#VQ7-2I !$? 50'1‚ 50'1]q $%23c\$yBwe^<'J=c„7-&'J f 523"Qgm.[1)2vJC7 / !"Ih-"#j-1)&"i& *wsdT ! # @=<' z.^"# 50_7 !

9N$'1&'{9N"#1)&m {$8B ('l"Q&'(! Y1)78Jl$8B "

"

(#$=T

(#Ž

)

N‘’x“W”)“ *+)Žp•–‘•–—!Ž?— )*%‘’8“23]—Y˜x‘•+™,+_š!›Yœ™x“ 43 *i—Yž25E•–Ÿ -x›]Ÿ`“3”?—!Ž?‘’x“ ./3—!Ž)š]“‡žr 5“‡Ž 0/3“^—1*€Ÿ``“3›5” 23;Ÿ``›5`“ ./3›!œ/‡™pœ–›5‘•–—YŽ-Ÿ3 ,|’8“ržr“3›]Ÿ•Ž

)46)*7

-•–w•–Ÿ4-x›]Ÿ`“3”?—!Ž --—5‘’z‘’8“ /3—!Ž)š]“‡žr 5“‡Ž 0/3“^—1* +D“3›!Ž\š!›Yœ™x“3Ÿ/›YŽx”R‘’x“;“3š!› &/‡™x›]‘•–—!ŽK— 1*ƒ›&/‰—Y™xŸ`‘•/-,|›!š5“3Ÿb•Ž ™)Ž8Ÿ``“3›5” 235/3›Yœ/r™pœ–›5‘•–—YŽ-Ÿ ¡ (!"Q@'VZ1\j878VQ<'1{$yB ! Y$OL=231Nj-1)&m G789N$%<'(Y !"#9zB`1N1NJn@=789Ng "Z(p* ! (./01 ! ' +! &m, 0 &„ 50'1G$m 50'1)2/0=7m&'J +‘$%2 ! ,I0=78(I !$¢@M1?9:0'$8q (!1)&?VQ7-23h-1D1)&'$%<'h'0? !$GL=231Nj-1)&m ƒ7IJC23"B` ƒ"Q&? 50'1^>?1)7-& jx78VQ<'1N(G+`L=231N(!(]<=231-H%>z7y(!(WFC$8.GTNTNT!,NTk^0'1/>?"Q&'"Q>?<=> 78JC>?"#(!(!"Q@'V#1nj878VQ<'1 ! ( ! $ "#(O&'$- ‹ $'1NJ @-co789N$%<'(3q !"#9N()T }~ n(! ]23$%&'h-V#c£J=1)L=1)&'J=(_$'&£ 50'1u9N$%>zL=<' 57pq !"#$%&GL=7-2Y7->?1N !1)23( 1 +`eD1Nc=&'$-V#J=(ƒ&'<=>X@M1)2NH # 2u1#)¤1N9N Y( (5<'9:0Š78(;(!.E"i2vV¥Hwh-1N$%>?1N 52vc-TNTNT!,NT 0 50'1)2; !1N(Y !(¢+`&'$m 231)L=$%2v !1NJl0'1)231),?(5<'hmh-1N(! ? 50=78 z9N0'$'$-(!"Q&'h ! V#$8.D1)2 50=7-& " " $8B¥ Y1)&?"Q&'9)231)78(!1N(D9N$%&mj-1)23h-1)&'9N1E !"Q>?1N([7-&'J (!$%>?1N !"Q>?1N(WJ=$'1N(ƒ&'$- ¦7yVZV#$8._>z78(!(AFd< $%1N(ƒ7m&'JXL=231N(3q (5<=231; !${2v1)789:0\7§(! !1)78J=cR(! 578 Y1-T

."Zh¨T ';$3)¤1)2v(ƒ7;(!"Q>zL'V#1;(5<=>z>P7-23cG$yB¤ ]0'1N(!1

‹

&'J'q "i&'hm(+~U^$- Y1z 50=78 4."#hCT 'a"#(z789N 5<=78V#V#c_7 2Iq‰L'V#$- 5,©T

1

1

2

0

0

Mean pressure may drift

0

0

σ

min

0.58

π

3π

5π

Mode f , f and f are not damped

Adequate range

Mode f and f are not damped

Mode f is not damped

σ

56789,:89/; %<= >4> ?@A!.BC)*'DEFGHI7EJ + =GHKJLM#H'C )*

!

C/J DEFGH ?/MNC@=/OPDQ7RMSOT?/JLUI>4HN?@J ,+ C - & = ?/JLDQ7EDQ7EHPO :

: ' V()*+ wªWX!«¬,-! Y )*.ZV;­d¬D­ [X!Y \ ® f 1N9N !"#$%& B(=T! 0=78([L=23$8j'"#J=1NJK7G ]0'1N$%231N !"#9)78V¨(! 57-@'"#V#"# ‰c 7-&=78V#c%(Y"Z(?$8B^ 50'1PJC<'9N X$yB ."#hCT &=T\}r& 50'1{L=231N(Y1)&- (!1N9N !"#$%&|H‚ 50'1N(!1n231N(]<'VZ Y(¯7m2316 !1N(! Y1NJ &'<=>?1)23"#9)78V#V#c-T k^0'1ƒJC<'9N W"Q&'VZ1N "#(7bBr<'VZV#c?231]FC1N9N !"Q&'h;9:0=7m2!789N !1)23"#(! Y"Z9 "i&'V#1N ;.^"# 50a"Q>zLM$-(!1NJ„j-1NV#$%9N"# 3c & T %…&'$'&mq‰231]FC1N9N !"Q&'h

; °|—YŽxŸ`“/0]™8“rŽ)‘œ3]8‘’x“6--—!™pŽx”)›Yž 73<5=> 1?@ •–ŸMŽx—] /rœ–“3›Yž`œ3^”)“ 8 +)Žx“3”I•ŽG±'•–  ¡ 9 ¡ 7 ²:/`’p•–“‰š]“3”n‘’pžr—!™8 !’ ; <:= (Ežr“‡œ–›]‘•–—!Ž³´ ->3K•+b˜-—]Ÿ‘•Žx  " 2 " 3

BC6<&D"#(E FG -'4*BC6HIJ-'():K$% L-'&'&D"#4M$.I%6F$%(E$./#(NIJ()4."#&'$%45 <G 4%()O)$%-' !P#4 !Q6HP#: 9R ST!67"#4M4%-U67PV8W(?IJ$."9I%X96<$%-' !PY-'4 4."98W(?IJ-UZN89 74M():N$% [$./#(\-DP#-'$%-U6F&]89IJ()4%4."9I^( _ ()&':N$M =( !"H` O)-'$%(a6<&b&c6FO) !"#4%$%-'O5Zd #:9()4?R

e\+\ YBC6F&U"#()4fGg !I 67IJ(h-UP7B7()4M$%-'T!6F$%():i$% K-'&'&U"#4J` $.I%6<$%(V$j/#(V-UPHkl"#(?P#O)(i !PmX9 H$./n$./#(Y:o67ZN8#-UP#T <G 6FO) !"#4M$%-'O)4 6HP#:p$./#([4./#-Gg$q <Gr$./#([()-bTH(?PH`gGsIJ()t9"#(?P#O)-'()4 FGu$./#(*:o"#O)$?R

!" ! v]wFx ""#$%&'()

y*+,-z{Pn$j/#-b4|4%()O)$%-' !P]}d$./#(~IJ()&U6 #"96F$%-' !PG{6FO)$% !Ih6F$0$./#( !"#$%&'()$ -'4 _ "!():i$%

"

$%&$. €+[/#-bO)/-'4p$% # /#-'T!/~$M Q()BF6FO?"96F$%(L$j/#( _ IJ4%$d()-'T7(?PH`‚Zd #:9()4p FG[$./#( :o"#O)$?R\e,/#(a89I^()4%4."9IJ(d89(?IJ$."9IMX96F$%-' !P#4?}]IJ()O) !IJ:9():06F$

-100

-50

0

50

100

Pressure (Pa)

0.20

0.15

0.10

0.05

0.00

Time (s)

"#$% & '/0() ! $*+,- - ./01234(56$7018 0#$9:;<=->?@?A4 <=- : -/<@564 <"#BC D 56$7018 C15 57E(-,$*8F2-/512 %" $'&3. 45(&

$./#(N-UP#&'()$54j/# C+ƒ6HP0( !"„8W !P#(?PH$%-U6F&…:9()O?6†‡;ˆ GHu-bT‰R $'& Š X9(jGg !IJ( !

"

&

"

$

O) #I%IJ()4.8W !P#:9-UP#TK$% 0$./#(L()BC6<O?"96†` $%-' !Pf FGu/#-bT#/EGsIJ()t9"#(?P#O)‡‹Zd #:9()4?R…e,/#(*4M-bT#P96F&]$./#(?P IJ(?ZN6<-DP#4K6F&UZd H4%$‹O) !P#4M$.67PH$‹4%-UP#O)(~&' F+\`gGsI^()t„"#(?P#O)‡ Zd #:9()4=67IJ(5I^(jko()O)$%():QXH‡p$j/#(5 !"#$%&'()$,O) #P#:9-b$M-b #P]R

Hu-'ToR $($ -'4L$./#( IHo #"9IJ-'(?If$.I%67P#4JGg !I%ZŒ FG*$./#(h-DPH` &'()$=89IJ()4%4j"9IJ(*4%-'T!P96F&gR\e,/#(*4.89()O)$jI%"9Z -'4,O) !ZN896HIJ(): $% h$./#(L( !"„8W()O)$%():~()-'T7(?PH`gG{IJ()t„"#(?P#O)-'()4‹67P96<&b‡#$%-'O?6F&'&'‡ :9(?IJ-'B7():n-UP€4%()O)$M-b #P !9R eA/#(

_

IJ4%$L$./9I^()(VZd #:9()4 89IJ():9-'O)$%():iX7‡ JKtlR )*$ 67P#:Y:9-'4.8#&U6†‡7():Ž-UP Hu-'ToR $($

0.001

0.01

0.1

1

10

100

Power

800

600

400

200

0

Frequency (Hz)

AVBP with LRM

Analytical LRM

Closed duct

"#$L%M& '1'N) !1:M-@O/56<@4 + 0#$ :;<=->?@?A4 <=- ? $%(8FC1262 &"

$'&7. 45 OP01+Q: CN<=-PR 560JC18FC12 $% 56$*OPC12 . C1234 -A? 0#$I?A-PO 56$7018 # C18SR " &'( ((&

IJ(?ZN6F-UP~-UP$./#(0O) !ZN89"#$j6F$%-' !P96F&,:9 #ZN6F-UPi6F4N4."#TF` T7()4%$%():G{IJ !Z Hu-bT‰R#$|G !I

"

$'&

uR !Q #IJ() FB7(?I)} $./#(\G{IJ()t„"#(?P#O)-'()4\ FG2$./#()4%([Zd !:9()4\67IJ(=P# H$u$./#(,$.I%"#( ()-bTH(?PH`gGsIJ()t9"#(?P#O)-'()46<4 89I^():9-bO)$M():‹X7‡ KtlR

)() +[/#-bO)/ T7-'B7()4L$./#(0()-'T7(?PH`gGsI^()t„"#(?P#O)-'()40 FGa67Pi6FO) !"#4M$%-'O?6F&'&b‡ O)&b H4%():0:o"#O)$‘ˆ’-UZN8W 74%():0B7()&' #O)-b$J‡‹6F$[$./#(>-DP#&'()$567P#: -DZE89 74%():L89IJ()4%4j"9IJ(6<$…$./#(, #"#$%&'()$.Š)Rue,/#-'4\O) !P _ I%Zd4 $./#(pX#-U6F4%-UP#TK( PT2()O)$%4>-DP#:o"#O)():ŽXH‡0$./#(f !"#$%&'()$5O) !PH` :9-b$M-b #PL+=/#(?P“&D6HIJT7(*BC6<&D"#()4[ FG 67IJ(5"#4M():2@

)

$./#(” !"#$%&'()$r-'4c4%$.IJ #P#T7&'‡5IJ(jko()O)$%-UP#T*4% [$./96F$2O)(?I•` $.6F-UPLZd #:9()4*67IJ(5P# 7$\:o67ZN8W():Q67P#:

)

$./#(“GsIJ()t9"#(?P#O)-'()4‹ FG*$./#(0Zd !:9()4L:9-T2(?I‘GsIJ #Z $./#(*$.IM"#(*()-bTH(?PH`gGsIJ()t9"#(?P#O)-'()45 FGu$./#(*:o"#O)$?R

U X7B#-' !"#4%&'‡7}o$j/#-b4\+\ !"#&':‹P# 7$,X9(56>T7 # #:pO)/# 7-'O)([G #I G #Iq89IM6FO)$%-'O?6F&2O) !ZE89"#$.6F$%-' !P#4?R

e,/#(a6FT#IJ()(?Zd(?PH$=X9()$J+\()(?P‹P#"9Zd(?IJ-'O?6F&u67P#:‹67P96†` &b‡#$%-'O?6F&5IJ()4j"#&b$M4L-b4LTH ! #:2},$./#(h4./#-G$‹XW()$J+q()(?P$./#( ()-bTH(?PH`gGsIJ()t9"#(?P#O)-'()4V <G‘$j/#(KO)&' 74%():m:o"#O)$K67P#:€$./#( :o"#O)$,+,-'$./‹1–3 *!—-'4,O) !I%I^()O)$%&'‡‹89IJ():9-'O)$%():2R

$% ! v]wCx %8#$%9'

-z{PL$j/#-b4A4%()O)$%-' !PL$j/#(5IJ()&U6 #"„6<$%-' !PEGs6<O)$% !I -'4,4%()$,$M u}o+[/#-bO)/0-'4*$./#(N/#-'T!/#()4%$*BF6F&U"#(N$./96F$[(?P967X#&'()45$./#( ()BC6FO?"96F$M-b #PL FG…6F&'&r6FO) !"#4M$%-'O5Zd !:9()4[ FGr$j/#(5:o"#O)$?R

-100

-50

0

50

100

Pressure (Pa)

0.20

0.15

0.10

0.05

0.00

Time (s)

!"# $ % &' "()*+,+ -./0123&45"6/07 /!"89:;<+=>?>@3 ;<+ 9 +.;?453 ;!"AB C 45"6/07 B04 46D&+*"(7E1+.4 ! #$ #"#F$ G !"#$% !" &'()*+,-&./0()121 #$345*67)1879/:!;<=>?@A1BC8<8DE*=FG/0()1 4AHI1B&:&'JAHI1K4A18HI/0JAHLMA<=/:!"*67)&8% %G/ $ & ! <=>">N<=CB*)J)&IO /:!"CPQR*6@A1B&?(A<STU1GT+<97)!V&'()1B@W%XYG*9/:1G/0(A<=/Z/0()18HI1G!"&?7)* &:*6JAH[CB1X/:18H:Q\!;7]/0()1X@A*6Q^<=!;7_<97)@_/'(A<=/W/0()1Z&:!;Q_J)>;<SO /:!"*67R!"&?`AJ)!V/L1a>"*67)#]CB*6QK4A<UHI1B@^/:* !" $ b%&0HcCBD)CB>"1B&0dZ&:* /0(A<=/-1BTU187e<fHI1'g$1BCB/L!V*)7hCB*)1 '(fCB!"1879/P&:>"!V#)(U/:>"Di>V*=,?18H /0(A<=/ bj/0()1K@A1 k 7)!"/:!"*67l*mFn/0()1KC8J)/[Oo* IpFqH[1B`3J)187)CBD " $ !"&P&:*K/0(A<m/ ' $ % # " %dK187A<UM)>"1B&P/:*2<=/:/:187)JA<=/:1 /0()1.QR*)@A1U%

0.1

2

3

4

5

6

7

1

2

3

4

5

6

7

10

Power

800

600

400

200

0

Frequency (Hz)

AVBP with LRM

Analytical LRM

Closed duct

!"# $ %&' ' $09J+?K.45;?3 ) /!" 9L;M+@>N> 3 ;M+O>@"#&7PBF1 ! ($ # # KN/0)Q9 B0;M+NRS45/ B07EB01() 45"6KNB01T- BF13 +@>*/!"*> +NK 45"6/07 ,B07ER % $) &$ &0JAHI1u&:!"#67A<=>v*=F G !"#$% !" MA1B/I,?1B187 $ & <U7)@ $ & ! & H %iwG()1f&'4A1BCB/0H:JAQ !V&RCB*6Q^4A<UHI1B@l/:*h/0()1 1#$3451BCB/:1B@K1B!"#U1879OxFoHI1B`3J)187)CB!"1B&a<97A<=>"D6/:!"C8<=>">"DR@A18HI!"TU1B@ !y7E&:1BCB/:!"*67 Hh<97)@z/:*i/0()1K/0HLJ)1f1B!"#U1879OxFoHI1B`AJ)187)CB!V1B& *=FX/0()1R@$J)CB/.#U!"TU187eM9D V{`|% ")" %PwG()1R&0451BCB/0H:JAQ !"& 7)*U/KTU18HIDl&0(A<UHL4p@$J)1h/L*}/0()1h1 #$A4A*)7)187U/:!;<=>c@A1BC8<SD *=Fa/0()1v&:!"#67A<m> <97)@pDU1B/8~/0()1h<=#6H[1B18QR187U/R!"&RFo<=!;HI>"D #U*6*)@W% %a#)<=!;7v/0()1R1B!"#U1879OxFoHI1B`3J)187)CB!"1B&R*=FX/0()1R@$J)CB/ <UHI1G&:!"#67)!k C8<U79/:>"Dv&0()!Fx/:1B@fCB*)QK4A<UHI1B@2/:*_<97K<=CB*6J)&IO /:!"C8<=>">VDiCB>"*U&:1B@h@$J)CB/S%

! W X €"#NY.:ZAW X

%p&:!;QK4)>"1a<=CB*6J)&:/L!VC?/0()1B*)HIDP,G<=&‚4AHI*645*U&:1B@R/:*GCB(A<UHƒO <=CB/:18HI!"„B1R/0()1f<=CB/'JA<=>NH[1'g$1BCB/:!"*67iCB*61 '(fCB!"1879/.*=FG76J9O QR18HI!"C8<=> [)7)*679O…HI1'g$1BCB/:!;7)# 0[†MA*)JA7)@$<UHID CB*67)@A!"/:!"*67 J)&:!;7)# ‡Wˆ*+ bx‡‰!;7)18<UHŠˆ?1B>;< *$3<m/:!"*67 +e1B/0()*)@$d‹<m& 4AHI*64A*9&:1B@KMUD.ˆGJ)@ADR<U7)@ Œ /0HI!;9,n18H[@$<

\A]

*)H‚Ž*U!;7)&:*9/ <U7)@‡1B>V1 \9%‘’q/K(A<=&KMA1B187l&0()*=,a7E/0(A<=/]>y<9HI#U12T+<=>O J)1B&‚*=F$/'()1 H[1B>y< $3<=/L!V*)7PCB*)1 '(fCB!"187U/

,

J)&L1B@_!;7]/0()1B&:1 QR1B/0()*)@A& /:*a>"!;7A]/0()1X!;7)#U*9!y7)#],G<8T91X<UQK4)>"!"/0J)@A1-.

^

/:*R/0()1_4AHI1B&L&0JAHI1P@A!IW18HI187)CB1hb %/01 / 2 dB~AQK<U=1P/0()1 MA*6JA7)@$<UH[D“CB*67)@A!"/:!"*67”FqJ)>">"D H[1'g$1BCB/:!;7)#”<97)@ /0(A<m/ , C8<U7lMA1v>"!;7A=1B@‹/L*•<iC8J)/IOo* _IpFqHI1B`AJ)187)CBD " $ M9D , $ 3*#&" $ % %G/_<–#U!"TU187}T=<=>;J)1f*mF , ~ <=>">nQ—*6@A1B& &0J)C˜(E/0(A<=/ '" $ " $ <UHI1f7)*9/_@$<9QK4A1B@W% % 4AH[*64A18H &:/0H:<=/L1B#UD}/:*zQ—!y7)!;QR!"„B1}<=CB*6J)&L/:!"CB&fCB*6JA4)>"!;7)#p!"&f/L* C˜()*)*U&:1 , &:*“/0(A<=/ " $ !"&}>"*=,?18Hh/'(A<=/}/0()1 k HI&L/ @$J)CB/.<=CB*6J)&:/:!"CKQR*)@A1U%.’o7i/0()1RC8<=&L1_*mFn< `™@$J)CB/ ,G!"/0(u!;QK4A*9&:1B@“TU1B>"*)CB!V/IDu!;7)>"1B/8~./'()!V&i!"&h*6M)/0<m!y7)1B@ MUDl,aHI!"/:!;7)# , $ b 1 ! \ d8b%& % (ZdR<U7)@pC˜()*)*U&IO !y7)# & ! " $ $ # % G$*6HvQ—*6HI1eCB*6QK4)>"1 #$uC8<=&:1B& b%& `K~WCB*)QK4)>"1 #$z#U1B*)QR1B/0HI!"1B&8~ CB*)Q_MAJ)&L/:!"*67‰%8%B%:dš<U7 1#$6/:187)&L!V*)7}*=FG/0()!"&R&:/0HL<=/:1B#UDi!V&R/:*i&:*U>"TU1K/'()1f#U1879O 18H:<=>)1B`3JA<=/L!V*)7R*=F|<=CB*6J)&L/:!"CB&X!;7R/0()1?@A*6QK<m!y7]/:* k 7)@ !V/L&‚1B!"#U1879OxFoHI1B`AJ)187)CB!V1B& "'(:%‚wG()1?QK< *$)!yQRJAQ\T+<=>;J)1?*=F !"&G/0()187v#9!VT9187hM9DfCB()*6*9&:!;7)# " $ $ ) + ! b*")(jdB% ! * +,-'+ ›.+ X €"+Z

^ VX7)#U`AJ)!"&:/8~ /c%"# +•<_a…@$<5~ %R%;~ [)%PM)&:*6H:M)!;7)# MA*6JA7)@$<UH[DfCB*67)@A!"/:!"*67)& Fs*6HZ/0()1.7)JAQR18HI!"C8<=>‰&:!;QRJ)>;<SO

! " ";5;<;<=: #$ ! !.> $#1 :

22?@$0"A&B3!$0 .5CDE: FG=128#105 H :53! %I$%J;&BKL-MNOPQR2$B&BK;"S> !"#$%&B3@)9$%N@K;"ANO12R2 73!"ATUJB8A-( !"#$%&B3*$V)WR2$%TX<;NY123!3!"AZB8#1[/B"#3O> R2$%JB3\]L$^+_345 !"#4' 3()*+,-. $0'$% &"#567' !`67$089: "#$ 5 5 ! ! ! 5;<;<=: "# "(> %! !;:

84a_"bNY3!Rcd=5 #$E:A5&' . *9: ;<=>?<")@ 3()*+,-. $A")$A=%()B (%& =>B C $A: ;<B4"#@D")B4EF: () $A: ;<B4")@ ' ( (G67'*+0$Bd;&

)

"#8#12P05 ! & &;: H ae-(f03! gNY$%Th5(DE: ia[-MNO"Ad;-0N!-0&=5 H :0jk:A5 !&'eR2R4J;N!-( !1 %I$%J;&BKL-0NOP *$`$B&BK;"l k"l$B&B3*)9$%N I()B !14NO"#$%N*?INO$%ZB8#14TU3`"A& , -V3JKEP;&;-0TU"#R05 ! ! ")$%& : *9")$A=%?L6M( %& 3()*+,-. $A")$A=%()B -! 67$089:31 5 ! & &m5%<;<=: 1 & >N1 ! ";:

O

Dn$%J;N!NO12 ! k105*G7:A5 !&'ENO !"o R2"A-(8 %I$%J;&BKL-MNOPQR2$B&BK;"S> !"#$%&B3\)p$%N\ gdB1qG="A&B14-0NY"lr212K $ $%TX<;NO123!3k"bZB8#1hs[-./B"#14Nt> H !$%uV123 IvwxJ;-( k"l$B&B345 ! #4'! 3()*+,-. $0'.% &"#567' !E67$089:

! "m5 ! ! ";5;<;<=: > ! ";:

P ae-4PBK;14N25 +y: I,: zD{J;Nku(12895 I_:A5)!Bs,$%&;NY1|]L12R2 !"A&Bf %I$%J;&BKL-0NOP $`$%&BK;"# !"#$%&B3U)9$%N /+012 9Q78#$^+ $`$BTX<;JB g-.> !"#$%&B345!$%,- %&%0#4' !x67$M8p: ! !;5 ! !)1;50<;<=: !&! $ "(> !&! " # :

R2D`3!P;&;u($(/=5 H :}6U:A5 !%seJ;TU14NO"#R4-(8 H $08AJB !"#$%& $()~?*NO$%ZB8#14TU3$B& 12e&BZ;$%J;&BK;12K KE$BTX-("A&B34: ' €`12/B"#12+5 ! % ,),-@S=>:<E "'T./*9:5;U=%?U"#@ ! ")$%&/:5* "#$A=>?767V

'W;X")BY6L")?5$A=>()B 6 (%&.-(! % !3!,67$089: ! ";5 ! ! &;5e<;<=: " $)1^>N1 ! ! : Z26*-(3!"#8#P012/=5 [

:76U: %*$(+[T‚-0&=5 /$\:A53!

H

12R2$%&BKB> f014&B14N!-V !"#$%&'+ -./0128#12 eR2$08#8#$%R4-( k"l$B&ƒTU12 gdB$BKh)9$%N_ gdB1 3!$08AJB !"#$%&„$V)\<;-0NY !"A-(8`K;" \]…14NO14&M !"A-(8,12w;J;-( !"#$%&B3.5 ! #4'

3()*+,-. $0'4%3&"#56 '5!,67$M8p: $)1;5 ! 5 !&#&#&# 57<;<=:'$ $ # > $ ! !m: ^ "e"ATh5 6+L: ) : G†12105 KX:/+L:A5_! , 14&B14Nk-(8#"lr212K $,d;-0NY12R2 !14NO"#3! !"#R %I$%J;&BKL-0NYP 0$`$%&BK;"# !"#$%&B3\)p$BN ,$`$%T‡> <;JB g-( k"l$B&;-(8 ('[14NY$%-(R2$%JB3! k"lR23.5 !)%,-*% %7# ' !767$089: *! &;5 !&#&# # 5;<;<=: !&# " # > !&# " !;:

`0a $`$M8l$B&B"bJB3.5%DE:A50G†128l105

H

: 1+ˆ$0"A&=5B?{:5 ! %I$%J;&BKB> -0NOPhR2$B&BK;"l k"l$B&B3e)p$%N_K;"ANO12R2 ER2$%TX<;JB |-( !"#$%&h$V)C-(14NO$(> K;P;&;-0TU"#R~3!$%J;&BK f014&B14Nk-( !"#$%&=5 !+%,-*% %8# (#./;<B4")@ 9! 67$089:2! 5(!;5 ! ! !m5x<;<=: 1 " "(> 1 & ! :

`b`g€`$(+,8#12P05 $E:

)

: 3$`$M8l$B&B"bJB3.5%DE:A5 #!)KE"#3!R4NO12 !128#P s_$%&;NO1|]L12R2 !"A&Bf )%*$BJ;&BKL-0NOP 4$`$%&BK;"# !"#$%&B3_)9$%N,G="A&B14-0N a_Px<m14N!Z;$08#"#R H P%3k !14TU345 !:# '" ()*+,-./$0' ;% &"#567' !=67$089: 1 "m5 ! #&#&# 5;<;<=:31 # # >N1 ! &;: `N2<Q=NO14J;&BK…5 <+L:*%e:A5+!%?*NO$B<;$03!12K~jt&M]L$^+ [ JB !8#$(+ %I$%J;&BKL-0NOP 5$,&BK;"# !"#$%&‰)9$%N KE"ANO12R2 6$`$%TX<;JB g-V !"#$%&X$V) 'e14NO$%K;P;&;-0TU"#R H $BJ;&BK…5 !$%,-*% %=#4'5!L67$M8p: 7!)1m5 ! ! ";5

`08 D,dB$%TX<B3k$%&=5 "U:

)

:A5 !%D "bTŠ1‹K;14<;14&BK;14&M Z;$%J;&BKL-0NYPƒR2$%&BK;"# !"#$%&B3U)9$%NdMP;<;14N!Zm$08#"lR'3!PB3! !14TU345! # '" 3(#* ,W./$0';% &"#567' !…67$089: $ &;5 ! & ";5;<;<=: >! "L:

`0H D,dB$%TX<B3k$%&=5 "U:

)

:A5 !%D "bTŠ1‹K;14<;14&BK;14&M Z;$%J;&BKL-0NYPƒR2$%&BK;"# !"#$%&B3U)9$%NdMP;<;14N!Zm$08#"lR'3!PB3! !14TU345! # ' 3()*+,-. $0';%3&"#56 '5!…67$M8p: & !;5 ! ! # 5B<;<=:7" ! !.>" $ :

`NO , N!-M<;<B"b&=5Œ€E:A5=G=$%N!-( .5 >+Œ: %`JB ! !"#f%dB$ c]†14N5$'Ž:5 !&'[8)9/?@14&

)

-./01X?*NO$%<;-Vf%-( !"#$%&'"A&ˆ gdB1Xa_"lfBd H $%&;-MN $`$%NO$%&;-m5 !,% 6 $A;X()BW' %T6L$A;X(<,4& "#56 '5!767$089: 8! $ ! 5 !&# #&# 5 <;<=:2! "

!

> !#1 &;:

`0P %`-0N!NOP05 *'Ž:A5 %I"l128A-0uŒ5 A+L: +ƒ-(R $_-0T‡P05…€E: 9$E:5 ! [ & 'ZB3!$BN!ZB"A&Bf +%*$BJ;&BKL-0NOP $ $%&BK;"# !"#$%&B37)9$%N

)

-./01 ?*NO$%<;-Vf%-( !"#$%&=5 ! # ' 3(#* ,W./$0'7%3&"#56 '5!‡67$089: " !m5

! & &;5;<;<=: 7" " !.> " $ &m:

`NR

+ƒJ;N25 , :A5#!&'ZB3k$%N!ZB"A&BfqZ;$BJ;&BKL-0NOP‰R2$%&BK;"# !"#$%&B3 )p$%N gdB1 o &B"# !1|>K;"]…14NO14&BR21‘$()„ gdB1 k"bTŠ1|>K;$%TX-("A& 128l12R2 |NO$%TX-(fB&B12 !"#R|>

o

128lK’12w;J;-( !"#$%&B34:A5! - ! ! $A;X")B 67' ()B ! @S:<? $A;X()*9"Ld/Be: $A=%? ?<()*+, "#$A=-.7=>@S=%$/#B!„67$089: ! !m5

! & 5;<;<=:2! " ".> ! & ! :

`0Z4a,$Œ5 $\::+“: ’a[JB14NkNO105*?@:A5 G!%?@14NO gJ;NkZ;12K”)•NO121 3gdB14-0Ne8A-.P014NO345 !8# ' f3@S./=%E0! :U?7& ' !*67$089: $;5 ! & "L5 <;<=:2! $)1;:

`0^ G…JBR4-(345XGv: €`$BRcuM+`128l895 _KX:A5g!

H

128S)–>1+)BR2"l k12K h—12 4˜J;<B3! |NO14-0T TU$BKLJB8A-( !"#$%&-0&BK TUJB8l k"b<B8#1 )•NO1|> wxJB14&BR2"#12345 ! #4'Tf3@S. =>E ! :<?L& ' !67$089: " ";5 ! & "L5 <;<=:2! ! !.> !#1 ! :

2Xa $,NO"#f%dM !$%&=5 KX:

,

:5DKE$(+,8#"A&BfL5 ,'Ž:A57a_12RcuM895 ;+y:5 G=14<;<B"A&Bf0 !$%&=5 Q,:

)

"#8l8#"A-0TU3.5 C+L:A51! ( E):5;UB * :5$%&)C ( E 6 =>B ")B4"#@# $A=%?U"#@ ")?<().)6L$A=%?L6 ' H <;NO"A&Bf014N™6714NO8A-(fL5

! !

!

:

2<`DE

-0&BfL5e6U: $,JB8#"#R2uL5 Q,: I,:<$E:5J! '&;-(8#PB3!"#3 $(),8l$(+`>9)NO12w;JB14&BR2PQR2$BTŽZ;JB3k !"#$%&ƒ"A&B3! |-0ZB"#8l"# !"#123‰"A&“-8b-MZ;$%N!-V !$%NOPXN!-0T h—12 `R2$%TUZ;JB3! !$BN25 !# ()*2.L.#6 $0' !-?5=i'

'W:U?7& '5!=67$089: 7"/1m5 ! & $m5%<;<=: > ! 1;:

2b2

?@$0"A&B3!$0 45‰DE:A5hD@NO$%JB/ ?@105 'U:A5‰6712P;&;-0&0 k105 KX:5 $,-0&BK;12895

H

: I73|<;$03!"# !$Œ5 I,:A5 !067$%NO !1 +)šKLNO"#/014& -(R2$%JB3! !"#R4-(8#8#PhR2$%J;<B8#12KˆR2$%TUZ;JB3! !"#$%&‰"A&B3! g-MZB"l8#"# !"#12345 ! # ' f3@S. =>E ! :<?L& ' !…67$089: " ";5 ! & ";5;<;<=: ! $)1.>! ! ! :

2X8 +ˆR +›-M&%JB345 "U:A5*?@$0"A&B3!$0 .5CDE: $,-M&BK;128p5

H

:5 !&'}NO12/%"#12+ $(),-(R2 !"#/01XR2$%&M gNO$M87$(),R2$%TUZ;JB3! !"#$%&ˆ"A&M> 3! g-0ZB"#8#"# !"#12345 !F% ;X(bd '"! Be: ;%d"# ()*2. ' !W? =A'5!L67$M8p: !m5

!*?$!.@A%$/ #'B0/#(5CD?.,-#$% $EF*F*=

G

H(IJ8"'(?K@AH925!":J2K3;L&M"%+IN> H9L+?K'O?.!"FP?KL&2"8":A@AH9LB% )*D%&L&@A:Q2"!RL&25IJ)*8":A@AH925!RH(!":A@S?T> )"H9'9HU@VHU%&:T/ +(,-./0 " !1234"# 5 6789: .07;$#<2=> ?@AB078 !"#$%&' ! !()

) !"!"! )*+(+,' "! !-. " " ' "#$

/01*2(3$) %&4'(56' 7 89:;<=>?@ABC:;2 D() ! ' E'=) '"#FG*#HG?. :;BIJ BKLK8M<=G*NO#H1*8PJ #Q@RST#H1(G*2 D$:;3ULK#HG*2(<V8M<V#HGWXY#H: Z91(S?. ZM#HG*<VL6[\D-]H<VBC:

7

8M#H>^BKZALCD^%VLC1*%=D^8M<V#HG*Z-) (' ")* !+,-. /01234* #+5 6 *$%_!"#$%&' !""!() $ #"$ ! )*+(+,' ! !Q. % ! '

"78 `a#H1(:M:;BK8b8MB$)4c"' 5\D^%=+(BC:MG,)Oc"'=) !"#96:, ;$#:<=> ? , 1 >)@ A ;9<=B:6 A >)@ %CA& B ;D6 %'( ,-. A-<E> (FBGH9, .I/ ,J6 <K 3L<=, > ) B 3=52,M@ 6 * ! / /NOP<=HGA-3L<E, > 6 3L, %CA ;=? B !NH9A OPB !+, .I/N1 3LA 3L<=, >N* d[0#^]^D 7 LK<VBCG*LKB$) "! !%$ '

"#QKef#$<=G*ZM#?8C)g`E' !"BK3hG(D$G?8MB$) &4'=)*+R52B9, ;7B 3L<=H9A O A >%@ >)1 .SB ;9<EH9A OTH9, . '#:1-6 3L<=, >0* /E'i`E'UVj2(@ D?:;2(ZC) "!"! $ '

"7W

#

Dk%l%&)$ E'=)\e"<VBC:;LKB$) % m'+&4' &n#$<=G,)Eeo')X' # 7 BCpq<.r<=p4+*%V<VLK<=8 '&nBK89s*#*2qXY#*:"/ABKZM#$%=1*8M<V#HGt#^X #gLK#*1*Z;. 8M<VL

#

DC]?BKZ4<=Guc,#^@ &vD^LKsw[g1(pqSTBC: UY"%V#^@0ZC) (' ")* !+, .I/N1 34*&# 5, M6Z*$%,!"#$%&'

$ $ $ ) "!"!" )*+(+,' ! (2!-.[!"! "T' "7\

[d<VLK#H1*2_) Y0'=) ' &EB x G*<=G*Ny@0D-]$BzD$p4+*%V<V891*2(Bn<=G LKs(D$:MD^LK8MBC:P<lZb8M<VL{S(#*1(G*2 D$:;3OLK#HG*2(<V8M<V#HG*ZC)' " *"!+,-.]K /N1 34*&# 5, M6Z*$%|!"#?%Y' $ ( #T) ) $ #"#"$()(+(+,' )( $ $-. ( " ' ^#_ ' <l%VBKZC) &w'=) 'H[d#HG?.}:;BIJ BKLK8b<~G*NwS(#*1(G*2 D$:;3LK#HG?. 2(<V8M<V#HG*Z€XY#*: V1*%VBC:€BK‚h1(D^8b<l#*G4LCD^%VLC1*%=D^8M<V#HG*Z-) (''!$* !-! "`, 12;9>)A-O%_!"#?%Y' $() $& ) $ #"# ! )(+(+,' "! ! ! . "! ! $('

^:a"#ED$1?Xrp4D$G(G,) #q'=)_[d<VLK#H1*2_) RY0' ef#$<=G*ZM#$8-)_`E'=) ' Y"%V#^@bYƒ#H:;LK<=G*N6`aBKL„s(G*<V‚(1*BKZX&#H:f[g1(p…BC:;<VLCD^% 7 <=p{1?. %=D^8M<V#HGW#^X A#HpqS(1*ZM8b<l#*GO†rG*ZM89D?S*<l%V<V8M<VBKZC) (' !+, .'#:1-6:34* c+OPA .SB (%_!"#$%&' $ " $ ) "!"!" )h+(+,' ""% $ . ""$-!('