Proceedings of the International Conference on Speedup Technology for Railway and Maglev Vehicles, Yokohama, Japan November 22-26, 1993 © Japan Society of Mechanical Engineers PS3.3 ‘Shock-Free Tunnel’forFuture Tralns PS3-3 ‘Shock-free Tunnel’ forHigh-Speed Future High-Speed Trains N.SUGIMOTO エ ∧ Z)印αΓzma4 a/'μed・znic・zl£殉μn,ari㎎,j4zcu哨/θ/'&lが7leer垣!7 一 びni‘17ersizy o/'θsa・z,θs・zjh ぶ卯,J卯an ln ABSTRACT This p奉pel inhibit aJI &cous吻=s!igck high-speed tza.il8. of intloduces a. ゛shock-frge wave in as sjde ud is bza.皿cbe8 ajso to sound its lise waves to to thi8 gf a.りIllael of 4 sho4 】:&thel to !t il wit4 is case, of tuune1. As the &n equ&laxial lol is to 4coustic evo1Ution of that 晦e ‘highgz-ozdel witlμhe is dispezsion' !lonnll9&l w&ve a.nd s4卵p- th&t・,in pxop&ptedinplaci in cu suc4. & 4u卵l is be a・pp!led But it subjected zeachiug,Qtcourse,i'i!;hout 刄9t Qldy the shock,flee action a of sko4 t!le to i8 uy wa.vebut tu耳el da早pi刄g祁dthe of it. be to less noted 8llo9k ln in of wave. 】;n gzdel as tht itづis A gshock-free tunneP ploposed in this p&per is such &tunnel tha,t the &bove two mech&nisms a,le embodied in the infra.sound !al so&s to inhibit &shock wave. lt consists ofづ&m&inp&ss&ge loz tla.ins,i.e・,&usu&l tunnel,a.nd m&nyc&vitie8&il&nged exteln&ny&8 8ide bl&nche8&longthe&xial di】:ectionof the tilnnel in a.Ilay a.nd a,lso along its dlcumieleiltial dilection.Role of the c&vities is to give rise to dispelsion &ndd&mping into the n6n- tQjlhibit to gpli)i4both dispezsion. INTRODUCTION 犬 ∧ lol future high-8pied train8 nke m&gneticdy the envizonmeat4l no;ise pl;oblem is a,d面4tgdly levit&ted on卵, ole ol tl!e ma,jol dispelsive infr&-sound. Suppose&single c&vitybeconnected to a.long tunnel&xld let &pl&ne sound w&ve be incident upon the civity,i.e・,pis8ing by l!! the tlinnel [11.Then the incident w&ve is tl&nsmitted a.nd ldφeted in palt. A degzee of leflection is is・sues to be solVed se11)us!y.lll,pa,ltlとul&z,Wkel&txaj,ntl&vels illside ofltu鳳lel,it kapμns that pleSsuze dis7tuZb&nces gex1ezate4 tul!el no other me&ns tha.n exploithlg intinsic&Uy with sound plop&i8,0f course, d&mping, whi(;h shock w&ve&ppe&ls leces・llny i豆&lolg tunnel. Ill this lespect, &usu&l tunnel is nothing but, so to spe&k,ajshock tube゛。 aJly a・ pezsl皐teltいpxopaga4i911, 8o A continuajl &ctk)n in propis necessaly a.nd essenti・al. into So desjgned to eventuany emelgeice of&8hod【・w&ve.The othez m6ch&nism is dispelsion,whkh caxl &ctlially ゛diipelse? the infz&・sou皿d t6 plevent芦lhock lolm&t16皿.lt ist=he vely point th&t the ilfl:arsouxld in the tunnel -daej s∂texkibit J廸 dispelsion and therefoze・th&t & th&tthe㎞fr&-8ouxld 4ampi早g l!ighe潭:-oldez it is important m&y be used to diininish the infr&-8ound before &shock w&veis lolmed. To inhibit &sh6ck w&ve,oie might think thit it only tube lexx!odele冪isti尊g扨uels to a.l芦osuch sho司(□)e all intzoduced sonton IShysicdy, su乖ces to m&ke the infr&-sound bed&mped. As it8 ples8ule level becomes high, howevel,it wm ha・ppen tha.t the d&mping a.lone fijls to comp包te with the no皿nle&lsteepening to &now aonnne&zinfra,-8ound a・ 8hock sonton be numelical ill lRhibition compete inhibit to The is ●' | , ・ − ● ・ ・。■ ・ ea.sny i94zgd byus畑g sniall aa(!clo8edsiφibza.lches tunnels. To this end, there seem to be physical mech&nismsa8sod&ted gation. 0ne 6bvious mechanism simpl一tex&m- spadx!g. this ploblem b e a,chieved by a loc&1 &ction only. aμtion,eveU though wei.kloCdy, cavities consideled coatx&stto&usualtunxlel ●♂ I I ・ as &shock tube. As the shock.fl:ee 1)I extexn4y the uderstanding noteth&t pressure dist!1rb&nces le&ding tt)shock folm&tion&le propa・ga.ted in the form of no4inea.1 1nfr&-sould. To contlol such a.nev&sive w&ve&nd to inhibit a. shock w&ve&re very difllcult to to tu皿皿elin&zr&y &・i(!ea。ofa,‘sgnto靫!ube'js&1芦o shai・e,lhei(1ea &s passage of tl呻 】:esonatgls sho7wl ploile connection, the ca.n ening畑p!:ess!1ze this main con!x冪neiictivene88ofthe&zza,y wa;ve. 9xtleme a &zΣ&x!ged dixectiox!。尽ole Helmholtz thaJり知da・mpinS so 芦8づw一&84&mpingftonon-dispelsive out tululel of dizectiQn dzcumlelential 8im・1&tions&zec&lzied in &xia,l dispel;Sion a.II&y conl!ected tke plopa・gating ple,&single consists ln&nyc&vities alol!g &long give designed 皐,tun軍1el ge皿ela,tgdbyt】;avehg Thj芦tulinel tlaj,xls,1.e。a.uslaltullnel,&nd tulneP Scie71,ce, give lise to emexgelceづof a尊 aμ)卿tic峰ock wa;veレi皿the even if the4xajll spe4l is weU below t!le sollnd 8peed, 4s this8hgck'w=a;ve iSladiate4,from仙e tu血聊!,9xit,itl)l恒p a.bgut a 8evele no≒e plol加mlikeaso111c=boo㎞by nl・4zs941ight. Evel at pl・s鴫t,wheil the=Sllin4lselz零skeり;x1!;oぁ!9lgt卵lel withcoaczetesl&b t!:ad【,a,buzSt is l皐4i&tedfroxx4theexil,.Bej:a琴9 tb・ tl4n畢peed − ● ■■ is subso!lic,!,he shock w為籾ニぁppe弗l:so喊y 細:ahe奉di刄4e tuae!,ia Qthez wQxdj,94yia,弗!9耳&t!りlx!el (1・・g・,jevelal toj1・9n kjloxagte!;s1一茜).\But一瞬etla緬,speed知cleues,レjtl4spl;Q111exlり)ecom邸sevelel a,xld!;e尊命to&ppe&zgien in4,shoztel tunnel. Unle8s suit&ble countezme&8uzes weze t&ken, こtk,ezel xliglltllappel the glavest sitl41ol tkat fltl!z・ kigk-・peed Uaj風wmづbelllable to opez&te o・ 畠legllaz ba・is,evetづtkQugh tk●:oth腱・edllical issls were de&zeddl of couis●i thisづpliblemi麹重ot64y:li加ited to\the iadiation φfthe sigck w・.yl,lyilt a!一八iid一一M4/1寮ldtnt回ly,loO11irl回ity illd lsetl,tniuce4f tt幽1S ・i ien aS tilinelS. ダI ダ…………… contlolled by the l:a.tio)ofthe c&vity゛s volume pez w&velength. tion t&kespl&ce. one a.xi&lsp&cing a・p&rt in &Ila,y,the 8ound w&ve wm be plop&g&ted with repetition of lellection &ndtl&nsmissio皿,which yields the del&y in plop&g&tion depend!lnt on its fzequency &nd thezefoze yields dispel8io皿. Especi&ny if the frequelcy coindde8 with & n&tuz&l frequency of the (;奉vity,then-aeside-bl&nch leson&皿ce t&ke8 pl&ce so th&t the inddent w&ve is t6tally relected with no ti&nsmis8ion,&8f&r&8 the pl&le-w&ve&pproxim&tion holds. Altelnatively if a.ii&l dist a,xlceco㎞ddes with multiple of a. hilf w&velength,!he so-canid BI4gg xeiection t&kes pl&ce lhis time so th&t the incident w畠ve is &lso totdy zeiected. Thu8 the a.zla,y ofc&vities c&n give rise to lot only dispelsion damping s91e4ively. \ , ノ As the sl畑ple8t sho政一free tu拿nel,u allay il ge皿el&l but ajso of Helmholtz zes- 9n&tol8(c&Ued sim1)lyies91!i・ol8 hele&ftel)is pzoposed to be connected to the tunnel with equ&1&xi&lsp&cillg&s shown in Fig.1(鼻),Alongリ19 −284。− to the tunnePs The l&堵elt¥1 r&tio becomes, the mole zdecWhen m&nyc&vities&ze connected with some ciz9lxxllezexltia,l d iz一tio!!9f the tullel, only one resonatol is connected fol the sake of simpncity. Since the cavity゛sshape is less important for the resona,tol, the c&vities c&n bea,rranged,technically,in a side tunnel a・s shown in Fig.1(b) by pa・ltitioning it into many connecting each compartment compa.rtments by bulkheads and with the ma・in tunnel. Froln eco- nomical sta,ndpoint, it is obvious Spedficany the cavity゛s volume tha,t a smd cavity is prefelable。 F should be chosen small com- paled with the tunnePsonepel a・xialspadng d,i.e・,y/Ad≪1。 Abeing the tunnePs cross-sectional area・. But the spa£ing must be taken smanel than a typical wavelength yLfol both damping and dispersion to be efrective,i.e・,d≪yl. The infra-sound makes it possible to assume this and also each cavity to be ゛acousticdy comPact゛ in the sense thajtits typical dimension, is much ln smanel tha,n the the fonowing, yl/3 say, three lields with distancesfrom the tlain,1.e・,an innelflow釦ld, a,near sound field a,nd a fa,rsound field. The innel flow field is a,field just a,lound the tlain in which plessure disturbances a,re genela.ted origindy. Since a trajn speed is well subsonic, it is a・ weakly compressible でflowfield. Although this is aバflow rather tha,n a sound field,in reanty, there a,re small but many sound soulcesdepending on the geometry of the train and the tunne1. Hence a very complicated flow/sound field of three dimensional natu】:e appears with a widerange of frequencies。 wavelength,i.e・,yl/3≪y1. a・tfilst,we review shock folmation in a tunnel. Next, resona・tols yields both damping and blidy physical process of to show how the a,rlay of dispersion, the dispelsion characteristics are discussed for nnea,r sound waves. FOl plopagation of nonnnearinfra-sound,we leca・pitulate the formulation pleviously ma・de a・nd give the results of the numerica.l simula・tions to see effectiveness of the a,Ilay. lt is demonstlated how both action of da皿ping and dispersion comes into play in plopa・gation of the infla・-sound to inhibit &shock wa.ve. ln this connection, a,n idea,of ajsonton tunnel a,sa,shock evolve into shock in the form of sound. Since the tunnel pla・ys a role of a waveguide for sound, they a・le propa・ga・ted along the tunnel without a・ny geometrical spleading. ln considering thesound field,it is important to note tha・t the l:egion in the tunnel is divided into tube゛ is a18o intloduced in contr&st to a usual tube. Plessure disturba,nces in this tube never wave8 but &sequence of&coustic sontons. Be- ca,usethe sonton tubec&nbe lea,lize(! by aJlanging smaJl ud dosed side bl&nches of any sh&pe,it is expected th&t this idea can be used to lemodel existing tunnels into shock-free tunnels. With distances away from the train,the pressule distulba・nces tend to propa・ga.te in the form of sound. Becausethe magnitude of pressure disturba,ncesis stm sma,n relatively to the a・tmospheric plessure, as we shall give its estimate, the nnea・「 a,coustic theory is a,ppncable there to the lowest approximation. Such a region is identilied a・sthenea,r sound field. As the tunnel forms a・ waveguide, thisnea,l sound field is described bya,superposition of the lowest non-dispersive mode and many higher dispersive modes. Since the group velocity of the higher mode is slower tha,n the sound speed, the fa,stest disturbances are propaga.ted in the lowest mode, while those in the highel modes fonow in the oscma・tory form to be dispersed eventu&ny. ln&ddition, high-frequency components involved a.redecayed out quickly due to signific&nt difusive efects so tha・t only low-frequency nents wm remain in the lowest mode. compo- R)r this nea.r sound ield, the train゛s motion m&y be modeUed by a pair of a,coustic monopoles with opposite sign [21. Such PHYSICAI,PROCESS OF SHOCK FORM:ATION ln the beginning,it‘is instructive to look&t&situ&tion physicany how &8hock wave is formed in a.tunnel. VVhen &tlain rushes into a.tunnel or when &tr&in st&rts to move rapidly in a long tunnel with &const&nt 8peed (impu18ively&8&nl!11t c&se), pres8uredi8turb&nce8 ale immedi&tely genel&ted&nd plop&g&ted sources give rise to prop&g&tion entry into the tunn61, it8 duration length l divided by a.tla・in speed of&hump in p】:essule. FOI time i8 given by a. tra・in゛s a ・xial U, whne for impulsive motion in&long tunnel, it is estimated to be l/αo,αo being thesound speed. ln a.tunnel of dia.meter .Z),¬her time 8cale DjUwm plovide&typica.l time a.ssociated with the pre8sure rise. This time is obviously shorter tha.n the dur&tion time of the hump・ On the other h&nd,a・ ma・gnitude of pressure disturba,nces△p excess over the a,tmospheric pres8ure po depends on at】:ain゛s motion and&lso the ratio x of the tra,in゛scros8-sectionala,rea to the tunnePs one. 7n)e8tim&te the m&gnitude for entry into thetunnel,we con8i4er the nneal sound field genera,ted by the 'p&ir of one-dimensional a.cou8tic monopole8 moving with 8peed び a・nd h&ving strength of士pox 17'&ver&gedoverthe tunnePs cro8ssection,po the density of the &ir in the a,tmosphere. Then △p/po is given by 7χM2/(1 − jf2)with the r&tio of spe ・ic hea.ts 7 a.sa numerical const&nt,whne for impu18ive motion in a・long tunne1, △p/po is estim&ted to be 0.57X,M'/(1−jf)[2]。 Suppose&train of length l ° 20 m be tr&venng with &speed び=150 m/s (540 km/h)in a,tunnel of dia,meter .Z)=10maJld with the cro88-sectiona,I r&tioχ=0.1. Then the dur&tion time become8 comparable witlμhe ri8e time a,nd both a,re e8tima,ted to be a,bout l/15 to 2/15 second a,pproxim&tely. Then &typica,I frequencyl i8 e8tim&ted to be 2.5 Hz to 5 Hz, while△畦po is estimated to be 0.034 to 0.0&5(corre8ponding in the 8ound pressure level (SPL)).The&bove 165 dB to 169 dB rough &rguments suggest propa名&tion of a.n infr&-8ounl of linite ma・gnitude, i.e・, nonnnea,r infr&-sound. For entry problem of the Shink&nsen,in fa.ct,thefrequency spectra, of pressure disturb&nce8 indica,te the infra-soilnd below 10 Hz a・tmos!;,whne the ma,ximum △p!po observed is a,b(jut 0.013 for び=62.5 m/8 IR)r ex●mple, ・uppo・e by●Gau・・ian a longer ●temporal function in the form tra.inin re●lity,●hump deflled闘2/ω●t which frequency. Fig. I A tunnel with aJ1&rr&y of Helmholtz reson&tors are given ina Then per second)・ −285− v●riation of the pulse be ●pproximated ofexp(−ω212),t be 11●t with the time, though,for plate●u. A the pulse dec●y・ by !;he f。ctor efrom ・while a typical frequency of the pulse wiU i・ ・imply defined similar lormof a typical frequency Ixl&gnitude (225 km/h) may here asω.But pulse width is the ㎞uimum, the Fourier spectra exp(−∂'2/4ω2),∂'beil!g an angul●「 be deflned as 2ω(in unit of radian and x =0.2 16 [3,41,which estimate 0.011. With further should be compared dista,nces awa,y from with DISP:ERSION the above CHARACTERISTICS FOR SHOCK-FREE TUNNEL The idea undellying the shock-free tunnel is to give rise to dispelsiona,swen a・sda・mping bya,rranging ma.ny cavitiesa,lound the tlain,the血onnnearity gives rise to steepening of pressure plofile and eventudy to formation of a discontinuity in Plofile,i.e・,a shock wa,ve. This is the tunnel. To confilm this qua・ntitatively, we examine the nnea・「 dispelsion relation fol soundwavespropa・gating in the tunnel the fa,rsound lield.Here a,lnecha,nismcountelacting the shock folmation is the difrusivity of sound but it is nmited only in a vely thin shock layel becausethea,coustic Reynolds numbel for shown in Fig.3. Thelesona,tols a,re connected with ,equal a,xial spa£ing infinitely so the tunnel is separated into infinite numbel the infla-sound is very large of older 108. 0utside of this layer, the difusivity ofsound is negngible. Also a bounda・ry layel a・t the tunnel wall is vely thin and the lesulting wall friction is neg ・ of intervals by neighboringlesona,tols.Here the throa,t゛scrosssectional a・rea・召is a,ssumed to be so smd compared with the tunnePsoneyl tha・t its splea・d may be negngible. ngible locally. But becauseits efect tends to beacculnulated in propagation,it becomes necessaly in evalua・ting the fal4eld behaviol. As fa・l a.s the magnitude of pressure disturba,nces remains rela,tively sma11, in fa.ct,the wa・n friction ca,n inhibit shock folmation. But as it becomes ¬¬/Z ¬¬ n十1 ¬¬ lalge, it fails to do so. Figure 2 shows the lesult of thenulnerical simula,tion fol farfield evolution of a pulsive plessule disturbance radia,ted flom the nea.r sound field. lt depicts the excess pressule from the a,tmosphelicone∫applopriately a,xial coordinate x along the normazed tunnel and lneasuled in a frame moving with the an initial pulse of a・ Ga・ussian fllnction the neal sound field. lf the nonnnearity ignored,it would be plopagated with a・ny change in form. file would appeal in as a・ function of the the retalded time θ sound speed. At x =0, is assigned a.sinfeHed by a.nd the wall friction wele thesound speed without Then the three-dimensional Fig. 2 to be unifolm lidge Let each interval be numbered a,s7z consecutively from minus to plus infinity (n=…,−1,0,1,…)and take the axial coordinate z along the tunnel with its origin at a midpoint in the interval shock wave a,t a celtain point of x. Quantitatively, the shock formation point colresponds to a・bout 5.4 km for this pressule disturba,ncehaving a typical frequency 5 Hz a・nd a・lelatively low dB ︲ Fig. 3 Geometlical configuration of a・tunnel with a.n alray of Helmholtz lesonators plessure proof the initial Gaussian-shaPed closs-section with its clest in parallel with the x&xis.But thenonlinearity now steepens the profile leftwa,rd as x inclea,ses so that a discontinuity appears in plofile,i.e・,a plessure level △pかo=0.0023(141 1 ︲ ︲ 1............』 n=O. FOl the sake of simpndty, 1ossless and plane wave plopagation is assumed in ea.ch intelva・1,which is desclibed by the nnea,l wave in SPL)・ equation as fonows: ∂2瓦 ぬt 一 一 2j ∂ │ぬit 一一 7αOs∂Z (い the tunnel exit, the the exit is estimated y こ恥 When this shock wave is radia,ted from excesspressure瓦ut a・ta dista・nce s fal from by the linear acoustic theoly to be 一 (1) where尺xit(Z)denotes theexcesspressule of the wave just about time. Hele oindicates the solid angle of ra,diation 【 μsuay□=27r fol semi-infinite f】:ee space but experimentdy 7 =4.6 for the Shinka,nsen [4D. Henceif尺xit involves a discontinuity (though smoothed by a very thin but finite shock layer),瓦ut becolnesvely large so tha・t a bulst is heard in spite of the infra-sound oliginally ∂2瓦 (2) (n α  ̄゜゛゜ ̄1タOタ1ヅ゛゛)7 ∂Z2 where pa denotes the excess p】:essure in the intelval n. Assuming a.tempoldy sinusoidal disturba,ncein the form of exp(−iωt),c4ノ beinga,nangular frequency,戻js given by a・superposition of two waves propa・ga・tingi nto the positive and negative dilectionsofz a,sfonows: to lea,ve the tunnel μthe f g 一 ∂Z2 瓦=みexp[i(h。−ωz)]十知exp[i(−£4− ・)], (3) where£is a・ wavenumber defined by ω/αo and zn ≡z− nd(−d/2<zn<d/2).Here y7l and gn leplesent the respective complex wave a,mplitudes,which a,reto be determined by relations a・mong the tunnel in the neighboling intervals and the resonator in between. Given the excess pressure (3),the axial velocity恥in the interval n is derived immediately by using the acoustic impedance poαofor plane lyaves a,sfonows: 刄 t4 一 一 一 exp[i(h。一畝)]− ρoα0 弘 一 ρOα0 exp[i(−h。−・・,ノz)] (4) Since the resona,torsa,rea,ssumed to be connected a,tz=(71十 1/2)d,the bounda,ly conditions there 】:equire t he continuity of mass flux and tha,t of pressule a・sfonows: pa°戻l十1, a,tz =(n十1/2)d where tらdenotes the】:esonatolfrom the tunne1. j j ″︲0 βny V V jpo(叫l−゛U,71,十1)=召pot4, the velocity directed into ’4 θ 12 一 Fig. 2 Fa・14eld evolution of a・Gaussia・n-sha4)edpressure pulse l&diated from the ne&r sound ield To specify t4, we must examine a response of the resonator to pressure fhlctuations at the orifice. The cavity゛s volume is much 28G − gleater tha・n the thloa・t゛sone,soa.motion of the a,irin the c&vity is negngible. Hence we considel m&8s fol the cavity &s fonows: tz・。, (7) the c&vity゛s volume tzjcdenotes &nd the the velocity lown into the c&vity,a.velaged ovel the thlo&t゛8 closs-section. FOl the thlo&t,the nne&lized inviscid equation of motion is a.s8!lmed a.nd integla,ted ilong the thioat’s length j; a.sfonows: ∂tzノ po£管=一尺十八, (8) whele瓦&nd瓦leplesent,le8pectively,the exces8 ple8sule a・t the orifice on the cavity side a・nd on the tunnel side. ln deriving axial velocity tzノis&ssumed a chalacteristic wavelength uniform a・long the is much longer tha.n the throat゛8 length so tha,t the compressibmty of the air is negngible in the throat. Thus uノc i8 equal to ゛UJ.By the adiaba・tic apploximation for the ajr in the cavity, use is made of the relabeing the me&n dpc/dpc s plessure in ag, Eqs.(7) a・ppears a frequency range in which g becolnes colnplex. ln fa・ct, Figs.4(&)and 4(b)show,respectively,the rea・l &nd ima・ginary parts of gd a.8 the a,bsds8a・ versus the frequency ω/ωo&s the ordinate. Here a tunnel of diamete】:10 m is &ssumed,to which a・spherical cavity of dia.metel 6 m is connected through a・throa・t of drcula,l closs-section of diametel 2 m a・nd of length 3 m with a・xi&lsp&cing 10 m. A na.tura・lfrequency of theresona,tor is then given by 5.2 Hz. 0nly the positive bra.nches of the real and imaginary p&rts a・re shown, but note that −9 is also a.solution. 2 1 (10) 了 ω Since the excess plessule.pS must be equ&l to p:s &nd瓦+l a.t z=(n十1/2)d,wec&n express tzj,s in terms of those plessure by using the&coustic imped&nceZ. The lel&tion between Cら,gn) &nd(。/;s+1,9,s+1)is then est&bnshed through a.tl&nsmission matrixyW'&s fonows: X。+1=W'X。, (11) 0 2 3 4 5 6 7 .Rり9djl ・ ●● ・ ` ・・ −・ ’ ●甲 ●● ・・ ●= ・ j ■ ●・ 向 タ ・ wjth児=ZIZA whele 瓦4(=poao!A)is the a,coustic impedance of the tu皿nel. Thus Eq・(11)ca.n be 8olved successively, lol ex- slf '(12) 7w ] 1/2児 (1十1/2児)exp(−iid) │.rDN9 − 1/2児)exp(iid) −1/2児 7 ︽`11︾ a,nd (1 w°[ 1 0 1 刄k L 一一 X with ㈲ │.rDt Cf:︶ j ぐ 一一c,ノ2 ρ0£ S slf With M valying sinusoidajlyin the folm of j°exp(−i・・ノt),the volume low jtzj from the tunnel into the thlo&t isinduced similaly-ln the folm of Q exp(−iω1),whele?&nd Q denote complex a血pntudes a.nd the latio Plq ddnes a.n &coustic imi)eda.nce of thelesona.toI Z depending on ω.By u81ng Eqs・(7)&nd(9),Zis given&s lonow8: 7 ︵O whereωo[=(.Sag/£y)1/2]is a.na.tul&l frequency of the leson&tol. Hele note th&t£畑u8uany lengthened by the so-caJled end corlections [11。 ㎜ 恥.(13)・ c&tion should be avoided because no energy sources are present, but wh&t doe8 the damping imply? This da・mping ha.s nothing 十‘φい岫4・ (9) ㎜ cha,lacteristics ba.sed on The ima・ginary p&rt gives rise to da・mping or ampnfica・tion of soundwa.ves in spite of the lossless ca.se. 0bviously the a,mpnfi- 諮k Z dispersion no longer &con8t&nt independent of Qノ. This me&n8 tha・t the sound w&ve8 exhibit the dispersion! When g is solved with a rea・l frequency ωgiven,9 is lisudy found to be le&1 but there ㎜● tion∂pcl∂t Qs(dpc/dpc)∂pc/∂ちpc the c&vity. Fillthermore&pproxim&ting a,nd(8)a,re combined into us examine Without the 8ide blanches, i.e・,児→cx),9 is given by ω/αo(= 1;),which i8 nothing but the disi)ersion rel&tion for the nondispersive lowest mode. For ajinite va・lue of児,dω/dg becomes ● this,the mean thro&t bec&use Let ●・ denote, lespectively, of the a,izin it,whne d. This elementa.ly 8olution fo】:p″ is ajB!och wave fundion゛ known genera・lly for prop&g&tion in a. 8pa・tidy periodic stlucture, whne g is cded ajBloch w&venumber゛[5]. 四・ &ndpc the tunnel can be expressed in the folm ofΦ(z)exp[i(9z−ωZ)] whereΦ(z)[=Φ(z十d)]is a periodic function of z with period ■ y mea.n density of ■㎜ whele the conselv&tion ㎜. y管=jpo only ゛・44- But we con8idel &n element&ly solution to Eq.(11)in the folm ・ 2 1 a,lilple,if(j,,9o)is given。 gf x,l=λ゛C whele C is a.n a.lbitr&ly column vectol. R)l this to be &solution,λtuln8 out to be eigenv&lues of‘VV. VVhen λ is set to be exp(igd),9 being aJlowed to be complex, it i8 found sa.tisflythe foUowing j J一恥 of n x・tio/。/g,sin j_ 2児 ぐ ・m ぐ The C08 j d2 o a cos(9d)= .02 dispe】:8ionlel&tion: - tha.t g' must 0.1 加/9d/ (13) Fig. 4 Dispelsi(ind!a.I&ctelistic8 fol sound w&vesin a,tunnel with&n a.zra・yof Helmholtz leson&tozs;(&)&nd(b)show,lespec- each iltelval is so thatノtheexcess .04 .06 pzess・re tively the le&l −287− and imagin&ly parts of gd With lespect to∇・4ノ/ωo・ to do with the dissipation of enelgy tra.nsformed into a,n evane8cent mode tha,t occuls due to the so-cded he&t. lt is r&diation of minus of ha・1foldel defined half ordel once with 一一 j ∂d 四 ぐ ∂h a玩 =[ d&mping and the renection of 8ound w&ves by the leson&tors. Ne&Iω/ωo=1,the resona,tors le8on&te with the incident wave a,nd the 8tlong r&dia・tion d&mping occurs. Thus the ima・gina・ly derivative ∂-1tj 一 ∂r1 1 ㎜ 7rl/2 by differentiating respect the deriva.tive to Z a.sfonows: 1 ∂u(t j/こ -- tl) dz″.(17) (t−が)112 ∂t’ p&it of gd diverge8 a.sω/ωo→1. But it8 real paJ:t is fixed a・t 7『 forω/ωo<l a,nd a,t zero for c4ノ/ωo>1.Nea,Γω/ωo Q1 3.3 and Along the periphery in the vicinity of the orifice,t7is given by 6.5,in&ddition,the imaginaly pa・lts al8o appe&r but they 】:emain −tzノ,the velodty directed into the throa.t with its sign revelsed. finite where the realpalts arefixed a,t 7r and 27r,respectively。 Thus the integra・l in Eq・(14)ca.n be a・pproxim&ted perunit&xial This da・mping is brought a・bout by the BI&腿leflection when the length of the tunnel as 4ial spacing becomes multiple of a half wavelength 7Γαo/ω,i.e・, parts exist,the (dp/dp)1/2]to into に of the &coustic nolma,l main flow &nd t, denotes the to it. 一 with the 8ign N RBj2A)/,R. 2Caoμ1/2∂ ̄1 一一 丑* ∂r1 the nonhe&I lisponse loss due tQ folm&tion modiied&8 fonows ∂悦 of t!1e ail in the c&vity aJldthe nonhea,『 of jets a,re t&ken into &ccoilnt,Eq.(9)is [6]: 2cぴ/1/2∂1瓦 「 7−1 27po ∂d ∂y 一 ∂i2 十司式 y + Equations(19)a・nd(20)describe 7+1 αO+ ㎜ 2 ∂・U, C =一一(7−1)/.Prl/2 whele z/ a,nd lj°rdenot9, !:espectiyely, the kinematic viscoiity of the a.iland the PI;aidilnumbei. Hele the jnteglaljs nothingbut the olie knoWn &i the deliね.tiveof minus half 6lden)f∂tl/∂z so the &bblevi&ted iolm oil tliel&st te1・ijlisu sed 【71. I血the fonowing, i'e wm ise fieqilentli・4 e −288 - 一一 .R‘ ∂r1 j ぐ with ㎜ ゐ・;,(2o) 一 ∂Z U 一 Z a a - (16) - 一 ∂t bi-directionalprop&g&tion j U り1・ 睨 into the positive &nd neg&tive of z. ;o the positive a,nd neg&tive di】:ections directionsof z. lo IFor prop&g&tion into the positive direction only, they c&n be simplified into ぐ + 。(1 − ∂u(2;, ∂Z が)1/2 吠 一 ∂Z B LepoalS 四 where r s4nds for the hydraunc ra.diusof the thloa,t &nd the derivative of three-hd!oldel is defined by(nferentiating the deriva,tiveof hjf ordel (17)farthel with le8pect to l once. Here 4[=(B(4jL。V)1/211s dehed by the efective throat's length £e on ma.king the end colrections and c£isthe r&tioL'!L, where £″sta,ndsf or the vi8cous end correction [61. 1″ j /j ぐ 四 au 一 a£ Cz/1/2 7rl/2 - 'Ub (19) poαoAd∂y &ccount, it is induded in the fo】:m of a, si畢na.l heledit&y integlal known &s the deliv&tive of thzge-half oldez. Fulthelnloleif Cαoz/1/2∂ ̄1 _! ≡(:71/1/2匹 ∂t-1 y ∂K 干 vertically ordeled where ljR゛ is defined a.s(1− These equations a,redosed togethel with Eq.(9), i!17tegzal&s 1 recast &8fa.r as the nnear and lossless response of the resonatol is assumed.But if the wd friction a,tthe throa,t゛swa,n is ta,ken into 恥百 heleditaJy 士 一 Along the peliphely &dj¢ to the bounda・ly layel, tjis given by the velodty on the edge of the bo!II!d&ly l&yel tjl,.!tis&Ilea.dy shown in[6]that t・&is lelated to the a.xial velocity t4 !)y the fonowi皿g p, Eqs.(14)a,nd(15)al:e . the difelence between both closs-sectional &le&8 18 vely sman so th&t the one of the ma.in皿owh&s&Ilea.dy been lepl&cgd by j4 in Eq・(14).But the integlaHs callied out a.long the periphezy pa.nd 2 U士 7−1 十(u=1=a)£ - a・lrタ enminate j of the NB a,ccounts a.le neglected, there exists 一I S SS a density efrects on sound ぐ (15) and the foUowing equ&tions: (14) ρ ∂Z゛ p denote, respectively,the velocity compQnenti皿w&ld the difusive ∂p ∂U +U ∂Z of the tnnnel a と州ds, radius adiabatic relation betwen p &ndpin the a,coustic ma,in flow, i.e・,p/po=(ρ/poyy where the subscript `O゛ impnes the qua,ntity in equmbrium state. lntlodudng the local sound speed α[= + 一 ∂忿2 the&xia.l velocity &nd the ples8ule, a.U&ver&ged ovel lhe ciosssection of the ma,in llow, not ovel the tunnePs c】:os8-section.But of the closs-8ection Since 1一 一 = wherep,t£,a,nd 豆お + 卵盲 ∂U ∂i (pu)= .R is the hydraunc j α FOl the infra・-sound,the difusive efects a,levely sma,U &nd neglected in the fonowing simulations. But the wa.U frictiondue to the bounda.ryl&yer is taken intoa,ccount in evaluating prop&gation ovel a long dist&nce. Using the 8&me not&tionza,ndta,s befoye,the equation of continuity &nd the equation of motion in the a・xia・l direction are given a.sfonows 【6】: whele (18) fol the total closs-sectional area of the or護ces pel unit axial length,y(=1/d)being the number density of the lesonato】:s。 ぐ 1 NUMERICAI,SIM:ULATIONS Formulation Let us now examine an efrectof the alr&y of re8onators on plopa・ga・tionof ngnnnealinfra-sound. Although both d&mping a,nd dispersion are blought &bout by the nnea.r mecha・nisms,they can stm be exploited in weakly nonlinearca,se concerend here. Since a typical wavelength of the infr&-sound is very long, the a,xialspa・dng ca・n be chosen smd enough for the resonators to be tegarded a・sbeing continuously di8tributed. This 4continuum a,pproximation゛enable8 us to folmulate the ploblem in a・framework of one-dimensional propaga・tion for &n`acoustic ma・ildlow゛ delined as a・region in the tunnel except for the vicinity of the or護ces a,nd the thin bounda,ry layer &djacent to the tunnel waJI。 トd∂s1[(?j−yダ)po'Ub-NBpo籾]く a心 sound wa.ves cannot be plopaga・ted folwa・rd. Such a・frequency range defines astopband incontrast to a p&8sband outside ofit。 For sound waves containing many frequendes, only components in the stopba.nd a,leindeed 4stoppe(P and the othels are pa,ssed but dispersed &s a whole of the waveform. The stopband due to the side-bra,nchlesona,nce wm be useful but tha,tdue to the Br&gg rellectionis too n&rrow and the damping is too small to be exploited in the present alray・ - the im&ginaly 1一Å ω=゛n,1r(1o/d(71 ° 1,2, ・・.).When y ∂瓦 一一 2poαoλd∂Z° (21) FOl the det&ns,see the l:elelence [6].Heze we nolm&nze the equtions by setting 【(i十1)/2]u/ao and [(7+1)/27]幻po to be り'&ndEg,lespectively,whele 6 (≪1)me&8ure8 the smaJlle8s ofthe m&gnitide of the illfr&-8ound,i.e・,the we&k nonline&zity, io that / and g a.ze zega.lded a8 tleing of ordez uity. Bec畠ue the excess ple8sule p″is giveil by 画aot1 1)r the unl-dilectiolal p!opa・g&tion within the plesent a・pploxim&tion,げmeasuzesajso − the m&gnitude the tunnel. Given of the plessule distulb&nce[(7+1)/27]p″/po in a typica,l frequency c4ノ,wei ntroduce the non-dimensional retarded time θ[=c4ノ(z−z/αo)]me&sured in a frame moving with thesound speed and aylong &xia・l coordinate5 X [゜(Ec4ノz/αo)] assodated with the smdness of nonnnea.lity. Then Eqs.(21)a.nd (20)a,le lewlit4en in the foUowing dimensionless form fol≒/゛and !7: Numerical Results We now show the 】:esults of the numelical simul&tions on the b&sis Qf Eq8.(22)a,nd(23)for spatia,1evolution of pressure distu】:ba,nces ple8c】:ibeda ,t x =O. When a,train rushes into a tunnel or when a,train stalts to move r&pidly with a constant speed in a long tunnel,the pressure pulse is radi&ted forwa.rd. Suppose this.pulse be given in the form of a・Gaussia・n fllnction with its typical frequency ω(see iootnote l)&nd the maximum excess pressule△p. Ta・king 6 to be [(7十1)/27]△p/po,the initial (physicdy boudary)condition for / is given by 荘一居=一如崇一瓦詰, (22) ∂り The initia.lcondition foり7 is to be determined Eq.(23)with / plesclibed by (28). 十 公f +ng= F j2/ 十δ。誹十j2g 函 ぐ ー + 2y 日諮 7+1 ∂θ2 wheleδ.R,jr,δΓ&ndj7&】:e ぐ / μ 4 一 (23) 瓦 一 2a・1d ㎜ EjZ* ゛ 一 タ j 叫7 2c禎φノ)1/2 一 (24) 『 Hele it should be noted th&t the nonnnea.l terms in Eq・(23)ale of higher ordel in Ebut they &】:e】:et&inedfol&c&sein which j2 becomes 8maJl ct)mp&lably with E.The efect of the w&n friction appeals through δ.RandδΓ,which me&8ure a,ssume that 6 =0.05 andω=107r raヅsec.so tha,t δR and 4.are lixed to be 4.0 ×10 ̄3 and l.4×10 ̄(i.e・,c£=1 for simpndty),respectively,whne瓦is taken to be l ol 10. TO see a・n efect of the a,rra・yof reson&tors,we ex&mine fir8t evolutions for thlee types of j? with 瓦fixed. Figures5,6 and 7 show the evolutions f()I・r?=0.1,1 and 10 with 瓦ニ1. 1n each iigure, y c・ノ)1/2 C &s a. 8olutiol! to Evolution in the tunnel without the a,rray of lesona.tols corlesponds to setting 瓦equal to zero so tha.t Eq.(22)is decoupled. By 8olving this equ&tion,Fig.2 ha.s been drawn. The foUowing simulations by ぐ = n″ 4・ 一 訃 匈 一 ∂θ (・y十1)BLe deined (28) /(e、X=o)=exp(一θ2)・ a typica・l thickne8sof the uppel figure (a)and the lower one (b)show theevolution of fud 9,lespectively,where the dilectioR of the x a,xisis chosen leversed in (b)to exhibit an initia.lplofile of !7. 1n pa.ssing,unity inx(;orrespond8 physicany to about 0.2 km for such ajlalge5 vμue ofE.Figure 5 shows emergence of a, shock wa,ve, whne Fig. 6 shows disinteglationsof the initial pulse into thlee shock the bound4lylayel(z//ω)1/2 relative to the tunnePs ladius &nd the throat`s radius, lespectively. The efect of the &rz&y ofles- w&ves up to x =10. The latte】:situation is obviously worse tha,n that without the a,rray of resonators. Even for ・r?=l where the onatols appeals thlough two pa・rametels 瓦レand tively,the gcoupnng pala.meter` a・nd th6 gtuning damping is the most enha.nced, itca,nnot suppress the shock formation for 瓦=1. But Fig.7 shows no evidence of shock w&ves at aU, though the initial pulse is not damped. Next we examine j2 cded, lespecpa.ra.meter゛.The follne】:lneasuresthe c&vity゛s sm&nnes8y/j4d zel&tive to thenonnne&rity,whne the lattel me&sl;Iles how fa.lthe typic&l frequency of the infra-sou(!is detued fl;om the.na,tura,l frequency of the i:6si)n&tol.FOl the a.Iray of reson&tols a・1re&dy&ssumed in the efect of the coupnng para皿etel瓦.As it increa,ses,both efects of da皿ping and d≒)ersion becolne enha,nced so tha,t it is expected that shock waves tend to be inhibited. For 瓦=10 these pa,lametels a.le evalu&ted&s δΓs l.4 ×10 ̄3cz。瓦=0.072μ&nd (5Hz)・ a・nd j? ゜ 1,in fa.ct, shock w&ves aJ:einhibited &s shown in Fi&.8 a・nd the initial pulse is decayed out signilic&ntly a・tx =10. For n=10, 0f course, no shockwa,ves elnerge a.sin theca,sewith fonows: δ.R s 2.0×1 ,r?s l.l for ω=107「 沢 rad/s lf ・Q is chosen large a・nd both the wd frictionsand the nonnnealterms are negngible、9c&n be approximated a・s 1 ∂リ ー /− − ∂θ2 j? 一 一 +○ These 6 8hows j l一が 一 1 ∂り /− − 芦’ j? ぐ !7 一 瓦=1. For ・r?=0.1, hgwever, it is found stm emerge. (25) Neglecting the order on/a2, KOltew41g-de vries eqil&tjon(cded simply K-dv equ&tion helea,ftez)isdelived lor /: ぶドj4卜41≠べぶレ \(26) tha,t two shock w&ves lesults sugge8t severa・l import&nt impncation8. Figure that the d&mping a,longca,nnot compete with thenon- nnea.l steepening to dow emelgence of two shockwaves. For .r2=0.1&nd j2 =10,0f coulse, no subst&ntial damping &ppea.rs. Although the dispe】iion lela.tion 8uggests the stopba・nd nea.Iω/ωos3,3,i.e・,・Q咄0.09,the continuum ipproximation h&8 smea,led out the disclete distribution of the leson&tors so that the damping due to the Blagg le恥ctionc&nnot be t&ken into a.ccount. Fulthelmgle, in deliving Eq.(21),propag&tion a.long the neg4tive dilectiOn of x h&8beensimpnied so tha.t the ra・4i&tion da.mping h&8&iso been disc&ided impncitly・ The witl! r =jr/j'2. The weU-k皿own plopeltie8 of this equation 8uggestth&tinitial distuzb&nces do no longel evolve.intgihock d&mping&t J7 s l in the numelic&1 results is solely due to the t&ken into a.c'wave8but&sequence of sontolls&symptotic&ny as x ¬゛cx)【8】・ wan fliction. lf the la.di&tiol! da.mpingwerefully count by solving Eqs・(19),the da.mping wm be enh&ncid。 Ea.ch sontoli is explessed in the lonowing fo】:m: wheze .4&n.dθo&ze lniti&l-v&lue ploblem wmnotgive θ一尽x+1.4x−θo cons4nts to be determined foI Eq・(26).Since lise to &bulst when タ ー ー ぐ 2 / 1 /=.4 sichリ(奈) (27) by solving &n its profile is smooth, it r&dia,ted from a,tunnel exit as faj・&s its pressule level is modez&te.But is(。4/12.r)1/2 be(;omes much gleatel tha尨unity,i.e・,tlle width of the solitol becomes tooll&i!low,&s th包 expiessio塁(1)stipul&te8,theze m&y&lise, i皿8tead of&buxsti奉皿ewelvilonmentaJloise plol?lem (though po88ibly i冪l&udible)assoda4ed with theJ&dia!lon of sontons a,8 1皿fra,・so・皿d. ノ ゲ −289− ・・ As瓦incre&ses,it is found that shock waves tend to be inhib- ited but only for j7 ≧1. FOI&glea.t va.lue of j2,a.sm&U vajlue of jr iseno!lgh to inhibit shockw&vesbut the i刄iti&l pulse is o!)viously free from damping.Th.is can be udersto6d by the K-dv equ&tion fol .Q ≫1.The dispelsion in this c&se a,ppears through the highel-ozdel deriva.tiye (the thil4 ordel)tha.n that in thenonnne&rtelm. This ghighel-oder dispelsion`c&n colnpete with the nonnne&z steepening to inhibit shockw&ve8. 0n the contra,ly,the dispelsion for J7≦l yields only the glowel-ozdel 4ispersion9,which fi,ilsto countel&ct the no皿11nearity to anow emelgence of shock wa,ves eventu4ly[61. Thus the dispel8ion a・ppeazs intwo dUrez9!lt Ways. 0n】,ythe !1ighel-oldel d18pirsion f f 15 −5 12 θ −8 θ g 10 g 10 奥8 θ Fig. 5 Spatial evolution for ・r? ニ 12 ’5 θ 15 0.1 a,nd瓦=1 Fig. 7 Spatial evolution fol j?=10 and 瓦=1 f f −5 −5 15 θ θ g l 0 g 10 ’5 θ 15 '5 θ 15 Fig. Fig. 8 Spatial evolution fol ・r?=l and 瓦=10 Spa・tial evolution for ・r2 =l and 瓦=1 - 6 290 − 15 is efredive in inhibition of the shock wave, but the damping cannot be expected in this case. ln older to exploit both mechanisms,・r?should be set in an intelmediate lange, 1<・r2<10. 1n fact,thecasewith ・r?=3∼5 is ploved to be very efrective. FRO:M SHOCK TUBE T0 4SOLITON TUBE゛ lt has been levealed tha,t the inhibition of a,shock wave can be achieved by the highel-ordel dispersion. Evolution in thiscaseis found to be described apploximately by the K-dv equation fol ー≫1.Thena,n acoustic sonton wm emerge in pla.ce of a shock wave. The tunnel in this ca,se may be called ajsonton tube゛ in contlast to a usual tunnel a・s a shock tube. Thus symbolica,Ily that inhibition of a shock wave can lemodennga,usua,1 tunnel into a solition tube。 lt is alleady shown ized bya,rlanging in [9]tha,t small and the dosed sonton it ma・y be said be achieved by tube side blanches can be leal- not only of the resonator type but a,lso of any shape as fal as they are a・coustically compa,ct. The lesponse of each side blanch can then be desclibed by the nneartheory. By using its acoustic impeda,nce Z, the complex volume flow Q is given by the complexexcess plessure ? a,tthe ol護ce onthe tunnel side a,sQ=?/Z.For an infla-sound,the inverse of the a,coustic impedance, i.e・,admittance y can be exPanded in telm of (一応ノ)as fonows: 召 Q=yj)= 一 [α(一心)−β(一心)3十‥1乃 (29) ρOα0 一 一 ∂ α ρOα0 Hereαandβhave − ∂Z づ箆 ∂Z3 +… Fig. 9 An aHa・y of eight Helmholtzlesona,tors fiUed on the tunnel wall:(a)and(b)show, lespectively,the tunnePs crosssection and a schematic view of each resonator. (3o) the dimension of time and cube j 一 b 召扨 p j ぐ 召 /1ヽ、 whele召/poαo is a typical admittance andαandβare lealconstants because ? a.nd Q should be 90°out of phase fol no net energy to flow into the side blanch when the dissipationisneglected. Also no quadratic term in こ4ノisp lesent because of the leality condition,i.e・, Z(−ω)=Z*(Qノ),Z*being the complex conjugate of Z. This expression impnes, since(一応ノ)corlesponds to the difrerentiationwith lespect to ちthat of time, respectively. AlsoB(巾)oαo gives the capacitanceof the side branch pel unit risein the pressule and β/αleplesentg a snght time-lag(squaled)in the response of the volume flow. For the resonatorjn fact, αa,ndβtake V7B(1o and α/ωg,respectively・ lf a・qualtel-wa・velength tube, i.e・,a・strajght tube of uniform cross-sectiontelminated a,ta dosed end,its acoustic impedance Z is given by i(poαo/ £the depth,so that αand βtake,respectively, 祠・ 対 With the volume flow (30),we employ the same equations (14)and(15)and foUow a similar way of reduction. Neglecting the dissipation due to the waU friction, we findy a,Ilivea,tthe same K-dv equa・ti(¥1(26)with the coe伍dents瓦=召αoα/264d a、nd r =痢4ノ2瓦/α. Sinceαhas the dim6nsion of time、Baoα impnes volume. lndeed,the compressibmtyp ̄ldp向)zl/poαg capacitance divided by the gas° gives the volume of the side branch. Thus the 】:atiojαoα/y 、measures the sma,nness of the side bra,nch,For theresonatol,瓦and r are given by y/264d a.nd瓦/・r2,respectively,whne for the qua・rter-wavelength tube, 瓦=召£/264d and r =7r2瓦/12j? whele (24)but Qノo is given by ωo=7Γαo/2£。 j? is defined a,stha,tin lt is the merit tha,t the sonton tube can be realized by sma11 side blanch gf a,ny shape so tha,t they can be fitted directly just j b in Fig. 9. Figure wheleeight identi- ぐ on the tunnel wa11. 0ne example is shown 9(a)depicts the cross-section of the tunnel caI Helmholtzresona,tors,ea・ch shown in Fig.9(b),are arranged a,round the peliphe】:y of the tunnel (and along the axial direction m by m. Typical the number of side branchesa,Ila,nged a.long the pe(a positive integel),then瓦is simply.multipned dimensions of the cavity a・re taken as foUows: 一 a,sweU).Let riphery be Fig. 10 An arra・yof four curved side blanches fitted on the tunnel wall:(a)and(b)show,respectively,the tunnePs crosssection a,nd a,schematic view of ea,chside branch. 291 − ふ `熟 μ 1 1 E I ` the cavity゛s volume y =4 召=47r/100 m2 (diameter £e °0.5 frequency 0.051μ。 Another m m3, the throat゛s cross-sectionala,rea 0.4 m),thethloa,t°s effectivelength and the axial spa・dng d ° 4 m. Then a.natulal of the resonato】:is about 14 Hz and 瓦is given by example is shown in Fig. 10. Figure ACKNOVVLEDGEMENT The a,uthor wishes to thank ProfessoI T. Kakutani for his comments on the manuscript a.nd MI. T. Horiokayfor preparing Fig. 4. He also a,cknowledges the support by The Kurata Foundations, Tokyo, Ja・pan. 10(a)depicts the cross-section of the tunnel where the four identical curved side branches, each shown in Fig.10(b),a,re a,rla,nged a・round the REFERENCES periphery of the tunnel. Here the cross-sectiona,l a,rea jand the depth £(a・long the circumference of the tunnel)are chosen to be 5 m2 and 5 m, respectively so tha,t the total volume of [1]Pierce, the side bra,nch is 25m3.When these fouf side bra,nches are connected with the a・xial dista・nce 10 m, 瓦is given by 0.064μ. lf the lesult for the quatel-wa・velength tube were a・ppned to this culved side branchjts na・tura.l frequency is given by 17 Hz. As the sonton tube ca,n ea,sily be implemented, it ca,n be appned to remodel existing tunnels into shock-fr6e tunnels.But it is a・gain empha.sized tha・t even if a shockwave ca,nbe inhibited, the infra-sound pelsists in propa・gation in the form of a sonton. A. j・lej anj 【21Sugimoto,N・,Sound field in generated by traveling relical anj C∂mptjlali∂nalj4c∂ujlicj,Mystic,Connecticut,USA, of a at jr?11ernali∂nal C∂n/rerence∂nrゐe∂- 1993. 【3】Ozawa,S・,Studies exit,Rajlway ways of micr(ypreisure '11chnic&I wa;ve radiated Research Report, from &tunnel Jap a,n Na.tion&I Rajl- N0. 1 12 1,pp.1-92,1979,【i】IJ&paJlese)。 【4】Oza.wa。S・,Maeda。T・,Ma,tsum11ra,T・,Uchida。K・,Kajiy&ma・, anj Ta,】lemoto,K・,Counterme&silres radiating from tQ reduce micro-pressure lexits of Shinka,nsentunnels,j4er∂jynamiEj l/enlilali∂n∂/14Aicle 7unnels,Elsevier SciencePublishers, pp.253-266,1991. 【5】Brinouin,L・,W'at・Ej°r叩aμfi∂ninj)eri∂・li・: SIru dt4re j, Dover Publishers,1953・, 【61Sugimoto,N・,Propaga.tion with a,n &rray of nonnnear of Helmholtz a,coulitic waves reson a,tols,J.Fhlid Meck・, in a, tunnel V()1. 244, pp.55-78,1992. [7]Sugimoto,N./Generalize(P culus,ln jV∂ Burgers equation 「inear iyat'e M・∂li∂n(id. A. a・nd fra£tiona・l ca・レ Jefrey),Longma!l/VVi- ley,pp. 1 62- 1 79, 1989. suited for a shock-free tunnel. But the da・mping is not expected. lno】:del to inhibit not only the shock wave but also the persis- [8]Drazin,P. G. Ca,mbridge university tent plopa・gation along the tunnel, an ultimate shock-free tunnel consists,in plindple, of two kinds of array,one f(jrthe da・mping 【91Sugimoto,N・,0n a.nd the other for the higher-older dispersion. Efrect8 of such a double(multiple)a・rray wm be presented in a forthcomi刄g paper. a・ tunnel tra・in,presented vvaves only by the a・ction of the higher-ordel dispersion. As fa・r as the inhibition alone is lequired, the idea of the sonton tube is wen ,PAyjicaj' ?rinci- high-speed H. and CONCLUSION This pa・per h&s introduced the idea of the shock-free tunnel a・nd demonstrated by the numerical simulations tha・temergence of a,n acoustic shock wave can be inhibited in this tunnel. By a,rranging many cavities as side blanches,it exploits the two physica・1mecha.nisms, the damping &nd the higher-order dispelsion. Ea,ch side blanch゛saction is very smd but its cummulative efect ca,ninhibit a shock wave eventuany. lt 峰ould be rema.lked aga・intha・tonly da.mping of the infra-sounddoes not necessarny lead to inhibition of a,shock wave a,nd that it can be a,chieved D・, 。・4c∂ujlicj,;j4n jnlr∂・ludi∂nl∂ilj jpμicali∂nj,MCGraw-Hm,1981. jV'∂ and Johnson, generation 「inear j‘1c∂ujliEj(ed. 550,1993. R. S・。S∂lil∂nj,; j,n瓦1r∂・ltjcli∂n, Press, 1989. oPacoustic H. Hobaek)World soUton゛, in ,4心ancejin Scientific,pp.545-
© Copyright 2024 Paperzz