PS3-3 `Shock-free Tunnel` for Future High

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. ﾀﾞ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)，&nother 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&cent 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ﾀﾞ)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ﾂﾞ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 /:
ぶﾄﾞ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-