Numerical Simulation of Acoustic Waveguides for Webster-Lokshin Model Using Diffusive Representations

Webster-Lokshin model using diusive

representations

ThomasHélie 1

andDenisMatignon 2?

1

IRCAM,Analysis-Synthesisteam,1,pla eIgorStravinsky,

F-75004Paris,Fran e

2

INRIA,Sossoproje t,domainedeVolu eau-Ro quen ourt,B.P.105,

F-78153LeChesnay,Fran e

1 Introdu tion

This paper deals with the numeri al simulation of a ousti wave propagation

inaxisymmetri waveguideswithvarying ross-se tionusingaWebster-Lokshin

model. Splittingthepipeinto pie esonwhi hthemodel oe ientsare nearly

onstant,analyti alsolutionsarederivedintheLapla edomain,enablingforthe

realization of thepropagation by on atenatings attering matri es oftransfer

fun tions (Ÿ 2). These fun tions involve standarddierential and delay

opera-tors, as well aspseudo-dierentialoperators ofdiusive type,indu edby both

thevis othermallossesandthe urvature.Theseoperatorsareexpli itly

de om-posedthankstoanasymptoti expansion,andthediusiveonesmaybedened

and lassied (Ÿ 3). Various equivalentdiusive realizations may be proposed,

that aredeeplylinkedto hoi esof utsin the omplexanalysisof thetransfer

fun tions.Then,niteorderapproximationsaregivenfortheirsimulation(Ÿ4).

2 Deriving the model

2.1 A ousti model

A mono-dimensional model of the propagation of the a ousti pressure p in

axisymmetri waveguides in luding vis othermal losses on the wall has been

derived in [1, hap 1℄, assuming the quasi-spheri ity of isobars near the wall.

Dening in the Lapla e domain b

(z;s) = R (z)p(z;b s) where s is the Lapla e

variable,z isthe urvilinearordinatemeasuringthear lengthof thewall,and

R (z)istheradiusof theguide,theWebster-Lokshinmodelmaybewritten:

 2 z b z (s)   s  2 +2"(z)  s  3 2 +(z)  b z (s)=0: (1)

is thesound speed, (z)=R 00

(z)=R (z)the urvature, and"(z)= p 1 R 0 (z) 2 R(z)

quanties the ee t of the vis othermal losses. Note that ontant urvatures

?

orrespondto ylindersor ones(=0),exponentialor atenoidalshapes(>0),

andsinusoidalshapes(<0),for the urvilinearordinatez.Forshortpie es of

guide on whi h  and " may be approximatedby their onstant mean value,

Eq.(1)hastheanalyti solution b z (s)=A(s)e z (s) +B(s)e z (s) where (s)is asquareroot of s
2 +2" s
3 2 +. 2.2 S attering matrix The waves pb  (z;s) = (bp(z;s)  z b

p)=2 dened in [2℄ are lo ally outwardly

(bp +

)andinwardly(bp )dire ted.ForaC 1

-regularproleR (z),their onne tion

at z  is simply given by pb  n (z  ;s) = pb  n+1 (z 

;s) where n and n+1 index two

on atenatedpie esofguide.Forthislastreason,weareinterestedinthe

time-domainsimulationofthes atteringmatrixdened for b



z

=R (z)p(z;b s).

For onvenien e, we onsider the adimensional problem ( = jj = 1, and

z2[0;1℄,/",and (s)isasquarerootofs 2

+2s 3

2

+sign()).Thisyields

" b + L (s) b 0 (s) # = " T(s) R 1 (s) R 0 (s) T(s) #" b + 0 (s) b 1 (s) # : (2)

wherethetransmissionT andthereexionR

z

attheinput(z=0)ortheoutput

(z=1)are rationnal fun tions withrespe tto s, (s) ,and e  (s)

. InŸ 3,we

investigatee  (s)

forthe ase 0whi hextendsthestandarddelayoperator

e s

foridealpipes( ==0)[1, hap.3℄.

3 Using diusive representations

Theimpulseresponseofapseudo-dierentialoperatorofdiusivetype anbe

de- omposedona ontinuousfamilyofpurelydampedexponentialswithweight.

Adiusiverealizationhelpstransforminganon-lo alintimepseudo-dierential

equation into arst-orderdierentialequationon aHilbert state-spa e,whi h

allowsforstabilityanalysis;thisapproa hrevealsusefulforboththeoreti aland

numeri al treatment of pseudo-dierential equations. We refer to [7, Ÿ 5.℄ for

the treatment of ompletely monotone kernels, [6℄ for diusiverepresentations

of pseudo-dierentialoperators, and[5℄for links betweenfra tionaldierential

operatorsanddiusiverepresentations.

For the simulation of e  (s)

, we pro eed as explained in Ÿ 3.4 : it is a

generalizationofdiusiverepresentationsoftherstandse ondkind,asre alled

in Ÿ3.2andŸ3.3;itisbasedonanasymptoti expansionpresentedin Ÿ3.1.

3.1 De ompositionof the transferfun tion

Theasymptoti expansionof (s)forjsj!+1reads (s)=s+ p s  2 2 + o(1).Thus,e  (s)

maybede omposedintoseveraltransferfun tions,namely:

e  (s) =e s e  p s H (s);wheree s isapuredelay,e  p s isdiusiveof

therstkind(seeŸ3.2),and e

( (s) s  p s + 2 =2) isdiusivein ageneralized

3.2 Diusiverepresentationof therst kind

Thetransferfun tionH

1 (s)=e

p

s

,understoodwitha utonR ,isknownto

bediusiveoftherstkind,withanextensiondenedin[6,Ÿ5.2℄,withdensity

 1 ()= sin( p ) 

for>0.Itfulllsthewell-posedness ondition R +1 0 j1()j 1+ d< 1.Thus,H 1 (s)=s R +1 0 1() s+

d givesrisetoanaturalrealizationofthe

trans-fer,asasuperposition ofbasi rst-orderdierentialsystems.

3.3 Diusiverepresentationof these ond kind

Adiusiverepresentationofthese ondkindmaybedire tlyusedinthe ase=

0.Then, e  (s) =e  p s 2 +1 , where p s 2 +1may bedened by p s i p s+i, p

zbeingstillunderstoodwitha utforz2R .Thetwobran hpoints+iand

iarethusasso iatedtothe utsi+R and i+R ,respe tively.Adiusive

representationofsu hanoperatormaybederivedfrom omplex-valueddensities

omputedonthesetwo uts,asdes ribedin [5,Ÿ3.3℄ for(s 2 +1) 1=2 .

−2

−1.5

−1

−0.5

0

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Figure1:PhaseofH2(s)denedforthe ut[s1;s2℄[R , =1,and=0:3.

Bran h-pointsarerepresentedbyÆandthepolesoftheapproximatingtransferfun tionby.

3.4 Generalized diusiverepresentation

When 6=0, somemore aremust betaken forH

2 (s)=e

( (s) s)

, and the

previousideasmustbeadapted.

Analysis. 8 >0, (s)has twozeross

1 and s 2 = s 1 with <e(s 1;2 )<0 and =m(s 1

)>0,thusleadingto4bran hpointsforH

2 ,namely0;s 1 ;s 2 andapoint

atinnity,su has 1.Thisimpliesthat (s)maybedenedforany utsmade

Diusive representation of e

( (s) s)

. Among the dierent ompatible

uts,we aninvestigateeither3parallel uts( on atenationofŸ3.2andŸ3.3),

ora rossmadeofR andtheverti alsegmentjoinings

1 ands

2

(seeFig.1).The

orrespondingdensity anbefoundanalyti allybylimitformulasoneitherof

these two uts,after somequite tedious omputations: in this ase, dire tly

a ountsforthedis ontinuityofH

2

(s)a rossthe ut.

Analternativetodiusiverepresentationofthese ondkindpioneeredby[7,

Ÿ6.℄anddeveloppedby[3℄istheso- alled - ontour.Aregular urve C

en ir les all the singularities (poles, bran hpoints, uts,...) of H

2

(s); thus, by

the inverse Lapla e transform, we get a representation with a omplex-valued

density , expresseddire tlyand ontinuouslyfromH

2

(s),fors2 .

Realization with a - ontour. Although nodieren es in prin ipleremain

between realizations for - ontours and for uts(a ut an even beseen asa

limitingpro essof - ontour),thisisnotthe aseforrealizationsapproximated

by rst-order linear onstant- oe ient systems.Moreover, - ontoursyield a

density ontainingalltheinformation arriedbytheen ir leddis ontinuity,

butunderasomewhataveragedform.Onthe ontrary, utsyieldamorefo used

information,sin ethedensityexa tlymeasuresthedis ontinuity.

Thepla ementofthepolesgivesrisetoanunlimited hoi e,andthequestion

ofndingasuitableifnotminimaldes riptionisaninterestingopenquestion,as

alreadynoti edin[7,Ÿ6.℄.Atrade-omustthenbefoundbetweenaveraged

in-formationwithadensity omputed ontinuouslyfromthetransferfun tion,and

lo alized anda urate informationwith adensity omputed asadis ontinuity

ofthesametransferfun tion.

4 Numeri al simulations with optimization

Wenowderiveapproximationsoftransferfun tionsbyrst-orderlinear

onstant- oe ient systems, stemming from the generalized diusive realizations

pre-sentedinŸ3.4.Theapproximatedmodelhastheform M (s):= P 1k K k s k ,

wherethesetof poles =f

k

g is hosenwithahermitiansymmetry.The

set may representeither a ut ora - ontour asso iated with the diusive

operatorunder onsideration.Theproblemnow onsitsofestimating=f

k g.

Arstmethod omputes

k

analyti ally: isdes ribedinthe omplexplane

by  7! (), and  k = R +1 0 ( ()) k () 0 ()d, where  k are interpolating

fun tions (seee.g. [4℄and[3℄).

These ond method, that is beingused herefor the ross- ut, onsists ofa

least-square regularizedoptimization of f

k

g,minimizingthedistan e between

H 2 (i!)and M (i!): for!2[! min ;! max

℄.Wetakethefollowing riterion:

C  ()=k M : H 2 k 2 L 2 (!min;!max) + 1 X fk = k 2[s 1 ;s 2 ℄g j k j 2 + 2 X fk = k 2R g j k j 2 (3) where 1 and 2

areregularizingparametersfortheos illatinganddiusiveparts,

respe tively. An exampleof result obtained for this riterion and for the pole

pla ement represented in Fig.1 is given in Fig. 2.Note that the optimization

DiusiveRepresentationsforWebster-LokshinSimulation 5

10

−4

10

−2

10

0

10

2

10

4

10

−4

10

−2

10

0

10

2

10

4

Bode'sdiagram(module) Bode'sdiagram(phase)

0 20 40 60 80 100 0  2 3 4 5 dB radians ! ! original original approximation approximation

Figure2: Bodediagramsoftheexa ttransferfun tionH

2 (i!)andof  M (i!):  for  =1,=0:3,  1 =0, 2 =10 10

,with4 omplex onjugatepairsofpoleson[s

1 ;s

2 ℄

and16polesonR ( min= 10 3

).Polesarespa edwithageometri sequen e.

5 Con lusion

Ourmethodsumsupasfollows.Wederivethetransferfun tions intheLapla e

domain.Weperforma omplexanalysis:asymptoti expansions,polesand

bran h-points, hoi e of uts inthe left-half omplexplane.Finally, theparametersof

theapproximatedmodel areoptimizedin aleast-squaresenseonapathin the

Lapla edomain,usingeitherthe utsora - ontouren ir lingallthe

singulari-ties;theerrorisevaluatedinthefrequen ydomain.Thetime-domainsimulations

anbedonestraightforwardly.

Referen es

1. Th.HélieModélisationphysiqued'instrumentsdemusiqueensystèmesdynamiques

etinversion. PhDthesis,UniversitédeParisXIOrsay,De ember2002.

2. E. Du asse. Modélisation et simulation dans le domaine temporel d'instruments

à vent à an he simple en situation de jeu : méthodes et modèles. PhD thesis,

ENSAM,BordeauxI,2001.

3. M. Dunau. Représentationsdiusivesdese ondeespè e :introdu tionet

expéri-mentation.DEAd'Automatique,Toulouse,2000.

4. D. Heles hewitz. Analyse et simulation de systèmes diérentiels fra tionnaires

et pseudo-diérentiels linéaires sousreprésentation diusive. PhDthesis, ENST,

De ember2000.

5. D. Matignon. Stability properties for generalized fra tional dierentialsystems.

ESAIM: Pro eedings,5:145158,De ember1998.(Available onweb)

6. G. Montseny. Diusive representation of pseudo-dierential time-operators.

ESAIM:Pro eedings,5:159175,De ember1998.(Available onweb)