Astudyofwavepropagationinvaryingcross-sectionwaveguides bymodaldecomposition.PartII ...perso.univ-lemans.fr/~vpagneux/Site_3/Publications_files/... · 2011-02-17 · Astudyofwavepropagationinvaryingcross-sectionwaveguides

A study of wave propagation in varying cross-section waveguidesby modal decomposition. Part II. Results

N. Amir, V. Pagneux, and J. KergomardLaboratoire d’Acoustique de l’ Universite du Maine, URA 1101 CNRS, BP 535, Avenue Olivier Messiaen,72017 Le Mans Cedex, France

~Received 25 March 1996; revised 13 December 1996; accepted 19 December 1996!

The full derivation of the equations governing the generalized impedance matrixZ, the pressure,and the velocity were presented in Part I of this series@Pagneuxet al., J. Acoust. Soc. Am.100,2034–2048~1996!#. Here only the results of that paper, i.e., the final set of equations which neededto be solved are repeated. Other factors influencing the solution are the boundary conditions at theend of the waveguide: Source and radiation conditions are also presented. Finally, the details of thenumerical implementation are also relevant, and will be discussed in some detail. ©1997Acoustical Society of America.@S0001-4966~97!02605-2#

PACS numbers: 43.20.Mv, 43.20.Ks, 43.75.Fg@JEG#

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LIST OF SYMBOLS

z axial coordinater radial cylindrical coordinatep acoustic pressureP column vector of coefficients ofp for modal de-

compositionv particle velocityvz axial component of particle velocityU column vector of coefficients of axial velocity fo

modal decompositionc speed of sound in free spacev angular frequencyk wave number in free spaceki wave number of thei th mode in the waveguideK diagonal matrix of modal wave numbers;Ki j

5ki2d i j

INTRODUCTION

This paper is the sequel to Part I,1 where the theoreticabackground was presented. Here in Part II we will studydepth two typical waveguides. The first is a trombone bellwhich the flare increases very rapidly toward the end.will calculate the input impedance and compare it tomeasured input impedance, in order to show how the mumodal analysis improves on the results of plane-wave ansis. We will also examine the calculated pressure and veity field along the horn. The second case is a waveguide wcircular cross section, and an elliptic longitudinal section,which we will study focalization effects and reflection proerties. In both cases we show the effects of varying therameters at our disposal on the solution, e.g., the numbemodes taken into account, error thresholds, etc. As a maof fact, the main goal of this part is the analysis of the fieinside the guide with varying cross section, taking advantof the specificity of the method. Incidentally, we compareresults with finite element results and experimental resubut this aspect is of secondary importance.

We will begin by reviewing in Sec. II the equations

2504 J. Acoust. Soc. Am. 101 (5), Pt. 1, May 1997 0001-4966/97/10

Z generalized impedance matrixZc characteristic matrix impedance of a cylindric

pipeZr value ofZ at the free end of the waveguideY generalized admittance matrixYr value ofY at the free end of the waveguidec column vector of eigenfunctions in the waveguidS cross-sectional area of waveguideR local cross-sectional radius of waveguideg i i th zero of Bessel functionJ1A diagonal matrix ofg i /Rf frequencye,e t,ev values used to calculate viscothermal lossesDi(t) function used to calculate radiation impedanceJm Bessel function of orderm.

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be solved numerically, followed by a description of thboundary conditions used for various types of radiation, athe numerical procedures used. Sections II and III then dwith the trombone horn and elliptic waveguide, respective

I. THEORETICAL BASIS

A. Equations

We wish to solve the matrical Riccati equation for thgeneralized impedance matrixZ:2

Z852 jkrc

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S~QZ1ZQ!. ~1!

SinceZ is a matrix of infinite order, we must truncate it tordern 3 n, wheren must somehow be determined.

OnceZ is known along the waveguide, we can find thpressure or axial velocity along the waveguide, integratone of the two following equations:3

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OnceP or U are found, the pressure or axial velocity agiven by

p5( c iPi , vz51

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In certain cases it may be inconvenient to use the Ricequation~1! for Z. If we define a generalized admittancmatrix Y so thatU5YP, we can obtain from~2! a Riccatiequation forY in the same way as~1! was obtained in Part I.1

This equation will be

Y852 jS

krcK2

S8

S~ tQY1YQ!1

jkrc

SYY. ~4!

A further refinement to these equations is the introdtion of viscothermal losses created at the boundaries. Keing in mind that the expressionK is simply a diagonal matrixof propagation constants for the various modes, we can sply replace it in Eqs.~1! and ~4! by an expression for thelossy propagation factors. This can be found in Ref. 4.the axisymmetric modes it is as follows:

ki2.k22S g i

R D 212k@ Im~e i !2 j Re~e i !#. ~5!

At atmospheric pressure and room temperature,e i can beapproximated as

e i5S 12g i2

R2k2D ev1e t , ~6!

where Re(ev)5Im(ev)52.0331025f 1/2, and Re(et)5Im(et)50.953 1025f 1/2 ~f in s21!.

B. Boundary conditions

The general problem we aim to solve in this paper isfollows: Given a rigid walled waveguide, we have a sourat one end, and a radiating termination at the other.source can be a pressure or velocity source, thus fixing eP orU at one end. The radiation condition must fix the valof the matrixZ at the other end, which we will callZr .

One simple case is a termination consisting of an infincylindrical tube. In this case,Zr is simply a diagonal matrixof characteristic impedances for each mode andZr5Zc . An-other simple case is the ‘‘ideal reflection’’ of an open eni.e., each mode has a pressure node at the open end, anZr is uniformly zero. Such a boundary condition for the impedance matrix is a very rough approximation of an opend. Even in the plane-wave approximation, a length cortion has to be taken into account. On the other hand, ifwish to model a closed end, the axial velocity is zero, thelements ofZr need to be infinite. In this case it is easiersolve the equation forY, Eq. ~4!, with a uniformly zeroboundary condition forYr . The value ofZ at the input canbe obtained by invertingY.

The most complicated case is the most realistic oneradiation into free space. This is quite difficult to solvtherefore we chose to adopt instead the radiation into a h

2505 J. Acoust. Soc. Am., Vol. 101, No. 5, Pt. 1, May 1997

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space; the open end of the waveguide is thus placed ininfinite baffle. This case has been treated before in Refand 6. Initially, the analytical expression forZr is

Zr5jvr

S2 ESESc~w!G~w,w8!Tc~w8!ds dS8, ~7!

whereG(w,w8) is the Green’s function for radiation into thhalf-space, andw is a vector onS. This integral is singularand thus difficult to evaluate directly. Therefore we choseuse the formulation presented by Zorumski,5 which seems tobe posed most efficiently for numerical evaluation. The epression forZr presented by the latter are

Zi j52 j E0

`

t~t221!21/2Di~t!Dj~t!dt. ~8!

Here,Zi j is a nondimensional form ofZr andDi is definedby

Di~t!5k2E0

r

rJ0~tkr !c i~r !dr. ~9!

Zorumski provides an exact analytical solution for~9!, andanother form of~8! which can be numerically evaluatemore readily:

Zi j5E0

1/2p

sin fDi~sin f!Dj~sin f!df

2 j E0

`

coshjDi~coshj!Dj~coshj!dj. ~10!

These results were compared to another calculation baseRef. 6, giving very good agreement.

Since this expression for the radiation impedance isclosest available to us, we compared the theoretical resulexperiments both with and without an ‘‘infinite’’ baffle, inorder to have an idea of how far this radiation conditionfrom radiation with no baffle. An extension of the calculatioof the case with infinite baffle to the case of no baffle coube made by iterative procedure~see Williams7!.

C. Numerical procedures

The basic ‘‘engine’’ used to solve the Riccati equatiofor Z or Y was a third- or fourth-order Runge–Kutta procdure. This proved to be relatively robust in most cases.vergence did occur when using simplified radiating contions where no energy was allowed to radiate fromwaveguides~the closed or ‘‘ideal open’’ terminating conditions!, since the impedance or admittance peaks becamefinite. In these cases it was imperative to incorporate bouary layer thermal and viscous losses into the calculationdescribed above.

It is also imperative to use an adaptive stepsize, ewith radiation or viscous losses, since the peaks inZ or Ycan be pronouncedly sharp, and there is no way to predicnecessary minimum stepsize. This type of solution requselection of an error threshold for determining when the stsize must be reduced; this threshold must be selected byand error. If it is not small enough, the computation wgenerally diverge. It may noticed that the effect of increas

2505Amir et al.: Varying cross-section waveguides

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the number of modes selected reduces the Runge–Kstepsize, because with more modes we have more peakthe impedance.

OnceZ or Y have been integrated from the radiating eto the source, it remains to calculate the pressure or velofield, integrating one of~2! in the other direction, i.e., fromthe source to the radiating end. This requires knowledgeZ or Y at each point of integration. This is a touchy point: Adescribed in Part I, an attempt to reintegrate either of thvalues back up the waveguide always fails. This meansZ or Y must be stored at each point during the numerisolution of the Riccati equations, to be used later when ingrating forP or U. This creates two problems: It can take umuch memory, and it fixes the integration points for thcomputation. The first problem has no easy remedy—RAM memory at the program’s disposal must be sufficienlarge, or disk space should be used~slowing down the cal-culation considerably!. The second problem means thatadaptive stepsize cannot be used for the second integra

In practice,Z was stored during the first integration ongrid of regularly spaced points along the axis, by ‘‘forcingthe adaptive stepsize algorithm to ‘‘visit’’ these pointsaddition to any intermediate points required by the algrithm. The second calculation was then performed withfixed stepsize. As a coherence check, the energy fluxcalculated along the waveguide, using lossless propagaWhen it came out nearly constant~relative error of less than0.1%! the calculation was judged to be sufficiently accuraThis procedure necessitates a very fine grid for storingZ,although apparently this is the only method possible.

We will demonstrate in the later sections how variochoices of parameters influence the results of the comption.

II. FIRST WAVEGUIDE: THE TROMBONE BELL

The first case to be studied in detail is the trombone bwhich comprises approximately the last 55 cm of the instment. It is characterized by a rapid flare at the end, whshould render one-dimensional analysis methods relatiinaccurate. In this section we will study the theoreticacomputed input impedance and compare it to resultstained experimentally. We will also examine the pressfield inside the horn.

The objective of this study is not only to verify the thoretical results by comparison to experiments, but alsostudy the influence of various parameters on the theoreresults. We wish to see the influence of various radiatconditions, the effect of the number of modes used onaccuracy, and the interaction between these parametersother point to be examined is how the assumption ofinfinite baffle can change the experimentally measured inimpedance.

We will begin with a description of the geometry of thhorn, and its relation to the frequency range considered. Twill be followed by a short description of the experimenapparatus, and then by comparisons of theoretically andperimentally obtained input impedances. The final part ofsection will present results concerning the pressure field


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A. Geometry and frequency

The part of trombone studied begins with a cylindricsection of 25.4-mm diameter. It then begins to flare moestly, terminating in an abrupt flare toR5115.4 mm Theradius as a function of distance appears in Fig. 1. Tabsummarizes the points that were measured. Intermedpoints were obtained by third order polynomial interpolatioa polynomial was calculated for four points at a time, aused for calculating values between the two middle poionly.

The frequency range to be studied was between 01200 Hz, this being approximately the musically userange of the trombone. It is interesting to note that for trange of frequencies, all of the higher order modes arecutoff, even at the mouth of the horn. At the throat of thorn this is 7.2% of the first cutoff frequency, and at tmouth it is 65% of the first cutoff frequency. The excitatiois assumed to be a plane piston, i.e., only the plane-wmode is excited at the throat, and any influence of higorder modes is due to mode coupling induced by the flaThese higher-order modes are assumed to be so deepcutoff at the throat that they can be ignored there, therefenabling us to treat the input impedance as a simple scvalue, which is the corner value of the matrixZ. The validityof this assumption will be examined later.

B. Experimental apparatus and results

The experimental apparatus available enabled us to msure the input impedance of the horn, but not the pressfield. This apparatus consisted of a velocity source an

FIG. 1. Radius of the trombone horn versus axial distance.

TABLE I. Radius of the trombone horn versus axial distance.

distance@cm# 0 5.5 5.6 5.7 5.8 5.9 8.3 12.4radius@cm# 12.25 12.25 12.25 12.25 12.65 12.65 13 14

distance@cm# 16.2 20.3 25.3 31.2 36 40 43.2 45.radius@cm# 15 16.5 18.5 21.5 24.5 28 32 37

distance@cm# 48.2 50 51.5 52.7 53.6 54.4radius@cm# 44.5 52 62 74.5 89.5 114.5


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FIG. 2. Measured input impedance with baffle, without baffle, and with rigid sealed end. Note the difference between the first two, at high freque
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pressure sensitive microphone at the throat of the horn, bconnected to a lock-in amplifier. The latter is connected tPC, which controls the frequency sweep. This sweep isformed adaptively, so that smaller increments are taken nthe impedance peaks, ensuring good accuracy in thesegions. The entire apparatus was placed in an anechoic chber, and performed at a controlled temperature. Calibrawas carried out using a cylindrical pipe of approximatelym length. For further details of this apparatus, refer to Ref

The measurements were carried out for three cases:where the horn radiated freely into the room, one wheremouth was placed in a large baffle (1.531.5 m), and onewhere the end of the horn was shut hermetically with aPlexiglass plate.

Figure 2 shows the results of these three measuremNote that there is a subtle but noticeable difference betwmeasurements made with and without the baffle, at higfrequencies. This is important, since the theoretical calctions use the baffled value of the radiation impedance. Thalso used often in the literature for the one-dimensional mels, although without always specifying whether the expements used a baffle also. Fortunately, at low frequenciestwo measurements give practically identical results; at sfrequencies a good approximation of an infinite baffle wohave to be very large. At frequencies where the two measments differ, the baffle used was in fact a good approximtion of an infinite one. The reasons for this behavior aexplained in Ref. 9, where it is shown that musical horns


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quite insensitive to the radiation impedance at low frequcies, and that even plane-wave calculations can give gresults at these frequencies.

Figure 2 also shows a big difference between measments with an open and closed horn. The closed horn haradiation losses, and being quite short the viscotherlosses are quite small also, therefore the impedance peaksimilar in height, and at different frequencies than thofound for the open horn.

C. Theoretical results for input impedance

Before comparing the theoretical and experimentalsults, we examine the purely theoretical results from sevpoints of view.

As stated above, the parameter we will be dealing wwill simply be the corner valueZ00 of the matrixZ. Beforecontinuing, the assumption that this parameter is the mone of importance must be justified. In Fig. 3 we show tdiagonal ofZ as a function of frequency, and the first rowZ as a function of frequency. These represent, respectivthe ‘‘self’’ impedances of each mode, and the ‘‘cross’’ impedance between the higher-order modes and the plane-mode. The very low values of all the other elements in retion to the corner value shows that their importance is mmal. They are also generally quite smooth, as opposed topeaks and dips found inZ00, thus giving no informationconcerning resonances or antiresonances.


FIG. 3. The diagonal and first row of the input impedance matrixZ ~arbitrary units!.

FIG. 4. The corner value ofZ calculated with progressively more modes. One mode5plane-wave mode only.

2508 2508J. Acoust. Soc. Am., Vol. 101, No. 5, Pt. 1, May 1997 Amir et al.: Varying cross-section waveguides

FIG. 5. Comparison of input impedance calculated with plane-wave mode and with six modes for four different end conditions:~a! closed end;~b! ideal openend; ~c! infinite pipe termination; and~d! baffled radiation model. Note that the largest difference between the two calculations is found in~d!.

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An interesting artifact found in the other elements ofZis the appearance of spurious peaks. These are simplyfacts of the numerical calculation, and can be made to dispear if a smaller stepsize is constrained, by using a smaerror threshold in the Runge–Kutta algorithm. This lengens the calculation time, with very little effect on the resuing value ofZ00. Therefore, to a certain extent these artifawere allowed. It is the adaptive stepsize which ‘‘returns’’ tdiverging curves to their normal values; using a fixed stsize allowed these artifacts to cause the whole algorithmdiverge.

The most important parameter of the calculation isnumber of modes used, i.e., the order to which the maZ is truncated. In Fig. 4 we show the corner value compuwith different sizes ofZ. The most radical change occuwhen adding just one mode to the basic plane-wave moAfterward the differences are small, becoming progressivsmaller, and more obvious at higher frequencies. Forcalculations we generally used six modes, even thoughthe higher order modes were evanescent. In general, thisrameter strongly depends on the frequency range being sied, and on the irregularity of the horn walls.

It is of primary interest to view how much the introdution of the higher-order modes changes the results, as cpared to plane-wave analysis, or spherical wave analysisthis context we studied a number of possible radiationpedances: the hole-in-infinite-baffle, zero radiation impance ~‘‘ideal open end’’!, infinite impedance~closed end!,


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and termination by an infinite cylindrical pipe. Not all othese, of course, could be implemented experimentally.

In Fig. 5 we see for each of these a comparison ofinput impedance calculated with one and six modes. Incases there is good agreement at low frequency, less shigher frequency. The difference is most marked for the mpractical case of radiation into a baffled space. For a clohorn, on the other hand, we can expect good results ewith a plane-wave model, as has already been shown in9.

Another interesting point is to compare the input impeances calculated with different boundary conditions. Thiscarried out in Fig. 6 for 1 and 6 mode calculations. As ashown in Ref. 9, the resonance frequencies for all casescept for the closed end are very near each other for the plwave calculation. This quality of musical horns is actuallyartifact of the plane wave model, as shown by experimentRef. 9, and borne out by the six mode calculation shohere; at low frequencies it remains true, but at higher fquencies the peaks for the baffled radiation case arelow, and at very different frequencies than the others.

D. Comparing theory and experiments

The most important comparison carried out here is tof the multimode calculation with the experimental resulThis appears for theoretical and experimental results in F7. A good agreement is observed, better than the result


ation.

FIG. 6. Comparison of input impedances for four end conditions:~a! with plane-wave mode only;~b! with six modes.

FIG. 7. Comparison of experimental input impedance with various theoretical curves. Best agreement is achieved with the multimodal calcul

2510 2510J. Acoust. Soc. Am., Vol. 101, No. 5, Pt. 1, May 1997 Amir et al.: Varying cross-section waveguides

the horn.
FIG. 8. Coefficients of the pressure series expansion along the horn. Note that the higher-order modes are significant only near the mouth of
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tained using only the lowest order mode. The validity of tsimplest model, with only the plane-wave mode, is cofirmed for the trombone to be limited to roughly 500 Hz,predicted, using simple arguments, by Kergomard.10 On theother hand we do not obtain perfect agreement, unfonately. A further study is necessary in order to verify tsource of this error.

E. Calculations of the pressure and velocity fields

In this section we present calculations of the pressand velocity fields along the horn for various frequenciusually with a plane piston~velocity source! excitation.

Our first example is a plane piston excitation at 600using 11 modes. In Fig. 8 we see the values of the coecientsP along the horn. From this figure it is clear that thhigher-order modes modify the pressure field only towthe very end of the horn, where the flare is largest. Tjustifies once more the use of a scalar input impedance athroat.

Next, in Fig. 9 we show graphs of equipressure surfatoward the end of the horn, using 2, 6, and 11 modes.spite the small coefficients inP for the higher order modesthe differences between the figures are clearly visible. Obously, to obtain a good visual representation of thesefaces, a relatively large number of modes must be used, ethough they are all evanescent.

Raising the frequency can create a much more comcated picture. In Fig. 10 all of the higher order modes


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evanescent. At this frequency the equipressure surfacesfar from being spherical.

One more comparison we made is the axial presscalculated by modal analysis to the axial pressure obtaiwith the horn equation, i.e., using only the plane-wave moThis appears in Fig. 11. A considerable difference is appent.

III. SECOND WAVEGUIDE: THE ELLIPSOIDALINDENTATION

In this section we show how the method can be applto a cylindrical duct with an ellipsoidal indentation. The gometry, which is axisymmetric, is characterized by the radof the cylindrical ductR0 , the maximum radius of the indentationRmax, and by the distance between the two focal poiL ~see Fig. 12!.

Our first objective is to observe how the transition frolow frequencies to higher frequencies is accompanied byincrease of the focalization at the right focal pointM f . Thisguide is excited by a monopole located at the left focal poM0 . Classical analysis of the radiation of monopole souinto a straight waveguide can be found in classical wo~see Morse and Ingard11 and Doak12!. The influence of thenumber of modes selected is also analyzed and compariwith a finite element method~FEM! code~ANSYS! are made.

In a second part, in the same geometry, the Riccati eqtion ~1! is integrated to find reflection properties of the gometry with two different maximum radius of the indenttion, Rmax.


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A. Study of the focalization

1. Procedure

According to the reasoning of geometrical acoustics, ifsignal is launched from the left focal pointM0 , a receiverlocated at the right focal pointM f will observe a focaliza-tion. This is due to the fact that each ray emitted fromM0

has to go toM f with the same phase shift. This is true incomplete ellipsoidal geometry and under the assumption

FIG. 9. Equipressure surfaces along the last 14 cm of the horn, at 600calculated with:~a! 2 modes;~b! 6 modes;~c! 11 modes.


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very high frequency. Here the effects may be modified duetruncation of the ellipsoidal geometry and to finite wavlengths.

Three different frequencies have been selected foranalysis. In the first case, wherekR051.85, only one modeis propagating, therefore it is possible to compare withclassical MPWA~matched plane-wave approximation! cal-culation and to observe the advantage of a multimodalproach even at low frequency. In the second case, whkR057.4, three modes are propagating. In the third cawherekR0514.8, five modes are propagating in the cylidrical duct. The peculiar interest here is that a sixth modepropagate in the indentation only, making this mode a locized one. We can therefore examine whether our methostill efficient in such a case.

Boundaries conditions at the right and at the left endthe guide are those of infinite propagation without reflectioi.e., Z5Zc ~resp.Z52Zc! at the right~resp. left! termina-tion. Calculations were made using the following procedufirst, starting from the right boundary condition (Z5Zc), the

z,

FIG. 10. Equipressure surfaces along the last 14 cm of the horn, at 1200calculated with 11 modes.

FIG. 11. Pressure amplitude along the axis of the horn, at 600 Hz, calcuwith 1 and 11 modes.


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Riccati equation is integrated down to the pointM0 . Thenwe haveP0

1 5 ZU01 valid at the right ofM0 , where1 and2

will be thereafter related to quantities at the right and llimit of M0 . On the other hand, knowing that the left termnation is infinite, we haveP0

252ZcU02 valid at the left of

M0 . Finally, it is possible to get the velocity coefficientsthe source by applying the conservation of pressureflowrate,

P012P0

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and

U012U0

25Q,

where vectorQ contains the coefficientQj5q0c j (0) of theprojection of the monopole of strengthq0 ~flowrate source!on the functioncm . This gives us

U015~Zc1Z!21ZcQ.

Finally, Eq. ~2! is integrated to yield the field across thwaveguide.

2. Frequency below the first cutoff of the waveguide

Figure 13 shows the modulus of the pressure for suclow frequency. Our method@Fig. 13~a!# achieves very goodagreement with the pattern obtained from the FEM co

FIG. 12. Geometry with ellipsoidal indentation. The two points,M 0 andM f , are focal points located at positions61.2R0 .

FIG. 13. Modulus of the pressure,kR051.85.


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@Fig. 13~b!#. In the two figures, a light focalization can beobserved even at this rather low frequency. In order to quatitatively compare the two computations, we have plotted thprofiles of the modulus and of the phase of the pressualong the central line (r50) in Fig. 14; four different calcu-lations with different numbers of modes~m51,2,3, and 15,wherem is the number of modes! are compared to the FEMprofile. It is very clear that selecting only one mode~m51,corresponding to the MPWA method! is not sufficient toyield the correct result. Adding just one mode to the modeing (m52) enables to improve the agreement. Eventuallywith 15 modes (m515) the agreement between the multi-modal and the FEM approaches is almost perfect.

Another view of the effect of the multimodal conver-gence is seen in Fig. 15, where the pattern of the modulusthe pressure is plotted for 1, 2, 3, and 15 modes. This showthe failure of the MPWA method (m51), but as soon as twomodes are used, a roughly correct pattern is obtained. A

FIG. 14. Modulus and phase of the pressure along the central liner50,kR051.85. ~* !: 1 mode;~1!: 2 modes;~3!: 3 modes;~s!: 15 modes; plainline: FEM method.


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m is increased, the pressure tends to achieve the bouncondition at the wall (]p/]n50), due to the increasingnumber of evanescent modes used.

We also measured the rate of convergence of the mumodal method in Fig. 16. The result obtained with 15 modis chosen as the reference, and three different quantitiesproposed for the errore:

e15*Vip2prefi22pr dr dz

*Viprefi22pr dr dz,

e25*Vip2prefi2 dr

*Viprefi2 dr,

e35* r50ip2prefi2 dz

* r50iprefi2 dz.

Here,e1 represents the actual error in the volume,e2 is theerror ignoring the axisymmetry of the geometry, and thstressing the difference along the central line (r50), e3 isthe error only along the central line.

FIG. 15. Modulus of the pressure with 1, 2, 3, and 15 modes, respectiv

FIG. 16. Error of the method.~1!: e1 , ~s!: e2 , ~* !: e3 .


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s

Figure 16 shows that all three error measures decremonotically as more modes are introduced in the calculatThe increased values ofe1 , e2 , ande3 , respectively, are dueto the fact that the source is a monopole located on the ctral line; e1 , e2 , and e3 take this progressively more intaccount.

The behavior ofe1 is almost exponential, except betweenm51 andm52 where the error is reduced. Thashows, once more, that even with only two modes a ladegree of accuracy is gained with the multimodal method

3. Higher frequencies

When the frequency is increased tokR057.85, threemodes are propagating and the focalization increases asbe seen in Fig. 17. Figure 17~a! and~b! show the agreemenbetween the multimodal and the FEM computations, as fathe pattern is concerned. On the other hand, when the prof the modulus of the pressure along the central liner50) is plotted, we discover some discrepancies. In fact,latter are an artifact of the FEM code since it is unablemodel the correct infinite termination condition at the etremities of the guide. We therefore repeated the compar~Fig. 18!, but this time applying in the multimodal calculation the same condition as in the FEM code (p5r0cuz) atthe extremities (z562R0). This is equivalent to imposingZnm5dnmr0c/S for the matrix impedance, rather than thcorrect infinite termination condition which isZnm5Zcnm5dnmr0ck/(Skn). The three profiles corresponding to thcorrect and incorrect termination conditions and to the FEcode are plotted in Fig. 18. We see that incorporatingsame boundary condition as the FEM code in our multimomethod enables us to obtain an excellent agreement. Netheless, one has to keep in mind that the actual correct stion is the multimodal one with correct boundary conditioObviously this difficulty of the FEM could be overcome b

ly.



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e iteftffi-ureforthens-

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using an integral formulation at the extremities, like th‘‘boundary element method’’ or the ‘‘infinite elemenmethod’’ ~see Ref. 13!.

The last frequency studied (kR0515.7) corresponds tofive propagating modes in the straight portions with a sixmode becoming propagating in the indentation. Apart frothe increasing frequency, this case is also interesting becathe sixth mode has a turning point and we wish to knowour method is able to deal with such a situation. Figureshows the pattern of the modulus of the pressure compato the FEM solution; the focalization has become more pnounced than with the two preceding frequencies. As inprevious case, incorrect termination conditions for the ipedance have to be taken into account in order to get a vgood agreement with the FEM code results.

In Fig. 20, the modulus and phase profiles along tcentral line show excellent agreement with the FEM solutiwhen the incorrect infinite termination condition is taken.

FIG. 18. Modulus and phase of the pressure along the central liner50,kR057.85. ~1!: correct boundary condition;~s!: incorrect boundary con-dition; plain line: FEM method.


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en

B. Reflection coefficient

Results for the reflection coefficient~defined in Part I!are studied in an ellipsoidal geometry whereL510 andRmax51.1 ~geometry 1! or Rmax51.3 ~geometry 2!. A com-mon way to characterize a singularity in a waveguide, hthe indentation, in a waveguide is to study the reflectcoefficient as a function of the frequency.

In this case, our method is especially convenient, sincis sufficient to calculate the impedance matrix from the lend to the beginning of the duct to get the reflection coecient. We do not have to integrate the velocity or pressequations. On the other hand, by using a FEM code,instance, we would have to specify exact properties ofsource, and the results obtained would have to be traformed to a suitable form to get the reflection coefficient.

In the two following examples, the boundary conditioat the right of the indentation is taken as the characterimatrix impedance, modeling the correct infinite terminatioIt has been assumed that a plane wave is incident at theBy knowing the impedance at the left of the indentation, itpossible to obtain the energy reflection coefficientR which isdefined asR5Wr /Wi , whereWr andWi are, respectively,the reflected and incident energy flux.

Results for the two geometries considered are showFig. 21. They display the behavior ofR againstf / f c1 where



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im-vity.

.ns

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f is the frequency andf c15(cg1)/(2pR0) is the cutoff fre-quency of the first higher-order mode of the duct. The twgeometries produce the same kind of results. The first madifference is the lower value ofR in the geometry that has alower radiusRmax in the indentation. It seems logical becausRmax measures the strength of the indentation. The secodifference is in the number of oscillations between each coff frequency. This is due to the relation of these oscillationto the length of the indentationsLS ; LS54.4 for geometryone and LS56.6 for geometry two. In fact, LS52AL214Rmax

2 A121/Rmax2 .

Three principal features can be observed for the globbehavior of the reflection coefficient.

First, the oscillations are due to quasiresonance of telliptical indentation viewed as an open cavity. In an expasion chamber of lengthLS they correspond to the conditionkLS5np under the plane-wave approximation. This cond

FIG. 20. Modulus and phase of the pressure along the central liner50,kR0515.7. ~1!: correct boundary condition;~s!: incorrect boundary con-dition; plain line: FEM method.


oor

ndt-s

al

e-

-

tion expresses the transparency of the cavity since thepedance is then the same at the two extremities of the caIt yields five and eight oscillations for geometry 1 (Rmax

51.1) and geometry 2 (Rmax51.3), respectively. From Fig21 it can be seen that it is roughly the number of oscillatiowe can observe.

Second, peaks correspond to each cutoff frequenThey can be understood by remarking that whenf5 f cn forsome moden, thenth component of the characteristic matrgoes to infinity. Then the end condition for this modeequivalent to a wall condition and the reflection is enhanc

Finally, the decrease ofR between each cutoff frequency can be understood by drawing an analogy witpotential barrier in quantum mechanics; for the plane-waapproximation the equation for the pressure is

~Rp!91S k22 R9

R D ~Rp!50,

FIG. 21. Reflection coefficient as a function of normalized frequenf / f c1 ~f c1 ; cutoff frequency of the first nonplane mode! for geometry 1 and2.


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and the indentation is equivalent to a potentialV(z)5R9/R. So when the frequency increases, so does theergy k2, and the wave is more capable to overcome thetential V. This kind of analogy was developed by Benaand Jansson.14 Nevertheless, we notice that it is entirely valif the junctions between the straight duct and the portwith varying cross section satisfy the continuity of both tfunctionR(z) and its first derivativeR8(z).15

IV. CONCLUSION

In this paper, we have shown how the multimodmethod can be used to solve various problems concerwaveguides with varying cross section, with no restrictionthe flare.

The convergence of the method has been investigaThe limit of the matched plane-wave approximation has bobserved, and it appears that adding only a few modeseven one mode in some cases, can be very effective inproving the accuracy of the modal approach.

The comparison with experiments for an baffled trobone shows better agreement than the plane-wave appmation, and we think that the remaining discrepancies cobe due to the finite size of the baffle in the experiments.plan to study this effect.

We also show that the multimodal method is able to dwith fairly high frequencies~five modes propagating! and toyield high-frequency results, like the focalization.

In conclusion, the multimodal method we have prsented in Part I and Part II of this series of papers seembe well adapted to the general form of waveguides with vaing cross section in a large range of frequencies.

ACKNOWLEDGMENTS

One of the authors~V.P.! would like to thank the GasEngineering Montrouge team of Schlumberger Industries


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scientific and part of financial supports. We also thank Niclas Joly~CTTM! for interesting discussions on the trombobell problem.

1V. Pagneux, N. Amir, and J. Kergomard, ‘‘A study of wave propagationvarying cross-section waveguides by modal decomposition. Part I: Thand validation,’’ J. Acoust. Soc. Am.100, 2034–2048~1996!.

2V. Pagneux, J. Kergomard, and G. Marquette,Propagation multimodaleen geometrie complexe~Congres Franc¸ais de Mecanique, Lille, 1993!, pp.1045–1049.

3J. Kergomard, ‘‘Une equation matricielle des pavillons acoustiques,’’R. Acad. Sci. Paris Ser. II316, 1691–1694~1993!.

4A. M. Bruneau, M. Bruneau, Ph. Herzog, and J. Kergomard, ‘‘Boundlayer attenuation of higher order modes in waveguides,’’ J. Sound V119, 15–27~1987!.

5W. E. Zorumski, ‘‘Generalized radiation impedances and reflection coficients of circular and annular ducts,’’ J. Acoust. Soc. Am.54, 1667–1673 ~1972!.

6Y. Kagawa, T. Yamabuchi, and T. Yoshikawa, ‘‘Finite element approato acoustic transmission-radiation systems and application to hornsilencer design,’’ J. Acoust. Soc. Am.69, 207–228~1980!.

7E. G. Williams, ‘‘Numerical evaluation of the radiation from unbafflefinite plates using the FFT,’’ J. Acoust. Soc. Am.74, 343–347~1983!.

8J. P. Dalmont and A. M. Bruneau, ‘‘Acoustic impedance measuremplane-wave mode and first helical mode contributions,’’ J. Acoust. SAm. 91, 3026–3033~1992!.

9N. Amir, G. Rosenhouse, and U. Shimoni, ‘‘Modal solution for the souwave equation in spherical coordinates for non-canonical horns,’’ Actica 82, 1–8 ~1996!.

10J. Kergomard, ‘‘Champ interne et champ externe des instruments a` vent,’’Doctoral thesis, Universite` Paris 6~1981!.

11P. Morse and U. Ingard,Theoretical Acoustics~McGraw-Hill, New York,1968!.

12P. E. Doak, ‘‘Excitation, transmission and radiation of sound from disbution source in hard-wall ducts of finite length,’’ J. Sound Vib.31, 137–174 ~1971!.

13T. W. Dawson and J. A. Fawcett, ‘‘A boundary integral equation methfor acoustic scattering matrix in a waveguide with nonplanar surface,Acoust. Soc. Am.87, 1110–1125~1990!.

14A. H. Benade and E. V. Jansson, ‘‘On plane wave and spherical wavehorns with nonuniform flare. Part I: Theory,’’ Acustica31, 79–98~1974!.

15C. Depollier, J. Kergomard, and F. Laloe, ‘‘Localisation d’Anderson dondes dans les re´seaux acoustiques unidimensionnels ale´atoires,’’ Ann.Phys.11, 457–492~1986!.


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Astudyofwavepropagationinvaryingcross-sectionwaveguides bymodaldecomposition.PartII ...perso.univ-lemans.fr/~vpagneux/Site_3/Publications_files/... · 2011-02-17 · Astudyofwavepropagationinvaryingcross-sectionwaveguides