Modeling of the multimodal radiation from an open-ended waveguide

Modeling of the multimodal radiation from an open-ended waveguide lix,1,a) Jean-Baptiste Doc,2 and Matthew A. Boucher3 Simon Fe 1

Laboratoire d’Acoustique de l’Universit e du Mans (LAUM), CNRS UMR 6613, Le Mans Universit e, avenue Olivier Messiaen, 72085 Le Mans, France 2 Laboratoire de M ecanique des Structures et des Syste`mes Coupl es, Conservatoire National des Arts et M etiers, 2 rue Cont e, Paris, 75003, France 3 KU Leuven, Department of Mechanical Engineering, Celestijnenlaan 300B, Heverlee, B-3001, Belgium

(Received 7 February 2018; revised 12 May 2018; accepted 22 May 2018; published online 13 June 2018) The multimodal radiation from the open end of a cylindrical waveguide with arbitrary wall thickness is solved by deriving algebraic solutions of the radiation impedance matrix, without restrictive hypothesis on the frequency range. The basic idea of the method is to turn the original unbounded problem into the problem of a cylindrical waveguide embedded in an infinite waveguide with an annular perfectly matched layer (PML) on its wall. Then, using a multimodal formalism of the guided wave propagation and a complex coordinate stretching PML, algebraic expressions are derived for the continuity and radiation conditions in this coupled system. C 2018 Acoustical Society of America. https://doi.org/10.1121/1.5041268 V [DES]

Pages: 3520–3528

I. INTRODUCTION

Most of the natural or man-made physical systems that guide waves also behave as sources, since they have at least one open end through which waves can radiate. In acoustics, one may mention the radiation through exhaust and engine systems,1 heating, ventilation and air conditioning networks,2 horn loudspeakers,3 musical wind instruments,4 or the vocal tract.5,6 These examples show the wide variety of practical situations where a complex coupling between waveguiding and radiation occurs, and more examples could be listed in other fields of wave physics whether optical, mechanical, etc. Despite the practical importance of this issue, the modeling of the radiation from an open-ended waveguide remains largely an open question. Most of the proposed solutions are based on strong assumptions (long wavelength7–14) or strongly idealized situations, such as a zero thickness duct wall15,16 or an infinite flange.17–20 Direct numerical simulations with finite or boundary elements, or finite differences, can of course be used, but they are not very conducive to physical insight or the implementation of optimization methods. The present paper aims at investigating the radiation from a cylindrical acoustic waveguide within the framework of a multimodal formulation of the propagation within the waveguide. Multimodal methods have been widely used to investigate guided wave propagation in structures with increasing complexity, and they have now proven to be very efficient and accurate in solving the coupled mode propagation in waveguides with complex geometry,21–23 finite admittance (liner) at the walls,24,25 heterogeneous media,26 or in the presence of flow.27,28 Zorumski,17 and later Kemp et al.,19 have formulated a multimodal radiation condition for a waveguide with circular and rectangular cross section, ending in an infinite flange. a)

Electronic mail: [email protected]

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J. Acoust. Soc. Am. 143 (6), June 2018

Using a complex coordinate stretching perfectly matched layer (PML),29–31 we derive algebraic solutions for the multimodal radiation impedance at the end of a waveguide open to free space. The basic idea of the method is to turn the original unbounded problem into the problem of a cylindrical waveguide embedded in an infinite waveguide with an annular PML on its wall. This method makes no assumption on the frequency range and can be applied to any cross-section geometry and with any wall thickness, or when the waveguide ends in an infinite flange. The paper is organized as follows: the formulation of the problem is given in Sec. II, where we give the radiation admittance matrix for the waveguide wall thickness being infinite, finite, or null. These results are then validated (Sec. III) by comparison with reference results of the literature. We finally show, by various examples (Sec. IV), how the proposed solution extends the range of application of previous methods toward higher frequency, higher order modes, and a large variety of waveguide geometries. II. MULTIMODAL RADIATION IMPEDANCE A. General formulation

Consider a semi-infinite cylindrical waveguide, denoted WGA, open at its output end to free space (Fig. 1). We let the waveguide inner radius be the reference length with respect to which all the lengths of the problem are scaled, including the wavelength. The wall thickness is denoted e. The cross section is assumed to be circular, but the same method would apply for any other geometry. In the same way, a homogeneous Neumann condition (rigid and perfectly reflecting boundary) is imposed at the wall, though a boundary condition with finite impedance (liner) could also be imposed.24 The pressure field within WGA satisfies the Helmholtz equation

0001-4966/2018/143(6)/3520/9/$30.00

C 2018 Acoustical Society of America V

It is denoted as the multimodal, or generalized,17 impedance matrix. ðeÞ The basic idea of the proposed method to calculate ZA is to turn the original unbounded problem into the problem of a cylindrical duct embedded in an infinite cylindrical waveguide with an annular PML on its wall (Fig. 1). The “perfect matching,” that is, the absence of reflection at the interface between the PML and the domain it encloses, as well as the wave damping in the PML, are achieved by means of a complex stretching30,31 of the radial coordinate r into ðr (5) r~ðrÞ aðr 0 Þ dr0 ; 0

FIG. 1. (Color online) Example of a cylindrical, open-ended, waveguide and equivalent problem with the “PML-waveguide.”

@2 2 þ D? þ k peilh ¼ 0; @x2

(1)

with D? the transverse Laplacian operator, k the wavenumber, and l the azimuthal index (l 2 Z). For reasons of symmetry, there is no coupling between solutions having different azimuthal dependences. Therefore, without loss of generality, we consider the particular solution pðk; x; rÞ expðilhÞ of the Helmholtz equation. The general solution for p within the waveguide is usually written as the infinite series pðk; x 0; rÞ ¼

þ1 X

pAn ðk; xÞuAn ðrÞ;

(2)

n¼0

where fuAn g is the orthonormal basis of the transverse 2 u with a homogeneous eigenmodes, solutions of D? u ¼ k? Neumann condition at the wall, du=dr ¼ 0. These are the classical Bessel functions uAn ðrÞ ¼ KAn Jl ðcln rÞ, where cln is the ðn þ 1Þth zero of J 0 l , the derivative of the lth-order Bessel function of the first kind, and KAn is a normalizing factor [Eq. (A6)], such Ð 1 that ðuAm ; uAn ÞA ¼ dmn , with the scalar product ðf ; gÞA 0 fg r dr, and f the complex conjugate of f. In order to solve the wave equation (1) within the waveguide, one needs to properly describe how the wave is radiated in the free space—or, in other words, what is the pressure/velocity relation at the output end. Formally, this can be enunciated as: finding the operator fðk; rÞ such that pðk; 0; rÞ ¼ fðk; rÞ@x pðk; 0; rÞ:

(3)

In the framework of a multimodal formulation of the wave propagation, as we are interested in, we solve this problem by ðeÞ calculating a modal estimate of f, ZA ðkÞ, whose terms are ðeÞ

ZAmn ¼ ðuAm ; fuAn ÞA : J. Acoust. Soc. Am. 143 (6), June 2018

(4)

with a a complex function filling argðaÞ 2 ½0; p=2½ [a time dependence expðixtÞ is understood throughout this paper], and aðr < 1 þ eÞ ¼ 1. The configuration shown in Fig. 1 actually displays two semi-infinite PML-waveguides: a circular one, denoted WGB for x 0, and an annular one, denoted WGC for x < 0. In order to characterize the radiation downstream and upstream from WGA’s output, we will first write that, in each semi-infinite PML-waveguide, no wave propagates backward from x ! 61 to x ¼ 0. This is done using the characteristic impedance matrix of each waveguide, and detailed in the following. B. Wave equation in WGB and reflectionless condition

Omitting the azimuthal dependence expðilhÞ, the Helmholtz equation (1) in the stretched coordinate system, ðx; r~; hÞ, reads @2 2 ~ (6) þ D ? þ k p~ ¼ 0; @x2 where p~ðk; x; r~Þ pðk; x; rÞ and 2 ~ ? ¼ 1 @ r~ @ l : D 2 r~ @~ r @~ r r~

(7)

Note that, from Eq. (5), the partial derivatives @r in the original transverse Laplacian simply transform to @r~ ¼ a1 @r . Writing r~ðrÞ ¼ rbðrÞ, Eq. (6) becomes32,33 @2 2 (8) ba þ k þ $? ðH$? Þ p ¼ 0; @x2 where the gradient and divergence operator are $? f ¼

@f il er þ f eh ; @r r

$? F ¼

1@ il ðrFr Þ þ Fh ; r @r r

(9) (10)

and H in cylindrical coordinates is ! b=a 0 H : 0 a=b

(11)

lix et al. Fe

3521

Then, as it was done in WGA, the problem is discretized using a basis of transverse functions. The solutions of the 2 u form a complete transverse eigenproblem D? u ¼ k? orthogonal basis in the disk ðr; hÞ 2 ½0; b ½0; 2p. Therefore we choose these functions, uBn ðrÞ ¼ KBn Jl ðcln r=bÞ, to discretize the transverse problem. They fulfill the Neumann boundary condition @r uBn jr¼b ¼ 0 that is imposed to the wavefield p. Note that the choice of the boundary condition at the back of the PML is arbitrary and has no significant consequence as soon as the amplitude of the wave is sufficiently small when it reaches the boundary r ¼ b. We have checked that the results given in the following are insensitive to the choice of a Neumann or Dirichlet boundary condition at the back of the PML. The projection of Eq. (8) over the basis fuBn g gives 2 p00 B ¼ ðM1 B KB k IÞpB ;

(12)

where pB is the vector of the modal components Ð b of p: pBn ¼ ðuBn ; pÞB , with the scalar product ðf ; gÞB ¼ 0 fg r dr. I is the identity matrix, and MBmn

ðb bauBn uBm r dr;

(13)

0

KBmn

ðb

ðHr? uBn Þr? uBm r dr:

(14)

0

These two matrices encapsulate the information about the PML, hence, the effect of the unbounded domain, through the scalar and matrix coefficients a, b, and H. Then, by diagonalizing the matrix coefficient M1 B KB 2 k I in Eq. (12) 2 2 1 M1 B K B k I ¼ UB C B UB ;

(15)

with C2B the eigenvalue matrix and UB the change of basis matrix, the general solution in WGB can be written as

FIG. 2. (Color online) (a) Exact free-field Green’s function, as given by Eq. (B1). (b) Modal Green’s function in a cylindrical domain as WGB, bounded by an annular PML (Appendix B). The wavefield is shown only where it differs from the exact solution by less than 1%. The computed wavefield in the PML naturally differs from the exact wavefield, and the discrepancy is also large very near the source point (x < 0.1) because of the singularity of the Green’s function. The source wavelength is such that k ¼ 4p, 30 modes are taken into account, and the PML parameters are d ¼ 1, ^ a ¼ 1 þ i; na ¼ 1 [cf. Fig. 1 and Eq. (26)], and e ¼ 0:5. 1 pB ¼ UB C1 B UB qB :

(18)

1 The matrix ZB UB C1 B UB can be interpreted as the characteristic impedance matrix of WGB.

C. Wave equation in WGC and reflectionless condition

pB ðxÞ ¼ UB ðeCB x c1 þ eCB x c2 Þ;

(16)

with Re½CBn 0 and Im½CBn 0, and where c1 and c2 are constant vectors. This series representation of the wavefield in a homogeneous free-space when bounded by a PML is somehow similar to the solutions proposed by Olyslager, De Zutter, and colleagues for the Green’s function.34–36 Indeed, if one assumes a source point in WGB, the coefficients c1;2 in Eq. (16) can be determined (Appendix B) and the resulting modal wavefield approximates the usual free-field Green’s function (Fig. 2). From the general solution (16), one can write the condition imposing that only right-going waves are propagating within WGB. If qB is the vector of the components of the axial gradient @x p in the basis fuBn g, then qB ðxÞ ¼ UB CB ðeCB x c1 eCB x c2 Þ: Therefore, imposing c2 ¼ 0 gives the relation, 8x 0, 3522


(17)

The pressure field in WGC is also governed by the modified wave equation (8). Then, the projection on a basis of functions fuCn g (again, solution of the eigenproblem 2 D? u ¼ k? u, in the annular domain r 2 ½1 þ e; b, see Appendix A) gives a matrix problem that reads exactly as Eqs. (12)–(14), with uCn replacing uBn and 1 þ e replacing zero as lower boundary in the integral terms. Then, the same calculation as made above for WGB leads to the following condition of left-going wave propagation in WGC: pC ðx < 0Þ ¼ ZC qC ðx < 0Þ;

(19)

1 where ZC UC C1 C UC , with CC and UC the eigenvalue and change of basis matrices in the diagonalization of 2 ðM1 C KC k IÞ.

D. Continuity conditions at x 5 0

The final step consists now in writing the continuity conditions for the pressure field, p, and its axial derivative, lix et al. Fe

ðeÞ

@x p, at x ¼ 0, in order to write ZA as a function of ZB and ZC . The continuity of p and @x p at x ¼ 0 and the rigid condition @x p ¼ 0 on the edge of the waveguide wall (1 r 1 þ e) lead to the following relations between the multimodal variables pg ; g ¼ A; B; C:

Fmn ðuAm ; uBn ÞA ;

(21)

particular finite values of the waveguide wall is also performed. In the following, the complex function aðrÞ determining the scaling and damping in the PML is written as na r ð1 þ eÞ aðr 1 þ eÞ ¼ 1 þ ðna þ 1Þðâ 1Þ ; d (26) Ð b where â ¼ d1 bd aðrÞ dr is the average value of aðrÞ inside the PML. The power parameter na is 1 or 2. The particular case of a constant parameter a ¼ â in the PML (for na ¼ 0) is not retained because it gives poorer results. Note that aðrÞ is independent of x to avoid the computation of the matrices Mg and Kg (g ¼ B; C) at each frequency.

Gmn ðuCm ; uBn ÞC ;

(22)

A. Waveguide open in an infinite flange: e fi 1‘

pA ¼ FpB ;

(20a)

pC ¼ GpB ;

(20b)

FT qA þ GT qC ¼ qB ;

(20c)

where

with the scalar product ðf ; gÞC

ðb

fgr dr:

(23)

1þe

Then, from the continuity equations (20) and the relations ðeÞ pA ¼ ZA qA , pB ¼ ZB qB , and pC ¼ ZC qC , we deduce ðeÞ

ZA ¼ FZB ðI þ GT YC GZB Þ1 FT ;

(24)

1 where YC ¼ Z1 C ¼ UC CC UC . This gives, algebraically, the multimodal radiation impedance, assuming a finite wall thickness e. We may remark, however, how e appears in the lower boundary of integral (23) and in the dispersion relation giving the eigenvalues cCln (Appendix A). Therefore, it can be put to zero and the method is still valid to investigate the limit case of a waveguide with zero wall thickness, as studied in many papers since the pioneering work by Levine and Schwinger.7 The other limiting case (e ! þ1), which corresponds to a waveguide ending in an infinite flange, can be solved by considering a sudden expansion junction between WGA and WGB. Hence, writing the continuity conditions between these two waveguides leads to

Figure 3 shows the top left 3 3 corner bloc of the radiað1Þ tion impedance matrix ZA as a function of the frequency in the range k 2 ½0; 10. An excellent agreement is found between Zorumski’s Helmholtz integral solution17 and our algebraic solution (25): the relative error is lower than 0.1% with the chosen parameters: d ¼ 1, na ¼ 1; â ¼ 20 expðip=4Þ; e ¼ 0, NB ¼ 60, NA ¼ 30 (Ng denotes the number of modes taken into account in WGg). However, Zorumski’s solution requires the computation ð1Þ of an infinite integral for each term ZAmn and at each frequency (Appendix C). By comparison, the characteristic impedance ZB in Eq. (25) deduces from simple algebraic operations on the matrices MB and KB . These two matrices, as well as F, are frequency independent and involve finite integrals. Therefore, the matrices MB , KB , and F are calculated once for all, and the ð1Þ computation of ZA is then very fast. B. Unflanged waveguide: e 5 0

One can also see that, if e ! þ1, WGC is replaced by a perfectly reflecting wall—the infinite flange—and the characteristic admittance YC in Eq. (24) must then be replaced by a zero admittance, leading to Eq. (25).

Levine and Schwinger’s work is illustrated in Figs. 1 and 2 of their paper,7 showing, as a function of the frequency, the reflection coefficient for a plane wave reaching the open end of a duct with zero wall thickness, and the socalled end correction that is, from Rayleigh, classically used to characterize the low frequency radiation. Within the framework of our formalism, if one decomposes the modal wavefield pA ðxÞ within WGA into a right going wave pþ A ðxÞ and a left-going wave pA ðxÞ, a reflection matrix RðeÞ ðxÞ can be defined that relates these waves: ðeÞ þ p A ¼ R pA . It generalizes the plane-wave reflection coefficient R calculated by Levine and Schwinger, the latter being actually the top- left corner term of the reflection matrix:

III. VALIDATION

R ¼ R00 ð0Þ. The reflection matrix at the output end is straightforwardly deduced from the radiation impedance

ð1Þ

ZA

¼ FZB FT :

(25)

ð0Þ

The first results given in this section concern the classical limit cases of a duct ending in an infinite flange and a duct with zero wall thickness in order to validate the proposed method by comparison with reference results of the literature (for instance, Zorumski17 and Levine and Schwinger7). A comparison with Ando’s calculations9 for J. Acoust. Soc. Am. 143 (6), June 2018

ðeÞ

matrix ZA through the algebraic relation ðeÞ

ðeÞ

RðeÞ ð0Þ ¼ ðZA CA þ IÞ1 ðZA CA IÞ; with CAmn ¼ i

(27)

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k2 c2ln dmn . The end correction ‘ is related

to the phase of R ¼ jRje2ik‘ . Again, an excellent agreement lix et al. Fe

3523

ð1Þ

FIG. 3. (Color online) Top left terms ZAmn (m; n ¼ 0; 1; 2) of the radiation impedance matrix of a semi-infinite waveguide with circular cross section ending in an infinite flange (solid and dashed lines), with comparison to Zorumski’s results (Ref. 17; full and empty circles). The waveguide has a unit radius. The PML interface is located at a distance e ¼ 0 from WGA’s wall, its width is d ¼ 1, and the damping function a is characterized by ^ a ¼ 20 expðip=4Þ and na ¼ 1. Sixty modes are taken into account in WGB and 30 in WGA.

is found between our results and Levine and Schwinger’s (Fig. 4), with a relative error lower than 0.1%. Following results (Fig. 6) will show how, compared with Levine and Schwinger’s work, our method extends to a larger frequency range and the higher order modes.

FIG. 4. (Color online) Amplitude of the reflection coefficient jRj at the output end of WGA when the wall thickness e is zero, and end correction ‘ (solid lines), with comparison to Levine and Schwinger’s results (Ref. 7; full circles). The PML parameters are d ¼ 1, e ¼ e ¼ 0; ^ a ¼ 20 expðip=4Þ; and na ¼ 1. Sixty modes are taken into account in WGB and 30 in WGA and WGC. 3524


C. Waveguide with a finite wall thickness

In his 1969 paper,9 Ando extended the work of Levine and Schwinger7 to the low frequency radiation of waveguides with finite wall thickness, and gave the variations with frequency of the plane wave reflection coefficient for

FIG. 5. (Color online) Amplitude of the reflection coefficient at the output section of WGA when the wall thickness e is finite (solid lines), with comparison to Ando’s results (Ref. 9) (full circles). The PML parameters are d ¼ 1, e ¼ e; ^ a ¼ 20 expðip=4Þ and na ¼ 1. 60 modes are taken into account in WGB and 30 in WGA and WGC. lix et al. Fe

selected values of e (Fig. 5). Again, Eqs. (24) and (27) allow very accurate recovery of these results. As in Levine and Schwinger’s, Ando’s method is restricted to the low frequency regime (plane wave approximation within the duct) and each new case—that is, each new value of the wall thickness e (providing, besides, that e is small, although not zero)—is solved through a sequence of calculations that is far from straightforward.9,37 Analytical formulas proposed by Dalmont et al.12 provide a more practical tool, but which is, still, limited to the plane wave mode and low frequencies. Figures 6 and 7 in the following give examples of computation at larger frequencies and with higher order modes, when e is finite. IV. RESULTS

As was shown in Fig. 3 for the case of an infinite flange (e ! þ1), higher order terms of the radiation impedance,

FIG. 7. (Color online) Variations with frequency of the first terms of the reflection matrix RðeÞ mn ð0Þ at the output end of WGA, as given by Eq. (27), for nonaxisymmetric modes (l ¼ 1). The PML parameters are d ¼ 1, e ¼ e ¼ 0:3; ^ a ¼ 20 expðip=4Þ; and na ¼ 1. Sixty modes are taken into account in WGB. The reflection coefficients are plotted with thick solid lines when both modes m and n are propagating in WGA, and with thin dashed lines when at least one is evanescent. ð1Þ

as well as the well-known “plane wave” term ZA00 , have a nonnegligible amplitude and a nontrivial dependence on frequency. Moreover, the thickness of the wall may have a significant influence on the radiation impedance, either on its ðeÞ first term ZA00 or on the higher order components (Fig. 6). As expected from theoretical and experimental estimates of the length correction in the long wavelength limit,9,12 the ðeÞ

FIG. 6. (Color online) Variations with frequency of three terms of the radiaðeÞ tion impedance matrix ZA for different values of the wall thickness, e ¼ 0, 0.3, and infinity. The PML parameters are d ¼ 1, e ¼ 0 in the “infinite flange” case and e ¼ e otherwise, ^ a ¼ 20 expðip=4Þ and na ¼ 1. Sixty modes are taken into account in WGB. J. Acoust. Soc. Am. 143 (6), June 2018

value of the plane wave term ZA00 at low frequencies varies monotonously with e between two bound values, corresponding to the asymptotic limits e ¼ 0 and e ! þ1. However, this is no longer true as soon as k 1, and the two asymptotic results no longer display an “envelope” bounding the variations of the impedance. The same observation can be made on higher order terms of the impedance matrix (Fig. 6). Note that the experimental results published by Dalmont et al.12 already showed this qualitative behaviour: the finite thickness cases going beyond the “bounds” of the limit cases. As already mentioned (Sec. II), for reasons of symmetry, there is no coupling between solutions having different azimuthal dependences. Therefore, the proposed solution has been derived assuming an angular dependence expðilhÞ with a given value of l 2 Z. Besides, for simplicity and to allow for comparison with reference results of the literature, all numerical results have been obtained taking l ¼ 0—that is, under the assumption of axisymmetry of the wavefield. However, the method still holds for other values of l, as illustrated by Fig. 7. Note that Fig. 7, obtained with e ¼ 0.3, can be qualitatively compared with Fig. 2(b) of Ref. 16, which was obtained assuming a zero wall thickness (a quantitative agreement is naturally not expected). The ability to compute the radiation impedance for higher values of l might be of importance in cases where a coupling occurs between modes with different azimuthal dependence. This is the case if all or part of the waveguide lix et al. Fe

3525

(WGA) is curved.38 Note that taking into account modes with different azimuthal dependences is straightforward in our method. One simply has to include, in the bases fugn g; g ¼ A; B; C, all the transverse functions that are needed, with angular dependence. Then, the calculation of ðeÞ ZA is done the same way as shown in the paper. The matrices F and G will still be scalar products between functions of two different bases, the characteristic impedances ZB;C will be defined and derived in the same manner, and the radiation impedance will still be given by Eq. (24).

APPENDIX A: MASS, STIFFNESS, AND DISCONTINUITY MATRICES WITH THE RIGID TRANSVERSE MODES 1. Rigid transverse modes

In WGA, WGB, and WGC, the rigid transverse modes, 2 that is, the solution of the eigenproblem D? u ¼ k? u with homogeneous Neumann boundary conditions at the walls, are uAn ¼ KAn EAn ðrÞ; KBn r EBn ; uBn ¼ b b KCn r uCn ¼ ECn ; b b

V. CONCLUSION

We have derived an algebraic solution for the multimodal radiation impedance at the open end of semi-infinite cylindrical waveguide, ending in a flange with either zero, finite, or infinite thickness. The method makes no assumption on the wavelength and is therefore not limited to the low frequency regime, hence, extending the range of application when compared to many classical solutions. Although the present paper focuses on the characterization of the radiation through the calculation of the impedance or reflection matrices, the proposed approach also makes it possible to calculate the radiated field in a straightforward manner (Fig. 8). Indeed, once the continuity conditions at x ¼ 0 are written, the wavefield outside the open-ended waveguide can be calculated using the algebraic solution (16) and thus directly known at any point x > 0, showing, for example, this nontrivial pattern with successive radiation lobes. Note that the optimization of the numerical computation, notably the search for optimal parameters of the PML, was not the purpose of this paper. Certainly the efficiency and numerical cost can be improved in the perspective of using this method as a common computational tool. Also, in order to account for more realistic situations as enumerated in the Introduction (such as modeling and control of radiation from horns), complex-shaped waveguides should be considered instead of the simple cylindrical WGA.

(A1) (A2) (A3)

where EAn ðsÞ ¼ EBn ðsÞ ¼ Jl ðcln sÞ; ECn ðsÞ ¼ Jl ðcCln sÞ

J 0 l ðcCln Þ Y 0 l ðcCln Þ

(A4) Yl ðcCln sÞ;

(A5)

with Jl the lth-order Bessel function of the first kind, Yl the lth-order Neumann function, cln the ðn þ 1Þth zero of J 0 l , and cCln the ðn þ 1Þth zero of J 0 l ðkÞY 0 l ðksÞ J 0 l ðksÞY 0 l ðkÞ, with s ¼ ð1 þ eÞ=b the ratio of the inner and outer radiis of the annular waveguide WGC. The normalizing factors are vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 u u 2 KAn ¼ KBn ¼ (A6) u Jl ðcln Þ t l2 1 2 cln and KCn

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 1 ¼u : uð 1 t 2 ECn ðsÞs ds

(A7)

s

2. Mass and stiffness matrices

The mass and stiffness matrices in WGB and WGC are (g ¼ B; C) ð1 ðba 1ÞEgn ðsÞEgm ðsÞs ds Mgmn ¼ dmn þ Kgm Kgn 1d=b

(A8) and Kgmn ¼ FIG. 8. (Color online) Pressure level (arbitray units) of a wave propagating inside and radiated from a semi-infinite cylindrical waveguide with unit radius and wall thickness e ¼ 0.2 (the wall appears in white in the figure). An incident plane wave is emitted from the left in the waveguide, with a frequency such that k ¼ 14.5. The PML parameters are d ¼ 2, e ¼ 1:4; â ¼ 20 expðip=4Þ, and na ¼ 1. Sixty modes are taken into account in WGB. The given axis is scaled with the wavelength k ¼ 2p=k. 3526


cglm b

2

Kgm Kgn dmn þ b2

"ð

1 1d=b

b 1 a

@s Egn ðsÞ@s Egm ðsÞs ds # ð1 a Egn ðsÞEgm ðsÞ 2 ds ; 1 þl s 1d=b b

(A9)

where cBlm ¼ clm . lix et al. Fe

0

CB jxx j g6 c1 ; l ðxÞ ¼ UB e

3. Discontinuity matrices

The discontinuity matrix F, defined by Eq. (21), is, for the rigid transverse modes, Fmn

KAm KBn f Jlþ1 ðfn Þ lJl ðfn Þ Jl ðclm Þ n ¼ ; b f2n c2lm

APPENDIX B: FREE-FIELD GREEN’S FUNCTION

Consider the free space Green’s function 0

(B1)

which we aim to approximate by X 0 0 0 eilh GðaÞ Gðr; r0 Þ ’ l ðx; r; x ; r ; h Þ;

eilh ðMB UB CB Þ1 uB ðr 0 Þ: c1 ¼ 4p

(A10)

s

eikjrr j ; 4pjr r0 j

where 0

with fn ¼ cln =b and, if l ¼ 0, F00 ¼ 1=b. The matrix G, defined by Eq. (22), is given by ð1 (A11) Gmn ¼ KCm KBn ECm ðsÞEBn ðsÞs ds:

Gðr; r0 Þ ¼

The first equation in Zorumski’s paper17 gives the overall pressure field outside the flanged waveguide and leads to the following relation between the pressure component pðk; 0; rÞ (as defined in the beginning of the paper) and its axial derivative at the waveguide opening (x ¼ 0): ð1 pðk; 0; rÞ ¼ k Iðk; r; r0 Þ@x pðk; 0; r 0 Þr0 dr 0 ; (C1) 0

where I ðk; r; r0 Þ ¼

(B2)

where GðaÞ l is the solution of the inhomogeneous equation @2 2 þ k þ $? ð H$? Þ GðlaÞ x;r;x0 ; r0 ;h0 ba 2 @x 0 eilh 1 dðx x0 Þdðr r0 Þ (B3) ¼ 2p r in an infinite circular PML-waveguide, as WGB. Upstream and downstream from the source plane, GðaÞ l is developed on the basis fuBn g, X 0 T þ gþ (B4) GðaÞ l ðx > x Þ ¼ ln ðxÞuBn ðrÞ ¼ uB gl ; 0