Scattering by inductive post in uniformly curved rectangular waveguide

Scattering by inductive post in uniformly curvedrectangular waveguide

I.V. Petrusenko and V.P. Chumachenko

Abstract: A rigorous solution to the problem of wave scattering by a single inductive post in acontinuously curved rectangular waveguide is presented. The position of the post and its radius arearbitrary. The waveguide curvature can be varied over wide limits. An efficient mathematicalmodel of the unit is based on the domain-product technique. The theory developed is applied to thenormalised reactances of the discontinuity and the induced surface current. The results obtainedlend support to the validity of the known approximate solution for a thin post as far as thisapproximation goes. It is found that data for a centrally placed post deviate slightly from those fora straight guide in a broad interval of curvature variation. The off-centre shift affects thecharacteristics appreciably. Over full waveguide bandwidth, the paper provides the data that areessential in the design of devices containing uniform bends, such as bandpass filters or transmissionresonators.

1 Introduction

The analysis of wave scattering by inductive obstacles in awaveguide has traditionally attracted considerable attentionin microwave theory [1–12]. A cylindrical post is a subject ofconstant interest in the design of manifold filters [1, 2, 4–12]and many other microwave devices [1–3]. Therefore, thecorrect analysis of inductive posts in various waveguidingstructures is of great practical importance.

The circular post in a straight rectangular waveguide hasbeen studied in detail [4–11]. The configuration has beenanalysed by a variety of methods. These include finite- andboundary-element methods [4, 5], the technique of equiva-lent sources [6, 7], the variational approach [8], the Rayleighhypothesis [9], the singular integral equation method [10],and the domain-product technique [11].

Scattering properties of the post in a curved guide are lesswell-known, although the problem is important forapplications as well. For example, transmission resonatorsand bandpass filters containing sections with inductive postsin both straight rectangular waveguides and continuouslycurved ones have been proposed. Such filters offer certainadvantages in the waveband away from the centralfrequency [12].

The cylindrical post in the curved waveguide is difficult toanalyse using traditional techniques in view of thecomplicated geometry of the problem. Owing to thesimultaneous presence of convex and concave boundaries,the use of full numerical techniques is not very efficient withrespect to computational effort and accuracy.

This paper presents a novel rigorous solution to theproblem of a circular post in a uniformly curved rectangular

waveguide. The domain-product technique (DPT) [13]is applied to calculate the normalised reactances of thediscontinuity and the induced surface current for a widerange of curvature variations, radii and arbitrary locationof the post in the interior of the waveguide bend. Thisapproach is highly efficient compared to full numericalmethods because DPT reduces the problem to anequation of the Fredholm type with a nuclear matrixoperator. This ensures a correct and accurate mathema-tical model, the absence of spurious solutions, thevalidity of the truncation procedure, the fast convergenceof the computed solution to the exact one and robustcalculations.

Here we deal only with a single inductive perfectlyconducting (PEC) cylinder, but the approach can easily beextended to the penetrable post with typical losses, or postarray placed parallel to the narrow or broad wall of theguide.

2 Problem specification and mathematicalmodelling

The configuration of interest and the co-ordinate systemsused are shown in Fig. 1. We consider a circular post ofradius r, placed across the guide parallel to the narrow PEC

Fig. 1 Geometry of the problem and pertinent co-ordinate systems

I.V. Petrusenko is with the Electronics Engineering Department, Gebze Instituteof Technology, Gebze, Kocaeli 41400, Turkey

V.P. Chumachenko is with the Zaporizhzhia National Technical University, 64Zhukovsky Str., Zaporizhzhia 69063, Ukraine

r IEE, 2003

IEE Proceedings online no. 20030773

doi:10.1049/ip-map:20030773

Paper first received 12th October 2002 and in revised form 16th June 2003.Online publishing date: 17 October 2003

498 IEE Proc.-Microw. Antennas Propag., Vol. 150, No. 6, December 2003

wall and centred at r¼Rp¼R1+d in the uniformly curvedwaveguide of the width 2a¼R2–R1. The geometry is a two-dimensional one and only the electric field has non-vanishing Ey-component. The guide is filled with ahomogeneous lossless medium and terminated in matchingloads. The wave incident upon the post is the dominantLM10 mode travelling in the positive y-direction. Theconvention of time dependence is exp(jot) and k¼ 2p/l isthe wavenumber.

Let us divide the interior of the bend into regularwaveguide regions I and III, and interaction region II(Fig. 1). In regions I and III, the field can be represented interms of LMm0–mode series expansions

Eð1Þy ¼ c1ðkrÞe�jn1ðyþaÞ þ

X1m¼1

CRmcmðkrÞejnmðyþaÞ ð1Þ

Eð3Þy ¼

X1m¼1

CTmcmðkrÞe�jnmðy�aÞ ð2Þ

with the reflection CmR and transmission Cm

T coefficients tobe found. The radial distributions of the modes are definedby the cross–eigenfunctions

cmðkrÞ ¼ Pnmðkr; kR2ÞjjPnm jj�1;m ¼ 1; 2; . . . ð3Þ

where

Pnðkr; kR2Þ ¼ JnðkrÞNnðkR2Þ � JnðkR2ÞNnðkrÞ ð4Þis the cross-product Bessel function with norm 77Pn77, Jnand Nn are respectively the Bessel and Neumann functions.The needed properties of cm(kr) are described in theAppendix, Section 6.

The symmetry of the geometry makes it possible tosubdivide the initial problem into two subproblemscorresponding to the symmetric and antisymmetric excita-tions, subsequently referred to as ‘s’ and ‘a’. Henceforth,both types of excitation are presented simultaneously.

According to DPT, let us imagine the interaction regionII as the common part of five domains, namely

� the semi-infinite sector {(r, y): r4R1,�aoyoa};

� the first guide {(r, y):R1oroR2, yoa};

� the sector of annulus {(r, y): *RoroR2, �aoyoa},

where the value of radius *RA(0, R1) is virtually of noimportance;

� the second guide {(r, y):R1oroR2, y4 �a};

� the exterior to the post {(r0, y0}: r04r, �poy0rp}.

The angle aop is assumed arbitrary, but greater than theviewing angle of the post.

Because of the linearity of the Helmholtz equation, itssolution Ey

(2) can be represented in the form

Eð2Þy ¼

X5

i¼1

ui ð5Þ

where all the functions satisfy the same equation. Separatingvariables, we obtain

u1 ¼X1n¼0

bð1Þn jnðyÞHHnðkrÞ ð6Þ

u3 ¼X1n¼0

bð3Þn jnðyÞWWnðkrÞ ð7Þ

jnðyÞ ¼ cos mnðyþ aÞ; HHn ¼H ð2Þ

mn ðkrÞH ð2Þ

mn ðkR1Þ; mn ¼

np2a

under qualification qu1/qy¼ qu3/qy¼ 0 at y¼7a and alsosbð1Þ2n�1 ¼ sbð3Þ2n�1 ¼ 0; abð1Þ2n ¼ abð3Þ2n ¼ 0. Here Hm

(2) symbo-

lises the Hankel function. In (7), #Wn may be any function,which is a regular solution of the Bessel equation in the

interval *RrrrR2. One appropriate choice of this functionis described in the Appendix, Section 6.

Assuming that u2 and u4 vanish at r¼R1, R2, we get

u2 ¼X1n¼1

bð2Þn cnðkrÞejnnðy�aÞ ð8Þ

u4 ¼X1n¼1

bð4Þn cnðkrÞe�jnnðyþaÞ ð9Þ

with sbð2Þn ¼ sbð4Þn and abð2Þn ¼ �abð4Þn . Lastly, u5 is taken inthe form

u5 ¼X1n¼0ð1Þ

xsn cos ny0

xan sin ny0

� �H ð2Þn ðkr0ÞH ð2Þn ðkrÞ

ð10Þ

where xns, n¼ 0, 1,y, and xn

a, n¼ 1, 2,y, are the expansioncoefficients sought.

On the PEC surfaces, the boundary conditions can bewritten as

u1 þ u3 þ u5 ¼ 0; r ¼ R1;R2 ð11Þ

X5

i¼1

ui ¼ 0; r0 ¼ r ð12Þ

In the interval R1oroR2 the continuity conditions for thetangential electric and magnetic fields are

X5

i¼1

ui ¼Eð1Þy ; y ¼ �a

Eð3Þy ; y ¼ þa

(ð13Þ

@

@yðu2 þ u4 þ u5Þ ¼

@Eð1Þy

@y; y ¼ �a

@Eð3Þy

@y; y ¼ þa

8>><>>: ð14Þ

On satisfying the conditions (11)–(14) and using theorthogonality of the functions jn, we find an infinite systemof linear algebraic equations (SLAE):

ðIþ AÞx ¼ t ð15Þ

with the nuclear matrix operator A (details of the procedureof such algebraisation are given in [11]). Here I is anidentity. The structure of A, the right-hand vector t, and theproof of correctness of the mathematical model obtainedare given in the Appendix, Section 6.

On truncating the systems (15) and subsequent inverting,

the xs, xa are found. Let AN ;M ¼deffaNmngMm;n¼1 be a truncation

of the nuclear operator (N is the number of modes retainedin the regions I, III) and xN,M be the solution of a‘truncated’ counterpart of SLAE (15). Then the relativeerror of approximation

dðN ;MÞ ¼ jjx� xN ;M jjjjxjj � jjðIþ AÞ�1jjjjA� AN ;M jj ð16Þ

tends to zero with M, N-N because the first factor in theright-hand part of inequality is a bounded constant, whilethe second one is decreasing [14]. The rate of decay of thefunction d(N, M) can be taken as the cost of the algorithm

IEE Proc.-Microw. Antennas Propag., Vol. 150, No. 6, December 2003 499

[15]. As an alternative estimate of truncation errors, thefunction

DðKÞ ¼ jjxK � xKþ1jjjjxK jj ;K ¼ N ;M ð17Þ

can be used too [16]. The behaviours of the functionsd(N, M) and D(K) are illustrated in next section.

Scattering parameters, with z¼�0,+0 as terminalplanes, can be expressed through expansion coefficients as

S11 ¼S22 ¼ ðsCR1 þ aCR

1 Þej2n1a

S12 ¼S21 ¼ ðsCT1 þ aCT

1 Þej2n1að18Þ

In a single-mode band, the normalised parameters ofan equivalent T-circuit of the post junction can be found asin [6]

XL ¼�j2S21

½ð1 � S11Þ2 � S221�

;XC ¼ j1 þ S11 � S21

1 � S11 þ S21ð19Þ

3 Numerical results

The algorithm proposed is a very efficient one in thenumerical implementation. Table 1 illustrates convergenceof the algorithm subject to geometrical parameters of theunit. The rate of stabilisation is extremely fast with respectto the truncation number M. It is also seen that the unitaryproperty of the S-matrix holds with high precision.

The error functions (16) and (17) are shown in Fig. 2aand 2b. In computing d, the data corresponding to M¼ 20,N¼ 60 have been fixed as the reference values. One can seethat it is enough to take a few equations to achieve anaccuracy sufficient for engineering needs. As a result,N¼ 14 and the 4� 4 SLAE can be proposed for theevaluation of physical characteristics. For this size oftruncated matrix the accuracy is do5� 10�4. Figure 2balso shows typical values of the condition number, whichare fairly close to unity. In all the cases, we cite the greatervalues of errors and condition numbers found in thenumerical process for the two subproblems considered.

According to known approximate approaches, a strip isused to simulate the thin post in a straight guide [10]. The

Table 1: Scattering characteristics subject to the truncationnumbers M, N and the parameters d, r and R1/R2 for k/

2a¼ 1.4286 (PCL¼ 7S1172+7S217

2, ORT¼S11%S21+S12

%S22)

d/a¼0.2, r/a¼0.1, N¼40

R1/R2 M 7S117 arg S11 PCL ORT

1 0.088746 1.705305 1.000003 0.000004

2 0.088546 1.706329 1.000003 0.000004

0.1 4 0.088536 1.706341 1.000003 0.000004

8 0.088536 1.706341 1.000003 0.000004

12 0.088536 1.706341 1.000003 0.000004

1 0.139468 1.730135 0.999999 �0.000001

2 0.138933 1.730920 0.999999 �0.000001

0.4 4 0.138932 1.730921 0.999999 �0.000001

8 0.138932 1.730921 0.999999 �0.000001

12 0.138932 1.730921 0.999999 �0.000001

1 0.151754 1.733333 1.000002 0.000002

2 0.151394 1.733933 1.000001 0.000001

0.9 4 0.151406 1.733935 1.000001 0.000001

8 0.151406 1.733935 1.000001 0.000001

12 0.151406 1.733935 1.000001 0.000001

d/a¼0.2, r/a¼0.1, M¼10

R1/R2 N 7S117 arg S11 PCL ORT

2 0.096980 1.531685 1.026847 0.029461

5 0.087857 1.695090 1.001607 0.001853

0.1 10 0.088511 1.705070 1.000170 0.000223

20 0.088533 1.706186 1.000025 0.000031

40 0.088536 1.706341 1.000003 0.000004

2 0.135502 1.728173 1.001399 �0.000119

5 0.138662 1.733175 0.999324 �0.000705

0.4 10 0.138918 1.731010 0.999976 �0.000030

20 0.138930 1.730933 0.999996 �0.000005

40 0.138932 1.730921 1.000000 �0.000001

2 0.148575 1.715790 1.004367 0.004387

5 0.151204 1.734106 0.999877 �0.000121

0.9 10 0.151390 1.733814 1.000034 0.000034

20 0.151396 1.733921 1.000005 0.000005

40 0.151396 1.733935 1.000001 0.000001

d/a¼ 1, r/a¼0.5, N¼ 40


1 0.999999 �1.994400 0.999998 0.000001

2 0.999770 �1.918815 0.999999 0.000001

0.1 4 0.999889 �1.911811 0.999999 0.000001

8 0.999889 �1.911795 0.999999 0.000001

12 0.999889 �1.911795 0.999999 0.000001

1 0.997720 �1.924362 0.999999 0.000000

2 0.999985 �1.851008 0.999999 0.000000

0.9 4 0.999999 �1.844055 0.999999 0.000000

8 0.999999 �1.844053 0.999999 0.000000

12 0.999999 �1.844053 0.999999 0.000000

d/a¼ 1, r/a¼0.5, M¼10

R1/R2 N 7S117 arg S11 PCL ORT

2 0.986429 �1.912672 0.973209 0.024541

5 0.999654 �1.911519 0.999518 0.000377

0.1 10 0.999844 �1.911772 0.999907 0.000079

20 0.999884 �1.911792 0.999988 0.000010

40 0.999889 �1.911795 0.999999 0.000001

2 0.992587 �1.847440 0.985337 0.005329

5 0.999732 �1.843906 0.999464 �0.000156

0.9 10 0.999963 �1.844051 0.999927 0.000008

20 0.999995 �1.844052 0.999990 0.000000

40 0.999999 �1.844053 0.999999 0.000000

Table 1: (continued)



admissibility of this substitution for a curved guide wasindicated in [12, 17]. Using Lewin’s method for the first-order approximation [10], the relations

XC �0

XL �n1

c1ðRpÞc1ðRp þ rÞX1m¼2

1

2jnmcmðRpÞcmðRp þ rÞ

ð20Þhave been obtained [17]. Comparison with the latter as wellas with the limiting case of a straight guide (R1-N,a¼ const) is given in Fig. 3. Here, the normalised reactances(19) are exhibited as a function of r/a for the centred post(d¼ a), various curvatures and a fixed operating frequency.One can take the view that the thin-strip approximation (20)is more suitable for a large curvature than a small one. Onecan see that a slight deviation from the straight guide datatakes place even for a sharp bend with R1oa (R1/R2o1/3).

Figure 4 presents the normalised reactances as a functionof r/a for the off-centred post (d¼ 0.6a) and various ratiosR1/R2. In contrast to the previous case, the reactance XC

depends strongly on the curvature and the curves begin todisperse for thin posts (rE0.05a). The effect is correlatedwith a noticeable shift of Ey-distribution from the post, as isillustrated in Fig. 5. In Fig. 4, the XL curve behaviourindicates that the range within which the approximation(20) is valid, is sufficiently smaller.

Fig. 2 The error functions d and D against number of modes N inthe curved waveguide, matrix truncation number M and conditionnumber of the matrix approximation.r¼ 0.25a, d¼ a, l/2a¼ 1.4, R1/R2¼ 0.5a Error functions against Nb Error functions against M (left-hand logarithamic axis) and condi-tion number (right-hand linear axis)

Fig. 3 Circuit parameters against radius size for centred postl/2a¼ 1.4

Fig. 4 Circuit parameters against radius size for off-centred postd¼ 0.6a, l/2a¼ 1.4

Fig. 5 Transverse distribution of the Ey-component of the incidentwave and relative position of off-centred posts of various sizesl/2a¼ 1.4

Fig. 6 Circuit parameters against position of the postr¼ 0.3a, l/2a¼ 1.2


Figures 6 and 7 present the dependence of reactances onposition of the post centre, operating wavelength andcurvature. The asymmetry of the characteristics about themedial line of the waveguide is clearly visible. It is seen thatthere are positions of the post and frequency points whereXL or XC is practically independent of the curvature.

Figures 8 and 9 deal with the current induced on the postsurface. The current can be derived from the electric field asfollows:

J ¼� 1

jom0

½uur0 � ðr� EyÞp0¼r

¼ uuy

j120pk

X5

i¼1

@ui@r0

� �r0¼r

ð21Þ

where #ur0 and #uy denote the unit vectors in the r0 and ydirections, respectively. Figures 8 and 9 show the currentdistributions against azimuth y0 for the centred and off-centred posts respectively. Here, the intervals (�1801, 0) and(0, 1801) correspond to the ‘illuminated’ portion of the postsurface and the ‘shadow’ one. Because of non-zerocurvature, the current has asymmetric distribution in bothregions even for the centred-post structure. In the case of asufficiently smooth bend, the results are in good agreementwith data from [7, 11].

4 Conclusions

The problem of a circular inductive post in a uniformlycurved rectangular waveguide has been solved using theDPT. All cases of thin and large (0oroa) arbitrarily placed(R1oRpoR2) posts have been considered over a broadrange of curvature variation (0.1rR1/R2o1). Taking

Fig. 7 Circuit parameters against l/2a for off-centred large postd¼ 0.6a, r¼ 0.3a

Fig. 8 Real and imaginary part of the surface current on thecentred postl/2a¼ 1.2a Realb Imaginary

Fig. 9 Real and imaginary part of the surface current on the off-centred postd¼ 0.6a, l/2a¼ 1.2a Realb Imaginary


advantage of the physical symmetry plane, the problem hasbeen partitioned into two independent subproblems differ-ing in the mode of excitation to get maximum efficiencyfrom the computational procedure.

The initial boundary value problem has been reduced toan infinite SLAE. It has been proved analytically that thematrix equation is of the Fredholm type with a nuclearoperator. A rapidly convergent numerical algorithm hasbeen developed to obtain the scattering matrix, theparameters of the equivalent T-network and the currentinduced on the post surface. The new data have beendetermined at low computational cost, and at the same timethe accuracies achieved are excellent with respect to anyengineering needs.

The results computed show good correlation with theapproximate solution, which has been derived by the Lewinmethod. All data obtained for a smooth bend (R1/R2Z0.9)agree well with data for the limiting case of a straight guide.

Over the full waveguide bandwidth, the method providesdata that are essential in the design of devices containinguniform bends with posts, such as transmission resonatorsor bandpass filters.

5 References

1 Marcuvitz, N.: ‘Waveguide handbook’ IEE Electromagnetic WavesSeries 21 (Peter Peregrinus, England, 1986)

2 Esteban, H., Cogollos, S., Boria, V.E., Blas, A.S. and Ferrando, M.:‘A new hybrid mode-matching/numerical method for the analysis ofarbitrarily shaped inductive obstacles and discontinuities in rectan-gular waveguides’, IEEE Trans. Microw. Theory Tech., 2002, 50, (4),pp. 1219–1224

3 Reiter, J.M., and Arndt, F.: ‘Rigorous analysis of arbitrarily shapedH- and E-plane discontinuities in rectangular waveguides by a full-wave boundary contour mode-matching method’, IEEE Trans.Microw. Theory Tech., 1995, 43, (4), pp. 796–801

4 Abdulnour, J., and Marchildor, L.: ‘Boundary elements and analyticexpansions applied to H-plane waveguide junction’, IEEE Trans.Microw. Theory Tech., 1994, 42, (6), pp. 1038–1045

5 Ise, K., and Koshiba, M.: ‘Numerical analysis of H-plane waveguidejunctions by combination of finite and boundary elements’, IEEETrans. Microw. Theory Tech., 1988, 36, (9), pp. 1343–1351

6 Leviatan, Y., Li, P.G., Adams A. T., and Perini, J.: ‘Single-postinductive obstacle in rectangular waveguide’, IEEE Trans. Microw.Theory Tech., 1983, 31, (10), pp. 806–811

7 Leviatan, Y., Shau, D.H., and Adams A. T.: ‘Numerical study of thecurrent distribution on a post in a rectangular waveguide’, IEEETrans. Microw. Theory Tech., 1984, 32, (10), pp. 1411–1415

8 Schwinger, J., and Saxon, D.S.: ‘Discontinuities in waveguides’(Gordon and Breach, New York, 1968)

9 Nielsen, E.D.: ‘Scattering by a cylindrical post of complex permittivityin a waveguide’, IEEE Trans. Microw. Theory Tech., 1967, 17, (3),pp. 148–153

10 Lewin, L.: ‘Theory of waveguides’ (Newnes Butterworths, London,1975)

11 Chumachenko, V.P., and Petrusenko, I.V.: ‘Domain product solutionfor scattering by cylindrical obstacle in rectangular waveguide’,Electromagnetics, 2002, 22, (6), pp. 473–486

12 Vol’man, V.I., and Knizhnik, A.I.: ‘Analysis of inductive postin continuously curved waveguide’, Radiotehnika, 1980, 35, (8),pp. 72–75 (in Russian)

13 Chumachenko, V.P.: ‘The integral equation method for solving a classof electrodynamics problems’, Izv. Vyssh. Ushebn. Zaced. Radiofiz.,1978, 21, (7), pp. 1004–1010 (in Russian)

14 Reed, M., and Simon, B.: ‘Methods of modern mathematical physics,v.1: Functional analysis’ (Academic Press, New York, 1978)

15 Nosich, A.I.: ‘The method of analytical regularisation in wave-scattering and eigenvalue problems: foundations and review ofsolutions’, IEEE Antennas Propag. Mag., 1999, 41, (3), pp. 34–49

16 Nosich, A.I., Okuno, Y., and Shiraishi, T.: ‘Scattering and absorptionof E- and H-polarised plane waves by a circularly curved resistivestrip’, Radio Sci., 1996, 31, (6), pp. 1733–1742

17 Petrusenko, I.V., Kodak, S.N., and Essert, V.E.: ‘Single-post inductiveobstacle in continuously curved waveguide’. Bulletin ‘‘Calculation andmeasurement of parameters of microwave components’’, Dniprope-trovs’k State University, 1988, pp. 62–66 (in Russian)

18 Morse, P.M., and Feshbach, F.: ‘Methods of theoretical physics’(McGraw-Hill, 1953)

19 Watson, G.N.: ‘A treatise on Bessel functions’ (Cambridge UniversityPress, 1944, 2nd edn.)

20 Petrusenko, I.V.: ‘High-order approximation of wave propagationconstants in the uniformly curved waveguide’, IEE Proc. Microw.Antennas Propag., 2001, 148, (5), pp. 280–284

6 Appendix

In a cylindrical frame (Fig. 1), the solution of the Helmholtzequation can easily be found via the variable separationmethod. Considering LMm0, m¼ 1, 2,y, modes ascirculating waves, an appropriate form for theirEy-component is

Ey ¼ unðkrÞe�jny ð22ÞHere un(kr) is an eigenfunction of the Sturm–Liouvilleproblem for Bessel’s equation of order n. On the finiteinterval 0oR1rrrR2, the set of these eigenfunctionsforms a complete orthonormal set of square integrablefunctions [18].

On the two curved walls Ey must vanish, and this leads tothe cross-product Bessel function

unðkrÞ ¼ c½JnðkrÞNnðkR2Þ � JnðkR2ÞNnðkrÞ�� cPnðkr; kR2Þ ð23Þ

and to the modal equation

PnmðkR1; kR2Þ ¼ 0 ð24ÞIn (23), the norming quantity c is found according to therelationship between solutions of the Bessel equation [19]Z

unðkrÞumðkrÞdrr

¼ rn2 � m2

@un@r

um � un@um@r

�ð25Þ

from which followsZ R2

R1

Pnðkr; kR2ÞPmðkr; kR2Þdrr

¼ dnmr2n

@Pnðkr; kR2Þ@r

@Pnðkr; kR2Þ@n

�r¼R1

ð26Þ

Here dnm is the Kronecker delta.In this paper, we use the special norm

jjPnjj ¼rsn

@Pnðkr; kR2Þ@r

@Pnðkr; kR2Þ@n

�1=2

r¼R1

s ¼ lnR2

R1

� ð27Þ

according to the orthogonality condition for the cross-eigenfunctions (3)Z R2

R1

cmðkrÞcnðkrÞdrr

¼ s2dmn ð28Þ

This provides the required quasistatic approximation formc1

cmðkrÞ ’ sinmps

lnR2

r

� �ð29Þ

The high-order approximation to the angular propagationconstant nm, m¼ 1,2,y, and k�n diagram for the modalequation (24) are given in [20].

In order to avoid division by zero during computations,let us construct the solution of the Bessel equation

WWnðkrÞ ¼ AJmnðkrÞ þ BNmnðkrÞ; ~RR � r � R2 ð30Þwhich does not vanish for any real value of the

wavenumber. For this purpose #Wn must satisfy

� boundary condition of the impedance type

WWnðk~RRÞ þ j~RR@WWnðk~RRÞ

@~RR¼ 0

� normalisation condition #Wn (kR2)¼ 1;


� additional requirement limn!1

WWnðkrÞ ¼ 0; ~RRoroR2 for

robust calculations.

The solution sought takes the form

WWnðkrÞ ¼WmnðkrÞWmnðkR2Þ

ð31Þ

WmðkrÞ ¼ Pmðk~RR; krÞ þ j~RR@Pmðk~RR; krÞ

@~RRð32Þ

Below the designations

p ¼2m

2m� 1

� �; q ¼

2n

2n� 1

� �; csnðyÞ

¼cos ny

sin ny

� �; f enðyÞ ¼ 2e�inna

cos nny

i sin nny

� �ð33Þ

are used to make formulae less cumbersome. Hereinafter,upper and lower symbols are connected with the symmetricand antisymmetric case respectively.

It can be shown in analogy to the result of paper [11] thatin (15) the matrix operator has the structure

A ¼ T1 �1

2TT2F1

� D1 þ T3 �

1

2TT2F3

� D3

� 1

2TT2ðF2 �GÞ ð34Þ

Let us introduce an auxiliary vector

sðnÞðr; yÞ ¼ ðHHqðkrÞjqðyÞ;cnðkrÞfenðyÞ;WWqðkrÞjqðyÞÞ

ð35Þ

then the elements of Ti, i¼ 1,2,3, and #T2 can be expressed inthe form

tðiÞmn ¼2

emp

Z p

0

½tðnÞi ðr; yÞ�r0¼rcsmðy0Þdy0;

ttð2Þmn ¼n1=2tð2Þmn ; em ¼2;m ¼ 0

1;m � 1

�ð36Þ

The elements of Fi, i¼ 1,2,3, and G are described by theformulae

f ð1Þmn ¼ 2m�1=2

sðm2q � n2

mÞ

� 2

pHHqðkR2ÞjjPnm jj

�1 þ r@cmðkrÞ

@r

�r¼R1

( )ð37Þ

f ð2Þmn ¼ 2

sm1=2

Z R2

R1

csnðy0ÞH ð2Þn ðkr0ÞH ð2Þn ðkrÞ

" #y¼�a

� cmðkrÞdrr

ð38Þ

f ð3Þmn ¼ 2m�1=2

sðm2q � n2

mÞ

� 2

pjjPnm jj

�1 þ r@cmðkrÞ

@r

�r¼R1

WWqðkR1Þ( )

ð39Þ

gmn ¼2m�1=2

jsnm

Z R2

R1

@

@ycsnðy0Þ

H ð2Þn ðkr0ÞH ð2Þn ðkrÞ

" #( )y¼�a

� cmðkrÞdrr ð40Þ

Matrices Di, i¼ 1,3, are defined by the elements

dð1Þmn ¼L�1p ½d�mn � WWpðkR1Þdþmn�

dð3Þmn ¼L�1p ½dþmn � HHpðkR2Þd�mn� ð41Þ

where

Lp ¼ 1 � HHpðkR2ÞWWpðkR1Þ ð42Þ

dþ;�mn ¼ � 2

epa

Z a

0

csnðy0ÞH ð2Þn ðkr0ÞH ð2Þn ðkrÞ

" #r¼R2;R1

jpðyÞdy

ð43Þ

Finally, the right-hand part of (15) takes the form

t ¼ f�tð2Þm1=2g.The proof of the correctness of the proposed model is

based on the Fredholm property of (15). It is sufficient toshow that A is a compact matrix operator in the Hilbertspace

h1 ¼def xn :X1n¼0

ðnþ 1Þjxnj2o1( )

ð44Þ

and the right-hand part tAh1 [14].Compactness of the operator (34) follows from asymp-

totic estimates of the integrals (36), (38), (40) and (43),which are Fourier coefficients of functions differentiableinfinitely many times. Namely, the estimation formulae

jtð1;3Þmn j ¼O e�mql1;3

m!mq

rRp

� m �

jttð2Þmn j ¼Oe�npl2

m!nmþ1=2 pr

sRp

� m �;m; n � 1; r � Rp

1 ¼ lnRpR1

� ;as; ln

R2

Rp

� � ð45Þ

give us tAh1 and lead to the inequalities

X1m;n

mjtð1;3Þmn j2 ¼ O½ð1 � z1;3Þ�b�o1

X1m;n

mjttð2Þmn j2 ¼ O½ð1 � z2Þ�t�o1

z ¼ rd;r

aRp;

r2a� d

� ; b4t40 ð46Þ

under conditions zio1, i¼ 1,2,3. Hence, T1,3, #T2 are theHilbert–Schmidt (H–S) operators h1-h1 [14] provided thatthe post does not touch the boundaries of the interactionregion II. Under the same condition, we get

X1m;n

jf ð1;3Þmn j2o1;

X1m;n

1

njxðm;nÞi j2o1; i ¼ 1; 2; 3

nðm;nÞ ¼ ðf ð2Þmn ; gmn; d

ð1;3Þmn Þ ð47Þ

Therefore, D1, D3, G, Fi, i¼ 1,2,3, present the H–Soperators as well. Thus, A: h1-h1 is the nuclear operatoras a sum of products of the H–S operators [14].


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Scattering by inductive post in uniformly curved rectangular waveguide