Numerical Implementation of Cross-Section Method for Irregular Waveguides

Wave scattering in irregular waveguides is investigated. The cross-section method is considered as a method for calculation of the field in a waveguide consisting of two regular waveguides with different cross-sections joined by an irregular domain. In the paper, a mathematically justified derivatio...

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Date:	2000
Main Authors:	Ramm, A.G., Voitovich, N.N., Zamorska, O.F.
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Language:	English
Published:	Радіоастрономічний інститут НАН України 2000
Series:	Радиофизика и радиоастрономия
Online Access:	https://nasplib.isofts.kiev.ua/handle/123456789/122197
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Cite this:	Numerical Implementation of Cross-Section Method for Irregular Waveguides / A.G. Ramm, N.N. Voitovich, O.F. Zamorska // Радиофизика и радиоастрономия. — 2000. — Т. 5, № 3. — С. 274-283. — Бібліогр.: 22 назв. — англ.

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spelling	nasplib_isofts_kiev_ua-123456789-1221972025-02-23T17:23:01Z Numerical Implementation of Cross-Section Method for Irregular Waveguides Численная реализация метода поперечных сечений для нерегулярных волноводов Числова реалізація методу поперечних перерізів для нерегулярних хвилеводів Ramm, A.G. Voitovich, N.N. Zamorska, O.F. Wave scattering in irregular waveguides is investigated. The cross-section method is considered as a method for calculation of the field in a waveguide consisting of two regular waveguides with different cross-sections joined by an irregular domain. In the paper, a mathematically justified derivation of the basic equations of the method is given. An iterative procedure for their numerical solution is proposed. The algorithm is applied to the problems with the smooth and nonsmooth irregularities. In particular, numerical results for a test problem having analytical solution, are presented. Исследуется рассеяние волн в нерегулярных волноводах. Рассматривается метод поперечных сечений для вычисления поля в волноводной системе, состоящей из двух регулярных волноводов, соединенных нерегулярной областью. В статье дается математически строгий вывод основных уравнений метода и предлагается итерационная процедура их решения. Алгоритм применяется к задачам с гладкими и негладкими неоднородностями. На примере модельной задачи, имеющей аналитическое решение, устанавливаются границы применимости метода. Досліджується розсіювання хвиль в нерегулярних хвилеводах. Розглядається метод поперечних перерізів для обчислення поля в хвилеводній системі, що складається із двох регулярних хвилеводів, з’єднаних нерегулярною областю. В статті дається математично строге виведення основних рівнянь методу і пропонується ітераційна процедура для їх розв’язування. Алгоритм застосовується до задач з гладкими та негладкими нерегулярностями. На прикладі модельної задачі, що має аналітичний розв’язок, встановлюються границі застосовності методу. The authors thank Prof. B. Z. Katsenelenbaum for useful discussions. 2000 Article Numerical Implementation of Cross-Section Method for Irregular Waveguides / A.G. Ramm, N.N. Voitovich, O.F. Zamorska // Радиофизика и радиоастрономия. — 2000. — Т. 5, № 3. — С. 274-283. — Бібліогр.: 22 назв. — англ. 1027-9636 https://nasplib.isofts.kiev.ua/handle/123456789/122197 en Радиофизика и радиоастрономия application/pdf Радіоастрономічний інститут НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	Wave scattering in irregular waveguides is investigated. The cross-section method is considered as a method for calculation of the field in a waveguide consisting of two regular waveguides with different cross-sections joined by an irregular domain. In the paper, a mathematically justified derivation of the basic equations of the method is given. An iterative procedure for their numerical solution is proposed. The algorithm is applied to the problems with the smooth and nonsmooth irregularities. In particular, numerical results for a test problem having analytical solution, are presented.
format	Article
author	Ramm, A.G. Voitovich, N.N. Zamorska, O.F.
spellingShingle	Ramm, A.G. Voitovich, N.N. Zamorska, O.F. Numerical Implementation of Cross-Section Method for Irregular Waveguides Радиофизика и радиоастрономия
author_facet	Ramm, A.G. Voitovich, N.N. Zamorska, O.F.
author_sort	Ramm, A.G.
title	Numerical Implementation of Cross-Section Method for Irregular Waveguides
title_short	Numerical Implementation of Cross-Section Method for Irregular Waveguides
title_full	Numerical Implementation of Cross-Section Method for Irregular Waveguides
title_fullStr	Numerical Implementation of Cross-Section Method for Irregular Waveguides
title_full_unstemmed	Numerical Implementation of Cross-Section Method for Irregular Waveguides
title_sort	numerical implementation of cross-section method for irregular waveguides
publisher	Радіоастрономічний інститут НАН України
publishDate	2000
url	https://nasplib.isofts.kiev.ua/handle/123456789/122197
citation_txt	Numerical Implementation of Cross-Section Method for Irregular Waveguides / A.G. Ramm, N.N. Voitovich, O.F. Zamorska // Радиофизика и радиоастрономия. — 2000. — Т. 5, № 3. — С. 274-283. — Бібліогр.: 22 назв. — англ.
series	Радиофизика и радиоастрономия
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fulltext	Ðàäèîôèçèêà è ðàäèîàñòðîíîìèÿ, 2000, ò. 5, ¹3, ñòð. 274-283 © Alexander G. Ramm, Nikolai N. Voitovich , and Olga F. Zamorska, 2000 Numerical Implementation of Cross-Section Method for Irregular Waveguides Alexander G. Ramm, Nikolai N. Voitovich1,2, and Olga F. Zamorska1, Kansas State University, Manhattan, KS 66506-2602,USA e-mail: ramm@math.ksu.edu 1Institute of Applied Problems in Mechanics and Mathematics, 79601, Ukraine, Lviv e-mail: voi@iapmm.lviv.ua 2 Bydgoszcz University im. Kazimiera Wielkiego, 85-064, Bydgoszcz, Poland e-mail: voi@wsp.bydgoszcz.pl Received August 8, 2000 Wave scattering in irregular waveguides is investigated. The cross-section method is considered as a method for calculation of the field in a waveguide consisting of two regular waveguides with different cross-sections joined by an irregular domain. In the paper, a mathematically justified derivation of the basic equations of the method is given. An iterative procedure for their numerical solution is proposed. The algorithm is applied to the problems with the smooth and nonsmooth irregularities. In particular, numerical results for a test problem having analytical solution, are presented. Index Terms: wave scattering, irregular waveguide, cross section method, iterative method 1. Introduction The idea of the cross-section method (CSM) was proposed several decades ago [1, 2]. The method was developed and investigated by different authors. The most essential contribution to its foundation is given in [3-5]. CSM is suitable for investigation of the waveguides with different kinds of small and smoothly varying irregularities such as smooth and slow change of the cross-section shape, jog and shift of the axis line, al- teration of the optical density of the filling etc. It is a useful technique for studying the wave scattering in closed and open irregular metallic, dielectric and im- pedance waveguides [6-10], field converters, cavity antennae [11-16], and other practically important prob- lems. An attempt to justify the above method for the 3D vector problem was undertaken in [17]. However, the range of the practical applicability of the method is still not examined theoretically and numerically. The purpose of this paper is to demonstrate the mathematical equivalence of the main equations of the method in the form presented in [3, 5] and in [1, 2], to suggest an iterative procedure for solving the main equations of CSM, and to investigate numerically the applicability of the method for the case of large enough irregularities. Application of the method is presented in the framework of the new general scheme for the investigation of the wave scattering in irregular waveguides proposed in [18]. Scattering problems in the domains with infinite boundaries were studied in [19]. A detailed analysis of the scattering by obsta- cles in regular waveguides is given in [20]. In [21] the waveguide theory is developed in application to opti- cal waveguides. Here we present the method for acoustical waveguides with soft walls; the pressure u on such walls is equal to zero. The waveguides with the con- tinuously varying cross-section shape are considered. The two-dimensional case is equivalent to the electro- magnetic problem for the E-polarization (u = E y ). Derivations of the main equations are made in the way similar to [1, 2]. The scattering problem is put in two forms: as a boundary value problem with inhomoge- neous equation and homogeneous conditions at infin- Ðàäèîôèçèêà è ðàäèîàñòðîíîìèÿ, 2000, ò. 5, ¹3 Numerical Implementation of the Cross-section Method for Irregular Waveguides 275 ity, as well as such a problem for homogeneous equa- tion with inhomogeneous conditions at infinity. The idea of CSM as a method for solving the prob- lems of wave propagation in irregular waveguides is not new, but in this paper a self-contained and rigorous der- ivation of its basic equations is given. The novel idea and novel result in the paper is the numerical implemen- tation of the method based on an iterative procedure. Numerical results are presented for two problems: for a test one having an exact solution and for a prob- lem with the geometrical parameters varying in a wide range. The results obtained for the test problem show the character of changing the errors of the computed solution versus geometry of irregularity as well as versus number of the normal modes taken into account. It is known that CSM is suitable for investigation of the waveguides with slowly varying irregularities (in the case considered in our test-problem the slow- ly varying waveguide means that the angle between the waveguide boundary and axis is not large). This limitation is necessary in order that the rate of con- vergence of the series representing the solution be satisfactory from the practical point of view. The nu- merical results obtained in the paper show that the method we use can be successfully applied for a wide range of the slopes of the waveguide irregular part (up to angles π/3 in our case). The results demon- strate also the efficiency of the proposed implemen- tation of the method. The time dependence of the form ( )tjω−exp is assumed. 2. Problem Statement Let us consider a waveguide which is a union of two regular waveguides, W 1 and W 2 , with the bound- aries S 1 and S 2 , respectively, joined by an irregular domain W 0 with the boundary S 0 (see Fig. 1). We as- sume that the cross-section D(z) of W 0 varies smooth- ly as a function of z, 0 ,z d≤ ≤ where z is directed along the waveguide. By x we denote the transversal to the z-axis coordinate in the cross-section D(z). In the 3D case the x-coordinate is two-dimensional, x = {x 1 ,x 2 }. We also assume that the boundary of the waveguide is such that there are no trapped modes in the waveguide, that is, there are no non-trivial qua- dratically integrable solutions to the homogeneous boundary value problem describing the waves in the waveguide. According to Theorem 2.1 in [20], p. 92, this is the case if the boundary is described by a mono- tone function of z. In the 3D case the geometrical condition on the boundary in [20], p. 92, is as follows: the exterior normal to the boundary forms an obtuse angle with the positive direction of the z-axis. The Helmholtz equation, ,)( 2 2 2 fuku z x =+ ∂ ∂+∆ (1) with a real wavenumber k holds in 1 0 2,W W W W= ∪ ∪ ∆ x is the Laplacian with respect to the x-variable, ( )zxff ,= is a compactly supported function, that is, the function vanishing outside a bounded region. We assume that the support of f is localized between some sections, 1z z= and 2zz = in 1W , 021 << zz : ( ) 0, =zxf if 1 2[ , ]z z z∉ . Here z 1 and z 2 are arbitrary negative numbers, so that the support of the source function f is located in W 1 . The support of f is the complement of the largest open set on which f van- ishes almost everywhere. The boundary condition 0=u at 201 SSSS ∪∪= (2) holds, and the radiation conditions at infinity are imposed: ∑ β− −−∞→ ≅ n n zj nz xvepu n ),()1()1( )1( (3) Fig. 1. Geometry of the problems Alexander G. Ramm, Nikolai N. Voitovich , and Olga F. Zamorska Ðàäèîôèçèêà è ðàäèîàñòðîíîìèÿ, 2000, ò. 5, ¹3 276 ∑ −β ∞→ ≅ n n dzj nz xvepu n ),()2()()2( )2( (4) where the coefficients )1( np− and )2( np are unknown. Here (and below) the summation is from n = 1 to ∞, 2/12)(2)( )( i n i n kk −=β , i = 1, 2; for driving modes ( )i nβ are the real positive constants and for damped modes they are imaginary ones )0( )( <β i nj ; 2)(i nk , )(i nv are the eigenvalues and eigenfunctions of the boundary value problem for the transversal Helmholtz equation 0)(2)()( =+∆ i n i n i nx vkv (5) with the boundary condition 0)( =i nv at the contours iD∂ of the cross-sections D i . The functions )(i nv are orthonormal in 2 ( ):iL D ,)()( )()( nm i m D i n dxxvxv i δ=∫ where the overbar stands for complex-conjugate (we assume that in general case mv are complex). Let us call the problem (1)-(4) Problem A. In prac- tice, instead of the force term ( )xf in (1), the excita- tion is often given in the form of the incident normal modes coming from .−∞ In this case f = 0, but the total field at −∞ is the sum of the incident and reflect- ed fields, while at +∞ the total field is the transmitted field. Problem A can be easily reduced to this form. One can present the solution of (1) in W 1 as 0 ,su U U= + where 0U is any partial solution of the problem (1)-(3) in W 1 and sU satisfies the homoge- neous Helmholtz equation in W 1 with the conditions (2), (3). The function 0U may be found as the solu- tion of the inhomogeneous problem in the regular waveguide W 1 extended to ∞ with the condition (3) at −∞ and the condition of type of (4) at ∞. This prob- lem can be easily solved through the separation of variables. Here z 2 has the same meaning as on the line above formula (2). Since f = 0 at ,02 << zz the func- tion 0U satisfies the homogeneous Helmholtz equa- tion in this region and has the form ∑ β= n n zj n xvepU n )()1()1(0 )1( (6) with the known coefficients )1( np . This function de- scribes the incident field, it represents the waves prop- agating in the positive direction of the z-axis. The general solution of homogeneous Eq. (1) in W 1 satisfying (2), (3) in W 1 is as follows: .)()1()1( )1(∑ β− −= n n zj n s xvepU n (7) So the solution of (1)-(3) in W 1 is .)(][ )1()1()1( 0 )1()1( 2 ∑ β− − β ≤< += n n zj n zj nzz xvepepu nn (8) Since k = const, the general solution of the prob- lem in W 2 follows from (4): .)()2()()2( )2(∑ −β ≥ = n n dzj ndz xvepu n (9) Putting z = 0 in (8) and z = d in (9) yields the conditions for u(x,z) at the vertical sides of W 0 (that is, on the sections z = 0 and z = d): ,)(][ )1()1()1( 0 ∑ −= += n nnnz xvppu (10) .)()2()2(∑== n nndz xvpu (11) In a similar way, differentiating (8), (9) with respect to z and putting z = 0 and z = d, respectively, yields two more conditions for zu ∂∂ at these boundaries: ∑ − = −β= ∂ ∂ n nnnn z xvppj z u ),(][ )1()1()1()1( 0 (12) Ðàäèîôèçèêà è ðàäèîàñòðîíîìèÿ, 2000, ò. 5, ¹3 Numerical Implementation of the Cross-section Method for Irregular Waveguides 277 ∑β= ∂ ∂ = n nnn dz xvpj z u ).()2()2()2( (13) Thus, the problem A is reduced to the non-stan- dard interior boundary problem (1), (2), (10)-(13) for the irregular domain W 0 . Let us call this problem Problem B. Here )1( np are the given magnitudes of the excitation modes calcu- lated in (6) and )1( np− , )2( np are the reflection and trans- mission factors, which are to be found. Often a statement of the problem, alternative to (1)-(4) is used based on the concept of the scatter- ing matrix. Namely, the source in the Eq. (1) can be taken in the form of one normal mode of the left waveguide W 1 coming from −∞. Then the prob- lem lies in finding the set of functions u m , m = 1, 2, ... satisfying the homogeneous Eq. (1) in W with the con- dition (2) and the following conditions at infinity: (1) (1)(1) (1)( ) ( )m nj z j z m m mn nz n u e v x r e v xβ − β →−∞ ≅ +∑ (14) ∑ −β →∞ ≅ n n dzj mnzm xvetu n )()2()()2( (15) Here { }mnr , { }mnt are the unknown reflection and transmission matrices, respectively. Let us call this problem Problem C. If this prob- lem is solved for all m = 1, 2, ..., then the solution of Problem A is given as ,)1(∑= m mm upu (16) where )1( mp are the same as in (6). If f(x,z) = 0, then the right-hand sides of (14), (15) give the solution of Problem C in W i , and the scatter- ing problem in W is reduced to the boundary value problem (1), (2), (10)-(13) in W 0 with u = u m , ,)1( mnnp δ= ,)1( mnn rp =− .)2( mnn tp = Thus, Problem C is reduced to Problem B. Hereafter we study Problem B. From the above arguments it follows that in general the scattering problems in irregular waveguides can be reduced to Problem B. 3. Mathematical Description of the Cross- Section Method The solution of Problem B can be found in the form ,),()(),( ∑= n nn xzvzczxu (17) where ),,( nn vuc = ∫= )( d),(),(),( zD nn xzxvzxuvu is the inner product in ( )( ),2 zDL ( )zxvn , are the eigen- functions of the equation of type of (5) in the cross- section ( )zD with the eigenvalues )(2 zkn and bound- ary condition 0=nv at )(zD∂ . The functions ( )zxvn , are orthonormal in ( )( )zDL2 : .),( nmnm vv δ= (18) The above formulation is valid in 2D and 3D cas- es. We assume that the eigenfunctions ( )zxvn , can be easily calculated for each cross-section ( )zD . Other- wise the practical application of the cross-section method is difficult. To obtain a set of ordinary differential equations for nc let us multiply (1) by nv and integrate over D(z) to get ,0),()(2 =′′+β nnn vucz (19) where ),()( 222 zkkz nn −=β ./ 22 zuu ∂∂=′′ Differentiating ( )zcn with respect to z yields: ).,(),(2),( nn vuvuvuc ′′+′′+′′=′′ (20) From (19), (20) one can obtain: .0),(),(22 =′′−′′−′′+β nnnnn vuvucc (21) Alexander G. Ramm, Nikolai N. Voitovich , and Olga F. Zamorska Ðàäèîôèçèêà è ðàäèîàñòðîíîìèÿ, 2000, ò. 5, ¹3 278 According to (17), the last term in (21) takes the form: ,),( ∑=′′ m mnmn cbvu (22) where ).,( nmnm vvb ′′= (23) If the boundary )(zD∂ is varied smoothly with respect to z, then the series (17) can be once differen- tiated termwise: .][∑ ′+′=′ m mmmm vccvu (24) Using (24) one writes the term ),( nvu ′′ as ,][),( ∑ ′+=′′ m nmmmnmn accdvu (25) where ),,( nmnm vvd ′′= ).,( nmnm vva ′= (26) From (21)-(25) one gets .0)2(22 ∑ ∑ =+−′−β+′′ m m mnmnmmnmnnn cdbcacc (27) Note that ,mnnm aa −= (28) as follows from differentiating the identity (18) with respect to z. Similarly, differentiating (18) twice, one gets: )(2 mnnmnm bbd +−= (29) and (27) takes the form .022 ∑ ∑ =+′−β+′′ m m mmnmnmnnn cbcacc (30) Eq. (30) must be satisfied for n = 1, 2, ... . It is useful to rewrite (30) in the matrix form: ,02 2 =+′−+′′ CBCACKC (31) where { },ncC = ( ),diag nK β= { },nmaA= }{ mnbB = is the matrix adjoint to B. Let us now eliminate the matrix B* from (31). For this purpose we differentiate the second equation in (26) and get nmnmnm dba +=′ or .DAB −′= (32) To eliminate the matrix D note that .∑=′ m mnmn vav (33) Then ∑ ∑=′=′′= p p mpnppmnpnmnm aavvavvd ,),(),( so that .AAD = (34) From (28) one gets A = �A. Therefore 2* AAB +′−= (35) and Eq. (31) can be written as .0)(2 22 =+′−+′−′′ CAAKCAC (36) Let us introduce a new pair of unknown vectors, C and G, in place of C; ( )zcn and ( )zgn are the components of the vectors C and G, respectively. Namely, let Ðàäèîôèçèêà è ðàäèîàñòðîíîìèÿ, 2000, ò. 5, ¹3 Numerical Implementation of the Cross-section Method for Irregular Waveguides 279 .ACCG −′= (37) Then =++′+′=′+′+′=′′ )( ACGACAGCACAGC CAAAGG )( 2+′++′= and (36) yields: =+−+′+−′ CBAKAAGG )( *22 .02 =+−′= CKAGG Thus we have the following set of equations for C, G: ,GACC +=′ .2CKAGG −=′ (38) The boundary conditions for Eqs. (38) can be ob- tained from (10)-(13) with account that ,)0,( )1( nn vxv = .),( )2( nn vdxv = From (10), (12) one gets ,)0( )1()1( nnn ppc −+= (39) ],[)0()0()0( )1()1()1( nnnn m mnmn ppjgcac −−β==−′ ∑ (40) where ( )0ng is the component of the vector ( )0G , G is defined in (37). One can eliminate the unknown np− from (39), (40) to obtain the following condition: ,2)0()0()0( )1(1 + − =− PGjKC (41) where }{ )1()1( npP =+ . Similarly, from (11), (13) we have .0)()()( 1 =+ − dGdjKdC (42) Eqs. (38) together with (41), (42) state the interior boundary value problem for the functions C, G in W 0 . We assume that 0)( ≠β i n for any n. The set of Eqs. (38) is stiff [22], because the func- tions ,nc ng contain both exponentially increasing and exponentially decreasing components (if ( ) 0Im ≠β j ). The computational methods developed for solving these equations are rather expensive. To eliminate this difficulty, let us introduce the new unknown functions { ( )}nP p z+ = and { ( )}nP p z− −= describing the magnitudes of the forward and back- ward normal modes in the irregular domain: ,−+ += PPC ).( −+ −= PPjKG (43) Then one has the new set of equations from (38): ,)( 21 −++ ++=′ PZPjKZP (44) ,)( 21 +−− +−=′ PZPjKZP (45) where ,2/)( 11 1 AKKKKAZ −− +′−= (46) .2/)( 11 2 AKKKKAZ −− −′+= (47) The boundary conditions for +P and −P can be easily obtained from (41), (42): ,)0( )1( ++ = PP (48) .0)( =− dP (49) Eqs. (44), (45) are equivalent to Eqs. (2.46) from [5]. Theoretical problems concerning the numerical solution of the problem (44)-(49) need further inves- tigation. Note that the functions +P do not contain exponentially increasing component, whereas −P do not contain exponentially decreasing component. This fact allows one to apply an iterative method for solv- ing this problem: at each iteration a Cauchy problem is solved for one subset of Eqs. (44), (45) in the for- ward or backward directions, respectively. Namely, the equation Alexander G. Ramm, Nikolai N. Voitovich , and Olga F. Zamorska Ðàäèîôèçèêà è ðàäèîàñòðîíîìèÿ, 2000, ò. 5, ¹3 280 ( )[2 1] [2 1] [2 ] 1 2( )q q qP Z jK P Z P+ + + + − ′ = + + (50) (the value in the square brackets denotes the serial number of iteration) with the initial condition of the type of (48) is solved with respect to ]12[ + + qP for dz ≤≤0 in the forward direction at each odd (2q + 1)-th iteration (q = 0, 1, 2, ...) with ]2[ qP− taken from the previous iteration. At the first iteration one takes 0]0[ ≡−P . Similarly, the equation ( )[2 2] [2 2] [2 1] 1 2( )q q qP Z jK P Z P+ + + − − + ′ = − + (51) with the initial condition of the type of (49) is solved with respect to ]22[ 1 +qP for dz ≤≤0 in the backward direction at each even (2q + 2)-th iteration with ]12[ + + qP taken from the previous iteration. Such a technique can be interpreted as taking into account successive transformations of the normal waveguide modes at the irregularities. The definition of the functions ( ),zpn ( )zp z− (43) is unique everywhere except the �critical sections� nzz = where 0)( =β zn . At these points the compo- nents of 1−K in Eqs. (44), (45) are not defined. In this paper we do not investigate the properties of the solution in the neighborhoods of �critical sections�. Let us note that the functions ( ),zpn ( )zp z− are in- troduced by (43) only for ( )δ+δ−∉ nn zzz , with some small δ. In the intervals ( )δ+δ− nn zz , Eqs. (38) should be used. The matching conditions at δ±= nzz for these equations follow from Eqs. (43) used at these points. Then the n-th Eq. (50) or (51) is solved at each iteration only for ,nn hzz >− where zh is a step size of the variable z in the numerical method for the Cauchy problem, and the equation is substituted by the n-th pair of Eqs. (38) at the last discretization point pre- ceding .nz At the first discretization point after nz the above pair of Eqs. (38) is substituted by the n-th Eq. of (50) or (51). At the points of substitution the functions ,nc ng and ,np np− are matched by for- mula (43). This technique for dealing with critical cross-sections was used in [3]. 4. Numerical Results The applicability of the CSM is defined by the rate of convergence of the series (17). There are no theo- retical estimates of this rate. Numerical results sug- gest that this rate decreases as the slope of the bound- ary of the waveguide irregular part increases. The numerical experiments were carried out to find out the above dependence and the practical limitation of the method. The numerical results presented refer to the 2D problems with the same regular waveguides ,1W 2W but different shapes of the irregular domain 0W (Fig. 1 (a), (b)). To demonstrate the dependence of the errors on the number of the terms kept in series (17) we show the numerical results for the test problem concerning the waveguide shown in Fig. 1 (a). In this case such errors are expected to be greater than in the second problem because of the nonsmoothness of the waveguide upper boundary. Next, the results of the numerical solution of both problems are presented for the case when the incident field is the first normal mode of the left waveguide. The first problem is a problem for the waveguide with iW of the height ,ih i=1,2, and the height of 0W given by the formula: dhhzhzh /)()( 121 −+= (see Fig. 1 (a)). As a test problem for the method, we choose the above one with the initial data allowing an exact analytical solution. These data are taken in the following way. First, an exact solution of the homoge- neous Eq. (1) with conditions (2) in 0W is analytically constructed. Then the magnitudes )1( np , )2( np− of inci- dent waves in ,1W ,2W respectively, are calculated, which excite jointly the field in 0W described by this exact solution. Next, these magnitudes are used as initial data in the Problem B, which is solved numer- ically. Finally, the obtained solution is compared with the exact one. Let us choose the solution of (1), (2) in 0W in the form of standing field: ),/sin()(),( / απϕ= απ krJzxu (52) where απ /J is the Bessel function of the first kind, Ðàäèîôèçèêà è ðàäèîàñòðîíîìèÿ, 2000, ò. 5, ¹3 Numerical Implementation of the Cross-section Method for Irregular Waveguides 281 ),/)arctan(( 12 dhh −=α )),/(arctan( 0zzx +=ϕ .))(( 2/12 0 2 zzxr ++= The function (52) satisfies homogeneous Eq. (1) in 0W as any function of the form )sin()( νϕν krJ does, and conditions (2) at ,0=ϕ α=ϕ in 0W . Denote ,)( )( izz i uu == i = 1, 2, ( ) ,01 =z ( ) .2 dz = Expand these functions and their derivatives with respect to z as the Fourier series with respect to the basis functions :)/sin()/2()( 2/1)( ii i n hxnhxv π= ,)()( )()()( ∑= n i n i n i xvcxu (53) ,)( ),( )()( )( ∑= ∂ ∂ = n i n i n zz xvg z zxu i (54) and calculate ( )0np and ( )dp n− by the formulas: ,2/)/()0( )1()1( nnnn jgcp β−= (55) ,2/)/()( )2()2( nnnn jgcdp β+=− (56) where 2/122)( ))/(( i i n hnk π−=β . Eqs. (55), (56) are the boundary conditions for Eqs. (44), (45). To reduce them to the form (48), (49) one should consider two problems with the incident wave coming from −∞ with the magnitudes of incident modes (55) and from ∞ with the magnitudes of incident modes (56), respec- tively, and add the solutions of these problems. But the numerical implementation of the above iterative procedure shows that it converges not only for the problem with the boundary conditions (48), (49), but also for the more complete conditions (55), (56). This fact allows one to apply the above procedure for solv- ing the problem (44), (45), (55), (56). To investigate the dependence of the calculating errors on the wall inclination in W 0 and the number of normal modes taken into account, the problem has been numerically solved at the different values of these parameters. In Fig. 2 the values )()()()( /)( iii N i N uuu −=ε (57) are given as functions of d and N for the waveguide with ,5.11 π=kd ,5.42 π=kd where ( )i Nu are the ap- proximate values of ( )iu calculated using the series (17) in which N first terms are kept. There are one driving mode (with 0Im )1( =βn ) in the left section and four such modes (with 0Im )2( =βn ) in the right one. One can see that a high accuracy (the error is less than 1 per cent) is achieved for 2 1tan ( ) / 0.5h h dα = − < (α < 25°) with ,6=N that is by taking into account only two decreasing modes (with 0Im )2( >βn ) in .2W With N = 25 this accuracy is achieved for the values of α up to tan 2α = (α < 60°). In all the variants of the input data the iterative procedure yielded the solution with the accuracy 0.01 per cent in 20÷30 iterations. In the second problem the height of 0W is taken in the form of a cubic spline: 2 1 2 1( ) ( / ) (3 2 / )( )h z h z d z d h h= + − − (see Fig. 1 (b)). Both problems are solved numerically by the above method with the same geometry of the regular waveguides: 1 1.5 ,kh = π 2 4.5 .kh = π In this case only one driving mode exists in the left waveguide, and four ones exist in the right waveguide. The length of irregular domain W 0 varies in the range .305÷=kd The number of retained normal modes in series (17) is taken in the inverse dependence of kd in the range Fig. 2. Relative computation accuracy of the field at z = 0 (solid lines) and z = d (dashed lines) keeping N terms in (17) for the waveguide shown in Fig. 1 (a) Alexander G. Ramm, Nikolai N. Voitovich , and Olga F. Zamorska Ðàäèîôèçèêà è ðàäèîàñòðîíîìèÿ, 2000, ò. 5, ¹3 282 .625÷=N The excitation is assumed to be of the form of the driving mode of W 1 coming from −∞. The magnitudes of the reflection and transmis- sion factors of driving modes in W 1 and W 2 for the both problems are shown in Fig. 3. As it was expect- ed, the reflection factor (1) 1p− is negligibly small in the second problem where the shape of the regular domain is smooth. But this factor is also not large in the first problem, even in the case of a large angle of the wall break. The difference between transmission factors (2) ,np n = 1, ..., 4, in the problems is visible and it varies not too strongly with the length of the irregular domain. 5. Conclusion The problem on wave scattering in irregular waveguide with different asymptotics of the bound- ary at −∞ and ∞ and the irregular domain with the continuously varied cross-section has been investi- gated by the cross-section method. Derivation of the main equations of the method has been applied, which does not use the differentiation of the non-uniformly converging series. The problem is reduced to a bound- ary value problem for a countable set of ordinary differential equations in the irregular part of the waveguide. An iterative procedure has been proposed for solv- ing these equations. It allows one to avoid the expo- nentially increasing errors in the stiff set of the dif- ferential equations to which the problem is reduced originally. Numerical results have been obtained for two 2D problems with smoothly and nonsmoothly varied cross- section of the irregular domain. In particular, a test problem with the two-side excitation forming a stand- ing field in the irregular part of the waveguide has been considered. The numerical results demonstrate high efficiency and stability of the method. For both problems the dependences of the reflection and trans- mission factors on the geometry are calculated and compared with each other. Acknowledgment The authors thank Prof. B. Z. Katsenelenbaum for useful discussions. References 1. A. F. Stevenson. J. Appl. Phys. 1951, 22, No. 12, pp. 1447- 1460. 2. A. F. Stevenson. J. Appl. Phys. 1951, 22, No. 12, pp. 1461- 1463. 3. Á. Ç. Êàöåíåëåíáàóì. Òåîðèÿ íåðåãóëÿðíûõ âîëíîâî- äîâ ñ ìåäëåííî ìåíÿþùèìèñÿ ïàðàìåòðàìè. Ìîñêâà, Èçä. ÀÍ ÑÑÑÐ, 1961, 171 ñ. 4. F. Sporleder and H. G. Unger. Waveguides Tapers Transitions and Couplers. Peregrinus, Stenvenage, U. K., 1979. 5. B. Z. Katsenelenbaum, L. Mercader del Rio, M. Pereyaslavets, M. Sorolla Ayza, and M. Tumm. Theory of Nonuniform Waveguides. The Cross-Section Method. IEE Electromagnetic waves series. 1998. 6. À. Ã. Ñâåøíèêîâ, À. Ñ. Èëüèíñêèé, È. Ï. Êîòèê. Ðàñ- ïðîñòðàíåíèå êîëåáàíèé â íåðåãóëÿðíîì âîëíîâîäå ñ áîêîâîé ïîâåðõíîñòüþ ñëîæíîé ôîðìû. Âû÷èñëèòåëü- íûå ìåòîäû è ïðîãðàììèðîâàíèå. Ìîñêâà, ÌÃÓ, 1965. 7. À. Ñ. Èëüèíñêèé. Ðàñïðîñòðàíåíèå ýëåêòðîìàãíèòíûõ âîëí â íåðåãóëÿðíûõ âîëíîâîäàõ ñ ïåðåìåííûì ïîïå- ðå÷íûì ñå÷åíèåì. Òðóäû Âû÷èñëèòåëüíîãî öåíòðà Ìîñ- êîâñêîãî óíèâåðñèòåòà. Ìîñêâà, ÌÃÓ, 1970. 8. Â. Â. Øåâ÷åíêî. Ðàäèîòåõíèêà è ýëåêòðîíèêà. 1967, 12, ¹1, ñ. 156-160. 9. V. V. Shevchenko. Continuous Transitions in Open Waveguides. Golem Press, Boulder, 1971. 10. V. G. Pavel�ev, S. H. Tsimring, and V. E. Zapevalov. Int. J. Electron. 1987, 63, No. 3, pp. 379-391. 11. H. Kumric, M. Thumm, and R. Wilhelm. Int. J. Electron. 1988, 64, No. 1, pp. 77-94. 12. Í. È. Âîéòîâè÷, Í. Í. Âîéòîâè÷, Î. Ô. Çàìîðñêàÿ, Á. Ç. Êàöåíåëåíáàóì. Ðàäèîòåõíèêà è ýëåêòðîíèêà. 1993, 38, ¹7, ñ. 1247-1255. 13. Í. Í. Âîéòîâè÷, Ð. È. Ãåðìàíþê, Î. Ô. Çàìîðñêàÿ. Ðàäèîòåõíèêà è ýëåêòðîíèêà. 1994, 39, ¹8-9, ñ. 1321- 1328. Fig. 3. Magnitudes of the reflection and transmission factors in the waveguides shown in Fig. 1 (a) (solid lines) and Fig. 1 (b) (dashed lines) with kh 1 = 1.5π, kh 2 = 4.5π excited by the dominant mode of the left waveguide with p 1 (1) = 1 Ðàäèîôèçèêà è ðàäèîàñòðîíîìèÿ, 2000, ò. 5, ¹3 Numerical Implementation of the Cross-section Method for Irregular Waveguides 283 14. N. N. Voitovich, O. F. Zamorska, and R. I. Germanyuk. Elecrtomagnetics. 1998, 18, No. 5, pp. 481-494. 15. E. Luneville, J.-M. Krieg, and E. Giguet. IEEE Trans. Microwave Theory Techn. 1998, No. 46, pp. 1-9. 16. O. F. Zamorska. Proc. of III Int. Seminar/Workshop on Direct and Inverse Problems of Elecrtomagnetic and Acoustic Wave Theory. Tbilisi, 1998, pp. 62-66. 17. Í. Êîâàëåâ. Ðàäèîòåõíèêà è ýëåêòðîíèêà, 1985, 30, ¹9, c. 17-29. 18. A. G. Ramm. Math. Sci. Research Hot-Line. 1997, 1, No. 3, pp. 1-2. 19. A. G. Ramm. Scattering by Obstacles, D. Reidel, Dordrecht, 1986. 20. A. G. Ramm, G. Makrakis. Scattering by Obstacles in Acoustic Waveguides. In: Spectral and Scattering Theory, Plenum Press, New York, 1998, pp. 89-110. 21. D. Marcuse. Theory of Dielectric Optical Waveguides, Acad. Press, New York, 1991. 22. J. D. Lambert. Computational Methods in Ordinary Differential Equations, John Wiley and Sons, 1973. ×èñëåííàÿ ðåàëèçàöèÿ ìåòîäà ïîïåðå÷íûõ ñå÷åíèé äëÿ íåðåãóëÿðíûõ âîëíîâîäîâ A. Ã. Ðàìì, Í. Í. Âîéòîâè÷, Î. Ô. Çàìîðñêàÿ Èññëåäóåòñÿ ðàññåÿíèå âîëí â íåðåãóëÿðíûõ âîëíîâîäàõ. Ðàññìàòðèâàåòñÿ ìåòîä ïîïåðå÷íûõ ñå÷åíèé äëÿ âû÷èñëåíèÿ ïîëÿ â âîëíîâîäíîé ñèñ- òåìå, ñîñòîÿùåé èç äâóõ ðåãóëÿðíûõ âîëíîâîäîâ, ñîåäèíåííûõ íåðåãóëÿðíîé îáëàñòüþ. Â ñòàòüå äàåòñÿ ìàòåìàòè÷åñêè ñòðîãèé âûâîä îñíîâíûõ óðàâíåíèé ìåòîäà è ïðåäëàãàåòñÿ èòåðàöèîííàÿ ïðîöåäóðà èõ ðåøåíèÿ. Àëãîðèòì ïðèìåíÿåòñÿ ê çàäà÷àì ñ ãëàäêèìè è íåãëàäêèìè íåîäíîðîäíîñ- òÿìè. Íà ïðèìåðå ìîäåëüíîé çàäà÷è, èìåþùåé àíàëèòè÷åñêîå ðåøåíèå, óñòàíàâëèâàþòñÿ ãðàíè- öû ïðèìåíèìîñòè ìåòîäà. ×èñëîâà ðåàë³çàö³ÿ ìåòîäó ïîïåðå÷íèõ ïåðåð³ç³â äëÿ íåðåãóëÿðíèõ õâèëåâîä³â Î. Ã. Ðàìì, Ì. Ì. Âîéòîâè÷, Î. Ô. Çàìîðñüêà Äîñë³äæóºòüñÿ ðîçñ³þâàííÿ õâèëü â íåðåãóëÿð- íèõ õâèëåâîäàõ. Ðîçãëÿäàºòüñÿ ìåòîä ïîïåðå÷íèõ ïåðåð³ç³â äëÿ îá÷èñëåííÿ ïîëÿ â õâèëåâîäí³é ñè- ñòåì³, ùî ñêëàäàºòüñÿ ³ç äâîõ ðåãóëÿðíèõ õâèëå- âîä³â, ç�ºäíàíèõ íåðåãóëÿðíîþ îáëàñòþ. Â ñòàòò³ äàºòüñÿ ìàòåìàòè÷íî ñòðîãå âèâåäåííÿ îñíîâíèõ ð³âíÿíü ìåòîäó ³ ïðîïîíóºòüñÿ ³òåðàö³éíà ïðîöå- äóðà äëÿ ¿õ ðîçâ�ÿçóâàííÿ. Àëãîðèòì çàñòîñîâóºòü- ñÿ äî çàäà÷ ç ãëàäêèìè òà íåãëàäêèìè íåðåãóëÿð- íîñòÿìè. Íà ïðèêëàä³ ìîäåëüíî¿ çàäà÷³, ùî ìàº àíàë³òè÷íèé ðîçâ�ÿçîê, âñòàíîâëþþòüñÿ ãðàíèö³ çàñòîñîâíîñò³ ìåòîäó.

Numerical Implementation of Cross-Section Method for Irregular Waveguides

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