Sourcewise represented Green’s function of the circular waveguide

Singular part of the Green’s function of unbounded space is singled out in explicit form and contains all singularities, including a delta-shaped singularity. The problem of construction of Green’s function for a field is solved, as a problem for diffraction of potential and rotational components el...

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Дата:	2007
Автори:	Prijmenko, S.D., Bondarenko, L.A.
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Мова:	English
Опубліковано:	Національний науковий центр «Харківський фізико-технічний інститут» НАН України 2007
Назва видання:	Вопросы атомной науки и техники
Теми:	Теория и техника ускорения частиц
Онлайн доступ:	http://dspace.nbuv.gov.ua/handle/123456789/110417
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Цитувати:	Sourcewise represented Green’s function of the circular waveguide / S.D. Prijmenko, L.A. Bondarenko // Вопросы атомной науки и техники. — 2007. — № 5. — С. 137-140. — Бібліогр.: 8 назв. — англ.

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Digital Library of Periodicals of National Academy of Sciences of Ukraine

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spelling	irk-123456789-1104172017-01-05T03:02:43Z Sourcewise represented Green’s function of the circular waveguide Prijmenko, S.D. Bondarenko, L.A. Теория и техника ускорения частиц Singular part of the Green’s function of unbounded space is singled out in explicit form and contains all singularities, including a delta-shaped singularity. The problem of construction of Green’s function for a field is solved, as a problem for diffraction of potential and rotational components electric field intensity of a point current source on the circular waveguide walls. The singling out of the electric field intensity singularity in an explicit form about a source enables to develop an effective algorithm of Green’s function calculation at any distance between the source point and observation point in a circular waveguide. Cінгулярна частина функції Гріна круглого хвилеводу у формі функції Гріна необмеженого простору виділена в явному вигляді й містить всі особливості, включаючи дельта-подібну особливість. Задача побудови функції Гріна для поля розв'яза як задача дифракції потенційної й вихрової частин напруженості електричного поля крапкового джерела струму на стінках круглого хвилеводу. Виділення особливості напруженості електричного поля в явному вигляді в околиці джерела дозволило розробити ефективний алгоритм розрахунку електричної функції Гріна при довільній відстані між крапками джерела й спостереження в круглому хвилеводі. Cингулярная часть функции Грина круглого волновода в форме функции Грина неограниченного пространства выделена в явном виде и содержит все особенности, включая дельта-образную особенность. Задача построения функции Грина для поля решена как задача дифракции потенциальной и вихревой частей напряженности электрического поля точечного источника тока на стенках круглого волновода. Выделение особенности напряженности электрического поля в явном виде в окрестности источника позволило разработать эффективный алгоритм расчета электрической функции Грина при произвольном расстоянии между точками источника и наблюдения в круглом волноводе. 2007 Article Sourcewise represented Green’s function of the circular waveguide / S.D. Prijmenko, L.A. Bondarenko // Вопросы атомной науки и техники. — 2007. — № 5. — С. 137-140. — Бібліогр.: 8 назв. — англ. 1562-6016 PACS: 29.17 + w, 02.30.Rs, 84.40.Sr http://dspace.nbuv.gov.ua/handle/123456789/110417 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
topic	Теория и техника ускорения частиц Теория и техника ускорения частиц
spellingShingle	Теория и техника ускорения частиц Теория и техника ускорения частиц Prijmenko, S.D. Bondarenko, L.A. Sourcewise represented Green’s function of the circular waveguide Вопросы атомной науки и техники
description	Singular part of the Green’s function of unbounded space is singled out in explicit form and contains all singularities, including a delta-shaped singularity. The problem of construction of Green’s function for a field is solved, as a problem for diffraction of potential and rotational components electric field intensity of a point current source on the circular waveguide walls. The singling out of the electric field intensity singularity in an explicit form about a source enables to develop an effective algorithm of Green’s function calculation at any distance between the source point and observation point in a circular waveguide.
format	Article
author	Prijmenko, S.D. Bondarenko, L.A.
author_facet	Prijmenko, S.D. Bondarenko, L.A.
author_sort	Prijmenko, S.D.
title	Sourcewise represented Green’s function of the circular waveguide
title_short	Sourcewise represented Green’s function of the circular waveguide
title_full	Sourcewise represented Green’s function of the circular waveguide
title_fullStr	Sourcewise represented Green’s function of the circular waveguide
title_full_unstemmed	Sourcewise represented Green’s function of the circular waveguide
title_sort	sourcewise represented green’s function of the circular waveguide
publisher	Національний науковий центр «Харківський фізико-технічний інститут» НАН України
publishDate	2007
topic_facet	Теория и техника ускорения частиц
url	http://dspace.nbuv.gov.ua/handle/123456789/110417
citation_txt	Sourcewise represented Green’s function of the circular waveguide / S.D. Prijmenko, L.A. Bondarenko // Вопросы атомной науки и техники. — 2007. — № 5. — С. 137-140. — Бібліогр.: 8 назв. — англ.
series	Вопросы атомной науки и техники
work_keys_str_mv	AT prijmenkosd sourcewiserepresentedgreensfunctionofthecircularwaveguide AT bondarenkola sourcewiserepresentedgreensfunctionofthecircularwaveguide
first_indexed	2025-07-08T00:35:30Z
last_indexed	2025-07-08T00:35:30Z
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fulltext	SOURCEWISE REPRESENTED GREEN’S FUNCTION OF THE CIRCULAR WAVEGUIDE S.D. Prijmenko∗, L.A. Bondarenko Institute for Plasma Electronics and New Methods of Acceleration, National Science Center ”Kharkov Institute of Physics and Technology”, 61108, Kharkov, Ukraine (Received March 15, 2007) Singular part of the Green’s function of unbounded space is singled out in explicit form and contains all singularities, including a delta-shaped singularity. The problem of construction of Green’s function for a field is solved, as a problem for diffraction of potential and rotational components electric field intensity of a point current source on the circular waveguide walls. The singling out of the electric field intensity singularity in an explicit form about a source enables to develop an effective algorithm of Green’s function calculation at any distance between the source point and observation point in a circular waveguide. PACS: 29.17 + w, 02.30.Rs, 84.40.Sr 1. INTRODUCTION Singular and hypersingular integral equations [1] with a kernel in the form of Green’s function are a highly efficient apparatus of mathematical physics. These equations are applied in problems of the mi- crowave electronics and accelerating engineering, for example, for calculation of electromagnetic fields in a coaxial girotron [2], an accelerating structure of H- type [3] and a bunch accumulator of the charged par- ticles [4] (p.80). Advantages of singular and hypersingular equa- tions are connected with the use of well stipulated matrixes providing a high accuracy and stability of calculations. However, then the Green’s function is to be calculated at short distances between the source points and the observation points. Let the accelerating structure is a system of metal radio-frequency (rf) electrodes in a circular waveg- uide. Then the integral equations use the electric Green’s function of a circular waveguide for the field Ĝe(k, r, r′) relative to the surface density of the force of the electric current which flows only on the elec- trode surface. Upon expansion using the system TE and TH waves, Ĝe(k, r, r′) is described by double se- ries which diverge. This is because Ĝe(k, r, r′) in an implicit form includes the electric Green’s function of unbounded space for a field ĜS e (k, r, r′) which has sin- gularities 1/\|~r − ~r′\|, 1/\|~r − ~r′\|2,1/\|~r − ~r′\|3, δ(~r − ~r′). The problem of construction of Green’s function for a field is solved, as a problem for diffraction of po- tential and rotational components of a tensor spheri- cal wave of the electric field intensity of the point cur- rent source (a current point source is delta-shaped lo- cation current) on the circular waveguide walls. Thus the singularity of electric field intensity was singled out in an explicit form about a current source that allowed us to create an effective algorithm of electric Green’s function calculation at any electric length of nonhomogeneities in the circular waveguide. The use of the Green’s function Ĝe(k, r, r′) with an explicit singularity enables the numerical solution of two-dimensional hypersingular integral equations instead of the three-dimensional equations. Hyper- singularity and two-dimensionality of the equations can provide an increased accuracy and reduced time of calculations respectively. 2. THE BASIC PART 2.1. ANALYTICAL RELATIONS The Green’s electric function for a field is defined by the formula of [5],[6] ĜS e (k, r, r′) = ( Î + 1 k2 grad(r)div(r) ) ĜE(k, r, r′), (1) where ĜE(k, r, r′) is the Green’s function for a vector potential. In case of a circular waveguide the function Ĝe(k, r, r′) is constructed by the system TE and TH waves in [7]. The Green’s function of a circular waveguide for a field is obtained in the form of TE and TH waves and in the form of superpositions Ĝe(k, r, r′) = Ĝp e(k, r, r′) + Ĝr e(k, r, r′), (2) Ĝe(k, r, r′) = ĜS e (k, r, r′) + ĜR e (k, r, r′), (3) where Ĝp e(k, r, r′) = Ĝsp e (k, r, r′) + ĜRp e (k, r, r′), (4) Ĝr e(k, r, r′) = Ĝsr e (k, r, r′) + ĜRr e (k, r, r′), (5) ∗Corresponding author. E-mail address: sprijmenko@kipt.kharkov.ua PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY, 2007, N5. Series: Nuclear Physics Investigations (48), p.137-140. 137 ĜS e (k, r, r′) = ĜSp e (k, r, r′) + ĜSr e (k, r, r′), (6) ĜR e (k, r, r′) = ĜRp e (k, r, r′) + ĜRr e (k, r, r′). (7) Here Ĝp e(k, r, r′) and Ĝr e(k, r, r′) are, respectively, the potential and rotational components of the electric field intensity of the point current source in the cir- cular waveguide, ĜS e (k, r, r′) and ĜR e (k, r, r′) are, re- spectively, the intensity of the electric field in the un- bounded space and of the field reflected from the walls of the circular waveguide of the point current source; ĜSr e (k, r, r′) and ĜSr e (k, r, r′) are, respectively, the potential and rotational components of the electric field intensity of the point current source in the un- bounded space; ĜRp e (k, r, r′) and ĜRr e (k, r, r′) are, re- spectively, the potential and rotational components of the intensities of the electric field of the point cur- rent source reflected from the waveguide walls. The potential and rotational components of the electric field intensity are stipulated by the scalar and vec- tor potentials, respectively or by the distributions of charges and currents in the source point. The tensor function ĜS e (k, r, r′) in an explicit form describes singularities of intensity of an electric field of a current point source. It is found 9 components of Ĝe(k, r, r′) in the form of TE and TH waves of a waveguide and in the form of (2)- (7). In particular,, by the system of TE and TH waves of a circular waveguide, Ĝe11′ (k, r, r′) is Ĝe11′ (k, r, r′) = +∞∑ m=−∞ +∞∑ n=−∞ eim(φ−φ′)× (Amn ( 1− (kh mn)2 k2 ) J ′m(kh mnρ)J ′m(kh mnρ′)× fmn(z, z′) + Bmn m2 (ke mn)2ρρ′ Jm(ke mnρ)× Jm(ke mnρ′)lmn(z, z′)) , (8) where Amn = 1 πJ2 m+1(ke mnR)R2 , Bmn = 1 πJ2 m(ke mnR)R2 ( 1− m2 (ke mnR)2 ) , lmn(z, z′) = e−γmn\|z−z′\|/2γmn , γmn = √ (ke mn)2 − k2 , ke mn = µmn R , fmn(z, z′) = e−βmn\|z−z′\|/2βmn , βmn = √ (kh mn)2 − k2 , kh mn = νmn R , γmn, µmn are the roots of the equations Jm(z) = 0 and J ′m(z) = 0, respectively, and R is the waveguide radius. It is shown, that div(r⊥)ĜE(ke mn, r⊥, r′⊥; z, z′) = 0, (9) i. e. TE or TH waves in the circular waveguide are ro- tational waves relative to the transverse coordinates. The singular components of GS e11′ (k, r, r′), Gsp e11′ (k, r, r′), Gsr e11′ (k, r, r′) are described in the form of (2-7) by the formulas GS e11′ (k, r, r′) = Gsp e11′ (k, r, r′) + Gsr e11′ (k, r, r′) , (10) Gsp e11′ (k, r, r′) = Gsp exx′ (k, r, r′) cos ϕ cos ϕ′+ Gsp eyx′ (k, r, r′) sin ϕ cos ϕ′ + Gsp exy′ (k, r, r′)× cos ϕ sin ϕ′ + Gsp eyy′ (k, r, r′) sin ϕ sin ϕ′, (11) Gsr e11 (k, r, r′) = Gsr exx′ (k, r, r′) cos ϕ cosϕ′+ Gsr eyy′ (k, r, r′) sin ϕ sin ϕ′ = Gsr E11 (k, r, r′) = 1 4π eik\|~r−~r′\| \|~r − ~r′\| cos(ϕ− ϕ′) , (12) Gsp exx′ (k, r, r′) = 1 4πk2 eik\|~r−~r′\|× { 3(ρ cosϕ− ρ′ cosϕ′)2 \|~r − ~r′\|5 − 1 \|~r − ~r′\|3− 4π 3 δ(r, r′)− 3ik(ρ cosϕ− ρ′ cos ϕ′) \|~r − ~r′\|4 − k2(ρ cos ϕ− ρ′ cosϕ′)2 \|~r − ~r′\|3 + ik \|~r − ~r′\|2 } , (13) Gsr exx′ (k, r, r′) = 1 4π eik\|~r−~r′\| \|~r − ~r′\| , (14) Gsp exy′ (k, r, r′) = Gsp eyx′ (k, r, r′) = 1 4πk2 eik\|~r−~r′\|× (ρ cos ϕ− ρ′ cos ϕ′)(ρ sin ϕ− ρ′ sin ϕ′)×{ 3 \|~r − ~r′\|5 − 3ik \|~r − ~r′\|4 − k2 \|~r − ~r′\|3 } , (15) Gsp eyy′ (k, r, r′) = 1 4πk2 eik\|~r−~r′\|× { 3(ρ sin ϕ− ρ′ sin ϕ′)2 \|~r − ~r′\|5 − 1 \|~r − ~r′\|3− 4π 3 δ(r, r′)− 3ik(ρ sin ϕ− ρ′ sin ϕ′)2 \|~r − ~r′\|4 − k2(ρ sin ϕ− ρ′ sin ϕ′)2 \|~r − ~r′\|3 + ik \|~r − ~r′\|2 } , (16) Gsr exx′ (k, r, r′) = Gsr eyy′ (k, r, r′) = 1 4π eik\|~r−~r′\| \|~r − ~r′\| , (17) The potential GRp e11′ (k, r, r′) and rotational GRr e11′ (k, r, r′) components of the regular Green’s func- tion of a circular waveguide are in the form of (2)-(7) GRp e11′ (k, r, r′) = i 8π +∞∑ m=−∞ eim(ϕ−ϕ′)× +∞∫ −∞ eiη(z−z′)ν2(η) k2 J ′m(ν(η)ρ)J ′m(ν(η)ρ′) H (1) m (ν(η)R) Jm(ν(η)R) dη , (18) 138 GRr e11′ (k, r, r′) = − i 8π +∞∑ m=−∞ eim(ϕ−ϕ′)× +∞∫ −∞ eiη(z−z′)J ′m(ν(η)ρ)J ′m(ν(η)ρ′) H (1) m (ν(η)R) J ′m(ν(η)R) + m2 ν2ρρ′ Jm(ν(η)ρ)Jm(ν(η)ρ′) H (1)′ m (ν(η)R) J ′m(ν(η)R) dη . (19) Notice, that the problem of construction of the Green’s function for a field is solved, as a problem of diffraction of the potential and rotational com- ponents of the intensity of electric field divergent spherical wave of a point current source on the circu- lar waveguide walls. As this takes place, the potential and rotational components correspond to the scalar and vector potentials, respectively. We used the representation of a spherical wave in the form of a spectrum of non-uniform cylindrical waves diverging in two opposite directions along the radius, i.e. the sourcewise representation of a spherical wave in the radial direction of [8](p.42) 2.2. NUMERICAL RESULTS 0 10 20 30 40 50 0 2 4 6 8 10 R e G E1 1' m 1 2 Fig.1. Rotational component of Green’s function Re Gr e11′ (k, r, r′) of a circular waveguide (k = 12.56 m−1; R = 0.0755 m; \|~r − ~r′\|/λ = 0, 02) The algorithm of calculation of Ĝe(k, r, r′) in the form of TE and TH waves of a circular waveguide and in the form of (2)- (7) is developed. Singulari- ties of the tensor Green’s function are singled out in an explicit form for representation of (2)- (7). The efficiency of calculations of Ĝe(k, r, r′) in the form of (2)- (7) is illustrated by the plots in Fig.1-4. The real part of the rotational component Re Gr e11′ (k, r, r′) of the Green’s function and its derivative ∂ReGr e11′ (k, r, r′)/∂x1 (x1 is the radial coordinate) for the cutoff circular waveguide ver- sus the number of an azimuthal harmonic m is shown in Fig.1 and Fig.3 for k = 12.56 m−1; R = 0.0755 m; ρ = 0.07 m; ρ′ = 0, 08 m; z = z′; ϕ = ϕ′; \|~r − ~r′\|/λ = 0, 02, and in Fig.2 and Fig.4 for k = 12, 56 m−1; R = 0, 0755 m; ρ = 0.07 m; ρ′ = 0.08 m; z = z′; ϕ = ϕ′ + π; \|~r − ~r′\|/λ = 0.26. 10 20 30 40 50 -0,8 -0,6 -0,4 -0,2 0,0 0,2 0,4 R e G E1 1/ m 1 2 Fig.2. Rotational component of Green’s function Re Gr e11′ (k, r, r′) of a circular waveguide (k = 12.56 m−1; R = 0.0755 m; \|~r − ~r′\|/λ = 0, 26) 0 10 20 30 40 50 0 400 800 1200 1600 2000 d( R e G E 11 /)/ dx 1 m 1 2 Fig.3. Rotational component of Green’s function Re∂Gr e11′ (k, r, r′)/∂x1 of a circular waveg- uide (k = 12.56m−1; R = 0.0755m; \|~r − ~r′\|/λ = 0, 2) 0 10 20 30 40 50 -120 -80 -40 0 40 80 d( G E 11 /)/ dx 1 m 1 2 Fig.4. Rotational component of Green’s function Re∂Gr e11′ (k, r, r′)/∂x1 of a circular waveg- uide (k = 12.56m−1; R = 0.0755m; \|~r − ~r′\|/λ = 0, 26) The cases when the singularity is singled out and not singled out are described by curves 1 and 2 re- spectively. As follows from Fig.1-4 curves of 1 flat- tens out at m > 20 as for ReGr e11′ (k, r, r′) and for a derivative of ∂ReGr e11′ (k, r, r′)/∂x1., i.e. series con- verges good at m > 20. Week oscillations take place for curves of 2 at m > 20 for Re Gr e11′ (k, r, r′) , i.e. series converges worse than for curves of 1. A siz- able oscillations take place for curves of 2 at m > 20 for a derivative of ∂ReGr e11′ (k, r, r′)/∂x1, i.e. series diverge. 139 3. CONCLUSIONS For the first time the problem of construction of Green’s function for a field is solved as a problem of diffraction of potential and rotational parts of the electric field intensity of a point current source on circular waveguide walls. The potential and rotational components of the electric field intensity are conditioned by the scalar and vector potentials or distributions of a charge and a current, respectively, in the source point. By singling out the singularity of the electric field intensity in an explicit form about of a source it is possible to develop the effective algorithm of calcu- lation of the electric Green’s function at any electric length of nonhomogeneities in a circular waveguide. REFERENCES 1. S.O. Dovgij, I.K. Lifanov. A method of singu- lar integral equations. Theory and an application // Kyiv: ” Naukova dumka”, 2004, 510 p. (in Ukrainian). 2. J.V. Gandel’, G.I. Zaginajlov, S.A. Steshenko. The rigorous electrodynamic analysis of cavity systems of coaxial girotrons // Journal of techni- cal physics. 2004, v.74, N.7, p.81-89. (in Russian). 3. S.D. Prijmenko, N.A. Khizhnyak. To calculation of electrodynamical characteristics of the H-type accelerating structure // Radiotechniques. 2001, N117, p.85-87. 4. R.S. Dikanskij, D.V. Pestrikov. Physics of inten- sive bunches in stores // Novosibirsk: ”Nauka”, Sib. Branch, 1989, 336 p. (in Russian). 5. H. Levin, J. Schwinger. On the theory of electro- magnetic wave diffraction by an aperture in an infinite plane conducting screen // Communica- tion on pure and appl. Mathematics. 1950, v.3, p.355-368. 6. C. Tai, P. Rozenfeld. Different representation of dyadic Grren’s functions for a rectangular cavity // IEEE Trans. Microwave Theory Tech. 1976, v.24, p.597-601. 7. S.D. Prijmenko, V.G. Papkovich, N.A. Khizh- nyak. Electric tensor Green’s functions of cylin- drical waveguides. // The review. M.: Atomin- form. 1988, 27 p. (in Russian). 8. G.T. Markov , E. N. Vasil’ev Mathematical meth- ods of applied electrodynamics //”Sov. Radio”. 1970, 120 p. (in Russian). ИСТОКООБРАЗНАЯ ФУНКЦИЯ ГРИНА КРУГЛОГО ВОЛНОВОДА С.Д. Прийменко, Л.А. Бондаренко Cингулярная часть функции Грина круглого волновода в форме функции Грина неограниченного пространства выделена в явном виде и содержит все особенности, включая дельта-образную особен- ность. Задача построения функции Грина для поля решена как задача дифракции потенциальной и вихревой частей напряженности электрического поля точечного источника тока на стенках круглого волновода. Выделение особенности напряженности электрического поля в явном виде в окрестности источника позволило разработать эффективный алгоритм расчета электрической функции Грина при произвольном расстоянии между точками источника и наблюдения в круглом волноводе. ДЖЕРЕЛОПОДIБНА ФУНКЦIЯ ГРIНА КРУГЛОГО ХВИЛЕВОДУ С.Д. Прийменко, Л.О.Бондаренко Ciнгулярна частина функцiї Грiна круглого хвилеводу у формi функцiї Грiна необмеженого про- стору видiлена в явному виглядi й мiстить всi особливостi, включаючи дельта-подiбну особливiсть. Задача побудови функцiї Грiна для поля розв’яза як задача дифракцiї потенцiйної й вихрової частин напруженостi електричного поля крапкового джерела струму на стiнках круглого хвилеводу. Видi- лення особливостi напруженостi електричного поля в явному виглядi в околицi джерела дозволило розробити ефективний алгоритм розрахунку електричної функцiї Грiна при довiльнiй вiдстанi мiж крапками джерела й спостереження в круглому хвилеводi. 140

Sourcewise represented Green’s function of the circular waveguide

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