/compile/output.dvi EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 9, No. 4, 2016, 360-366 ISSN 1307-5543 – www.ejpam.com On Energy Waves Via Airy Functions in Time-Domain Emre Eroğlu 1,∗,Özlem I̧sık Department of Mathematics,Faculty of Arts and Sciences, Kırklareli University, Kırklareli, Turkey Abstract. The main idea is to solve the system of Maxwell’s equations in accordance with the causality principle to get the energy quantities via Airy functions in a hollow rectangular waveguide. Evolutionary Approach to Electromagnetics which is an analytical time-domain method is used. The boundary- value problem for the system of Maxwell’s equations is reformulated in transverse and longitudinal coordinates. A self-adjoint operator is obtained and the complete set of eigenvectors of the operator initiates an orthonormal basis of the solution space. Hence, the sought electromagnetic field can be presented in terms of this basis. Within the presentation, the scalar coefficients are governed by Klein- Gordon equation. Ultimately, in this study, time-domain waveguide problem is solved analytically in accordance with the causality principle. Moreover, the graphical results are shown for the case when the energy and surplus of the energy for the time-domain waveguide modes are represented via Airy functions. 2010 Mathematics Subject Classifications: 35Q60, 35Q61, 83C50 Key Words and Phrases: Maxwell’s equations, wave boundary operators, Airy functions, surplus of energy 1. Introduction Time-domain waveguide problem deals with the propagation of a signal in a waveguide and this problem can be solved numerically or analytically. Significant publications on ap- proaching time-domain solutions of electromagnetic fields are based on different techniques (see, e.g. [2, 5, 7, 9, 10, 16]). In [11] finite difference time domain method which is a powerful numerical method is studied for time-harmonic fields by Taflove and Hagness. One of the analytical methods depends on integral transforms such as Fourier and Laplace. In this study, the analytical method ’Evolutionary Approach to Electromagnetics’ (EAE) is con- sidered (see, e.g. [1, 3, 4, 8, 14, 15]). As the name suggests, this method deals with solving evolution equations, which contain time derivative. The main idea is to obtain some self- adjoint operators from the system of Maxwell’s equations via decomposition. These are called ’Wave Boundary Operators’ (WBO) and act on transverse coordinates (see, e.g. [1, 12, 13]). ∗Corresponding author. Email addresses: emreeroglu@klu.edu.tr (E. Eroğlu), ozlem.isik@klu.edu.tr (Ö. I̧sık) http://www.ejpam.com 360 c© 2016 EJPAM All rights reserved. E. Eroğlu, Ö. I̧sık / Eur. J. Pure Appl. Math, 9 (2016), 360-366 361 Their eigenvector set initiates an orthonormal basis set in the solution space, i.e., Hilbert space L2(S). Due to the elements of the basis two kinds of solutions, in other words transverse elec- tric (TE) and transverse magnetic (TM) time domain modes occur. Each field component of the modes is a product of the WBO eigenvector function which is the solution of Nuemann or Dirichlet boundary value problem depending on the mode and the modal amplitude depend- ing on axial coordinate z and time t. An evolutionary formulation for the modal amplitudes is obtained by writing the system of Maxwell’s equations in terms of basis (see, e.g. [12, 13]). This evolution equation can be solved in compliance with the causality principle. 2. Material and Method A perfect electric conducting and hollow waveguide with its cross-section domain S, bounded by a closed singly connected contour L is considered. A right-handed triplet of the mutually or- thogonal unit vectors (z, l,n) is used where z× l= n and so on. z is oriented along the Oz axis, l is tangential to L and n is the outer normal to S. The three-component vector R = (x , y, z) is written in the form of R = r + zz where r = (x , y). The electric and magnetic fields are determined by solving the system of Maxwell’s equations ∇× #» E (R, t) =−µ0∂t #» H(R, t), ∇ · #» E (R, t) = 0, (1) ∇× #» H(R, t) =ε0∂t #» E (R, t), ∇ · #» H(R, t) = 0. (2) Due to the perfect electric conductor surface of the waveguide, the field components are sub- jected to the following boundary conditions n · #» H(R, t) � � L = 0, l · #» E (R, t) � � L = 0, z · #» E (R, t) � � L = 0. (3) In addition, as being a hyperbolic type of PDE, the solution to (1)-(2) should satisfy some given initial conditions #» E (R, 0) = 0, #» H(R, 0) = 0. (4) Decomposition of the field vectors and the nabla operator, respectively, into transverse and longitudinal parts as #» E (R, t) = #» E (r, z, t) + zEz(r, z, t), #» H(R, t) = #» H(r, z, t) + zHz(r, z, t), (5) ∇ =∇⊥ + z∂z , (6) yields two subsystems of equations ∇⊥Ez =µ0∂t � #» H × z � + ∂z #» E , ε0∂t Ez =∇⊥ · � #» H × z � , ∂z Ez = −∇⊥ · #» E , (7) ∇⊥Hz =ε0∂t � z× #» E � + ∂z #» H, µ0∂t Hz =∇⊥ · � z× #» E � , ∂zHz = −∇⊥ · #» H. (8) The subsystems (7) and (8) can be rewritten in a 4× 4 matrix form respectively, WH #» X = � 0 ε−1 0 � z×∇⊥ �∇⊥ µ−1 0 ∇⊥ � z×∇⊥ � 0 �� #» E #» H � , (9) E. Eroğlu, Ö. I̧sık / Eur. J. Pure Appl. Math, 9 (2016), 360-366 362 WE #» X = � 0 ε−1 0 ∇⊥ �∇⊥ × z � µ−1 0 �∇⊥ × z �∇⊥ 0 �� #» E #» H � . (10) where 0 is 2× 2 zero matrix and #» X = ( #» E , #» H)T = (Ex , Ey , Hx , H y) T . The operators WH and WE are called wave boundary operators (WBO). Together with the boundary conditions (3), n · #» H = 0, l · #» E = 0 holds for r ∈ L [1, 12, 13]. Because of the physical principle that the electromagnetic field energy is always finite, the initial-boundary value problem (1)-(4) should be solved in a class of integrable vector functions of coordinates and time. This suggests an inner product which is used in [1, 4] for the vector #» X = ( #» E , #» H)T = (Ex , Ey , Hx , H y) T as #» X1, #» X2 � = 1 2 ∫ S � ε0 #» E 1 · #» E 2 +µ0 #» H1 · #» H2 � ds. (11) According to the inner product (11) it can be shown that WH #» X1, #» X2 � = #» X1,WH #» X2 � and WE # » X1, #» X2 � = #» X1,WE #» X2 � which means the operators WH and WE are both self-adjoint. Therefore, the eigenvalue equations WH #» Y m(r) = pm #» Y m(r) and WE #» Z n(r) = qn #» Z n(r) hold where pm and qn are the real eigenvalues, respectively. All the eigenvalues are situated sym- metrically on the real axis and they can be put in order as p+m = −p−m > 0, q+n = −q−n > 0. The formulation for these eigenvalues are obtained in [12, 13] as p±m = ±υ2 m/ p ε0µ0, q±n = ±κ2 n/ p ε0µ0 by solving Neumann and Dirichlet boundary eigenvalue problems, respec- tively, �∇2 ⊥ + ν 2 m � ψm(r) =0, ∂ ∂ n ψm � � � � L = 0, ν2 m S ∫ S � �ψm � � 2 ds = 1, (12) �∇2 ⊥ + κ 2 n � φn(r) =0, φn � � L = 0, κ2 n S ∫ S � �φn � � 2 ds = 1. (13) The WBO eigenvectors #» Y m(r) and #» Z n(r) corresponding to the eigenvalues pm and qn, respec- tively, are presented by the scalar potentialsψm(r), φn(r) in [12, 13] which are eigensolutions to problems in (12)-(13), as #» Y ±m(r) = �q ε−1 0 �∇⊥ψm × z � ,± q µ−1 0 ∇⊥ψm �T , #» Z ±n(r) = �q ε−1 0 ∇⊥φn,± q µ−1 0 � z×∇⊥φn � �T . (14) Consequently elements of the orthonormal basis is specified on the cross section S via � #» Y ±m(r) ∞ m=1 , � #» Z ±n(r) ∞ n=1 in Hilbert space L2(S). Then the vector #» X = � #» E , #» H �T can be pre- sented in terms of the basis elements. Due to the completeness of ¦ #» ψm(r) ©∞ m=0 and ¦ #» φn(r) ©∞ n=0 in L2(S), the field components Ez and Hz can be written in terms of ψm and φn respectively [1]. The basis set implies that (7) and (8) have two kinds of solutions as TE and TM time domain waveguide modes. The solutions of the Neumann and Dirichlet boundary value problems E. Eroğlu, Ö. I̧sık / Eur. J. Pure Appl. Math, 9 (2016), 360-366 363 (12)-(13), generate the TE and TM time-domain modal fields, respectively, with the following components #» E m(r, z, t) =− q ε−1 0 ∂c thm(z, t) �∇⊥ψm(r)× z � , Ezm(r, z, t) = 0, #» Hm(r, z, t) = q µ−1 0 ∂zhm(z, t)∇⊥ψm(r), Hzm(r, z, t) = q µ−1 0 υ 2 mhm(z, t)ψm(r), (15) and #» E n(r, z, t) = q ε−1 0 ∂zen(z, t)∇⊥φn(r), Ezm(r, z, t) = q ε−1 0 κ 2 nen(z, t)φn(r), #» Hn(r, z, t) =− q µ−1 0 ∂c t en(z, t) � #»z ×∇⊥φn(r) � , Hzm(r, z, t) = 0, (16) where ∂c t = (1/c)∂t and c = 1/ p ε0µ0. The potentials hm(z, t) and en(z, t) in eq. (15)-(16) are governed by Klein-Gordon equations � ∂ 2 υmc t − ∂ 2 υmz +υ 2 m � hm(z, t) = 0, � ∂ 2 κnc t − ∂ 2 κnz + κ 2 n � en(z, t) = 0 (17) which are obtained by projecting the system of Maxwell’s equations on to the basis [15]. The KGE in (17) can be written in the general form � ∂ 2 τ − ∂ 2 ξ + 1 � f (ξ,τ) = 0 (18) where f (ξ,τ) is either hm(z, t) provided that ξ= υmz and τ= υmc t for TE-modes or en(z, t) provided that ξ = κnz and τ = κnc t for TM-modes (τ is the scaled time and ξ is the scaled coordinate) [15]. Depending on W. Miller’s idea [6] the solution f (ξ,τ) for KGE is interpreted as a function of new variables as f (u(ξ,τ),ν(ξ,τ)). This idea propose 11 suitable functions which enables the factorization of the solution as f (u,ν) = U(u)V (ν). In this study, the func- tion pairs u + ν = (ξ + τ)/2, u − ν = ±pξ+τ are considered and substitution in equation (18) yields f (ξ,τ) =      0 ,τ < 0 � c1Ai(u) + c2Bi(u) � � c3Ai(ν) + c4Bi(ν) � , 0≤ ξ≤ τ 0 ,ξ > τ . (19) In accordance with the causality principle where c1,2,3,4 are arbitrary constants, Ai and Bi are Airy functions. Their arguments are the functions of time, τ and axial coordinate, ξ. All possible combinations of the Airy functions are f1(ξ,τ) =Ai(u)Ai(ν), f2(ξ,τ) = Ai(u)Bi(ν) f3(ξ,τ) =Bi(u)Ai(ν), f4(ξ,τ) = Bi(u)Bi(ν). (20) On the other hand, energy waves are propagating along the waveguide together with the elec- tromagnetic waves. So the energy density stored in electric and magnetic fields and the differ- ence of energy densities are investigated in terms of the source functions, i.e. Airy functions. Surplus of the energy is the difference between stored energies. E. Eroğlu, Ö. I̧sık / Eur. J. Pure Appl. Math, 9 (2016), 360-366 364 The energy and surplus of energy is given in [4] with the following formulas, respectively, W (ξ,τ) = � A2(ξ,τ) + B2(ξ,τ) + f 2(ξ,τ) � /2, sW (ξ,τ) = � A2(ξ,τ)− B2(ξ,τ) � /2 where A(ξ,τ) = − ∂ ∂ τ f (ξ,τ), B(ξ,τ) = ∂ ∂ ξ f (ξ,τ) and f (ξ,τ) is the Airy function. In this work, energetic quantities are specially discussed for the Airy functions. In Figure 1 and Figure 2 dependence on time, τ of energy density, W3,4(ξ,τ) and surplus of the energy, sW3,4(ξ,τ) are exhibited for on fixed position, ξ= τ− 0.05 of the cross-section. (a) 0≤ τ≤ 15 (b) 0≤ τ≤ 50 Figure 1: Time dependence of W3 and sW3. (a) 0≤ τ≤ 15 (b) 0≤ τ≤ 50 Figure 2: Time dependence of W4 and sW4. REFERENCES 365 3. Conclusion In this study, the time-domain waveguide modes are expressed analytically by a method of Evolutionary Approach to Electromagnetics (EAE). A hollow waveguide is considered with perfect electric conductor surface. Energy quantities for the time-domain fields are analysed in details and the energy waves which are also propagating accompanying the electromagnetic field waves are obtained. Especially the energy and surplus of the energy are presented via Airy functions. Thus, the energetic wave process of exchange by energy stored in the longitudinal and transverse field components is introduced in the time-domain, directly. In further studies, the other possible solutions proposed from the Miller’s eleven cases will be considered for the solution of different problems such as partially filled lossless and lossy waveguides. ACKNOWLEDGEMENTS This work is supported by Kırklareli University [KUBAP 08] and Uludağ University [UAPF-18]. References [1] S. Aksoy and O.A. Tretyakov. Evolution equations for analytical study of digital signals in waveguides. Journal of Electromagnetic Waves and Applications, 17, 2003. [2] V.V. Borisov. Transient Electromagnetic Waves. 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