- Email: [email protected]
Transient response of a vertical electric dipole above a two-layer medium
Applied Mathematics and Computation 151 (2004) 401–410 www.elsevier.com/locate/amc
Transient response of a vertical electric dipole above a two-layer medium Samira T. Bishay a, Adel A.S. Abo Seliem a
b,*
Department of Mathematics, Faculty of Science, Ain Shams University, Abassia, Cairo, Egypt b Department of Mathematics, Faculty of Education, Tanta University, Kafer El Sheiek Branch, Egypt
Abstract The paper presents a method which allows the calculation of the atmospheric distortion of radar pulses, provided that the influence of the atmosphere is to transfer the transmitted signal through a duct. The study thus deals with the investigation of the pulses signal from a vertical electric dipole above an evaporation duct. Two integral transforms of the wave equation of Hertzian vector––a Laplace transform in time and a two dimensional Fourier transform in the horizontal coordinates in space––are applied. This leads to an integral representation of the solution of the wave equation in transform space, considering initial, boundary and transition conditions. This integral representation determines the electromagnetic field anywhere in the ionosphere. Saddlepoint and residue methods are used to compute the integral. Ó 2003 Elsevier Inc. All rights reserved. Keywords: Vertical dipole; Laplace transform; Transient response
1. Introduction Radiometeorlogists have extensively studied the correlation between the propagation of electromagnetic waves in the atmosphere and meteorological conditions. Today, the interest is mainly concerned with the effect of the atmosphere on target identification and imaging. Historically, in the problem of the electromagnetic radiation from a vertical dipole situated at a certain * Corresponding author. Address: Department of Mathematics, Jizan Teachers College, Abo Areesh, P.O. Box 203, Jizan, Saudi Arabia. E-mail address: [email protected] (A.A.S. Abo Seliem).
0096-3003/$ - see front matter Ó 2003 Elsevier Inc. All rights reserved. doi:10.1016/S0096-3003(03)00350-3
402
S.T. Bishay, A.A.S. Abo Seliem / Appl. Math. Comput. 151 (2004) 401–410
height h above a plane earth, all field quantities are usually assumed to vary harmonically in time. One of the two well-known methods for solving this steady-state problem is due to Sommerfeld [1], the other is due to Weyi [2]. An accurate and efficient method for computing Sommerfeld integrals is crucial in the analysis of the electromagnetic fields due to radiators and scatterers embedded in layered media. These integrals are encountered in problems involving microstrip antennas, planar transmission lines, microwave devices, geophysical applications, and wave propagation in stratified media. In several recent publications, however, the case is considered where the time dependence of the current in the dipole is impulsive rather than harmonic, e.g. the studies by Banos [3], Kuester and Chang [4], Lindell and Alanen [5] and Dvorak and Mechaik [6]. These techniques can be grouped in the following categories: quasi-analytical solutions which include asymptotic approximations, series expansions and image representations, direct numerical integration, and methods which use numerical techniques. This paper treats the special problem of propagation of an electromagnetic impulse in an abnormal stratification of the upper medium. i.e. in the ionosphere, the so-called surface duct. Kahan and Eckart [7] studied the electromagnetic field due to a dipole placed within a uniform-height duct, and at an arbitrary height from the conducting earth. The duct here is the middle medium of the three (earth–air–ionosphere) media. The model assumes a discontinuous drop of the usually constant relative permittivity at the upper duct boundary. The earth is assumed to be quite plane and ideal conductor, and thus corresponds to the sea in the range of microwaves. In a recent paper [8], a theoretical treatment of the electromagnetic pulse propagation, using this model of the evaporation duct, was carried out for a vertical electric dipole source. The aim of the present work is to extend the steady-state duct propagation theory of Kahan and Eckart to transient excitation when no restrictions on the distance between receiving and transmitting ends are made. Two integral transforms are applied to analyze, the transient field of a vertical electric dipole above a dielectric layer. A Laplace transform in time and a two-dimensional Fourier transform in the horizontal coordinates in space are applied for the Hertz vector in the wave equation. This leads to an integral representation of the wave equation in the transform space. The integral representation of the field is studied from the physical point of view. Also, the integral is evaluated by two mathematical methods: residue and saddle-point methods.
2. Formulation and solution of the problem As shown in Fig. 1, the vertical electric dipole is placed in the upper layer of a two-layer medium. The problem is set up with the interface in die x–y plane,
S.T. Bishay, A.A.S. Abo Seliem / Appl. Math. Comput. 151 (2004) 401–410
403
Fig. 1. Geometrical configuration of the problem.
where the upper and lower layers are denoted as mediums 1 and 2, respectively. The relative dielectric constant of the media are e1 and e2 while the relative permeability, l, is assumed to be the same for all successive media. The earth is assumed to be an infinitely conducting medium which is confined by the plane z ¼ 0. The source of the field is placed in medium 1 at the point x ¼ y ¼ 0. z ¼ d > h, whose moment is given by CF ðtÞez . The vector ez denotes the unit vector in the z-direction, t is the time variable, C is some arbitrary constant to which we give the value ll0 ; l0 being the vacuum permeability. Regarding F ðtÞ, we assume that F ðtÞ ¼ 0 for t 6 0. This guarantees the uniqueness of our solution. The starting point is the wave equation for the z-component of the Hertz vector pðx; y; z; tÞ which we denote by pi ðx; y; z; tÞ, i ¼ 1; 2 as: 1 o2 for t ¼ 2 2 ð1Þ r 2 2 pi ðx; y; z; tÞ ¼ 0 dðx; y; z dÞF ðtÞ for i ¼ 1 Vi ot where vi denotes the phase velocity in medium i. The electric and magnetic-fied generated by this Hertzian dipole can be derived from a Hertzian vector p through the relations. E ¼ grad divp l0 le0 e
o2 p ot2
ð2Þ
and H ¼ e0 e curl
op ot
ð3Þ
In the region z > h, we write p ¼ ðp0 þ p1 Þez
ð4Þ
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S.T. Bishay, A.A.S. Abo Seliem / Appl. Math. Comput. 151 (2004) 401–410
where p0 yields the primary stimulation, while p1 accounts for the secondary stimulation, Similarly, in the region 0 < z < h, we write p ¼ p2 ez
ð5Þ
where p2 yields the refracted field and consists of an incident and reflected waves from the conducting plane z ¼ 0. At any interior point of the appropriate half-space p1 ¼ p1 ðx; y; z; tÞ and in medium p2 ¼ p2 ðx; y; z; tÞ are assumed to be continuous together with their first and second order partial derivatives. The application of Laplace transform in time and a two dimensional Fourier transform in the horizontal coordinates x and y leads, under consideration of the initial, boundary and transition conditions for pi ðx; y; z; tÞ to an integral representation for pi ðx; y; z; sÞ, the Laplace transform of pi ðx; y; z; tÞ, s being variable in die transform space. We get for h < z1 [8] Z Z 2sc1 ðhdÞ sf ðsÞ 1 1 esc1 jzdl Þ sc1 ðzþdÞ ð1 þ C12 e pðx; y; z; sÞ ¼ e 8p2 1 1 c1 ð1 þ C12 e2sc1 h Þ c1 ejsðaxþbyÞ da db
ð6Þ
c1 ða; bÞ c2 ða; bÞ c1 ða; bÞ þ c2 ða; bÞ
ð7Þ
where C12 ða; bÞ ¼ and 1=2 ci ða; bÞ ¼ ða2 þ b2 þ v2 i Þ
ð8Þ
with Reci P 0, i ¼ 1; 2. Here a and b are the variables in the transform space of die two-dimensional Fourier transform f ðsÞ is the Laplace transform of F ðtÞ. A similar expression can be derived for p2 ðx; y; z; sÞ, however, it will not be given here since we restrict our attention to the field in the first medium. The first term of the integrand in (6) is the potential due to the primary field, the second term denotes the diffracted field, and C12 corresponds to Fresnel reflection coefficient.
3. Computation of the integral We are going to discuss the function pðx; y; z; sÞ which is stated in (6) from the mathematical and physical points of view. It follows that by using polar coordinates, i.e. X ¼ q cos /; where
y ¼ q sin /
and
a ¼ k cos /0 ;
b ¼ k sin /0
ð9Þ
S.T. Bishay, A.A.S. Abo Seliem / Appl. Math. Comput. 151 (2004) 401–410
405
q2 ¼ x2 þ y 2 and da db ¼ k dk d/0 ð10Þ Z 1 Z 1 sc1 jzdl sf ðsÞ e esc1 ðzþdÞ ð1 þ C12 e2sc1 ðhdÞ Þ pðq; /; z; sÞ ¼ 8p2 1 1 c1 ð1 þ C12 e2sc1 h Þ c1
k2 ¼ ða2 þ b2 Þ;
0
ejsk cosð// Þ k dk d/0
ð11Þ
The evaluation of the double integral in Eq. (11) is a very difficult task. Therefore, using Bessel integral representation then Z sf ðsÞ 1 esc1 jzdl esc1 ðzþdÞ ð1 þ C12 e2sc1 ðhdÞ Þ pðq; /; z; sÞ ¼ 8p2 1 c1 ð1 þ C12 e2sc1 h Þ c1 kJ0 ðkpsÞ dk
ð12Þ
J0 ðkspÞ is BesselÕs function of order zero, ci ¼ ðk2 K12 Þ1=2 , and v1 i ¼ jKi, i ¼ 1,2. Our aim is to determine the potential pðq; /; z; sÞ at some fixed point ðq; /; zÞ above the duct layer as a function of time. Only the second term in (12) will be delt with, as it represents the secondary field. Therefore Z sf ðsÞ 1F ðksÞ ps ðq; /; z; sÞ ¼ J0 ðkpsÞk dk ð13Þ p 0 MðksÞ where F ðksÞ ¼ esc1 ðzþdÞ ðc1 þ c2 Þ þ ðc1 c2 Þe2sc1 ðhdÞ
ð14Þ
MðksÞ ¼ c1 ðc1 þ c2 Þ þ ðc1 c2 Þe2sc1 h
ð15Þ
and
In order to discuss the integral (13) we have to investigate the singularities in the denominator of the integral, i.e. Eq. (15). The integrand in (12) is not single-valued as it involves square roots of ci ði ¼ 1; 2Þ. The integrand has four values that correspond to the four combinations of signs of ci , and its Riemann surface has also four sheets. By our rule of signs that the real parts of ci are positive, one of the four sheets is singled out as a permissible sheet. To insure the convergence of the integrals, we demand that the path of integration, at infinity, should be on the permissible sheet only. As previously demonstrated by Kahan and Eckart [7], the poles, the branchcuts and the branch-points, which are suitable for operating the integration will be also determined. Besides, the integral, will be evaluated about its poles with the residue method and along the contour W using the saddle-point method. If we observe the singularities of MðksÞ, we find two branch-points at k ¼ K1 and k ¼ K2 and an infinite number of poles on the upper Riemann sheets of the branch-points at k ¼ K1 and k ¼ K2 .
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Fig. 2. Branch-cuts, the steepest descent paths, and the poles in the positive k-plane.
The branch-cuts are chosen such that the upper Riemann sheets have the positive real parts as mentioned above. Fig. 2. illustrates these two branch-cuts and the steepest descent paths. ð1Þ Next, J0 ðkpsÞ is transformed into a Hankel function H0 ðkpsÞ to change the semi-infinite integral in (13) into a fully infinite integral: Z sf ðsÞ 1 F ðksÞ ð1Þ H0 ðkpsÞk dk ps ðq; /; z; sÞ ð16Þ p 1 MðksÞ The previous integral (16) can be evaluated along the contour integration W . from 1 to 1, and its value goes around the poles and the branch-cuts. Eq. (16) then takes the form: X F ðksÞ ð1Þ H ðkpsÞ; kk ps ðq; /; z; sÞ ¼ 2jsf ðsÞ; MðksÞ 0 Z sf ðsÞ F ðksÞ ð1Þ H0 ðkpsÞk dk ð17Þ þ 2p MðksÞ w where nðaÞÕs are eigenvalues of the poles of the integrand and kk is the solution of the pole equation Mðkk sÞ ¼ c1 ðkk sÞfðc1 ðkk sÞ þ c2 ðkk sÞÞ þ ðc1 ðkk sÞ c2 ðkk sÞÞe2sc1 ðkksÞhg ¼ 0 ð18Þ This determines the poles of the integral, substituting the value k ¼ kk in the first term of (17), where " # ð1Þ H0 ðkpsÞF ðkk sÞkk k D ðkk sÞ ¼ 2jsf ðsÞ ð19Þ Mðkk sÞ M 0 ðkk sÞ ¼
dMðksÞ dk
ð20Þ k¼kk
S.T. Bishay, A.A.S. Abo Seliem / Appl. Math. Comput. 151 (2004) 401–410
407
Next, we can estimate the second term in (17) by using the saddle-point method. In this work we treat the far field so that the Hankel function can be transformed into the asymptotic expansion, as is well known [9] sffiffiffiffiffiffi 2 jðqðnþ1=2Þp=2Þ ð1Þ e Hn ðqÞ ¼ ð21Þ pq From (14) and (15), we can get the following: sffiffiffiffi Z sffiffiffiffiffiffiffiffi 1 jðkpsp=4Þ k e AðksÞesc1 ðzþdÞ dk I¼ pqs c1
ð22Þ
where AðksÞ ¼
½ðc1 þ c2 Þ þ ðc1 c2 Þesc1 2ðhdÞ ½ðc1 þ c2 Þ þ ðc1 c2 Þe2sc1h
ð23Þ
we let q ¼ R sin a; ðz þ dÞ ¼ R cos a
and
c2i ¼ k2 ki2
Z rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 I¼ eiRsgðksÞ /ðksÞejp=4 dk psR sin a
ð24Þ
where gðksÞ ¼ k sin a j
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k2 k12 cos a 1=2
and /ðksÞ ¼ AðksÞðk=ðk2 K12 ÞÞ . Thus the saddle-point k ¼ ks for the integral is determined by [10] jk cos a g0 ðksÞ ¼ sin a qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k2 k12 Therefore k ¼ k1 sin a Then, the integral (24) is evaluated as follows: I ¼ Aðks sÞ
2ejsk1 R SR
ð25Þ
408
S.T. Bishay, A.A.S. Abo Seliem / Appl. Math. Comput. 151 (2004) 401–410
4. Numerical results In (25) we, arrange I by using the inverse Laplace transform and the denominator ð1 þ C12 e2sc1 h Þ 1 is expanded using the geometrical series, we get: " # 1 1 X 1 X n nþ1 n n ð 1Þ C12 dðt t1 Þ þ ð 1Þ C12 dðt t2 Þ I¼ pR n¼0 n¼0
Fig. 3. (a)–(c): Variation of I with time at q ¼ 5000 m.
S.T. Bishay, A.A.S. Abo Seliem / Appl. Math. Comput. 151 (2004) 401–410
409
where t1 ¼ fV11 R þ ð2ðn þ 1Þh dÞc1 g t2 ¼ fV11 R þ ð2nhc1 g 1 c2 =c1 C12 ¼ ; 1 þ c2 =c1
c1 ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k2 ki2
ð26Þ
we take e1 ¼ 1 and e2 ¼ 1:0004. The height of the primary source and the point of observation has been taken to be z ¼ d ¼ 20 m and the duct height h ¼ 15 m, where q denotes the spherical distance between the source and the point of observation i.e. q ¼ 5000 m. Numerically integrated for both type of sources n ¼ 0, 5 and 10 are given in Fig. 3(a)–(c), which show the integral along the contour ‘‘w’’ as a function of time in different observation points. Thus, we have estimated the upward wave from from the upper surface of the duct to the observing point. By the same method we can determine die field in the duct.
5. Conclusion A theoretical study for computing the electric and magnetic field from a Hertzian vector in die ionosphere is presented. The solution is valid for arbitrary distances between receiving and transmitting ends for a source position in the medium of lesser refractive index. The residue and saddle-point method are used to compute the problem.
References [1] A. Sommerfeld, Uber die Ausbreitung der Wellen in der drahtlosen telegraphic, Ann. d. Phys. 28 (1909) 665–736. [2] H. Weyl, Ausbreitung eiektromagnetischer, Wellen uber einem ebenen Leiter, Ann. d. Phys. 60 (1919) 481–500. [3] A. Banos, Dipole Radiation in the Presence of a Conducting Half-space, Pergamon Press, New York, 1966. [4] E.F. Kuester, D.C. Chang, Evaluation of Sommerfeld integrals associated with dipole sources above Earth, Sci, Rep.43, Electromagnetic Laboratory, Department of Electrical Engineering, University of Colorcedo, Boulder, 1979. [5] I.V. Lindell, E. Alanen, Exact image theory for the Sommerfeld half-space problem. Part II. Vertical electric dipole, IEEE Trans. Antennas Propag. Ap-32 (1984) 841–847. [6] S.L. Dvorak, M.M. Mechaik, Application of the contour transformation method to a vertical electric dipole over earth, Radio Sci. 28 (3) (1993) 309–317. [7] T. Kahen, G. Eckart, Theorie de la propagaon des ondes elictromagnetiques dans le giude dÕonde atmospherique, Annales de Physique 5 (1950) 641–705.
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[8] S.T. Bishay, Electromagnetic fields of transient signals above evaporation duct, Proc. Indian Natn. Sci Acad. 51A (6) (1985) 954–958. [9] A. Sommerfeld, Partial differential equations in physics, Academic Press, New York and London, 1964. [10] D.S. Jones, The Theory of Electromagnetism, Pergamon Press, London, England, 1964.
Transient response of a vertical electric dipole above a two-layer medium Samira T. Bishay a, Adel A.S. Abo Seliem a
b,*
Department of Mathematics, Faculty of Science, Ain Shams University, Abassia, Cairo, Egypt b Department of Mathematics, Faculty of Education, Tanta University, Kafer El Sheiek Branch, Egypt
Abstract The paper presents a method which allows the calculation of the atmospheric distortion of radar pulses, provided that the influence of the atmosphere is to transfer the transmitted signal through a duct. The study thus deals with the investigation of the pulses signal from a vertical electric dipole above an evaporation duct. Two integral transforms of the wave equation of Hertzian vector––a Laplace transform in time and a two dimensional Fourier transform in the horizontal coordinates in space––are applied. This leads to an integral representation of the solution of the wave equation in transform space, considering initial, boundary and transition conditions. This integral representation determines the electromagnetic field anywhere in the ionosphere. Saddlepoint and residue methods are used to compute the integral. Ó 2003 Elsevier Inc. All rights reserved. Keywords: Vertical dipole; Laplace transform; Transient response
1. Introduction Radiometeorlogists have extensively studied the correlation between the propagation of electromagnetic waves in the atmosphere and meteorological conditions. Today, the interest is mainly concerned with the effect of the atmosphere on target identification and imaging. Historically, in the problem of the electromagnetic radiation from a vertical dipole situated at a certain * Corresponding author. Address: Department of Mathematics, Jizan Teachers College, Abo Areesh, P.O. Box 203, Jizan, Saudi Arabia. E-mail address: [email protected] (A.A.S. Abo Seliem).
0096-3003/$ - see front matter Ó 2003 Elsevier Inc. All rights reserved. doi:10.1016/S0096-3003(03)00350-3
402
S.T. Bishay, A.A.S. Abo Seliem / Appl. Math. Comput. 151 (2004) 401–410
height h above a plane earth, all field quantities are usually assumed to vary harmonically in time. One of the two well-known methods for solving this steady-state problem is due to Sommerfeld [1], the other is due to Weyi [2]. An accurate and efficient method for computing Sommerfeld integrals is crucial in the analysis of the electromagnetic fields due to radiators and scatterers embedded in layered media. These integrals are encountered in problems involving microstrip antennas, planar transmission lines, microwave devices, geophysical applications, and wave propagation in stratified media. In several recent publications, however, the case is considered where the time dependence of the current in the dipole is impulsive rather than harmonic, e.g. the studies by Banos [3], Kuester and Chang [4], Lindell and Alanen [5] and Dvorak and Mechaik [6]. These techniques can be grouped in the following categories: quasi-analytical solutions which include asymptotic approximations, series expansions and image representations, direct numerical integration, and methods which use numerical techniques. This paper treats the special problem of propagation of an electromagnetic impulse in an abnormal stratification of the upper medium. i.e. in the ionosphere, the so-called surface duct. Kahan and Eckart [7] studied the electromagnetic field due to a dipole placed within a uniform-height duct, and at an arbitrary height from the conducting earth. The duct here is the middle medium of the three (earth–air–ionosphere) media. The model assumes a discontinuous drop of the usually constant relative permittivity at the upper duct boundary. The earth is assumed to be quite plane and ideal conductor, and thus corresponds to the sea in the range of microwaves. In a recent paper [8], a theoretical treatment of the electromagnetic pulse propagation, using this model of the evaporation duct, was carried out for a vertical electric dipole source. The aim of the present work is to extend the steady-state duct propagation theory of Kahan and Eckart to transient excitation when no restrictions on the distance between receiving and transmitting ends are made. Two integral transforms are applied to analyze, the transient field of a vertical electric dipole above a dielectric layer. A Laplace transform in time and a two-dimensional Fourier transform in the horizontal coordinates in space are applied for the Hertz vector in the wave equation. This leads to an integral representation of the wave equation in the transform space. The integral representation of the field is studied from the physical point of view. Also, the integral is evaluated by two mathematical methods: residue and saddle-point methods.
2. Formulation and solution of the problem As shown in Fig. 1, the vertical electric dipole is placed in the upper layer of a two-layer medium. The problem is set up with the interface in die x–y plane,
S.T. Bishay, A.A.S. Abo Seliem / Appl. Math. Comput. 151 (2004) 401–410
403
Fig. 1. Geometrical configuration of the problem.
where the upper and lower layers are denoted as mediums 1 and 2, respectively. The relative dielectric constant of the media are e1 and e2 while the relative permeability, l, is assumed to be the same for all successive media. The earth is assumed to be an infinitely conducting medium which is confined by the plane z ¼ 0. The source of the field is placed in medium 1 at the point x ¼ y ¼ 0. z ¼ d > h, whose moment is given by CF ðtÞez . The vector ez denotes the unit vector in the z-direction, t is the time variable, C is some arbitrary constant to which we give the value ll0 ; l0 being the vacuum permeability. Regarding F ðtÞ, we assume that F ðtÞ ¼ 0 for t 6 0. This guarantees the uniqueness of our solution. The starting point is the wave equation for the z-component of the Hertz vector pðx; y; z; tÞ which we denote by pi ðx; y; z; tÞ, i ¼ 1; 2 as: 1 o2 for t ¼ 2 2 ð1Þ r 2 2 pi ðx; y; z; tÞ ¼ 0 dðx; y; z dÞF ðtÞ for i ¼ 1 Vi ot where vi denotes the phase velocity in medium i. The electric and magnetic-fied generated by this Hertzian dipole can be derived from a Hertzian vector p through the relations. E ¼ grad divp l0 le0 e
o2 p ot2
ð2Þ
and H ¼ e0 e curl
op ot
ð3Þ
In the region z > h, we write p ¼ ðp0 þ p1 Þez
ð4Þ
404
S.T. Bishay, A.A.S. Abo Seliem / Appl. Math. Comput. 151 (2004) 401–410
where p0 yields the primary stimulation, while p1 accounts for the secondary stimulation, Similarly, in the region 0 < z < h, we write p ¼ p2 ez
ð5Þ
where p2 yields the refracted field and consists of an incident and reflected waves from the conducting plane z ¼ 0. At any interior point of the appropriate half-space p1 ¼ p1 ðx; y; z; tÞ and in medium p2 ¼ p2 ðx; y; z; tÞ are assumed to be continuous together with their first and second order partial derivatives. The application of Laplace transform in time and a two dimensional Fourier transform in the horizontal coordinates x and y leads, under consideration of the initial, boundary and transition conditions for pi ðx; y; z; tÞ to an integral representation for pi ðx; y; z; sÞ, the Laplace transform of pi ðx; y; z; tÞ, s being variable in die transform space. We get for h < z1 [8] Z Z 2sc1 ðhdÞ sf ðsÞ 1 1 esc1 jzdl Þ sc1 ðzþdÞ ð1 þ C12 e pðx; y; z; sÞ ¼ e 8p2 1 1 c1 ð1 þ C12 e2sc1 h Þ c1 ejsðaxþbyÞ da db
ð6Þ
c1 ða; bÞ c2 ða; bÞ c1 ða; bÞ þ c2 ða; bÞ
ð7Þ
where C12 ða; bÞ ¼ and 1=2 ci ða; bÞ ¼ ða2 þ b2 þ v2 i Þ
ð8Þ
with Reci P 0, i ¼ 1; 2. Here a and b are the variables in the transform space of die two-dimensional Fourier transform f ðsÞ is the Laplace transform of F ðtÞ. A similar expression can be derived for p2 ðx; y; z; sÞ, however, it will not be given here since we restrict our attention to the field in the first medium. The first term of the integrand in (6) is the potential due to the primary field, the second term denotes the diffracted field, and C12 corresponds to Fresnel reflection coefficient.
3. Computation of the integral We are going to discuss the function pðx; y; z; sÞ which is stated in (6) from the mathematical and physical points of view. It follows that by using polar coordinates, i.e. X ¼ q cos /; where
y ¼ q sin /
and
a ¼ k cos /0 ;
b ¼ k sin /0
ð9Þ
S.T. Bishay, A.A.S. Abo Seliem / Appl. Math. Comput. 151 (2004) 401–410
405
q2 ¼ x2 þ y 2 and da db ¼ k dk d/0 ð10Þ Z 1 Z 1 sc1 jzdl sf ðsÞ e esc1 ðzþdÞ ð1 þ C12 e2sc1 ðhdÞ Þ pðq; /; z; sÞ ¼ 8p2 1 1 c1 ð1 þ C12 e2sc1 h Þ c1
k2 ¼ ða2 þ b2 Þ;
0
ejsk cosð// Þ k dk d/0
ð11Þ
The evaluation of the double integral in Eq. (11) is a very difficult task. Therefore, using Bessel integral representation then Z sf ðsÞ 1 esc1 jzdl esc1 ðzþdÞ ð1 þ C12 e2sc1 ðhdÞ Þ pðq; /; z; sÞ ¼ 8p2 1 c1 ð1 þ C12 e2sc1 h Þ c1 kJ0 ðkpsÞ dk
ð12Þ
J0 ðkspÞ is BesselÕs function of order zero, ci ¼ ðk2 K12 Þ1=2 , and v1 i ¼ jKi, i ¼ 1,2. Our aim is to determine the potential pðq; /; z; sÞ at some fixed point ðq; /; zÞ above the duct layer as a function of time. Only the second term in (12) will be delt with, as it represents the secondary field. Therefore Z sf ðsÞ 1F ðksÞ ps ðq; /; z; sÞ ¼ J0 ðkpsÞk dk ð13Þ p 0 MðksÞ where F ðksÞ ¼ esc1 ðzþdÞ ðc1 þ c2 Þ þ ðc1 c2 Þe2sc1 ðhdÞ
ð14Þ
MðksÞ ¼ c1 ðc1 þ c2 Þ þ ðc1 c2 Þe2sc1 h
ð15Þ
and
In order to discuss the integral (13) we have to investigate the singularities in the denominator of the integral, i.e. Eq. (15). The integrand in (12) is not single-valued as it involves square roots of ci ði ¼ 1; 2Þ. The integrand has four values that correspond to the four combinations of signs of ci , and its Riemann surface has also four sheets. By our rule of signs that the real parts of ci are positive, one of the four sheets is singled out as a permissible sheet. To insure the convergence of the integrals, we demand that the path of integration, at infinity, should be on the permissible sheet only. As previously demonstrated by Kahan and Eckart [7], the poles, the branchcuts and the branch-points, which are suitable for operating the integration will be also determined. Besides, the integral, will be evaluated about its poles with the residue method and along the contour W using the saddle-point method. If we observe the singularities of MðksÞ, we find two branch-points at k ¼ K1 and k ¼ K2 and an infinite number of poles on the upper Riemann sheets of the branch-points at k ¼ K1 and k ¼ K2 .
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Fig. 2. Branch-cuts, the steepest descent paths, and the poles in the positive k-plane.
The branch-cuts are chosen such that the upper Riemann sheets have the positive real parts as mentioned above. Fig. 2. illustrates these two branch-cuts and the steepest descent paths. ð1Þ Next, J0 ðkpsÞ is transformed into a Hankel function H0 ðkpsÞ to change the semi-infinite integral in (13) into a fully infinite integral: Z sf ðsÞ 1 F ðksÞ ð1Þ H0 ðkpsÞk dk ps ðq; /; z; sÞ ð16Þ p 1 MðksÞ The previous integral (16) can be evaluated along the contour integration W . from 1 to 1, and its value goes around the poles and the branch-cuts. Eq. (16) then takes the form: X F ðksÞ ð1Þ H ðkpsÞ; kk ps ðq; /; z; sÞ ¼ 2jsf ðsÞ; MðksÞ 0 Z sf ðsÞ F ðksÞ ð1Þ H0 ðkpsÞk dk ð17Þ þ 2p MðksÞ w where nðaÞÕs are eigenvalues of the poles of the integrand and kk is the solution of the pole equation Mðkk sÞ ¼ c1 ðkk sÞfðc1 ðkk sÞ þ c2 ðkk sÞÞ þ ðc1 ðkk sÞ c2 ðkk sÞÞe2sc1 ðkksÞhg ¼ 0 ð18Þ This determines the poles of the integral, substituting the value k ¼ kk in the first term of (17), where " # ð1Þ H0 ðkpsÞF ðkk sÞkk k D ðkk sÞ ¼ 2jsf ðsÞ ð19Þ Mðkk sÞ M 0 ðkk sÞ ¼
dMðksÞ dk
ð20Þ k¼kk
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407
Next, we can estimate the second term in (17) by using the saddle-point method. In this work we treat the far field so that the Hankel function can be transformed into the asymptotic expansion, as is well known [9] sffiffiffiffiffiffi 2 jðqðnþ1=2Þp=2Þ ð1Þ e Hn ðqÞ ¼ ð21Þ pq From (14) and (15), we can get the following: sffiffiffiffi Z sffiffiffiffiffiffiffiffi 1 jðkpsp=4Þ k e AðksÞesc1 ðzþdÞ dk I¼ pqs c1
ð22Þ
where AðksÞ ¼
½ðc1 þ c2 Þ þ ðc1 c2 Þesc1 2ðhdÞ ½ðc1 þ c2 Þ þ ðc1 c2 Þe2sc1h
ð23Þ
we let q ¼ R sin a; ðz þ dÞ ¼ R cos a
and
c2i ¼ k2 ki2
Z rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 I¼ eiRsgðksÞ /ðksÞejp=4 dk psR sin a
ð24Þ
where gðksÞ ¼ k sin a j
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k2 k12 cos a 1=2
and /ðksÞ ¼ AðksÞðk=ðk2 K12 ÞÞ . Thus the saddle-point k ¼ ks for the integral is determined by [10] jk cos a g0 ðksÞ ¼ sin a qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k2 k12 Therefore k ¼ k1 sin a Then, the integral (24) is evaluated as follows: I ¼ Aðks sÞ
2ejsk1 R SR
ð25Þ
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4. Numerical results In (25) we, arrange I by using the inverse Laplace transform and the denominator ð1 þ C12 e2sc1 h Þ 1 is expanded using the geometrical series, we get: " # 1 1 X 1 X n nþ1 n n ð 1Þ C12 dðt t1 Þ þ ð 1Þ C12 dðt t2 Þ I¼ pR n¼0 n¼0
Fig. 3. (a)–(c): Variation of I with time at q ¼ 5000 m.
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409
where t1 ¼ fV11 R þ ð2ðn þ 1Þh dÞc1 g t2 ¼ fV11 R þ ð2nhc1 g 1 c2 =c1 C12 ¼ ; 1 þ c2 =c1
c1 ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k2 ki2
ð26Þ
we take e1 ¼ 1 and e2 ¼ 1:0004. The height of the primary source and the point of observation has been taken to be z ¼ d ¼ 20 m and the duct height h ¼ 15 m, where q denotes the spherical distance between the source and the point of observation i.e. q ¼ 5000 m. Numerically integrated for both type of sources n ¼ 0, 5 and 10 are given in Fig. 3(a)–(c), which show the integral along the contour ‘‘w’’ as a function of time in different observation points. Thus, we have estimated the upward wave from from the upper surface of the duct to the observing point. By the same method we can determine die field in the duct.
5. Conclusion A theoretical study for computing the electric and magnetic field from a Hertzian vector in die ionosphere is presented. The solution is valid for arbitrary distances between receiving and transmitting ends for a source position in the medium of lesser refractive index. The residue and saddle-point method are used to compute the problem.
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[8] S.T. Bishay, Electromagnetic fields of transient signals above evaporation duct, Proc. Indian Natn. Sci Acad. 51A (6) (1985) 954–958. [9] A. Sommerfeld, Partial differential equations in physics, Academic Press, New York and London, 1964. [10] D.S. Jones, The Theory of Electromagnetism, Pergamon Press, London, England, 1964.