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ELECTRODYNAMICS AND WAVE PROPAGATION

The Electromagnetic Field of a Horizontal Antenna under the Interface between Two Media E. D. Tereshchenkoa, * and P. E. Tereshchenkoa, b a

Polar Geophysical Institute, Russian Academy of Sciences, Murmansk, 183010 Russia Pushkov Institute of Terrestrial Magnetism, the Ionosphere, and the Radio-Wave Propagation, St. Petersburg Branch, Russian Academy of Sciences, St. Petersburg, 199034 Russia *e-mail: [email protected]

b

Received December 22, 2016

Abstract⎯Excitation of electromagnetic waves in a two-layer medium by a horizontal antenna or an antenna with flooded electrodes that is situated on the water surface is investigated. The region below the interface between two media is considered. The general solution of the problem is presented in the form of the well studied modified Bessel functions within the framework of a quasi-stationary approximation. In contrast to the scheme, which is widely applied in geoelectricity and connected with the calculation of the field on an interface, the Watson rather than Fok integrals are used. The Watson integrals make it possible to determine the field both on the interface and outside of it. The limit passage is made to the values of the potential and fields, when the interface is approached from below, as well as in the lower conducting medium at the distance equal to the thickness of several skin layers from a source. DOI: 10.1134/S1064226918040125

INTRODUCTION Consider the structure of the field formed by a linear horizontal antenna, which is grounded or situated on the water surface and has flooded electrodes. In practical applications (communication, operation, geological exploration), the information about the character of the field change in the region under the interface is often necessary. We use the two-layer model of a medium with a plane interface to simplify calculations. The main attention is given to finding formulas that are valid for the whole lower half-space including the neighborhood of a source. Consider also the limit passage to the distance exceeding the thickness of several skin layers of the conducting medium. We rely on the classic approach used in works [1, 2] for calculating the medium interface field, which is the main quantity applied in the geological exploration. 1. THE ELECTRIC VECTOR POTENTIAL. THE QUASI-STATIONARY APPROXIMATION Consider the radiation of a grounded antenna with flooded electrodes in a two-layer medium. The antenna has the length 2L and is fed by the current with the harmonical dependence on time exp( −i ωt ). The system of coordinates is chosen as follows. The center of Cartesian coordinates is placed in the antenna center, and axes z , x , and y are oriented upwards, along the antenna, and across the antenna, respectively. We denote the distance to the observation

point R , the distance on the plane ( x, y,0) from the antenna center ρ, and the distance from an arbitrary antenna point ρη . We consider that the medium in the region z > 0 is practically nonconducting (i.e., σ = +0 and the presence of + at the zero indicates a small absorption) with the permittivity ε0 10−9 36π F/m and the permeability μ0 = 4π × 10−7 H/m. We suppose that the electromagnetic parameters of the region of z < 0 are ε , μ0 , and σ1 . The problem of excitation the electromagnetic field by exterior current J is reduced to the solution of the Helmholtz equations for electric vector potential A with the corresponding boundary conditions [2–4]. Since we consider the radiation of monochromatic waves, it is convenient hereinafter to use the equations for complex amplitudes corresponding to monochromatic components, i.e., , , and A ∼ Aexp( −i ωt ) E ∼Eexp( −i ωt ) H ∼ Hexp( −i ωt ) , where E and H are the electric and magnetic fields, respectively. Taking into account that the source is oriented along axis x (Fig. 1), we can rep resent vector A in the form of two components j j j A( ) = Ax( )ex + Az( )ez , where the sign j = 0,1 indicates the medium and ex and ez are the unit vectors oriented along axes x and z , respectively.

335

336

E. D. TERESHCHENKO, P. E. TERESHCHENKO

ez ey

ε0, μ0, σ0

(x, y, 0)

ρ ρη η

0

–L ε1, μ0, σ1

L

ex

z=0

R (x, y, z) Fig. 1. Geometry of the problem.

The subsequent step is the solution of the system of equations j j (1) ∇2 A( ) + k 2j A( ) = −J ext, j = 0,1 with the boundary conditions 0 1 ( 0) (1) ∂Ax( ) ∂A ( ) A =A = x , , z =0 z =0 ∂z z = 0 ∂z z = 0 (2) (1) 1 divA (0) 1 . = 2 divA z =0 z =0 k02 k1 Wave numbers k0 and k1 entering the system of Eqs. (1) and (2) are determined by the expressions ε σ k0 ω (1 + i 0) ω , k1 = ω 1 + i 1 = ω ε1'. c c c ε0 ωε0 c Let us find the solution to system (1) with boundary conditions (2) for a point grounded (flooded) horizontal source located at the coordinate origin. In this case, ( 0) J ext = J Δ x δ ( x ) δ ( y ) δ ( z − 0) ex , (3) (1) J ext = J Δ x δ ( x ) δ ( y ) δ ( z + 0) ex , where δ is the delta function, J is the current, J Δ x is the dipole moment and Δ x is the dipole length tending to an infinitely small quantity.

Ax( j )

J Δx = 4π

∞

×

∫ 0

(

k02

−λ + 2

∞

∫

The transition to the fields corresponding to the excitation by a linear antenna of length 2L is realized as the result of summation over the antenna length. In ( j ) particular, electric field % excited by a linear antenna is L ( j ) ( j) % = E (ρη, z ) = E ( j ) (ρη, z ) d η, (4)

∑

(

) J (λρ) λd λ,

−λ + − λ2 J Δx ∂ 2 Az( j ) = k0 − k12 4 π ∂x 2 exp i k 2j − λ 2 z

k12

0

2

( − λ )( k 2

2 1

k12

(

k02

J Δ x →J

where E ( j ) (ρη, z ) is the field of the dipole located at point η of the antenna. The magnetic field satisfies the analogous relationship. It is convenient to obtain the solution of system (1) with boundary conditions (2) in cylindrical coordinate system ( z, ρ, ϕ) in the form of the decomposition of the functions cos (m 2)ϕ, m = 0,1..., which form a complete system on the interval ( 0,2π] [6]. The further steps are well known [2, 5] and connected with the definition, taking into account the boundary conditions, and the corresponding functions, which enter the decomposition and depend on ρ and z . Therefore, we omit the intermediate transformations and present the final result of calculation of Ax( j ) and Az( j ) for source (3)

2i exp i k 2j − λ2 z

k02

∫

−L

)

0

)

− λ 2 + k02 k12 − λ2

j = 0,1,

(5)

)

J 0 ( λρ) λ d λ,

where J0(λρ) is the Bessel function. JOURNAL OF COMMUNICATIONS TECHNOLOGY AND ELECTRONICS

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2018

THE ELECTROMAGNETIC FIELD OF A HORIZONTAL ANTENNA

The branch of the square root is fixed on the basis of the condition (6) Im k 2j − λ 2 > 0. We transform formulas (5) for a more compact form and for having an analogy with the works concerning geoelectricity [2]. To this end, we introduce the new denotation

k j = i û j , ν j = −i k j − λ = û j + λ , (7) J Δ x ( j) J Δ x ∂ ( j) ( j) ( j) Ax = F (ρ, z ) . Π (ρ, z ) , Az = 4π 4 π ∂x It follows from relationship (6) that Reν j > 0 , and relationships (5) and (7) imply that 2

Π( j ) (ρ, z ) =

∞

∫ν 0

F

( j)

(ρ, z ) = (

0

2

2

2 exp −ν z J λρ λ d λ, ( j ) 0( ) + ν1 ∞

û12

2

−

û02

) ∫ (ν 0

0

2 2 2 + ν1 ) û0ν1 + û1 ν0

(

(8)

)

× exp ( −ν j z ) J 0 ( λρ) λ d λ, j = 0,1. System (8) is basic when the electric vector potential and electromagnetic fields are calculated. Representation (8) has the form of integrals containing oscillating function J 0 ( λρ). There are well developed methods, in particular, the Longman method [5], for calculating such integrals. However, for practically important case of radiation of extremely low-frequency and lower frequency wave ranges, formulas (5) and (8) can be simplified and represented with the use of well studied functions. Let us consider the quasi-stationary approximation. It is considered in this approximation that û0 = 0 . It physically corresponds to the case that the Maxwell displacement current is neglected in comparison with the conduction current. When the waves of a very low frequency are used, the condition û20 û12 ! 1 is well fulfilled. This makes it possible to replace û 0 by zero and use the replacement ν0 → λ in the region under interface (8). Hence, we obtain

Π (ρ, z ) ≡ Π(1) (ρ, z )

∞ û0 = 0

=

2 exp(ν1z) J 0 ( λρ) λ d λ, ν + λ) 0 ( 1

∫

(9) ν 2 exp( z ) (1) 1 F (ρ, z ) ≡ F (ρ, z ) = J 0 ( λρ) d λ. û0 = 0 ν + λ ( ) 1 0 Integrals (9) determining Π (ρ, z ) and F (ρ, z ) are the derivatives of two Watson integrals. One of these integrals is connected with the incident field ∞

∫

∞

exp (ν1z ) exp(−û1R) J 0 ( λρ) λ d λ = ν R 1 0 exp(ik1R) = , R = ρ2 + z 2 , R

∫

(10)

337

and the other is connected with the product of the modified Bessel functions [6] ∞

exp(ν1z) J 0 ( λρ) d λ = I 0 ( r+ ) K 0 ( r− ) , ν 1 0

∫

(11)

where r+ = ( û1 2 ) ( R + z ), r− = ( û1 2 ) ( R − z ) , and I 0 and K 0 are the modified Bessel functions. With allowance for (10) and (11), the calculations give from (9) that

⎡ 2 exp(−û1R) Π (ρ, z ) = 22 ⎢ ∂ 2 R û1 ⎣∂z 3 ⎛ ⎞ ⎤ − ⎜ ∂ 3 − û12 ∂ ⎟ I 0 ( r+ ) K 0 ( r− )⎥ , ∂z ⎠ ⎝ ∂z ⎦ exp(−û1R) ∂ 2 ⎡ ⎤ + 2 I 0 ( r+ ) K 0 ( r− )⎥ . F (ρ, z ) = 22 ⎢− ∂ R û1 ⎣ ∂z ∂z ⎦ Let us introduce the denotations Ax (ρ, z ) ≡ Ax

(1)

, Az (ρ, z ) ≡ Az

(1)

û0 = 0

û0 = 0

.

Then, expanding the partial derivatives with respect to z and ρ and taking into account (7), we obtain the following expressions for the electric vector potential in the lower medium:

Ax (ρ, z ) =

2 J Δx ⎛ 2 ⎛ z d ⎞ 1 d exp (−û1R) ⎜ 2 ⎜1 + ⎟ 4π ⎝ û1 ⎝ R dR ⎠ R dR R

2 ⎛ ⎞ ρ2 z − z2 ⎜1 − 3 z 2 ⎟ I 0 ( r+ ) K 0 ( r− ) + 3 4 I1 ( r+ ) K1 ( r− ) R ⎝ R ⎠ R (12) 2 ⎡ ⎛ ⎞ 2 2 z⎤ 1 z z 1 − ⎜1 − 3 2 ⎟ − û1 ρ ⎥ I 0 ( r+ ) K1 ( r− ) + R ⎝ R⎦ R ⎠ û1R 2 ⎢⎣ 2 ⎞ ⎡ ⎛ ⎞ ⎤ + 1 2 ⎢ 1 + z ⎜1 − 3 z 2 ⎟ + û12ρ2 z ⎥ I1 ( r+ ) K 0 ( r− ) ⎟ , R ⎝ R⎦ R ⎠ û1R ⎣ ⎠

( ) ( )

Az (ρ, z ) =

J Δ x x ⎛ −2z d 1 d exp (−û1R) ⎜ 4π R ⎝ û12 dR R dR R

2 2ρ2 − z 2 I1 ( r+ ) K1 ( r− ) − 3 z 3 I 0 ( r+ ) K 0 ( r− ) − R R3 ⎡3z ( R − z ) û1z 2 ⎤ +⎢ − 2 ⎥ I 0 ( r+ ) K1 ( r− ) 4 R ⎦ ⎣ û1R ⎡3z ( R + z ) û1z 2 ⎤ ⎞ +⎢ + 2 ⎥ I1 ( r+ ) K 0 ( r− ) ⎟ , 4 R ⎦ ⎣ û1R ⎠

(13)

where I1 and K1 are the modified Bessel functions with the index equal to unity. Formulas (12) and (13) are valid for the whole lower half-space and describe the change of the electric vector potential components for any values of x , y , and z . Consider two limit cases. The first one is connected with the interface, i.e., with the value z = 0, and the second case is connected with moving away

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E. D. TERESHCHENKO, P. E. TERESHCHENKO

from the source at the distance r+ @ 1 . Let us start from the first case. So, we substitute z = 0 into (12). As a result, we obtain

Ax (ρ, z = 0 ) =

J Δ x ⎛ 2 d 1 d exp (−û1ρ) ⎜ ρ 4π ⎝ û12 d ρ ρ d ρ

ûρ ûρ ûρ ûρ ⎞ + 1 2 ⎡I 0 ⎜⎛ 1 ⎟⎞ K1 ⎜⎛ 1 ⎟⎞ + I1 ⎜⎛ 1 ⎟⎞ K 0 ⎜⎛ 1 ⎟⎞⎤ ⎟ . ⎢ ⎝ 2 ⎠ ⎝ 2 ⎠⎦⎥ ⎠ û1ρ ⎣ ⎝ 2 ⎠ ⎝ 2 ⎠ Taking into account that the expression in the square brackets is equal to 2 û1ρ [6], we obtain opening the derivative with respect to ρ that

J Δx 1 − (1 + û1ρ) exp (−û1ρ)]. (14) 2 3[ 2πû1 ρ The other component of the electric vector potential at z = 0 is described by the following formula: Ax (ρ, z = 0 ) =

J Δ x x ⎛ û1 ⎞ ⎛ û1 ⎞ I1 ⎜ ρ ⎟ K1 ⎜ ρ ⎟ . (15) 4 π ρ2 ⎝ 2 ⎠ ⎝ 2 ⎠ Expressions (14) and (15) are often given in geoelectricity literature [2], where the main measured quantities are the fields at an interface. According to its structure, expression (14) describes the combination of two waves. One of them propagates above the interface, and the other one propagates below the interface. The absence of the second component in (14) is due to the fact that we consider the quasi-stationary approximation according to which exp ( −û0ρ) is equal to unity. The interpretation of (15) is rather more complex. However, if the region û1 2ρ @ 1 is considered, we can use the asymptotic representations for I1 and K1 [6] Az (ρ, z = 0) = −

−û ρ exp ⎛⎜ 1 ⎞⎟ û ⎝ 2 ⎠ ⎡1 − 3 + …⎤ I1 ⎛⎜ 1 ρ ⎞⎟ ∼ ⎥ ⎝2 ⎠ πû1ρ ⎢⎣ 4û1ρ ⎦ −û1ρ π ⎛ ⎞ exp ⎜ +i ⎟ ⎝ 2 2 ⎠ ⎡1 + 3 + …⎤ , (16) − ⎢ 4û ρ ⎥ πû1ρ ⎣ ⎦ 1 ⎤ û −û ρ ⎡ K1 ⎜⎛ 1 ρ ⎟⎞ ∼ π exp ⎜⎛ 1 ⎟⎞ ⎢1 + 3 + …⎥ , ⎝2 ⎠ ⎝ 2 ⎠ ⎣ 4û1ρ û1ρ ⎦ where … corresponds to the terms with the order 2 (1 (û1ρ)) . Then, we can obtain the expression for Az (ρ, z = 0) as the result of substituting (16) into (15) J Δx x 2π û1ρ3 ⎡ ⎛ ⎞ ⎤ × ⎢(1 + …) − i ⎜1 + 3 + …⎟ exp(−û1ρ)⎥ , û 2 ρ ⎣ ⎝ ⎠ ⎦ 1 û1ρ @ 1. 2 Az (ρ, z = 0) ∼ −

Therefore, as we move away from the source, the z component of the electric vector potential acquires the form similar to (14). We take into account that the conductivity in the lower medium is nonzero, i.e., Reû1 > 0 . Then, if the condition Reû1ρ @ 1 is fulfilled, we obtain from (14) and (15)

J Δx , 2 3 2πû1 ρ (18) J Δx x , R e û @ 1. Az (ρ, z = 0) ∼ − ρ 1 2πû1 ρ3 Consider the behavior of the potential in the region z ≤ 0 when Rer+ @ 1 . Since z ≤ 0, then, Rer− @ 1. Let us substitute the asymptotic expansions of the modified Bessel functions [6] into expressions (12) and (13). Then, Ax (ρ, z = 0) ∼

J Δx exp(û1z ), 2πû12ρ3 (19) J Δx x exp(û1z ), Rer+ @ 1. Az (ρ, z = 0) ∼ − 2πû1 ρ3 Comparing expressions (18) and (19), we see that, in the region Rer+ @ 1 , the components of the electric vector potential are mainly formed by the field in the upper medium. This field penetrates deep and exponentially decays. Ax (ρ, z = 0 ) ∼

2. THE ELECTROMAGNETIC FIELD UNDER THE INTERFACE The electric field in the region z ≤ 0 is connected 1 with electric vector potential A( ) by the following relationship: (1) (1) (1) A div E = i ωμ0 A − grad , i ωε ' × ε 1

0

which implies the representations for the field compo1 1 nents expressed via functions Π( ) (ρ, z ) and F ( ) (ρ, z )

⎡ (1) ⎤ i ωμ0 (1) J Δ x ⎢Π (ρ, z ) − ∂ 12 ∂ P (ρ, z )⎥ , 4π ∂x û1 ∂x ⎣ ⎦ i ωμ0 1 1 (20) E y( ) = − J Δ x ∂ 12 ∂ P ( ) (ρ, z ) , ∂x û1 ∂y 4π i ωμ0 1 1 E z( ) = J Δ x ∂ S ( ) (ρ, z ) , 4π ∂x where the denotations (1)

Ex =

P (17)

S

(1)

(1)

(ρ, z ) = Π(1) (ρ, z ) + ∂ F (1) (ρ, z ) ,

(ρ, z ) = F

(1)

∂z 1 (ρ, z ) − 2 ∂ P (1) (ρ, z ) û1 ∂z

(21)

are introduced.

JOURNAL OF COMMUNICATIONS TECHNOLOGY AND ELECTRONICS

Vol. 63

No. 4

2018

THE ELECTROMAGNETIC FIELD OF A HORIZONTAL ANTENNA

We use relationships (8) and represent the func1 tions determining E ( ) via the integrals containing the Bessel functions

P

(1)

(ρ, z ) =

∞

û12

∫û ν

2 0 1

0 ∞

2 exp(ν1z )J 0 ( λρ) λ d λ, 2 + û1 ν 0

2ν 0 S (ρ, z ) = − 2 exp(ν1z )J 0 ( λρ) λ d λ. 2 0 û0ν1 + û1 ν 0 (1)

(22)

∫

It is possible to use integrals (10) and (11) and to integrate with respect to λ in (22) in the quasi-stationary approximation, i.e., when û0 → 0 . Then, we have

P (ρ, z ) ≡ P

(1)

S (ρ, z ) ≡ S

(1)

(ρ, z ) û =0 0

(ρ, z ) û =0 0

= 2 ∂ I 0 ( r+ ) K 0 ( r− ) , ∂z exp(−û1R) = − 22 ∂ . R û1 ∂z

length replacing ρ = x 2 + y 2 by ρη = ( x − η)2 + y 2 , JΔ x → Jd η, and ∂ ∂x → −∂ ∂η . Let us denote (1) and take into account that %≡ % û0 = 0

∂ = ∂ρη ∂ = x − η ∂ , and ∂ = y ∂ , ∂x ∂x ∂ρη ρη ∂ρη ∂y ρη ∂ρη i ωμ0 4π

η= L ⎡L ⎤ − η x ∂ ⎥, × J ⎢ Π (ρη, z ) d η + 2 P (ρη, z ) û1 ρη ∂ρη ⎢⎣−L ⎥ η=−L ⎦

∫

η= L

i ωμ0 y ∂ , J P (ρη, z ) 2 4πû1 ρη ∂ρη η=−L η= L

i ωμ0 z d exp(−û1Rη ) J %z = . 2 Rη 2πû1 Rη dRη η=− L Here, we denote Rη = ( x − η) + y + z . The derivative of function P (ρ, z ) with respect to ρ enters the first two equations. It can be written as 2

2

2

1 ∂ P ρ, z = 2 ⎡− r+ K r I r ( ) ⎢⎣ R3 0 ( − ) 1 ( + ) ρ ∂ρ r − − I 0 ( r+ ) K1 ( r− ) R 2 ⎤ ûz + 1 2 ( I 0 ( r+ ) K 0 ( r− ) − I1 ( r+ ) K1 ( r− ))⎥ . 2R ⎦

1 ∂ P ρ, z ( ) ρ ∂ρ z =0 û1 ⎡ ⎛ û1 ⎞ ⎛ û1 ⎞ û û = − 2 K 0 ⎜ ρ ⎟ I1 ⎜ ρ ⎟ + I 0 ⎜⎛ 1 ρ ⎞⎟ K1 ⎛⎜ 1 ρ ⎟⎞⎤ . ⎢ ⎝ 2 ⎠ ⎝ 2 ⎠⎦⎥ ρ ⎣ ⎝2 ⎠ ⎝2 ⎠ As it has been shown previously, the expression in the square brackets is equal to 2 ( û1ρ) . Therefore,

1 ∂ P ρ, z ( ) = − 23 . ρ ∂ρ ρ z =0

% x z =0 =

i ωμ0 2πû12

η= L ⎡ L [1 − (1 + û ρ ) exp (−û ρ )] ⎤ x−η 1 η 1 η ⎢ ⎥, d ×J η − ρ3η ρ3η η=−L ⎥ ⎢⎣−L ⎦

∫

η= L

% y z =0

i ωμ0 y , =− J 3 2 2πû1 ρη η=−L

(24)

% z z =0 = 0.

then, the integration over η yields

%y =

Let us pass to two limit cases as we have done when considering the potential. We begin with the definition of the field on the interface z = 0. When z = 0, (23) implies that

The determination of Π (ρ,0) with the help of (7) and (12) yields

To determine the field of a horizontal linear antenna, it is necessary to sum the components of dipole field (20), i.e., integrate these expressions over the antenna

%x =

339

The comparison of the first two expressions in (24) with the results obtained for the field on the interface when it is approached from above [2] shows their identity. This fact should be expected because of the continuity of the tangential field components on the interface between two media. Now, we consider that the condition Rer+ @ 1 is fulfilled. In this case, the horizontal field components can be calculated with the use of the asymptotical decompositions of the modified Bessel functions [6]. The substitution of the asymptotic decompositions into (21) yields the following result for the principal term of the asymptotic representation of the field: η= L L ⎤ i ωμ0 ⎡⎢ 1 x−η ⎥ exp(û1z ), %x ∼ J d η − 2 3 3 2πû1 ⎢−L ρη ρη η=−L ⎥ ⎣ ⎦

∫

η= L

i ωμ0 y %y ∼ − J 3 exp(û1z), 2 2πû1 ρη η=−L

û1 ( R + z ) @ 1. 2 Comparing expressions (25) and (24), we see that, accurate to the exponential small term, the horizontal components of the electric field at the depth of z and large distances from the source are determined by the Re

(23)

(25)

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2018

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field measured on the interface at point ρη and multiplied by the attenuation function exp(û1z) . Let us proceed to the analysis of the magnetic field. 1 Magnetic field H ( ) (ρ, z ) below the interface is connected with the electric vector potential as follows:

and the asymptotic value at a rather large distance from the source η= L

*x ∼ J y3 exp(û1z ), 4π û1ρη η=−L

û1 ( R + z ) @ 1. 2 It is seen from formula (27) that the field determined at a large distance from the source is the product of the asymptotical decomposition of the field on boundary (26) and the exponential function. The calculations similar to the above ones can be 1 1 performed for * y ≡ *(y ) (ρ, z ) and * z ≡ *(z ) (ρ, z ) . Then, we obtain that, on the interface, Re

1 1 H ( ) (ρ, z ) = curlA( ). 1 1 1 Since A( ) = Ax( )ex + Az( )ez , we have 1 1 1 1 1 H x( ) (ρ, z ) = ∂ Az( ), H y( ) (ρ, z ) = ∂ Ax( ) − ∂ Az( ), ∂y ∂z ∂x 1 1 H z( ) (ρ, z ) = − ∂ Ax( ). ∂y

* y z =0

Let us denote the magnetic field excited by a linear grounded (flooded) antenna of the length 2L by *( j ) (ρ, z ) . We can write by analogy with (4) that

( j )

L

* (ρ, z ) =

∫H

( j)

−L

⎡û û û = J ⎢ 1 ⎛⎜ I 0 ⎛⎜ ρη 1 ⎞⎟ K 0 ⎛⎜ ρη 1 ⎞⎟ 4π ⎢⎣ 4 −L ⎝ ⎝ 2 ⎠ ⎝ 2 ⎠ 2 L

∫

û û ⎞⎞ − I 2 ⎜⎛ ρη 1 ⎟⎞ K 2 ⎜⎛ ρη 1 ⎟⎟ dη ⎝ 2 ⎠ ⎝ 2 ⎠⎠

(ρη, z ) JΔ →J d η.

ρη

x

1

(1)

(1)

Hx

(

η

)

and, respectively, for distances exceeding the thickness of the skin layer for the lower medium,

⎡ L d η x − η η= L ⎤ J ⎢ ⎥ exp(û1z ), *y ∼ − 3 2πû1 ⎢−L ρ3η ρη η=−L ⎥ ⎣ ⎦

)

∫

Therefore, integrating in the antenna limits and performing transformations similar to those we have made when considering the electric field, we can 1 obtain the expression for *(x) (ρ, z ) ≡ * x η= L

⎛ ⎞ , * x = − J ⎜ y Az (ρη, z ) ⎟ 4π ⎝ x − η ⎠ η=−L here, we have Az (ρη, z ) = Az (ρ, z ), where the replacements x → xη and ρ → ρη are made and Az (ρ, z ) is determined by formula (13). Let us present two limit expressions for * x : on the boundary z = 0, η= L

* x z =0

(

1 ⎡3 − 3 + 3û ρ + û2ρ2 exp(−û ρ )⎤ d η, 1 η 1 η 1 η ⎦ 5 ⎣

∫ρ

−L

(ρ, z ) = ∂ y Az (ρ, z ) . ∂x x

y ⎛û ρ ⎞ ⎛û ρ ⎞ = J 2 I1 ⎜ 1 η ⎟ K1 ⎜ 1 η ⎟ , 2π ρη ⎝ 2 ⎠ ⎝ 2 ⎠ η=−L

⎝

2π û1

×

We change the order of differentiation taking into account that ∂ ∂y = y ρ × ∂ ∂ρ. As a result, we have

2⎠

* z z =0 = J y2 L

J Δ x ∂ ∂ (1) F (ρ, z ). 4 π ∂y ∂x

⎝

η= L

⎤ ⎥, 2 ⎠ η=−L ⎥ ⎦

( x − η) ⎛ û1 ⎞ ⎛ û1 ⎞ −2 I1 ⎜ ρη ⎟ K1 ⎜ ρη ⎟

Consider H x( ) (ρ, z ) . We represent this quantity in a somewhat different form depending on formula (7)

H x (ρ, z ) =

(27)

(26)

* z ∼ J y2

L

3 d η exp( û z ), 1 2π û1 −L ρ5η û Re 1 ( R + z ) @ 1. 2

∫

CONCLUSIONS The potential and electromagnetic fields excited in a two-layer medium by a linear horizontal antenna or an antenna with flooded electrodes that is situated on the water surface have been calculated. The general formulas for the potential and fields in the region below the interface between the media have been presented within the quasi-stationary approximation with the use of the modified Bessel functions, which are well studied special functions. Consideration of the limit regions, which are the medium interfaces and regions situated at the distance of several skin layers from a source, has shown that the potential, magnetic

JOURNAL OF COMMUNICATIONS TECHNOLOGY AND ELECTRONICS

Vol. 63

No. 4

2018

THE ELECTROMAGNETIC FIELD OF A HORIZONTAL ANTENNA

field, and horizontal components of the electric field can be represented in distant regions in the form of the product of the field in the corresponding point on the interface and the exponential function.

341

2. A. V. Veshev, Electric Profiling Using Direct and Alternating Currents (Nedra, Leningrad, 1980) [in Russian].

4. R. W. P. King, M. Owens, and T. T. Wu, Lateral Electromagnetic Waves. Theory and Applications to Communications, Geophysical Exploration and Remote Sensing (Springer-Verlag, New York, 2011). 5. G. I. Makarov, V. V. Novikov, and S. T. Rybachek, Distribution of Electromagnetic Waves over the Earth Surface (Nauka, Moscow, 1991) [in Russian]. 6. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Fizmatgiz, Moscow, 1962; Academic, New York, 1980). 7. A. B. Kuvarkin and E. I. Novikova, J. Vych. Mat. Mat. Fiz. 21, 1091 (1981).

3. J. R. Wait, Electromagnetic Waves in Stratified Media (Pergamon, New York, 1962).

Translated by I. Efimova

REFERENCES 1. V. Fock, Ann. Phys. (New York) 17, 401 (1933).

JOURNAL OF COMMUNICATIONS TECHNOLOGY AND ELECTRONICS

Vol. 63

No. 4

2018