ISSN 1063-7842, Technical Physics, 2017, Vol. 62, No. 3, pp. 475–479. © Pleiades Publishing, Ltd., 2017. Original Russian Text © E.D. Tereshchenko, P.E. Tereshchenko, 2017, published in Zhurnal Tekhnicheskoi Fiziki, 2017, Vol. 87, No. 3, pp. 453–457.
RADIOPHYSICS
Electric Field of the Horizontal Linear Flooded Antenna E. D. Tereshchenkoa * and P. E. Tereshchenkob a
b
Polar Geophysical Institute, Murmansk, 183010 Russia Pushkov Institute of Terrestrial Magnetism, Ionosphere and Radio Wave Propagation of the Russian Academy of Sciences, St. Petersburg, 199034 Russia *e-mail:
[email protected] Received June 7, 2016; in final form, September 8, 2016
Abstract—Electric field excitation by a horizontal flooded source placed at the interface between two media has been considered. A solution to the problem has been represented in the form of integrals that contain a quickly oscillating Bessel function. Under the quasi-static approximation, general functions that describe a field in water have been represented using the Watson integrals through the well-studied modified Bessel functions. It has been shown that, in the region at a distance more than a skin-layer from the antenna, the vertical component is determined by the field that propagates exclusively in the lower medium, and components perpendicular to the antenna have the form of the waves that propagate in the upper medium without absorption and then penetrate deeply, changing by the exponential law. DOI: 10.1134/S1063784217030240
INTRODUCTION The problem of the excitation of electromagnetic waves by a source placed at the interface of two media has rich history starting with works by Sommerfeld [1], Weil [2], and Bursian [3], as well as later works by Wait [4], Makarov et al. [5], Veshev [6], King et al. [7], etc. However, some questions have remained unconsidered in the known literature and, thanks to the development of the control systems for deeply buried objects, as well as of the electromagnetic methods of studies of the ocean lithosphere [8], and interest in these problems has again arisen. Unlike the experiments on the ground surface, where the measured value is the field at the interface, in the sea experiments, the information about the change in the field with the depth plays an important role. In this work, the analytical expressions that describe the structure of the electric field excited by the horizontal linear antenna with the flooded electrodes and the change in the field with the depth will be obtained. For brevity, the antenna will be referred to as a flooded antenna. Within the quasi-stationary method, which is a good approximation when describing the waves of extremely low frequency (ELF) and lower ranges, let us represent the field components in the form of the well-known functions. Unlike the widespread approach to calculations of the field at interface using the Fock integrals [3, 6], let us use two Watson integrals [9], which allows us to determine the field at the interface, as well as the change in the field with the depth.
1. ELECTRICAL VECTOR-POTENTIAL AND ITS RELATION WITH THE ELECTRIC FIELD Let us consider the radiation of a flooded antenna with a length of 2L, which is fed by the current with a harmonic time dependence exp(–iωt) in the doublelayer medium. Let us choose the coordinate system as follows (see figure). Let us place a center of the Cartesian coordinates in the center of the antenna, direct axis z upwards, axis x along the antenna, and axis y across the antenna. Let us designate a distance to observation point as R and a distance on plane (x, y, 0) as ρ. The medium in the region z > 0 is regarded to be almost nonconducting (σ = +0, the presence of + indicates weak absorption) with permittivity e 0 ≈ (10–9)/(36π) F/m and magnetic permeability μ0 = 4π × 10–7 H/m. Let us assume that the region z < 0 has the electromagnetic parameters e1 , μ0, and σ1. The problem of electromagnetic field excitation by the foreign current J is reduced to solving Helmholtz equations for an electric vector-potential A with the corresponding boundary conditions [4, 6]. Since we consider the radiation of monochromatic waves, let us use the equations for complex amplitudes A of the corresponding monochromatic components (A → Aexp(–iωt), E → Eexp(–iωt), E is the electric field). Taking into account that the source is directed along axis x (see figure), let us represent the vector A in the following form:
475
A ( j ) = A x( j )e x + Az( j )e z ,
(1)
476
E. D. TERESHCHENKO, P. E. TERESHCHENKO
(x, y, 0)
ε0, μ0, σ0 = +0 ρ
ρ1 −L
ρη ρ2
η
0
L
z=0
R ε1 , μ 0 , σ1 (x, y, z) Geometry of the problem.
where, j = 0, 1 indicates the medium, ex and ez are the unit vectors directed along axes x and z, respectively. The further problem was reduced to solving the equation system
∇ 2 A ( j ) + k 2j A ( j ) = −J,
j = 0,1
∂ A x(0) ∂ A (1) A | z = 0 = A | z = 0, , = x ∂ z z =0 ∂ z z =0 1 div A (0)| = 1 div A (1)| . z =0 z =0 k 02 k12
∞
× (3)
(4)
Let us consider solving the system Eq. (2) with the boundary conditions Eq. (3) for a point flooded horizontal source placed at origin of coordinates. In this case,
J (1) = J Δ x δ( x)δ( y)δ(z + 0)e x ,
( j)
=
∑E
( j)
(ρ η, z),
∫
2i exp(i k 2j − λ 2 | z|)
J 0(λρ)λ d λ, k 02 − λ 2 + k12 − λ 2 JΔx ∂ 2 ( j) 2 (k 0 − k1 ) Az = (7) 4π ∂ x 0
2 exp(i k j − λ | z|) 2
2
k 0 − λ + k1 − λ )(k1 k 0 − λ + k 0 k1 − λ ) × J 0(λρ)λ d λ, 2
2
2
2
2
2
2
2
2
2
Im k 2j − λ 2 > 0.
(8)
For notation compactness, as well as to have the analogy with works on geoelectrics [3, 6], let us transform Eq. (7) and introduce the new designations k j = i κ j , ν j = −i k j − λ = κ j + λ , (9) j Δ x ( j) J Δ x ∂ ( j) ( j) ( j) Ax = F (ρ, z). Π (ρ, z ), Az = 4π 4π ∂ x It follows from Eq. (8) that Reνj > 0, while from Eqs. (9) and (7), we can obtain 2
2
2
2
∞
Π (ρ, z) = ( j)
(5)
∫ν 0
where, δ is the delta-function, J is the current, JΔx is the dipole moment, Δx is the dipole length tending to infinitesimal quantity. The passing to the values corresponding to excitation by a linear antenna with a length of 2L is carried out using the summations over antenna length. In particular, % is the electric field excited by a linear antenna will be equal to
%
∞
where j = 0,1. The branch of square root was fixed taking into account that
The wave numbers k0 and k1 that enter the system of equations (2) and (3) are determined by the expressions
J (0) = J Δ x δ( x)δ( y)δ(z − 0)e x ,
∫( 0
(1)
k 0 = ω (1 + i 0) = ω , c c ε σ k1 = ω 1 + i 1 = ω ε 1' . ωε 0 c c ε0
JΔx = 4π
A x( j )
(2)
with the boundary conditions (0)
where E(j)(ρη, z) is the field of a dipole placed at point η of the antenna. The system Eq. (2) with the boundary conditions Eq. (3) is solved in the cylindrical coordinate system (z, ρ, φ) in the form of expansion in the basic functions cos m φ (m = 0, 1, …) that form a complete system at 2 interval (0, 2π] [9]. The further steps are well known [5, 6] and associated with determination of the corresponding functions that enter the expansion and depend on ρ and z taking into account the boundary conditions. Therefore, omitting rather simple transformations, we present the final result of calculations of A x( j ) and Az( j ) for a source Eq. (5) as follows
(6)
F
0
( j)
2 exp(−ν | z|)J (λρ)λ d λ, j 0 + ν1
(ρ, z) = (κ1 − κ 0 ) 2
2
(10)
∞
×
∫ (ν 0
0
2 exp(−ν j | z|)J 0(λρ)λ d λ, + ν1)(κ 02ν1 + κ12ν 0 ) j = 0,1.
The system Eq. (10) is the basic system at calculations of the electric field. Let us consider a practically important case, i.e., the field is in lower medium (in water). Taking into account that the electric field E(j) is TECHNICAL PHYSICS
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related with an electrical vector-potential by the following expression
∞
Π(ρ, z) = Π (ρ, z)| κ 0 =0 = (1)
(1)
E (1) = i ωμ 0 A (1) − grad div A , i ωε 1' ε 0
∫
A(1) =
+
∫
P (ρ, z) = P (1)(ρ, z )| κ 0 =0 = 2 exp(ν1z)J 0(λρ)d λ, (15) 0 ∞
Az(1)ez,
S (ρ, z) = S (1)(ρ, z)| κ 0 =0 = − 12 2exp(ν1z )J 0(λρ)λ d λ. κ1
∫
we can obtain making allowance for Eq. (9)
0
⎡ (1) ⎤ i ωμ 0 (1) J Δ x ⎢Π (ρ, z ) − ∂ 12 ∂ P (ρ, z)⎥ , 4π ∂ x κ1 ∂ x ⎣ ⎦ ⎡ ⎤ ωμ i 0 (12) E y(1) = − J Δ x ∂ ⎢ 12 ∂ P (1)(ρ, z)⎥ , 4π ∂ x ⎣ κ1 ∂ y ⎦ i ωμ 0 E z(1) = J Δ x ∂ S (1)(ρ, z), 4π ∂x where it is denoted that
Ex = (1)
P (1)(ρ, z) = Π (1)(ρ, z) + ∂ F (1)(ρ, z), ∂z (1) (1) 1 S (ρ, z ) = F (ρ, z ) − 2 ∂ P (1)(ρ, z ). κ1 ∂ z
∞
Π (ρ, z) =
∫ν 0
2 exp(ν z)J (λρ)λ d λ, 1 0 + ν1
0
∞
P (ρ, z) = (1)
κ12
∫
0 ∞
S (1)(ρ, z) = −
2 exp(ν1z)J 0(λρ)λ d λ, (14) 2 2 κ 0 ν 1 + κ1 ν 0
∫κ ν 0
2ν 0 exp(ν1z )J 0(λρ)λ d λ. + κ12ν 0
2 0 1
Thus, we have the improper integrals containing the oscillating function J0(λρ). To calculate these integrals, there are well-developed numerical methods, in particular the Longman method [10, 11]. For a practically important case of ELF and lower range of Eq. (14), it is possible to simplify and represent them using the well-known functions. 2. QUASI-STATIONARY APPROXIMATION In the quasi-stationary approximation it is assumed that κ0 = 0. In physical sense, it corresponds to neglect of the Maxwell displacement current with respect to the conduction current. When using the waves with very low frequency, the condition | κ 20 / κ12 | ≪ 1 is well satisfied, which allows one to replace κ0 by zero and ν0 by λ in formulas Eq. (14). Then, we obtain TECHNICAL PHYSICS
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The integrals that determine Π(ρ, z), P(ρ, z), and S(ρ, z) are derived from two Watson integrals. The first determines the incident field [5] as ∞
∫ 0
exp(ν1z ) exp(−κ1R) exp(ik1R) J 0(λρ)λ d λ = , = (16) R R ν1
R= ρ +z , while the other is associated with the product of the modified Bessel functions [9] as 2
∞
(13)
Let us use Eq. (10) and, hence, the functions that determine E(1) can be expressed using the integrals with the Bessel functions (1)
2 exp(ν1z ) J 0(λρ)λ d λ, ν1 + λ
0 ∞
(11)
and
A x(1)ex
477
∫ 0
2
exp(ν1z) J 0(λρ)d λ = I 0(r+ )K 0(r− ), ν1
(17)
where κ1 κ (R + z), r– = 1 (R – z) 2 2 and I0 and K0 are the modified Bessel functions. Let us calculate the functions Π(ρ, z), P(ρ, z), and S(ρ, z) using the Watson integrals Eqs. (16) and (17)
r+ =
⎡ 2 exp(−κ1R) Π(ρ, z) = 22 ⎢ ∂ 2 R κ1 ⎣∂ z 3 ⎛ ⎞ ⎤ − ⎜ ∂ 3 − κ12 ∂ ⎟ I 0(r+ )K 0(r− )⎥ , ∂z ⎠ ⎝ ∂z ⎦ (18) ∂ P (ρ, z) = 2 I 0(r+ )K 0(r− ), ∂z exp(−κ1R) 2 S (ρ, z) = − 2 ∂ . z R ∂ κ1 To determine the field of a linear antenna, it is necessary to integrate Eq. (12) with respect to antenna length replacing ρ = x 2 + y 2 by ρη = ( x − η) 2 + y 2 , JΔx → Jdη, and ∂ → – ∂ . ∂x ∂η Let us denote %(1)| x0 =0 = % and take into account that
∂ = ∂ρ η ∂ = x − η ∂ , ∂x ∂ x ∂ρ η ρ η ∂ρ η ∂ = ∂ρ η ∂ = y ∂ , ∂ y ∂ y ∂ρ η ρ η ∂ρ η
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E. D. TERESHCHENKO, P. E. TERESHCHENKO
then, as a result of integration with respect to η, we can obtain L ⎤ i ωμ 0 ⎡ x−η ∂ η= L J ⎢ Π(ρ η, z)d η + 2 P (ρ η, z )| η=− %x = L⎥, 4π ⎢ ∂ρ η κ ρ ⎥⎦ 1 η ⎣− L i ωμ 0 y ∂ L (19) J P (ρ η, z )| η= %y = η=−L , 2 4πκ1 ρ η ∂ρ η i ωμ 0 L JS (ρ η, z)| η= %z = − η=−L . 4π Equation (19) is a formal solution to the problem. Here, transformations of the well-known modified Bessel functions are used. Let us take some additional steps to demonstrate physical clarity. Let us start with determining the field at interface z = 0. To do this, it is necessary to know Π(ρ, 0) and
∫
1 ∂ P (ρ, z )| . z =0 ρ ∂ρ Let us consider
1 ∂ P (ρ, z ) = 2 ∂ ∂ I (r )K (r ). 0 + 0 − ρ ∂ρ ρ ∂ρ ∂ z For a passage to the limit z = 0, it is a good idea to differentiate with respect to ρ and z a product of the modified Bessel functions. Differentiating in sequence with respect to ρ and z, we can obtain 1 ∂ P (ρ, z) = ⎡2 − r+ K (r )I (r ) − r− I (r )K (r ) 1 − 3 0 + ⎢⎣ R 3 0 − 1 + ρ ∂ρ R (20) ⎤ κ12 z + (I 0(r+ )K 0(r− ) − I 1(r+ )K 1(r− ))⎥ . 2R 2 ⎦ It follows from this that, at z = 0,
1 ∂ P (ρ, z )| z =0 ρ ∂ρ κ κ κ κ κ = − 21 ⎡K 0 ⎜⎛ 1 ρ ⎞⎟ I 1 ⎛⎜ 1 ρ ⎞⎟ + I 0 ⎛⎜ 1 ρ ⎞⎟ K 1 ⎛⎜ 1 ρ ⎟⎞⎤ . ⎢ ⎝ 2 ⎠ ⎝ 2 ⎠⎦⎥ ρ ⎣ ⎝2 ⎠ ⎝2 ⎠ It is shown in [9] that the expression in the square brackets is equal to 2/(κ1ρ) and, correspondingly, we have
1 ∂ P (ρ, z )| = − 2 . z =0 ρ ∂ρ ρ3 Hence, at interface from below, the component % y of the electromagnetic field will be equal to η= L
% y | z =0
i ωμ 0 y . =− 2 3 2πκ1 ρ η η=−L
(21)
Using the aforesaid procedure, after a number of simple but cumbersome transformations, we can obtain a value of Π(ρ, 0) as follows:
Π(ρ,0) =
2 [1 − (1 + κ ρ) exp(−κ ρ)] 1 1
κ12ρ3
and, hence, the expression for % x | z =0 as
% x | z =0
(22) ⎤ x −η i ωμ 0 ⎡ 1 − (1 + κ1ρ η )exp(−κ1ρ η ) ⎢ ⎥. J d = η − ρ 3η ρ3η η=−L ⎥ 2πκ12 ⎢ ⎣−L ⎦ Analyzing Eqs. (21) and (22), we see that, unlike % y , the component % x has an exponential term associated with the wave that propagates in the lower medium. A comparison of Eqs. (21) and (22) with the results obtained for the field at interface from above [6] shows their identity. This should be expected because of continuity of the tangential components of the electric field at interface between two media. Let us analyze the field behavior at some distance from the antenna and interface. Taking into account that the component % z has a simple structure and is almost determined by the wave that propagates from a source to a receiving point, let us consider the component % y in detail. Let us assume that the condition |r+| ≫ 1 is satisfied or η= L
L
∫
κ1 1. 2 Since z ≤ 0, we also have (R + z )
(23)
κ1 1. 2 If the conditions |r+| ≫ 1, |r–| ≫ 1 are satisfied, asymptotic expansions for the modified Bessel functions and the following asymptotic representations for products of the modified Bessel functions can be used | r+ | = (R − z)
exp(κ1z) ⎡ κz ⎤ 1 − 12 2 ⎥ , ⎢ κ1ρ ⎣ 2κ1 ρ ⎦ exp(κ1z) ⎡ κz κR⎤ I 0(r+ )K 1(r− ) ~ 1 + 12 2 + 21 2 ⎥ , ⎢ κ1ρ ⎣ 2κ1 ρ κ1 ρ ⎦ (24) κ1 z κ1R ⎤ exp(κ1z) ⎡ − I 1(r+ )K 0(r− ) ~ 1+ , κ1ρ ⎢⎣ 2κ12ρ 2 κ12ρ 2 ⎥⎦ exp(κ1z ) ⎡ 3 κ1z ⎤ I 1(r+ )K 1(r− ) ~ 1+ . κ1ρ ⎢⎣ 2 κ12ρ 2 ⎥⎦ Substituting expansions Eq. (24) into Eq. (20) and the obtained result into Eq. (19), we get I 0(r+ )K 0(r− ) ~
%y = −
i ωμ 0 y ⎡ z ⎛1 + 3z 2 ⎞ 1 ⎤ exp(κ z),(25) − 1 ⎜ ⎟ ⎢ ⎥ 1 ρ 2η ⎟⎠ κ1Rη ⎦ 2πκ12 ρ 3η ⎣ 2Rη ⎜⎝
where Rη = ρ 2η + z 2 . Let us take into account that z/Rη ≤ 1 and, at long distances, which are much more TECHNICAL PHYSICS
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than the skin-layer, the terms with an order of |l/(κ1Rη)| can be neglected. Then, from Eq. (25), we obtain the following result
% y % y | z =0 exp(κ1z).
(26)
Thus, unlike the component % z described by the wave that only propagates in the lower medium, we have the wave that first propagates in the upper medium without absorption, then penetrates deeply, changing by the exponential law. If the similar transformations for % x are carried out, one can see more the complicated structure of the % x formation, which will be a superposition of two waves, one of them, as during the formation of % z , is only determined by propagation in the lower medium, while the other propagates in the upper medium later upon penetrating below. Taking into account that the direct field in the lower medium is the magnitude of about truncation terms in the asymptotic expansion Eq. (24), it can be neglected. Therefore, the regularity of changes in % x with depth will be similar to % y . CONCLUSIONS Thus, as a result of the calculations, the expressions for components of the electric field excited by a linear flooded antenna have been obtained in the form of integrals that contain quickly oscillating functions. In the range of frequencies, for which the condition of smallness of a wave number in vacuum compared with the magnitude of the wave number in the lower medium is satisfied, a solution to the problem has been found in the form of well-known special functions. In practice, this case is valid for seawater in the frequency range of zero to several kilohertz.
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It has been shown that the vertical component of the electric field is formed by a wave that propagates exclusively in the lower medium. In this case, at the interface, a horizontal field component perpendicular to the antenna is determined by the field that propagates in the upper medium, and the longitudinal component is determined by the sum of the waves in the lower and upper media. At the same time, with increasing distance from the antenna, the components % x and % y change with depth by the exponential law. REFERENCES 1. I. A. Sommerfeld, Ann. Phys. 28, 665 (1909). 2. H. Weil, Ann. Phys. 60, 481 (1919). 3. V. R. Bursian, Theory of Electromagnetic Fields Applied in Electrical Exploration (Nedra, Leningrad, 1972). 4. J. R. Wait, Electromagnetic Waves in Stratified Media (Pergamon, New York, 1962). 5. G. I. Makarov, V. V. Novikov, and S. T. Rybachek, Propagation of Electromagneic Waves over the Earth Surface (Nauka, Moscow, 1991). 6. A. V. Veshev, Electrical Profiling Using Direct or Alternating Current, 2nd ed. (Nedra, Leningrad, 1980). 7. R. W. P. King, M. Owens, and Wu Tai Tsun, Lateral Electromagnetic Waves. Theory and Applications to Communications, Geophysical Exploration and Remote Sensing (Springer, New York, 2011). 8. M. S. Zhdanov, Geophysical Inverse Theory and Regularization Problems (Nauchnyi Mir, Moscow, 2007). 9. I. S. Gradshtein and I. M. Ryzhik, Tables of Integrals, Series, and Products (Fizmatgiz, Moscow, 1962). 10. P. E. Tereshchenko, Tech. Phys. 55, 1062 (2010). 11. A. B. Kuvarkin and E. I. Novikova, Vychisl. Mat. Mat. Fiz. 21, 1091 (1981).
Translated by M. Astrov
