Far-Field Radiated From a Vertical Magnetic Dipole in the Sea With a ...
IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 44, NO. 8, AUGUST 2006
2135
Far-Field Radiated From a Vertical Magnetic Dipole in the Sea With a Rough Upper Surface Osama M. Abo-Seida, Samira Tadros Bishay, and Khaled Mohamed El-Morabie
Abstract—The problem of communication in the sea has been considered as propagation of radio waves in a three-layered medium (air, sea, and ground). With the aid of the perturbation calculus, this paper analyzes the influence exerted onto the electromagnetic field of arrangements radiating a pure transverse electric field in the sea. The sea height varies continuously with the distance from the transmitting end due to sea waves. Knowledge of the solution for the case of uniform sea height is presumed. The problem of the transition conditions at the upper boundary of the sea is solved in the imaging space of a Hankel transformation. Its reversal produces an integral presentation of the interfering field, which was previously quite difficult to evaluate. In this study, closed-form expressions for the far-field of a vertical magnetic dipole embedded below the sea surface are obtained by using a new technique to evaluate this integral with the aid of the complex image theory. The results obtained are compared with those mentioned elsewhere. Index Terms—Far-field, radiation in the sea, rough surface, stratified media.
I. I NTRODUCTION
T
HE PROPAGATION of radio waves in the sea is of great importance in many practical applications and has been treated by many authors [1], [2]. Those authors considered the problem as due to a two-layered medium (air–seawater). Afterward, Arutaki and Chiba [3] have investigated the consequence of considering the sea to be of finite depth, and it must be assumed instead as a three-layered conducting medium (air, sea, and ground). Lately, Bishay has studied the effect of sea waves that occur on the upper sea surface [4], [5] and on its lower surface [6], [7]. The deviations of the two rough, upper and lower, sea surfaces from flat ones are small and have small slopes. Moreover, it is worthy of note that the relevance of this work to very low frequency submarine communications is evident in the formulation and treatment of our model because the alternative high-frequency systems have the disadvantage of large attenuation in saline water. In this paper, the perturbation technique used by Becker [8] and Bishay [9] is employed here. Also, knowledge of the solution for the case of uniform sea height (flat one), which was derived by Arutaki and Chiba [3] and Long et al. [10], is presumed. Previous studies [3]–[7] have obtained the formulas
Manuscript received May 24, 2005; revised January 4, 2006. O. M. Abo-Seida is with the Department of Mathematics, Faculty of Education, Kafer El-Sheikh Branch, Tanta University, Kafer El-Sheikh 33516, Egypt. S. T. Bishay is with the Department of Mathematics, Faculty of Science, Ain Shams University, Cairo 11566, Egypt (e-mail: [email protected]). K. M. El-Morabie is with the Department of Mathematics, Faculty of Science, Tanta University, Tanta 31511, Egypt (e-mail: [email protected]). Digital Object Identifier 10.1109/TGRS.2006.872135
of the field in the region of the seawater due to a vertical magnetic dipole in a three-layered conducting medium by resolving the problem using the residue and saddle-point methods. However, these methods involve lengthy algebra and several transformations, which are very tedious and complicated. Also, in [4], the disturbed field in the sea was not estimated and did not include numerical results. In this paper, a new technique was used to obtain a closed-form expression of the far-field. This approach, which was presented first by Chew [11] and recently by Long et al. [10], has benefited from the bases of the stationary phase method and derives the far-field approximation in a few easily remembered steps. Besides, in a previous study [10], the air–sea interface is considered as a flat one, whereas in this study, we pay attention to the waves that occur in the upper surface of the sea, i.e., the upper surface is considered as a rough one. We were faced by many difficulties such as the estimation of Hankel transformations, which were not included in the special case (flat surface) as presented by Long et al. [10]. First, the form solutions of the far-field due to a vertical magnetic dipole in a sea (three-layered conducting medium) with variable interface are expanded as an infinite series. Then with the aid of the complex image theory [12], closed-form expression of the far-field in the seawater due to the dipole is obtained. Besides, the far-field expressions in the air above the sea surface due to the dipole are similarly obtained, and their physical meanings are presented. Because the slowly varying part in the Sommerfeld integrals has no singularities at the stationary phase point, this approach can be followed in other similar problems to calculate the far-field. II. G ENERAL D ESCRIPTION The model used in the calculation comprise three layers (air, sea, and ground), as illustrated in Fig. 1. The sea is of finite mean thickness a. The ground–sea interface is taken to be planar, whereas the air–sea interface varies slightly from its mean value, as shown in the figure. The propagation constant ki (i = 1, 2, 3) in the three regions can be determined from the material constants in the usual way [13]. The transmitter and receiver are located in the sea at heights h and z from the bottom, respectively. The former is a horizontal small electric current loop antenna of magnetic moment IS0 . It can therefore be regarded as a vertical magnetic dipole. The geometry of the system is described by using cylindrical polar coordinates (ρ, ϕ, z). The sea bottom (ground–sea interface) is chosen to be the plane z = 0, and the z axis is taken vertically upward through the small antenna. The perturbation ∆z in the sea surface (air–sea interface) can be expressed as
0196-2892/$20.00 © 2006 IEEE
∆z = γf1 (ρ) + γ 2 f2 (ρ) + · · ·
(1)
2136
IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 44, NO. 8, AUGUST 2006
According to Sommerfeld [14], the primary field at the general observation point P (ρ, z) takes the form IS0 πp (ρ, z) = 4π
∞
λ J0 (λρ)e−u|z−h| dλ u
(6)
0
√
where u = λ2 − k 2 and its real part is positive, k is the propagation constant of the medium under consideration, and J0 (λρ) is the Bessel function of zero order. At the same time, the secondary field at the general observation point P (ρ, z) takes the form IS0 πs (ρ, z) = 4π
∞
A(λ)euz + B(λ)e−uz J0 (λρ)dλ.
(7)
0
Fig. 1. Geometrical configuration of the problem.
where the functions fn (ρ), n = 1, 2, . . . are definite and continuous and γ is the disturbance parameter such that
|γ n fn (ρ)| a.
(2)
n
1 πsg
We restrict our attention to the first approximation. Accordingly, the first-order terms in γ will only be retained. Also, we shall drop the subscript for simplicity and denote f1 (ρ) by f (ρ). The unit normal to the roughness interface between the two media (air–sea) is given by ∂f (ρ) , 0, 1 . n = −γ ∂ρ
2
∂ π Hρ = ∂ρ ∂z
Hz = −
IS0 = 4π
2
∂ 1 ∂ + ∂ρ2 ρ ∂ρ
π.
2 πsg =
E is
=
E is0
+
γE is1 ,
H is
=
H is0
+
γH is1
(i = 1, 2, 3) (5)
where E s0 (ρ, z) and H s0 (ρ, z) are the secondary fields for the planar interface, i.e., for the case where γ = 0, which was considered by Arutaki and Chiba [3]. E is1 (ρ, z) and H is1 (ρ, z) are the secondary fields due to the variation of the interface between the two media (air–sea).
Ag (λ)e−u1 z J0 (λρ)dλ
IS0 4π
∞
Bg (λ)eu2 z + Cg (λ)e−u2 z J0 (λρ)dλ
0
3 = πsg
IS0 4π
∞
Dg (λ)e−u3 z J0 (λρ)dλ
(8)
0
where g = 0, 1 in the case of the planar and rough interfaces, respectively. The nonvanishing components of the electric and magnetic fields can be deduced without much labor by substituting from (6) and (7) in (4). However, due to the length of the obtained results, we only give here as an example the expressions found for the secondary fields in the sea, i.e., 2 Esgϕ =
−jωµ0 IS0 8π (2)
(4) As the primary fields (the incident or direct field from the source) E p (ρ, z) and H p (ρ, z) are independent of ϕ, the total fields E t (ρ, z) and H t (ρ, z) for the three media will be also independent of ϕ. Employing the perturbation technique by Bishay [4] and Becker [8], we can represent the secondary fields (reflected and transmitted fields) as
∞ 0
(3)
In the case of the horizontal small electric current loop antenna, it works as a vertical magnetic dipole. Accordingly, the magnetic Hertz vector has only a z component πz , and it satisfies the Helmholtz wave equation for a time factor exp(jωt), as is well known. The fields are given by the following formulas [13]: ∂π Eϕ = jωµ ∂ρ
It is evident that the first term of the integral represents the downward wave from the surface of the sea to the field point. The second term is the upward wave from the sea bottom to the field point. Also, the secondary part of π in the three media may be expressed as
2 Hsgρ
∞
Bg (λ)eu2 z + Cg (λ)e−u2 z
−∞
× H1 (λρ)λ dλ ∞ −IS0 = u2 Bg (λ)eu2 z − Cg (λ)e−u2 z 8π −∞ (2)
2 Hsgz
× H1 (λρ)λ dλ ∞ −IS0 Bg (λ)eu2 z + Cg (λ)e−u2 z = 8π −∞ (2)
× H0 (λρ)λ2 dλ,
g = 0, 1
(9)
where H02 (λρ) and H12 (λρ) are the second-kind Hankel functions of order zero and one, respectively. The fields at g = 0 are the solutions for the case where γ = 0 [3], B1 (λ) and C1 (λ) (i.e., g = 1) should be determined by the boundary conditions, and the tangential components of the electric and magnetic fields are continuous at the boundaries z = 0 and z = a + ∆z.
ABO-SEIDA et al.: FAR-FIELD RADIATED FROM A VMD IN THE SEA
2137
Neglecting the higher order terms of O(γ 2 ), it can be shown after a fair amount of manipulative algebra that −4π ωµ0 IS0 − ju1 u− 23 H1 {U (ρ)}−ωµ0 u23 H1 {V (ρ)} × e2u2 a + + 2u2 a − u− 12 u23 + u12 u23 e u− C1 (λ) = 23 B1 (λ) (10) u+ 23
If the integral in the foregoing has a closed-form expression, (16) is an analytic expression for the far-field approximation of (12). The error in the approximation in (16) is
B1 (λ) =
where u± ij = (ui ± uj ) (i, j = 1, 2, 3) and H1 {x} is the Hankel transform of the first order, which is represented as Hν {x} = ∞ 0 Jν (ρσ)xρ dρ, also ∂ 2 2 1 Epϕ + Es0ϕ − Es0ϕ U (ρ) = f (ρ) ∂z z=a ∂ 2 2 1 Hpρ + Hs0ρ − Hs0ρ V (ρ) = f (ρ) ∂z ∂f 2 2 1 + Hpz + Hs0z − Hs0z . (11) ∂ρ z=a The values of B1 (λ) and C1 (λ) in (10) are the same as those obtained by Bishay [4]. III. V IEW ON THE M ETHOD OF S OLUTION
∞ h(t, α)dt
[S(t) − S(t0 )] g(t, α)dt.
Ie =
(12)
−∞
where α is a large parameter. The first step in the procedure is to factorize the integrand h(t, α) into two parts, i.e., 1) a slowly varying part and 2) a rapidly varying part. In other words, we let
Ie is of higher order compared with I intuitively, because in (17), the slowly varying part of the integrand vanishes right at the place from where most of the contributions to the integral have come. In the next section, we will use this technique to obtain the far-fields due to a loop antenna in a three-layered conducting medium with rough upper surface. IV. E VALUATION OF THE H ANKEL T RANSFORMATION To evaluate the electric and magnetic fields in the sea with a rough upper surface, (9), the functions B1 (λ) and C1 (λ) have to be first evaluated from (10). This also must be proceeded by evaluating the Hankel transformation of U (ρ) and V (ρ), which are stated in (11). Then we can write it as
∞ ∂ 2 2 1 EP ϕ + Es0ϕ H1 {U (ρ)} = − Es0ϕ f (ρ) ∂z Z=a × J1 (λρ)ρ dρ
(18)
where the first term in the right-hand side is the primary field. The second term represents the electric field in the sea with a flat surface at z = a, the case which was considered by Arutaki and Chiba [3]. However, the third term represents the electric field in the air (also with a flat surface, at z = a), the case which was considered by Long et al. [10]. Thus, we can write (18) as H1 {U (ρ)} =
∞
3
Mi
(19)
I=1
S(t)g(t, α)dt
I=
(17)
−∞
0
In this section, we give a picture of the new approach for evaluating the far-field expression in a rapid way [11]. Let us first consider an integral of the type I=
∞
(13)
−∞
where S(t) is the slowly varying function, whereas g(t, α) is the rapidly varying function when α → ∞. Therefore, we may assume that g(t, α) ≈ ejαf (t) ,
α → ∞.
(14)
The second step then is to find the stationary phase point of g(t, α), which is given by the value of t as ∂f (t) |t=t0 = 0. ∂t
(15)
∞ −∞
g(t, α)dt.
∞ ∂ 2 J1 (λρ)ρ dρ f (ρ) Epϕ M1 = ∂z Z=a 0 ∞ ∂ 2 M2 = J1 (λρ)ρ dρ f (ρ) Es0ϕ ∂z Z=a 0 ∞ ∂ 1 M3 = − J1 (λρ)ρ dρ. f (ρ) Es0ϕ ∂z Z=a
(20)
0
Most of the contributions to the integral in (13) will be from the vicinity of the stationary phase point t = t0 . Hence, we replace the slowly varying part of the integrand S(t) by its value at the stationary phase point t0 , then (13) becomes I ≈ S(t0 )
where
(16)
First, we calculate the value of the primary field EP2 ϕ from (4) and (6), as follows: EP2 ϕ
−jωµ0 IS0 = 4π
∞
λ −u2 |z−h| e J1 (λρ)λ dλ. u2
0 (1)
(2)
Noting that J1 (λρ) = (1/2)[H1 (λρ) + H1 (λρ)] (1) (2) (1) H1 (λρ) = −e−iπ H1 (e−iπ λρ), where H1 (λρ)
and and
2138
IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 44, NO. 8, AUGUST 2006
(2)
H1 (λρ) are the first and the second kinds of Hankel function of order one, respectively, then EP2 ϕ
−jωµ0 IS0 = 8π
∞ −∞
λ −u2 |z−h| (2) e H1 (λρ)λ dλ. u2
Hence M1 =
(21)
Now, we will use our new technique, which was presented by Chew [11] as stated in Section III ∞ (2) −jωµ0 IS0 H1 (λρ) 2 λ (2) EP ϕ = 8π H0 (λρ) −∞ λ −u2 |z−h| (2) e H0 (λρ) dλ. (22) × u2 When λρ → ∞, there is an approximate formula for Hankel (2) function as Hm (λρ) ≈ 2/(πλρ)e−j(λρ−m(π/2)−(π/4)) . The first bracket in (22) is slowly varying part, whereas the second bracket is rapidly varying as λρ → ∞. The location of the stationary phase point is given by ∂ 2 2 j −λρ − (z − h) k2 − λ = 0 z h. (23) ∂λ λ=λ0
where
where r =
ρ2 + (z − h)2 . Then (22) becomes ωµ0 IS0 k2 ρ −jk2 r 2 e EP ϕ = . 4π r2
(24)
f (ρ0 ) .
2 Es0ϕ
−jωµ0 IS0 = 8π
∞
B0 (λ)eu2 z + C0 (λ)e−u2 z
−∞ (2)
× H1 (λρ)λ dλ = I1 + I2
(29)
where I1 =
jωµ0 IS0 8π
∞ −∞
λ u2
+ − − −2u2 h u− 12 u23 + u12 u23 e + + − −2u2 a u12 u23 + u− 12 u23 e
× e−u2 (2a−z−h) H1 (λρ)λ dλ (2)
−jωµ0 IS0 I2 = 8π
∞
(30)
− −u2 h − −u2 (2a−h) λ u+ −u− 12 u23 e 12 u23 e + − − −2u2 a u2 u+ 12 u23 + u12 u23 e (31)
(25)
(27) where ρ = ρ0 is the location of the stationary phase point, which is given by
∂ j −λρ − k2 ρ2 + d21 =0 ∂ρ ρ=ρ0
jd1 k2 . u2
(2)
0
[R]ρ=ρ0 =
2u2 d1 k22
× e−u2 z H1 (λρ)λ dλ.
(26) where R = ρ2 + d21 and d1 = (a − h). This leads to −u2 d1
e −jk2 −jωµ0 IS0 2 k2 d1 − 3 ρf (ρ) M1 = 4π u2 R2 R ρ=ρ0
jλd1 u2
1−
(28)
2 Now, to evaluate M2 , we must first evaluate Es0ϕ , which is the secondary electric field for the planar interface [3], i.e.,
−∞
It is clear that EP2 ϕ represents the direct wave from the source to the field point P (ρ, z), and this value tends to zero as ρ → ∞ (Appendix A). This coincides with Arutaki and Chiba [3] and Long et al. [10], as the propagation path is in the sea and the attenuation is very large. Substituting (25) in the first equation of (20) and repeating the same technique to find M1 , we have ∞ −jk2 ωµ0 IS0 2 k2 d1 ρf(ρ) M1 = − 4 e−jk2 R J1 (λρ)ρ dρ 4π R3 R
ρ0 =
M0 (λ) = λ
The solution of (23) is given by k2 ρ λ0 = r
−jωµ0 S0 I −u2 d1 e M0 (λ) 4π
The integral I1 had been previously evaluated by Long et al. [10] as −jωµ0 IS0 ρk12 −jk2 (d1 +a−z) N 2 e 2π k2 m+1 ∞ 1 − n3 e−jk1 rm+1 × (m + 1) e−jk2 2ma 3 1 + n3 rm+1 m=0
I1 =
(32)
where N = [((1 + n3 )/(1 − n3 )) + e−jk2 2h ], rm+1 = 2 ρ + (m + 1)2 d2 , d = −2j/k2 , and n3 = k3 /k2 . Using the same approach presented in the foregoing integral, we can evaluate the integral I2 as (Appendix B) I2 =
m+1 ∞ −jωµ0 IS0 k12 ρ −jk2 (z+h) 1 − n3 e 2π k22 1 + n3 m=0 −jk1 rm me e−jk1 rm+1 × e−jk2 2ma + (m + 1) . (33) 3 3 rm rm+1
Finally, from (32) and (33), we can find the electric field 2 Es0ϕ , where the series in I1 and I2 are uniformly convergent 2 in the sec(Appendix C). Substituting by the value of Es0ϕ ond equation of (20), arranging properly and using the same approach, we have M2 =
jωµ0 IS0 −jk2 d1 λe 4π × (N − e−jk2 2a )Gm+1 (λ) − e−jk2 2h Gm (λ) . (34)
ABO-SEIDA et al.: FAR-FIELD RADIATED FROM A VMD IN THE SEA
2139
1 To find M3 , let us use the formula Es0ϕ , which is the electric field in the air in the case of a planar interface. This value has been obtained before by Long et al. [10] as
1 Es0ϕ
m+1 ∞ ωµ0 IS0 1 − n3 −jk2 d1 Ne = 4π 1 + n3 m=0 × e−jk2 2ma
ρe−jk2 rm ρe−jk2 rm+1 − 2 2 rm rm+1
. (35)
Substituting in M3 and following the same approach, we find −jωµ0 IS0 λN e−jk2 d1 4π × [(Gm+1 (λ) − 2Cm+1 (λ) − Gm (λ) + 2Cm (λ) ]
M3 =
Lm (λ) = − λ (N + e−jk2 2a + ju1 N )Gm+1 (λ) + (e−jk2 2h − ju1 N )Gm (λ) + 3λjN u1 (Cm+1 (λ) − Cm (λ)) k2 λ2 + + jN + je−jk2 2a Gm+1 (λ) k2 u1
k2 − je−jk2 2h Gm (λ) . (42) − u1 The prime Gm and Gm+1 denote differentiation of f (ρ) with respect to ρ, and ρm+1 = jλ(m + 1)d and d = (−2j/k2 ).
(36)
V. D ISTURBED F IELD IN THE S EA In fact, the main purpose of this paper is to find the values of the secondary fields in the sea, i.e., g = 1 in (9). These will represent the changes that occur in the electromagnetic field of the wave propagation in the sea due to the perturbation applied on the upper surface. We will show here the process 2 2 of estimating Es1ϕ , and the other two components Hs1ρ and 2 Hs1z are also estimated in the same way. It is clear that the first 2 term in Es1ϕ represents the downward wave from the surface of the sea to the observing point. The second term is the upward wave from the sea bottom to the observing point. We focus our 2 attention on the first term. From (10), we can calculate Es1ϕ by using the following approach, in which the denominator can be expanded as an infinite series inasmuch as
where Gm (λ) =
and
m+1 ∞ 1 − n3 1 + n3 m=0 × e−jk2 2ma e−u1 md f (ρm )
m+1 ∞ 1 − n3 Gm+1 (λ) = 1 + n3 m=0 × e−jk2 2ma e−u1 (m+1)d f (ρm+1 ) m+1 ∞ 1 − n3 u1 Cm (λ) = 1 + n3 mdk12 m=1
− − u21 u23 −2u a 2 e u+ u+ ≺1 21 23
× e−jk2 2ma e−u1 md f (ρm ) Then
m+1 ∞ 1 − n3 u1 Cm+1 (λ) = 1 + n3 (m + 1)dk12 m=0 × e−jk2 2ma e−u1 (m+1)d f (ρm+1 ).
2 Es1ϕ
H1 {U (ρ)} =
−jωµ0 IS0 −u2 d1 e M0 (λ) + e−jk2 d1 Mm (λ) 4π (38)
where Mm (λ) = λ e−jk2 2a Gm+1 (λ) + (e−jk2 2h − N )Gm+1 (λ) + 2N Cm+1 (λ) − 2N Cm (λ)} . (39) With the same method, we can find H1 {V (ρ)}. But for the length of the derivation, we give here only the result as H1 {V (ρ)} =
−jIS0 −u2 d1 e L0 (λ) + e−jk2 d1 Lm (λ) (40) 4π
where L0 (λ) =
−jλ d1
3u22 λ2 − u d + 1 f (ρ0 ) − f (ρ0 ) (41) 2 1 2 k2 u2
∞
n=0
φ(n) +
∞
ψ(n)
= E1 + E2
n=0
(43)
(37)
Adding (27), (34), and (36), we can find the value of the Hankel transformation H1 {U (ρ)}. Then
−jωµ0 IS0 = 8π
∞ 1 = xm . 1 − x m=0
where − − n ∞ u1 u21 u23 φ(n) = λ e−2u2 na + + u21 u+ 21 u23 −∞
j (2) × M0 (λ) + L0 (λ) e−u2 (d1 +d2 ) H1 (λρ)dλ (44) u1 − − n ∞ u1 u21 u23 ψ(n) = λ e−u2 (2na+d2 ) + + u21 u+ 21 u23 −∞
j (2) × Mm (λ) + Lm (λ) e−jk2 d1 H1 (λρ)dλ. (45) u1 To determine φ(n) and ψ(n), we use the procedure used before; thus, we have − n u23 −u1 nd −u1 d φ(n) = j u2 e (1 − e ) u+ 23 λ=λ1 e−jk2 R j (46) × M0 (λ) + L0 (λ) u1 R λ=λ1
2140
IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 44, NO. 8, AUGUST 2006
where λ1 = ρk2 /R = k2 sin α and R= ρ2 + (d1 + d2 + 2na)2 . Then u1 is transformed as |2 is satisfied). Also, k2 sin α (when the condition |k2 |2 |k1 u2 = jk2 cos α, u3 = jk2 N3 , and N3 = φ(n) takes the form
n23 − sin2 α. Then
φ(n) = −k2 cos α n cos α − N3 −k2 d sin α −nk2 d sin α )e × (1 − e cos α + N3 −jk2 R e j × M0 (λ1 ) + L0 (λ1 ) . (47) k2 sin α R
Fig. 2. Variation of the absolute value of the electric field E1 versus radial distance for frequencies of 10 and 50 kHz.
Similarly ψ(n) = −k2 cos β n cos β −N3 × e−jk2 d1 (1−e−k2 d sin β )e−nk2 d sin β cos β +N3 × Mm (λ2 ) +
−jk2 R0 e j Lm (λ2 ) k2 sin β R0
(48)
where R0 = ρ2 + (d2 + 2na)2 and λ2 = ρk2 /R0 = k2 sin β. From (47) and (48), we can find the disturbed field in the sea without approximation. The physical meaning of the first term in (43) indicates a series of waves that propagate upward from the source and make n round trips between the rough sea surface and the bottom and finally arrive at the field point in the sea, where [(cos α − N3 )/(cos α + N3 )] is the reflection coefficient at the sea bottom. The second term represents another series of waves that first propagate downward from the source, then reflected upward at the sea bottom and travel along the same bath as aforementioned waves in [3] and [10]. These results also show that φ(n) and ψ(n) are proportional with e−jk2 R /R, and this means that the electric field in the sea takes the form as the Sommerfeld integral. Also, the result coincides with the result obtained by Long et al. [10], when the disturbed factor fn (ρ) is neglected. VI. N UMERICAL R ESULTS The perturbation electric field in the sea due to the vertical magnetic dipole is computed for different cases. The part of the sea under investigation is taken for different thicknesses, and the height of the source is taken to be 4 m. We choose the sea waves represented by the roughness cosine profile with a period of 10 m and of different frequencies. We provide numerical plots of |E1 | in Figs. 2 and 3, which show that the multilayered waves are decreasing rapidly with radial distance ρ and the vertical distance of the field point, respectively, for different frequencies. Also, numerical plots of |E2 | in Figs. 4 and 5 indicate that these multilayered waves are more important than |E1 |. VII. C ONCLUSION Closed-form expressions for the far-fields in the sea are obtained by utilizing the three-layered model. Analytical expres-
Fig. 3. Variation of the absolute value of the electric field E1 versus vertical distance of the field point for frequencies of 10 and 50 kHz.
Fig. 4. Variation of the absolute value of the electric field E2 versus radial distance for frequencies of 10 and 50 kHz.
Fig. 5. Variation of the absolute value of the electric field E2 versus vertical distance of the field point for frequencies of 10 and 50 kHz.
sions were found for the first-order corrections of the electric and magnetic fields due to a small perturbation applied in the upper surface due to sea waves. The complex image theory helped, in a rapid way, to approximate the Sommerfeld Integral. The results presented here indicate that a series of waves propagate either upward or downward from the source. These waves undergo cyclic motions between the rough sea surface and the bottom and finally arrive at the field point in the sea where [(cos α − N3 )/(cos α + N3 )] is the reflection coefficient at the sea bottom. The results obtained in this paper agree with those results obtained hitherto for special cases as [3] and [10]. Long et al. [10] have dealt with lateral waves, whereas in this study, we have considered the reflected waves. Numerical results are also given in this paper. It is also worth noting that our current research work is being carried out on lateral waves, too.
ABO-SEIDA et al.: FAR-FIELD RADIATED FROM A VMD IN THE SEA
A PPENDIX A
Then we have
The aim of the appendix is to verify that (25) matches the previous results, i.e., EP2 ϕ =
ωµ0 IS0 4π
k2 ρ −jk2 r e , r2
ωµ0 IS0 k2 e−jk2 ρ , 4πρ
2j(m + 1)k12 ρd Y2 (m) = 3 k2 rm+1
From (31), we can calculate I2 by using the following approach. The denominator in (31) can be expanded as an infinite series inasmuch as − − ∞ u21 u23 −2u a 1 2 ≺ 1 = e xm . u+ u+ 1 − x 21 23 m=0 Then ∞
T2 (m) +
m=0
∞
Y2 (m)
m=0
where T2 (m) = −∞
1 − n3 1 + n3
m+1
Then m+1 ∞ −jωµ0 IS0 k12 ρ −jk2 (z+h) 1 − n3 e I2 = 2π k22 1 + n3 m=0 −jk1 rm me e−jk1 rm+1 ×e−jk2 2ma + (m + 1) . 3 3 rm rm+1
A PPENDIX C The series in the formula of I2 can be written as a sum of two series
A PPENDIX B
λ u2
e−jk2 (z+h+2ma) e−jk1 rm .
0 ≤ ρ ≤ ∞.
2 The aforementioned result of Epϕ coincides with that of Sommerfeld [14].
∞
m+1
× e−jk2 (z−h+2(m+1)a) e−jk1 rm+1 .
In this paper, we deal with the first case, and this is clarified in the numerical results. The second case (ρ → ∞) is refused because on substituting (25) in the first equation of (20), calculating M1 integral with respect to ρ from −∞ to ∞ (i.e., ρ ∈ (−∞, ∞)) leads to an unacceptable result. Also, the last case is refused. Noting that ρ ∼ = r, then
1 − n3 1 + n3
Similarly
1) 0 ≤ ρ ≤ ∞, λ → ∞; 2) ρ → ∞, 0 ≤ λ ≤ ∞; 3) ρ → ∞, λ → ∞.
−jωµ0 IS0 I2 = 8π
2jmk12 ρd T2 (m) = 3 k2 rm
0 ≤ ρ ≤ ∞.
Inasmuch as the previous equation is obtained from (22) under the condition integral λρ → ∞, this condition gives three cases, namely:
2 = Epϕ
2141
− u− 21 u23 + + u21 u23
m
u− 23 u+ 23
× e−u2 (z+h) e−jk2 2ma H1 (λρ)λ dλ ∞ − m+1 λ u− 21 u23 Y2 (m) = + u2 u+ 21 u23 (2)
g(m) =
m+1 ∞ 1−n3 m=0
g(m + 1) =
1+n3
m+1 ∞ 1−n3
m=0
1+n3
me−jk1 rm 3 rm
e−jk2 2ma
(m + 1)e−jk1 rm+1 3 rm+1
where rm = ρ2 + m2 d2 . To prove that I2 is a uniformly convergent series, it is satisfactory to show that one of these series, e.g., g(m), is convergent. Therefore, we apply a “Weiertrass test” and the theorem from [15], i.e., “If | m i=0 fi (x)| = |Sm (x)|, Sm is bounded in the interval (a, b), i.e., |Sm (x)| ≤ M ∀ m, x ∈ (a, b) and {am } is a bounded sequence, then ∞ m=0 am fm (x) converges uniformly in the interval (a, b).” It is clear that g(m) = am fm (ρ), where am = m fm (ρ) =
−∞
× e−u2 (a+d1 +h) e−jk2 2ma H1 (λρ)λ dλ.
e−jk2 2ma
1 − n3 1 + n3
e−jk1 rm , 3 rm
m+1
e−jk2 2ma
0≤ρ≤∞
(2)
According to the complex image theory [12], when the condition |k2 |2 |k1 |2 is satisfied, we have u− 21 ≈ e−u1 d u+ 21
2u1 ≈ 1 − e−u1 d . u+ 21
1) {am } is a bounded sequence inasmuch as am+1 am =
m+1 1 − n3 rm ∀ m, then e−jk1 rm+1 < e−jk1 rm ∀ m, leading to r1 − r0 ≥
d2 2ρ
r2 − r0 ≥
2d2 4d2 ≥ 2ρ 2ρ
r3 − r0 ≥
3d2 9d2 ≥ 2ρ 2ρ
Osama M. Abo-Seida was born in Tanta, Egypt, on October 21, 1968. He received the B.Sc. degree in mathematics and the M.Sc. and Ph.D. degrees in electrodynamics from Tanta University, Tanta, in 1990, 1994, and 1997, respectively. Since 1997, he has been an Assistant Professor with the Department of Mathematics, Faculty of Education, Tanta University, Kafr El-Sheikh Branch, Egypt. His research interests include computational electromagnetic, wave propagation, antenna, and applied mathematics. Dr. Abo-Seida is a member of the Mathematics Society of Egypt and of the Scientific Council of Egypt.
.... Then e−jk1 r0 Sm (ρ) ≤ r03
1 − e−jk1 (
m+1 2 2ρ d d2
1 + e−jk1 ( 2ρ )
)
.
Therefore 2 2 , ρ = 0, |Sm (ρ)| ≤ 1d ρ3 sin k4ρ
2
k1 d 2nπ
where n = ±1, ±2, . . . . Hence, the sequence of partial sum Sm (ρ) is bounded, and g(m) is a uniformly convergent series. R EFERENCES [1] R. K. Moor and W. E Blair, “Dipole radiation in a conducting half space,” J. Res. Nat. Bur. Stand., vol. 65D, no. 5, pp. 515–522, May 1961. [2] S. H. Durrani, “Air to undersea communication with magnetic dipole,” IEEE Trans. Antennas Propag., vol. AP-12, no. 4, pp. 464–469, Jun. 1964. [3] A. Arutaki and J. Chiba, “Communication in a three-layered conducting media with a vertical magnetic dipole,” IEEE Trans. Antennas Propag., vol. AP-28, no. 4, pp. 551–556, Jul. 1980. [4] S. T. Bishay, “Communication of radio waves in sea with a rough upper surface,” AEÜ Archiv Elektronik Übertrag., vol. 38, no. 3, pp. 186–188, 1984. [5] L. E.-S. Rashid, S. T. Bishay, and O. M. Abo-Seida, “Electromagnetic field produced by a vertical magnetic dipole above a rough surface,” in Proc. ICEAA, Sep. 12–15, 1995, pp. 231–240. [6] S. T. Bishay, “Communication in a rough three-layered conducting medium,” Radio Sci., vol. 22, no. 5, pp. 781–786, 1987. [7] ——, “Estimation of the electromagnetic field in a sea with rough upper and lower surfaces,” Can. J. Phys., vol. 66, no. 4, pp. 319–322, 1988.
Samira Tadros Bishay received the B.Sc. and M.Sc. degrees from Ain Shams University, Cairo, Egypt, in 1962 and 1967, respectively, and the Ph.D. degree from Saarland University, Saarbrücken, Germany, in 1973. In 1962, she joined the Department of Applied Mathematics, Faculty of Science, Ain Shams University, where she has been a Professor since 1989. Her research interests include the development of communication systems in stratified-media rough surface, wave propagation in the sea, and antenna and waveguide boundary-value problems. Prof. Samira is a member of the Mathematics Society of Egypt and the Scientific Council of Egypt.
Khaled Mohamed El-Morabie received the B.Sc. degree from Tanta University, Tanta, Egypt, in 2002. He is currently working toward the M.Sc. degree at Tanta University. In 2002, he joined the Department of Applied Mathematics, Faculty of Science, Tanta University, as a Demonstrator. His research interests include wave propagation in the sea, antenna and waveguide boundary-value problems, and communications systems in stratified-media rough surfaces. Mr. Khaled is a member of the Mathematics Society of Egypt.
2135
Far-Field Radiated From a Vertical Magnetic Dipole in the Sea With a Rough Upper Surface Osama M. Abo-Seida, Samira Tadros Bishay, and Khaled Mohamed El-Morabie
Abstract—The problem of communication in the sea has been considered as propagation of radio waves in a three-layered medium (air, sea, and ground). With the aid of the perturbation calculus, this paper analyzes the influence exerted onto the electromagnetic field of arrangements radiating a pure transverse electric field in the sea. The sea height varies continuously with the distance from the transmitting end due to sea waves. Knowledge of the solution for the case of uniform sea height is presumed. The problem of the transition conditions at the upper boundary of the sea is solved in the imaging space of a Hankel transformation. Its reversal produces an integral presentation of the interfering field, which was previously quite difficult to evaluate. In this study, closed-form expressions for the far-field of a vertical magnetic dipole embedded below the sea surface are obtained by using a new technique to evaluate this integral with the aid of the complex image theory. The results obtained are compared with those mentioned elsewhere. Index Terms—Far-field, radiation in the sea, rough surface, stratified media.
I. I NTRODUCTION
T
HE PROPAGATION of radio waves in the sea is of great importance in many practical applications and has been treated by many authors [1], [2]. Those authors considered the problem as due to a two-layered medium (air–seawater). Afterward, Arutaki and Chiba [3] have investigated the consequence of considering the sea to be of finite depth, and it must be assumed instead as a three-layered conducting medium (air, sea, and ground). Lately, Bishay has studied the effect of sea waves that occur on the upper sea surface [4], [5] and on its lower surface [6], [7]. The deviations of the two rough, upper and lower, sea surfaces from flat ones are small and have small slopes. Moreover, it is worthy of note that the relevance of this work to very low frequency submarine communications is evident in the formulation and treatment of our model because the alternative high-frequency systems have the disadvantage of large attenuation in saline water. In this paper, the perturbation technique used by Becker [8] and Bishay [9] is employed here. Also, knowledge of the solution for the case of uniform sea height (flat one), which was derived by Arutaki and Chiba [3] and Long et al. [10], is presumed. Previous studies [3]–[7] have obtained the formulas
Manuscript received May 24, 2005; revised January 4, 2006. O. M. Abo-Seida is with the Department of Mathematics, Faculty of Education, Kafer El-Sheikh Branch, Tanta University, Kafer El-Sheikh 33516, Egypt. S. T. Bishay is with the Department of Mathematics, Faculty of Science, Ain Shams University, Cairo 11566, Egypt (e-mail: [email protected]). K. M. El-Morabie is with the Department of Mathematics, Faculty of Science, Tanta University, Tanta 31511, Egypt (e-mail: [email protected]). Digital Object Identifier 10.1109/TGRS.2006.872135
of the field in the region of the seawater due to a vertical magnetic dipole in a three-layered conducting medium by resolving the problem using the residue and saddle-point methods. However, these methods involve lengthy algebra and several transformations, which are very tedious and complicated. Also, in [4], the disturbed field in the sea was not estimated and did not include numerical results. In this paper, a new technique was used to obtain a closed-form expression of the far-field. This approach, which was presented first by Chew [11] and recently by Long et al. [10], has benefited from the bases of the stationary phase method and derives the far-field approximation in a few easily remembered steps. Besides, in a previous study [10], the air–sea interface is considered as a flat one, whereas in this study, we pay attention to the waves that occur in the upper surface of the sea, i.e., the upper surface is considered as a rough one. We were faced by many difficulties such as the estimation of Hankel transformations, which were not included in the special case (flat surface) as presented by Long et al. [10]. First, the form solutions of the far-field due to a vertical magnetic dipole in a sea (three-layered conducting medium) with variable interface are expanded as an infinite series. Then with the aid of the complex image theory [12], closed-form expression of the far-field in the seawater due to the dipole is obtained. Besides, the far-field expressions in the air above the sea surface due to the dipole are similarly obtained, and their physical meanings are presented. Because the slowly varying part in the Sommerfeld integrals has no singularities at the stationary phase point, this approach can be followed in other similar problems to calculate the far-field. II. G ENERAL D ESCRIPTION The model used in the calculation comprise three layers (air, sea, and ground), as illustrated in Fig. 1. The sea is of finite mean thickness a. The ground–sea interface is taken to be planar, whereas the air–sea interface varies slightly from its mean value, as shown in the figure. The propagation constant ki (i = 1, 2, 3) in the three regions can be determined from the material constants in the usual way [13]. The transmitter and receiver are located in the sea at heights h and z from the bottom, respectively. The former is a horizontal small electric current loop antenna of magnetic moment IS0 . It can therefore be regarded as a vertical magnetic dipole. The geometry of the system is described by using cylindrical polar coordinates (ρ, ϕ, z). The sea bottom (ground–sea interface) is chosen to be the plane z = 0, and the z axis is taken vertically upward through the small antenna. The perturbation ∆z in the sea surface (air–sea interface) can be expressed as
0196-2892/$20.00 © 2006 IEEE
∆z = γf1 (ρ) + γ 2 f2 (ρ) + · · ·
(1)
2136
IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 44, NO. 8, AUGUST 2006
According to Sommerfeld [14], the primary field at the general observation point P (ρ, z) takes the form IS0 πp (ρ, z) = 4π
∞
λ J0 (λρ)e−u|z−h| dλ u
(6)
0
√
where u = λ2 − k 2 and its real part is positive, k is the propagation constant of the medium under consideration, and J0 (λρ) is the Bessel function of zero order. At the same time, the secondary field at the general observation point P (ρ, z) takes the form IS0 πs (ρ, z) = 4π
∞
A(λ)euz + B(λ)e−uz J0 (λρ)dλ.
(7)
0
Fig. 1. Geometrical configuration of the problem.
where the functions fn (ρ), n = 1, 2, . . . are definite and continuous and γ is the disturbance parameter such that
|γ n fn (ρ)| a.
(2)
n
1 πsg
We restrict our attention to the first approximation. Accordingly, the first-order terms in γ will only be retained. Also, we shall drop the subscript for simplicity and denote f1 (ρ) by f (ρ). The unit normal to the roughness interface between the two media (air–sea) is given by ∂f (ρ) , 0, 1 . n = −γ ∂ρ
2
∂ π Hρ = ∂ρ ∂z
Hz = −
IS0 = 4π
2
∂ 1 ∂ + ∂ρ2 ρ ∂ρ
π.
2 πsg =
E is
=
E is0
+
γE is1 ,
H is
=
H is0
+
γH is1
(i = 1, 2, 3) (5)
where E s0 (ρ, z) and H s0 (ρ, z) are the secondary fields for the planar interface, i.e., for the case where γ = 0, which was considered by Arutaki and Chiba [3]. E is1 (ρ, z) and H is1 (ρ, z) are the secondary fields due to the variation of the interface between the two media (air–sea).
Ag (λ)e−u1 z J0 (λρ)dλ
IS0 4π
∞
Bg (λ)eu2 z + Cg (λ)e−u2 z J0 (λρ)dλ
0
3 = πsg
IS0 4π
∞
Dg (λ)e−u3 z J0 (λρ)dλ
(8)
0
where g = 0, 1 in the case of the planar and rough interfaces, respectively. The nonvanishing components of the electric and magnetic fields can be deduced without much labor by substituting from (6) and (7) in (4). However, due to the length of the obtained results, we only give here as an example the expressions found for the secondary fields in the sea, i.e., 2 Esgϕ =
−jωµ0 IS0 8π (2)
(4) As the primary fields (the incident or direct field from the source) E p (ρ, z) and H p (ρ, z) are independent of ϕ, the total fields E t (ρ, z) and H t (ρ, z) for the three media will be also independent of ϕ. Employing the perturbation technique by Bishay [4] and Becker [8], we can represent the secondary fields (reflected and transmitted fields) as
∞ 0
(3)
In the case of the horizontal small electric current loop antenna, it works as a vertical magnetic dipole. Accordingly, the magnetic Hertz vector has only a z component πz , and it satisfies the Helmholtz wave equation for a time factor exp(jωt), as is well known. The fields are given by the following formulas [13]: ∂π Eϕ = jωµ ∂ρ
It is evident that the first term of the integral represents the downward wave from the surface of the sea to the field point. The second term is the upward wave from the sea bottom to the field point. Also, the secondary part of π in the three media may be expressed as
2 Hsgρ
∞
Bg (λ)eu2 z + Cg (λ)e−u2 z
−∞
× H1 (λρ)λ dλ ∞ −IS0 = u2 Bg (λ)eu2 z − Cg (λ)e−u2 z 8π −∞ (2)
2 Hsgz
× H1 (λρ)λ dλ ∞ −IS0 Bg (λ)eu2 z + Cg (λ)e−u2 z = 8π −∞ (2)
× H0 (λρ)λ2 dλ,
g = 0, 1
(9)
where H02 (λρ) and H12 (λρ) are the second-kind Hankel functions of order zero and one, respectively. The fields at g = 0 are the solutions for the case where γ = 0 [3], B1 (λ) and C1 (λ) (i.e., g = 1) should be determined by the boundary conditions, and the tangential components of the electric and magnetic fields are continuous at the boundaries z = 0 and z = a + ∆z.
ABO-SEIDA et al.: FAR-FIELD RADIATED FROM A VMD IN THE SEA
2137
Neglecting the higher order terms of O(γ 2 ), it can be shown after a fair amount of manipulative algebra that −4π ωµ0 IS0 − ju1 u− 23 H1 {U (ρ)}−ωµ0 u23 H1 {V (ρ)} × e2u2 a + + 2u2 a − u− 12 u23 + u12 u23 e u− C1 (λ) = 23 B1 (λ) (10) u+ 23
If the integral in the foregoing has a closed-form expression, (16) is an analytic expression for the far-field approximation of (12). The error in the approximation in (16) is
B1 (λ) =
where u± ij = (ui ± uj ) (i, j = 1, 2, 3) and H1 {x} is the Hankel transform of the first order, which is represented as Hν {x} = ∞ 0 Jν (ρσ)xρ dρ, also ∂ 2 2 1 Epϕ + Es0ϕ − Es0ϕ U (ρ) = f (ρ) ∂z z=a ∂ 2 2 1 Hpρ + Hs0ρ − Hs0ρ V (ρ) = f (ρ) ∂z ∂f 2 2 1 + Hpz + Hs0z − Hs0z . (11) ∂ρ z=a The values of B1 (λ) and C1 (λ) in (10) are the same as those obtained by Bishay [4]. III. V IEW ON THE M ETHOD OF S OLUTION
∞ h(t, α)dt
[S(t) − S(t0 )] g(t, α)dt.
Ie =
(12)
−∞
where α is a large parameter. The first step in the procedure is to factorize the integrand h(t, α) into two parts, i.e., 1) a slowly varying part and 2) a rapidly varying part. In other words, we let
Ie is of higher order compared with I intuitively, because in (17), the slowly varying part of the integrand vanishes right at the place from where most of the contributions to the integral have come. In the next section, we will use this technique to obtain the far-fields due to a loop antenna in a three-layered conducting medium with rough upper surface. IV. E VALUATION OF THE H ANKEL T RANSFORMATION To evaluate the electric and magnetic fields in the sea with a rough upper surface, (9), the functions B1 (λ) and C1 (λ) have to be first evaluated from (10). This also must be proceeded by evaluating the Hankel transformation of U (ρ) and V (ρ), which are stated in (11). Then we can write it as
∞ ∂ 2 2 1 EP ϕ + Es0ϕ H1 {U (ρ)} = − Es0ϕ f (ρ) ∂z Z=a × J1 (λρ)ρ dρ
(18)
where the first term in the right-hand side is the primary field. The second term represents the electric field in the sea with a flat surface at z = a, the case which was considered by Arutaki and Chiba [3]. However, the third term represents the electric field in the air (also with a flat surface, at z = a), the case which was considered by Long et al. [10]. Thus, we can write (18) as H1 {U (ρ)} =
∞
3
Mi
(19)
I=1
S(t)g(t, α)dt
I=
(17)
−∞
0
In this section, we give a picture of the new approach for evaluating the far-field expression in a rapid way [11]. Let us first consider an integral of the type I=
∞
(13)
−∞
where S(t) is the slowly varying function, whereas g(t, α) is the rapidly varying function when α → ∞. Therefore, we may assume that g(t, α) ≈ ejαf (t) ,
α → ∞.
(14)
The second step then is to find the stationary phase point of g(t, α), which is given by the value of t as ∂f (t) |t=t0 = 0. ∂t
(15)
∞ −∞
g(t, α)dt.
∞ ∂ 2 J1 (λρ)ρ dρ f (ρ) Epϕ M1 = ∂z Z=a 0 ∞ ∂ 2 M2 = J1 (λρ)ρ dρ f (ρ) Es0ϕ ∂z Z=a 0 ∞ ∂ 1 M3 = − J1 (λρ)ρ dρ. f (ρ) Es0ϕ ∂z Z=a
(20)
0
Most of the contributions to the integral in (13) will be from the vicinity of the stationary phase point t = t0 . Hence, we replace the slowly varying part of the integrand S(t) by its value at the stationary phase point t0 , then (13) becomes I ≈ S(t0 )
where
(16)
First, we calculate the value of the primary field EP2 ϕ from (4) and (6), as follows: EP2 ϕ
−jωµ0 IS0 = 4π
∞
λ −u2 |z−h| e J1 (λρ)λ dλ. u2
0 (1)
(2)
Noting that J1 (λρ) = (1/2)[H1 (λρ) + H1 (λρ)] (1) (2) (1) H1 (λρ) = −e−iπ H1 (e−iπ λρ), where H1 (λρ)
and and
2138
IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 44, NO. 8, AUGUST 2006
(2)
H1 (λρ) are the first and the second kinds of Hankel function of order one, respectively, then EP2 ϕ
−jωµ0 IS0 = 8π
∞ −∞
λ −u2 |z−h| (2) e H1 (λρ)λ dλ. u2
Hence M1 =
(21)
Now, we will use our new technique, which was presented by Chew [11] as stated in Section III ∞ (2) −jωµ0 IS0 H1 (λρ) 2 λ (2) EP ϕ = 8π H0 (λρ) −∞ λ −u2 |z−h| (2) e H0 (λρ) dλ. (22) × u2 When λρ → ∞, there is an approximate formula for Hankel (2) function as Hm (λρ) ≈ 2/(πλρ)e−j(λρ−m(π/2)−(π/4)) . The first bracket in (22) is slowly varying part, whereas the second bracket is rapidly varying as λρ → ∞. The location of the stationary phase point is given by ∂ 2 2 j −λρ − (z − h) k2 − λ = 0 z h. (23) ∂λ λ=λ0
where
where r =
ρ2 + (z − h)2 . Then (22) becomes ωµ0 IS0 k2 ρ −jk2 r 2 e EP ϕ = . 4π r2
(24)
f (ρ0 ) .
2 Es0ϕ
−jωµ0 IS0 = 8π
∞
B0 (λ)eu2 z + C0 (λ)e−u2 z
−∞ (2)
× H1 (λρ)λ dλ = I1 + I2
(29)
where I1 =
jωµ0 IS0 8π
∞ −∞
λ u2
+ − − −2u2 h u− 12 u23 + u12 u23 e + + − −2u2 a u12 u23 + u− 12 u23 e
× e−u2 (2a−z−h) H1 (λρ)λ dλ (2)
−jωµ0 IS0 I2 = 8π
∞
(30)
− −u2 h − −u2 (2a−h) λ u+ −u− 12 u23 e 12 u23 e + − − −2u2 a u2 u+ 12 u23 + u12 u23 e (31)
(25)
(27) where ρ = ρ0 is the location of the stationary phase point, which is given by
∂ j −λρ − k2 ρ2 + d21 =0 ∂ρ ρ=ρ0
jd1 k2 . u2
(2)
0
[R]ρ=ρ0 =
2u2 d1 k22
× e−u2 z H1 (λρ)λ dλ.
(26) where R = ρ2 + d21 and d1 = (a − h). This leads to −u2 d1
e −jk2 −jωµ0 IS0 2 k2 d1 − 3 ρf (ρ) M1 = 4π u2 R2 R ρ=ρ0
jλd1 u2
1−
(28)
2 Now, to evaluate M2 , we must first evaluate Es0ϕ , which is the secondary electric field for the planar interface [3], i.e.,
−∞
It is clear that EP2 ϕ represents the direct wave from the source to the field point P (ρ, z), and this value tends to zero as ρ → ∞ (Appendix A). This coincides with Arutaki and Chiba [3] and Long et al. [10], as the propagation path is in the sea and the attenuation is very large. Substituting (25) in the first equation of (20) and repeating the same technique to find M1 , we have ∞ −jk2 ωµ0 IS0 2 k2 d1 ρf(ρ) M1 = − 4 e−jk2 R J1 (λρ)ρ dρ 4π R3 R
ρ0 =
M0 (λ) = λ
The solution of (23) is given by k2 ρ λ0 = r
−jωµ0 S0 I −u2 d1 e M0 (λ) 4π
The integral I1 had been previously evaluated by Long et al. [10] as −jωµ0 IS0 ρk12 −jk2 (d1 +a−z) N 2 e 2π k2 m+1 ∞ 1 − n3 e−jk1 rm+1 × (m + 1) e−jk2 2ma 3 1 + n3 rm+1 m=0
I1 =
(32)
where N = [((1 + n3 )/(1 − n3 )) + e−jk2 2h ], rm+1 = 2 ρ + (m + 1)2 d2 , d = −2j/k2 , and n3 = k3 /k2 . Using the same approach presented in the foregoing integral, we can evaluate the integral I2 as (Appendix B) I2 =
m+1 ∞ −jωµ0 IS0 k12 ρ −jk2 (z+h) 1 − n3 e 2π k22 1 + n3 m=0 −jk1 rm me e−jk1 rm+1 × e−jk2 2ma + (m + 1) . (33) 3 3 rm rm+1
Finally, from (32) and (33), we can find the electric field 2 Es0ϕ , where the series in I1 and I2 are uniformly convergent 2 in the sec(Appendix C). Substituting by the value of Es0ϕ ond equation of (20), arranging properly and using the same approach, we have M2 =
jωµ0 IS0 −jk2 d1 λe 4π × (N − e−jk2 2a )Gm+1 (λ) − e−jk2 2h Gm (λ) . (34)
ABO-SEIDA et al.: FAR-FIELD RADIATED FROM A VMD IN THE SEA
2139
1 To find M3 , let us use the formula Es0ϕ , which is the electric field in the air in the case of a planar interface. This value has been obtained before by Long et al. [10] as
1 Es0ϕ
m+1 ∞ ωµ0 IS0 1 − n3 −jk2 d1 Ne = 4π 1 + n3 m=0 × e−jk2 2ma
ρe−jk2 rm ρe−jk2 rm+1 − 2 2 rm rm+1
. (35)
Substituting in M3 and following the same approach, we find −jωµ0 IS0 λN e−jk2 d1 4π × [(Gm+1 (λ) − 2Cm+1 (λ) − Gm (λ) + 2Cm (λ) ]
M3 =
Lm (λ) = − λ (N + e−jk2 2a + ju1 N )Gm+1 (λ) + (e−jk2 2h − ju1 N )Gm (λ) + 3λjN u1 (Cm+1 (λ) − Cm (λ)) k2 λ2 + + jN + je−jk2 2a Gm+1 (λ) k2 u1
k2 − je−jk2 2h Gm (λ) . (42) − u1 The prime Gm and Gm+1 denote differentiation of f (ρ) with respect to ρ, and ρm+1 = jλ(m + 1)d and d = (−2j/k2 ).
(36)
V. D ISTURBED F IELD IN THE S EA In fact, the main purpose of this paper is to find the values of the secondary fields in the sea, i.e., g = 1 in (9). These will represent the changes that occur in the electromagnetic field of the wave propagation in the sea due to the perturbation applied on the upper surface. We will show here the process 2 2 of estimating Es1ϕ , and the other two components Hs1ρ and 2 Hs1z are also estimated in the same way. It is clear that the first 2 term in Es1ϕ represents the downward wave from the surface of the sea to the observing point. The second term is the upward wave from the sea bottom to the observing point. We focus our 2 attention on the first term. From (10), we can calculate Es1ϕ by using the following approach, in which the denominator can be expanded as an infinite series inasmuch as
where Gm (λ) =
and
m+1 ∞ 1 − n3 1 + n3 m=0 × e−jk2 2ma e−u1 md f (ρm )
m+1 ∞ 1 − n3 Gm+1 (λ) = 1 + n3 m=0 × e−jk2 2ma e−u1 (m+1)d f (ρm+1 ) m+1 ∞ 1 − n3 u1 Cm (λ) = 1 + n3 mdk12 m=1
− − u21 u23 −2u a 2 e u+ u+ ≺1 21 23
× e−jk2 2ma e−u1 md f (ρm ) Then
m+1 ∞ 1 − n3 u1 Cm+1 (λ) = 1 + n3 (m + 1)dk12 m=0 × e−jk2 2ma e−u1 (m+1)d f (ρm+1 ).
2 Es1ϕ
H1 {U (ρ)} =
−jωµ0 IS0 −u2 d1 e M0 (λ) + e−jk2 d1 Mm (λ) 4π (38)
where Mm (λ) = λ e−jk2 2a Gm+1 (λ) + (e−jk2 2h − N )Gm+1 (λ) + 2N Cm+1 (λ) − 2N Cm (λ)} . (39) With the same method, we can find H1 {V (ρ)}. But for the length of the derivation, we give here only the result as H1 {V (ρ)} =
−jIS0 −u2 d1 e L0 (λ) + e−jk2 d1 Lm (λ) (40) 4π
where L0 (λ) =
−jλ d1
3u22 λ2 − u d + 1 f (ρ0 ) − f (ρ0 ) (41) 2 1 2 k2 u2
∞
n=0
φ(n) +
∞
ψ(n)
= E1 + E2
n=0
(43)
(37)
Adding (27), (34), and (36), we can find the value of the Hankel transformation H1 {U (ρ)}. Then
−jωµ0 IS0 = 8π
∞ 1 = xm . 1 − x m=0
where − − n ∞ u1 u21 u23 φ(n) = λ e−2u2 na + + u21 u+ 21 u23 −∞
j (2) × M0 (λ) + L0 (λ) e−u2 (d1 +d2 ) H1 (λρ)dλ (44) u1 − − n ∞ u1 u21 u23 ψ(n) = λ e−u2 (2na+d2 ) + + u21 u+ 21 u23 −∞
j (2) × Mm (λ) + Lm (λ) e−jk2 d1 H1 (λρ)dλ. (45) u1 To determine φ(n) and ψ(n), we use the procedure used before; thus, we have − n u23 −u1 nd −u1 d φ(n) = j u2 e (1 − e ) u+ 23 λ=λ1 e−jk2 R j (46) × M0 (λ) + L0 (λ) u1 R λ=λ1
2140
IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 44, NO. 8, AUGUST 2006
where λ1 = ρk2 /R = k2 sin α and R= ρ2 + (d1 + d2 + 2na)2 . Then u1 is transformed as |2 is satisfied). Also, k2 sin α (when the condition |k2 |2 |k1 u2 = jk2 cos α, u3 = jk2 N3 , and N3 = φ(n) takes the form
n23 − sin2 α. Then
φ(n) = −k2 cos α n cos α − N3 −k2 d sin α −nk2 d sin α )e × (1 − e cos α + N3 −jk2 R e j × M0 (λ1 ) + L0 (λ1 ) . (47) k2 sin α R
Fig. 2. Variation of the absolute value of the electric field E1 versus radial distance for frequencies of 10 and 50 kHz.
Similarly ψ(n) = −k2 cos β n cos β −N3 × e−jk2 d1 (1−e−k2 d sin β )e−nk2 d sin β cos β +N3 × Mm (λ2 ) +
−jk2 R0 e j Lm (λ2 ) k2 sin β R0
(48)
where R0 = ρ2 + (d2 + 2na)2 and λ2 = ρk2 /R0 = k2 sin β. From (47) and (48), we can find the disturbed field in the sea without approximation. The physical meaning of the first term in (43) indicates a series of waves that propagate upward from the source and make n round trips between the rough sea surface and the bottom and finally arrive at the field point in the sea, where [(cos α − N3 )/(cos α + N3 )] is the reflection coefficient at the sea bottom. The second term represents another series of waves that first propagate downward from the source, then reflected upward at the sea bottom and travel along the same bath as aforementioned waves in [3] and [10]. These results also show that φ(n) and ψ(n) are proportional with e−jk2 R /R, and this means that the electric field in the sea takes the form as the Sommerfeld integral. Also, the result coincides with the result obtained by Long et al. [10], when the disturbed factor fn (ρ) is neglected. VI. N UMERICAL R ESULTS The perturbation electric field in the sea due to the vertical magnetic dipole is computed for different cases. The part of the sea under investigation is taken for different thicknesses, and the height of the source is taken to be 4 m. We choose the sea waves represented by the roughness cosine profile with a period of 10 m and of different frequencies. We provide numerical plots of |E1 | in Figs. 2 and 3, which show that the multilayered waves are decreasing rapidly with radial distance ρ and the vertical distance of the field point, respectively, for different frequencies. Also, numerical plots of |E2 | in Figs. 4 and 5 indicate that these multilayered waves are more important than |E1 |. VII. C ONCLUSION Closed-form expressions for the far-fields in the sea are obtained by utilizing the three-layered model. Analytical expres-
Fig. 3. Variation of the absolute value of the electric field E1 versus vertical distance of the field point for frequencies of 10 and 50 kHz.
Fig. 4. Variation of the absolute value of the electric field E2 versus radial distance for frequencies of 10 and 50 kHz.
Fig. 5. Variation of the absolute value of the electric field E2 versus vertical distance of the field point for frequencies of 10 and 50 kHz.
sions were found for the first-order corrections of the electric and magnetic fields due to a small perturbation applied in the upper surface due to sea waves. The complex image theory helped, in a rapid way, to approximate the Sommerfeld Integral. The results presented here indicate that a series of waves propagate either upward or downward from the source. These waves undergo cyclic motions between the rough sea surface and the bottom and finally arrive at the field point in the sea where [(cos α − N3 )/(cos α + N3 )] is the reflection coefficient at the sea bottom. The results obtained in this paper agree with those results obtained hitherto for special cases as [3] and [10]. Long et al. [10] have dealt with lateral waves, whereas in this study, we have considered the reflected waves. Numerical results are also given in this paper. It is also worth noting that our current research work is being carried out on lateral waves, too.
ABO-SEIDA et al.: FAR-FIELD RADIATED FROM A VMD IN THE SEA
A PPENDIX A
Then we have
The aim of the appendix is to verify that (25) matches the previous results, i.e., EP2 ϕ =
ωµ0 IS0 4π
k2 ρ −jk2 r e , r2
ωµ0 IS0 k2 e−jk2 ρ , 4πρ
2j(m + 1)k12 ρd Y2 (m) = 3 k2 rm+1
From (31), we can calculate I2 by using the following approach. The denominator in (31) can be expanded as an infinite series inasmuch as − − ∞ u21 u23 −2u a 1 2 ≺ 1 = e xm . u+ u+ 1 − x 21 23 m=0 Then ∞
T2 (m) +
m=0
∞
Y2 (m)
m=0
where T2 (m) = −∞
1 − n3 1 + n3
m+1
Then m+1 ∞ −jωµ0 IS0 k12 ρ −jk2 (z+h) 1 − n3 e I2 = 2π k22 1 + n3 m=0 −jk1 rm me e−jk1 rm+1 ×e−jk2 2ma + (m + 1) . 3 3 rm rm+1
A PPENDIX C The series in the formula of I2 can be written as a sum of two series
A PPENDIX B
λ u2
e−jk2 (z+h+2ma) e−jk1 rm .
0 ≤ ρ ≤ ∞.
2 The aforementioned result of Epϕ coincides with that of Sommerfeld [14].
∞
m+1
× e−jk2 (z−h+2(m+1)a) e−jk1 rm+1 .
In this paper, we deal with the first case, and this is clarified in the numerical results. The second case (ρ → ∞) is refused because on substituting (25) in the first equation of (20), calculating M1 integral with respect to ρ from −∞ to ∞ (i.e., ρ ∈ (−∞, ∞)) leads to an unacceptable result. Also, the last case is refused. Noting that ρ ∼ = r, then
1 − n3 1 + n3
Similarly
1) 0 ≤ ρ ≤ ∞, λ → ∞; 2) ρ → ∞, 0 ≤ λ ≤ ∞; 3) ρ → ∞, λ → ∞.
−jωµ0 IS0 I2 = 8π
2jmk12 ρd T2 (m) = 3 k2 rm
0 ≤ ρ ≤ ∞.
Inasmuch as the previous equation is obtained from (22) under the condition integral λρ → ∞, this condition gives three cases, namely:
2 = Epϕ
2141
− u− 21 u23 + + u21 u23
m
u− 23 u+ 23
× e−u2 (z+h) e−jk2 2ma H1 (λρ)λ dλ ∞ − m+1 λ u− 21 u23 Y2 (m) = + u2 u+ 21 u23 (2)
g(m) =
m+1 ∞ 1−n3 m=0
g(m + 1) =
1+n3
m+1 ∞ 1−n3
m=0
1+n3
me−jk1 rm 3 rm
e−jk2 2ma
(m + 1)e−jk1 rm+1 3 rm+1
where rm = ρ2 + m2 d2 . To prove that I2 is a uniformly convergent series, it is satisfactory to show that one of these series, e.g., g(m), is convergent. Therefore, we apply a “Weiertrass test” and the theorem from [15], i.e., “If | m i=0 fi (x)| = |Sm (x)|, Sm is bounded in the interval (a, b), i.e., |Sm (x)| ≤ M ∀ m, x ∈ (a, b) and {am } is a bounded sequence, then ∞ m=0 am fm (x) converges uniformly in the interval (a, b).” It is clear that g(m) = am fm (ρ), where am = m fm (ρ) =
−∞
× e−u2 (a+d1 +h) e−jk2 2ma H1 (λρ)λ dλ.
e−jk2 2ma
1 − n3 1 + n3
e−jk1 rm , 3 rm
m+1
e−jk2 2ma
0≤ρ≤∞
(2)
According to the complex image theory [12], when the condition |k2 |2 |k1 |2 is satisfied, we have u− 21 ≈ e−u1 d u+ 21
2u1 ≈ 1 − e−u1 d . u+ 21
1) {am } is a bounded sequence inasmuch as am+1 am =
m+1 1 − n3 rm ∀ m, then e−jk1 rm+1 < e−jk1 rm ∀ m, leading to r1 − r0 ≥
d2 2ρ
r2 − r0 ≥
2d2 4d2 ≥ 2ρ 2ρ
r3 − r0 ≥
3d2 9d2 ≥ 2ρ 2ρ
Osama M. Abo-Seida was born in Tanta, Egypt, on October 21, 1968. He received the B.Sc. degree in mathematics and the M.Sc. and Ph.D. degrees in electrodynamics from Tanta University, Tanta, in 1990, 1994, and 1997, respectively. Since 1997, he has been an Assistant Professor with the Department of Mathematics, Faculty of Education, Tanta University, Kafr El-Sheikh Branch, Egypt. His research interests include computational electromagnetic, wave propagation, antenna, and applied mathematics. Dr. Abo-Seida is a member of the Mathematics Society of Egypt and of the Scientific Council of Egypt.
.... Then e−jk1 r0 Sm (ρ) ≤ r03
1 − e−jk1 (
m+1 2 2ρ d d2
1 + e−jk1 ( 2ρ )
)
.
Therefore 2 2 , ρ = 0, |Sm (ρ)| ≤ 1d ρ3 sin k4ρ
2
k1 d 2nπ
where n = ±1, ±2, . . . . Hence, the sequence of partial sum Sm (ρ) is bounded, and g(m) is a uniformly convergent series. R EFERENCES [1] R. K. Moor and W. E Blair, “Dipole radiation in a conducting half space,” J. Res. Nat. Bur. Stand., vol. 65D, no. 5, pp. 515–522, May 1961. [2] S. H. Durrani, “Air to undersea communication with magnetic dipole,” IEEE Trans. Antennas Propag., vol. AP-12, no. 4, pp. 464–469, Jun. 1964. [3] A. Arutaki and J. Chiba, “Communication in a three-layered conducting media with a vertical magnetic dipole,” IEEE Trans. Antennas Propag., vol. AP-28, no. 4, pp. 551–556, Jul. 1980. [4] S. T. Bishay, “Communication of radio waves in sea with a rough upper surface,” AEÜ Archiv Elektronik Übertrag., vol. 38, no. 3, pp. 186–188, 1984. [5] L. E.-S. Rashid, S. T. Bishay, and O. M. Abo-Seida, “Electromagnetic field produced by a vertical magnetic dipole above a rough surface,” in Proc. ICEAA, Sep. 12–15, 1995, pp. 231–240. [6] S. T. Bishay, “Communication in a rough three-layered conducting medium,” Radio Sci., vol. 22, no. 5, pp. 781–786, 1987. [7] ——, “Estimation of the electromagnetic field in a sea with rough upper and lower surfaces,” Can. J. Phys., vol. 66, no. 4, pp. 319–322, 1988.
Samira Tadros Bishay received the B.Sc. and M.Sc. degrees from Ain Shams University, Cairo, Egypt, in 1962 and 1967, respectively, and the Ph.D. degree from Saarland University, Saarbrücken, Germany, in 1973. In 1962, she joined the Department of Applied Mathematics, Faculty of Science, Ain Shams University, where she has been a Professor since 1989. Her research interests include the development of communication systems in stratified-media rough surface, wave propagation in the sea, and antenna and waveguide boundary-value problems. Prof. Samira is a member of the Mathematics Society of Egypt and the Scientific Council of Egypt.
Khaled Mohamed El-Morabie received the B.Sc. degree from Tanta University, Tanta, Egypt, in 2002. He is currently working toward the M.Sc. degree at Tanta University. In 2002, he joined the Department of Applied Mathematics, Faculty of Science, Tanta University, as a Demonstrator. His research interests include wave propagation in the sea, antenna and waveguide boundary-value problems, and communications systems in stratified-media rough surfaces. Mr. Khaled is a member of the Mathematics Society of Egypt.
