electromagnetic macro modeling of propagation in mobile ... · for the fi elds of the ground wave,...

IEEE Antennas and Propagation Magazine, Vol. 54, No. 6, December 2012 17

Electromagnetic Macro Modeling of Propagation in Mobile Wireless

Communication: Theory and Experiment

Tapan K. Sarkar1, Walid Dyab1, Mohammad N. Abdallah1, Magdalena Salazar-Palma2, M. V. S. N. Prasad3, Sio Weng Ting4, and Silvio Barbin5

1Department of Electrical Engineering and Computer ScienceSyracuse University

Syracuse, New York 13244-1240, USAE-mail: [email protected], http://lcs.syr.edu/faculty/sarkar/

2Dept. of Signal Theory & CommunicationsUniversidad Carlos III de Madrid

Avenida de la Universidad, 30, 28911 Leganés, Madrid, SpainE-mail: [email protected]

3National Physical LaboratoryDr. K. S. Krishnan Road, New Delhi-110012, India

E-mail: [email protected]

4Dept. of Science and TechnologyUniversity of Macau

Av. Padre Tomas Pereira, Taipa, Macau, ChinaE-mail: [email protected]

5Departamento de Engenharia de Telecomunicações e ControleEscola Politécnica da Universidade de São Paulo

São Paulo, BrazilE-mail: [email protected]

Abstract

The objective of this paper is to illustrate that electromagnetic macro modeling can properly predict the path-loss exponent in mobile cellular wireless communication. This represents the variation of the path loss with distance from the base-station antenna. Specifi cally, we illustrate that the path-loss exponent in cellular wireless communication is three, preceded by a slow-fading region, and followed by the fringe region, where the path-loss exponent is four. The sizes of these regions are determined by the heights of the base-station transmitting antennas and the receiving antennas. Theoretically, this is illustrated through the analysis of radiation from a vertical electric dipole situated over a horizontal imperfect ground plane, as fi rst considered by Sommerfeld in 1909. To start with, the exact analysis of radiation from the dipole is made using the Sommerfeld formulation. The semi-infi nite integrals encountered in this formulation are evaluated using a modifi ed saddle-point method for fi eld points moderate to far distances away from the source point, to predict the appropriate path-loss exponents. The refl ection-coeffi cient method is also derived by applying a saddle-point method to the semi-infi nite integrals, and this is shown to not provide the correct path-loss exponent that matches measurements. The various approximations used to evaluate the Sommerfeld integrals are described for different regions. It is also important to note that Sommerfeld’s original 1909 paper had no error in sign. However, Sommerfeld overlooked the properties associated with the so-called “surface-wave pole.” Both accurate numerical analyses, along with experimental data, are provided to illustrate the above statements. In addition, Okumura’s experimental data, and extensive data taken from seven different base stations in urban environments at two different frequencies, validate the theory. Experimental data revealed that a macro modeling of the environment, using an appropriate electromagnetic analysis, can accurately predict the path-loss exponent for the propagation of radio waves in a cellular wireless communication scenario.

ISSN 1045-9243/2012/$26 ©2012 IEEE

AP_Mag_Dec_2012_Final.indd 17 12/9/2012 3:49:54 PM

18 IEEE Antennas and Propagation Magazine, Vol. 54, No. 6, December 2012

1. Original Sommerfeld Formulation for a Vertical Electric Dipole Over an

Imperfect Ground Plane

Consider an elementary electric dipole of moment Idz , oriented along the z direction, and located at ( ), , x y z′ ′ ′ . The dipole is situated over an imperfect ground plane, char-acterized by a complex relative dielectric constant, ε , as seen in Figure 1. The complex relative dielectric constant is given by

0r jε ε σ ωε= − , where rε represents the relative permit tivity of the medium, 0ε is the permittivity of vacuum, σ is the conductivity of the medium, ω stands for the angular fre-quency, and j is the imaginary unit, i.e., 1j = − . It is possible to formulate a solution to the problem of radiation from the dipole operating in the presence of the imperfect ground in terms of a single Hertzian vector, zΠ , of the electric type. A time variation of ( )exp j tω is assumed throughout the analy sis, where t is the time variable. The Hertzian vector ˆz zu Π in this case satisfi es the wave equation

( ) ( ) ( ) ( )2 21 1

0z

Édzk x x y y z zj

δ δ δωε− ′ ′ ′∇ + Π = − − − ,

(1)

( )2 2

2 2 0zk∇ + Π = , (2)

where

2 21 0 0k ω µ ε= , (3)

2 22 0 0k ω µ ε ε= , (4)

and δ represents the delta function in space. The primed and unprimed coordinates are for the source and fi eld points, respectively. Subscript 1 denotes the upper half space, which is air, and subscript 2 denotes the lower half space, which is the imperfectly conducting Earth characterized by a complex relative dielectric constant, ε . The electric and the magnetic fi eld vectors are derived from the Hertzian vector using

( ) 2i i i iE k= ∇ ∇ Π + Π

, (5)

and

( )0i i iÇ jωε ε= ∇×Π

, (6)

respectively, with 1i = , 2.

In medium 1, 1 1ε = , and for medium 2, 2ε ε= . The propagation constants in medium 1 and 2, called 1k and 2k , are thus related by 2 1k k ε= . At the interface 0z = , the tangential electric- and magnetic-fi eld components must be continuous, conditions which in terms of the Hertzian vector components can be written as

1 2z zy y

ε∂Π ∂Π

=∂ ∂

, (7a)

1 2z zx x

ε∂Π ∂Π

=∂ ∂

, (7b)

1 2z zy z y z

∂Π ∂Π∂ ∂ = ∂ ∂ ∂ ∂ , (7c)

1 2z zx z x z

∂Π ∂Π∂ ∂ = ∂ ∂ ∂ ∂ . (7d)

Since all the boundary conditions must hold at 0z = for all x and y, the x and y dependence of the fi elds on either side of the interface must be the same. Therefore,

1 2z zεΠ = Π , (8a)

1 2z zz z

∂Π ∂Π=

∂ ∂. (8b)

Keywords: Propagation; imperfect ground; surface wave; ground wave; Norton surface wave; cellular wireless communication; path loss exponent; macro modeling; Sommerfeld formulation; land mobile radio cellular systems; land mobile radio propagation factors

Figure 1. A vertical dipole over a horizontal imperfect ground plane.

The complete solutions for the Hertz vectors satisfying the wave Equations (1) and (2) and the boundary conditions of Equation (8) have been derived by many researchers over the last century. A partial list [1-18] that will be important to our discussions is provided in chronological order, starting with Sommerfeld [1]. The solutions are

( )1 11

1

expz

jk RP

R−Π =

( ) 2 2 2 21 202 2 2 2 2 2

0 1 1 2

J k k

k k k

ε λ λλρ

λ ε λ λ

∞ − − −+

− − + −∫

( ) }2 21exp k z z dλ λ λ ′− − +

(9)

and

( ) ( )2 2 2 20 2 12 2 2 2 2

0 1 2

J exp2z

k z k zP d

k k

λρ λ λλ λ

ε λ λ

∞ ′− − −Π =

− + −∫

(10)

for ( )2 21,2Re 0 kλ − > . ( )0J x represents the zeroth-order Bessel function of the fi rst kind of argument x. Here,

04I dzP

jω π ε= , (11)

( ) ( )2 2x x y yρ ′ ′= − + − , (12)

( )221R z zρ ′= + − , (13)

and λ is the variable of integration. For 1zΠ , the fi rst term inside the brackets can be interpreted as the particular solution or the direct line-of-sight (LOS) contribution from the dipole source, and the second term can be interpreted as the comple-mentary solution or a refl ection term (refl ection from the imperfect ground plane). We will call this potential responsi ble for the fi elds of the ground wave, as per IEEE Standard Defi nitions of Terms for Radio Wave Propagation [19]. Observe that the second term of this potential for the ground wave in Equation (9) is the strongest near the surface of the Earth, and exponentially decays as we go away from the inter face. In Appendix 1, we illustrate the different waves and their specifi c properties that we refer to in this paper, for clarifi ca tion.

Similarly, the solution for 2zΠ can be interpreted as a partial transmission of the wave from medium 1 into medium 2. With these thoughts in mind, 1zΠ , or equivalently, the potential responsible for the ground wave, can be split up into two terms:

( )1 1 01reflecteddirect

z z sz P g gΠ = Π +Π = + , (14)

where

( )1 1 1 1 0expdirectz P jk R R P gΠ = − = , (15)

2 2 2 2

1 21 2 2 2 2

0 1 2

reflectedz

k kP

k k

ε λ λ

ε λ λ

∞ − − −Π =

− + −∫

( ) ( )2 20 12 2

1

J exps

k z zd Pg

k

λρ λλ λ

λ

′− − + =−

.

(16)

The path of integration for the semi-infi nite integral is labeled 2C and is depicted in Figure 2, along with the singularities of

the multivalued function, two branch points at 1k and 2k , and a pole, p, arising from the ratio of two functions in Equa-tion (16). A physical explanation to the two components of the Hertz potential, 1zΠ , can now be given. The fi rst component,

1directzΠ , can be explained as a spherical wave originating from

the source dipole. This term is easy to deal with. The diffi cult problem lies in the evaluation of 1

reflectedzΠ . Therefore,

1reflectedzΠ can be interpreted as a superposition of plane waves

resulting from the refl ection of the various plane waves into which the original spherical wave was expanded. This arises from the identity

( ) ( ) ( )2 2

0 11 22 22 0 1

J expexp k z zjk Rd

R k

λρ λλ λ

λ

∞ ′− − +− =−

∫

(17)

for

( )2 21Re 0kλ − > and

( )222R z zρ ′= + + . (18)

The term under the integral sign in Equation (16) can be rec-ognized as a multiple plane-wave decomposition of the origi nal spherical-wave source. Upon refl ection of the plane waves from the dipole source as expressed in 1

reflectedzΠ , the ampli tude of

each wave must be multiplied by the refl ection coeffi cient, ( )R λ . The complex refl ection coeffi cient, ( )R λ , takes into

account the phase change as the wave travels from the source



1. Original Sommerfeld Formulation for a Vertical Electric Dipole Over an

Imperfect Ground Plane

Consider an elementary electric dipole of moment Idz , oriented along the z direction, and located at ( ), , x y z′ ′ ′ . The dipole is situated over an imperfect ground plane, char-acterized by a complex relative dielectric constant, ε , as seen in Figure 1. The complex relative dielectric constant is given by

0r jε ε σ ωε= − , where rε represents the relative permit tivity of the medium, 0ε is the permittivity of vacuum, σ is the conductivity of the medium, ω stands for the angular fre-quency, and j is the imaginary unit, i.e., 1j = − . It is possible to formulate a solution to the problem of radiation from the dipole operating in the presence of the imperfect ground in terms of a single Hertzian vector, zΠ , of the electric type. A time variation of ( )exp j tω is assumed throughout the analy sis, where t is the time variable. The Hertzian vector ˆz zu Π in this case satisfi es the wave equation

( ) ( ) ( ) ( )2 21 1

0z

Édzk x x y y z zj

δ δ δωε− ′ ′ ′∇ + Π = − − − ,

(1)

( )2 2

2 2 0zk∇ + Π = , (2)

where

2 21 0 0k ω µ ε= , (3)

2 22 0 0k ω µ ε ε= , (4)

and δ represents the delta function in space. The primed and unprimed coordinates are for the source and fi eld points, respectively. Subscript 1 denotes the upper half space, which is air, and subscript 2 denotes the lower half space, which is the imperfectly conducting Earth characterized by a complex relative dielectric constant, ε . The electric and the magnetic fi eld vectors are derived from the Hertzian vector using

( ) 2i i i iE k= ∇ ∇ Π + Π

, (5)

and

( )0i i iÇ jωε ε= ∇×Π

, (6)

respectively, with 1i = , 2.

In medium 1, 1 1ε = , and for medium 2, 2ε ε= . The propagation constants in medium 1 and 2, called 1k and 2k , are thus related by 2 1k k ε= . At the interface 0z = , the tangential electric- and magnetic-fi eld components must be continuous, conditions which in terms of the Hertzian vector components can be written as

1 2z zy y

ε∂Π ∂Π

=∂ ∂

, (7a)

1 2z zx x

ε∂Π ∂Π

=∂ ∂

, (7b)

1 2z zy z y z

∂Π ∂Π∂ ∂ = ∂ ∂ ∂ ∂ , (7c)

1 2z zx z x z

∂Π ∂Π∂ ∂ = ∂ ∂ ∂ ∂ . (7d)

Since all the boundary conditions must hold at 0z = for all x and y, the x and y dependence of the fi elds on either side of the interface must be the same. Therefore,

1 2z zεΠ = Π , (8a)

1 2z zz z

∂Π ∂Π=

∂ ∂. (8b)

Keywords: Propagation; imperfect ground; surface wave; ground wave; Norton surface wave; cellular wireless communication; path loss exponent; macro modeling; Sommerfeld formulation; land mobile radio cellular systems; land mobile radio propagation factors

Figure 1. A vertical dipole over a horizontal imperfect ground plane.

The complete solutions for the Hertz vectors satisfying the wave Equations (1) and (2) and the boundary conditions of Equation (8) have been derived by many researchers over the last century. A partial list [1-18] that will be important to our discussions is provided in chronological order, starting with Sommerfeld [1]. The solutions are

( )1 11

1

expz

jk RP

R−Π =

( ) 2 2 2 21 202 2 2 2 2 2

0 1 1 2

J k k

k k k

ε λ λλρ

λ ε λ λ

∞ − − −+

− − + −∫

( ) }2 21exp k z z dλ λ λ ′− − +

(9)

and

( ) ( )2 2 2 20 2 12 2 2 2 2

0 1 2

J exp2z

k z k zP d

k k

λρ λ λλ λ

ε λ λ

∞ ′− − −Π =

− + −∫

(10)

for ( )2 21,2Re 0 kλ − > . ( )0J x represents the zeroth-order Bessel function of the fi rst kind of argument x. Here,

04I dzP

jω π ε= , (11)

( ) ( )2 2x x y yρ ′ ′= − + − , (12)

( )221R z zρ ′= + − , (13)

and λ is the variable of integration. For 1zΠ , the fi rst term inside the brackets can be interpreted as the particular solution or the direct line-of-sight (LOS) contribution from the dipole source, and the second term can be interpreted as the comple-mentary solution or a refl ection term (refl ection from the imperfect ground plane). We will call this potential responsi ble for the fi elds of the ground wave, as per IEEE Standard Defi nitions of Terms for Radio Wave Propagation [19]. Observe that the second term of this potential for the ground wave in Equation (9) is the strongest near the surface of the Earth, and exponentially decays as we go away from the inter face. In Appendix 1, we illustrate the different waves and their specifi c properties that we refer to in this paper, for clarifi ca tion.

Similarly, the solution for 2zΠ can be interpreted as a partial transmission of the wave from medium 1 into medium 2. With these thoughts in mind, 1zΠ , or equivalently, the potential responsible for the ground wave, can be split up into two terms:

( )1 1 01reflecteddirect

z z sz P g gΠ = Π +Π = + , (14)

where

( )1 1 1 1 0expdirectz P jk R R P gΠ = − = , (15)

2 2 2 2

1 21 2 2 2 2

0 1 2

reflectedz

k kP

k k

ε λ λ

ε λ λ

∞ − − −Π =

− + −∫

( ) ( )2 20 12 2

1

J exps

k z zd Pg

k

λρ λλ λ

λ

′− − + =−

.

(16)

The path of integration for the semi-infi nite integral is labeled 2C and is depicted in Figure 2, along with the singularities of

the multivalued function, two branch points at 1k and 2k , and a pole, p, arising from the ratio of two functions in Equa-tion (16). A physical explanation to the two components of the Hertz potential, 1zΠ , can now be given. The fi rst component,

1directzΠ , can be explained as a spherical wave originating from

the source dipole. This term is easy to deal with. The diffi cult problem lies in the evaluation of 1

reflectedzΠ . Therefore,

1reflectedzΠ can be interpreted as a superposition of plane waves

resulting from the refl ection of the various plane waves into which the original spherical wave was expanded. This arises from the identity

( ) ( ) ( )2 2

0 11 22 22 0 1

J expexp k z zjk Rd

R k

λρ λλ λ

λ

∞ ′− − +− =−

∫

(17)

for

( )2 21Re 0kλ − > and

( )222R z zρ ′= + + . (18)

The term under the integral sign in Equation (16) can be rec-ognized as a multiple plane-wave decomposition of the origi nal spherical-wave source. Upon refl ection of the plane waves from the dipole source as expressed in 1

reflectedzΠ , the ampli tude of

each wave must be multiplied by the refl ection coeffi cient, ( )R λ . The complex refl ection coeffi cient, ( )R λ , takes into

account the phase change as the wave travels from the source



( ), , x y z′ ′ ′ to the boundary and then to the point of observation,

( ), , x y z . The refl ection coeffi cient, ( )R λ , is then defi ned as

( )2 2 2 2

1 22 2 2 2

1 2

k kR

k k

ε λ λλ

ε λ λ

− − −=

− + −, (19)

where the semi-infi nite integral over λ in 1reflectedzΠ takes into

account all the possible plane waves. As ε → ∞ , i.e., a perfect conductor for the Earth, then sg of Equation (16) reduces to Equation (17), and represents a simple spherical wave originating at the image point. This physical picture will later be applied in the derivation of the refl ection-coeffi cient method. The refl ection coeffi cient takes into account the effects of the ground plane in all the wave decomposition of the spherical wave and sums it up as a ray originating from the image of the source dipole, but multiplied by a specular refl ection coeffi cient,

( )R θ , where θ is interpreted as the angle of the incident wave to the ground.

It is now important to point out that there are two forms of

1reflectedzΠ that may be used interchangeably, as the two

expressions are mathematically identical in nature (but have different asymptotic properties, as we shall see). These are defi ned as

( )1 21

2

expreflectedz

jk RP

R−Π =

( ) ( )2 22 2 0 122 2 2 2 2 2

0 1 1 2

J exp2

k z zkd

k k k

λρ λλλ λ

λ ε λ λ

∞ ′− − + − − − − + −

∫

[ ]1 sVP g g− , (20)

where 1g represents the spherical wave originating from the image of the source, and sVg represents the correction factor to accurately characterize the effects of the ground. Equiva lently, one can rewrite the same expression as

( )1 21

2

expreflectedz

jk RP

R−Π = −

( ) ( )2 20 12 2 2 2

0 1 2

J exp2

k z zd

k k

λρ λε λ λ

ε λ λ

∞ ′− − + + − + −

∫

[ ]1 sVP g G− + . (21)

The image from the source now has a negative sign, along with the correction factor. This expansion is useful when both the transmitter and the receiver are close to the ground, and since the refl ection coeffi cient is 1− for a grazing angle of incidence, where 2θ π≈ . The direct term, 0g , then cancels the image term, 1g , leaving only the correction factor, sVG . It is very confusing to appreciate what the IEEE Standard Defi nitions for Radio Wave Propagation means by the term Nor ton surface wave, given as “The Norton wave consists of the total fi eld minus the geometrical-optics fi eld.” The Norton surface wave can therefore be associated with both sVg and sVG ! The question is, which one to use? Now, for grazing incidence, the fi elds are obtained using sVG . The question then is how does the Norton surface wave manifest itself in either of these two representations, as the difference between sVG and sVg is the geometrical-optics fi eld!

2. Properties Related to the Exact Contour of Integration

We now look at the properties of Equations (20) and (21). In general, the total fi eld in the upper half-space ( 0)z > consists of the direct fi eld from the source dipole situated at z z′= , its image situated at z z′= − , and the correction terms of sVg used in Equation (20). The correction term, sVg , takes into account the nature of the imperfect ground, because as ε → ∞ , 0sVg → . The problem of determining the refl ected fi eld amounts to evaluating the integral for sVg , since 0g and

1g are easy to calculate.

In Sommerfeld’s 1909 paper, he changed the limits of the integration from 0 to ∞ in Equation (20) to an integral from −∞ to +∞ by transforming the Bessel function of the fi rst kind and zeroth order to Hankel functions of the fi rst and sec ond kinds and zeroth order through the use of the following identity:

( ) ( ) ( ) ( ) ( )1 20 0 0

12

J x x x = Η +Η , (22a)

and also utilizing

Figure 2. The contour of integration along the real axis from 0 to ∞ in the complex λ plane.

( ) ( ) ( ) ( )1 20 0jxe xπΗ = −Η , (22b)

where (1)0H and (2)

0H are the Hankel functions of zeroth order and of the fi rst and second kinds, respectively. Sommerfeld thus closed the path integral, 2C , of Figure 2 by using the contour as shown in Figure 3a, plotting the locations of the branch points,

0k and 1k , and the pole, p, for the term 1reflectedzΠ . Also shown

in Figure 3a are the two appropriate branch cuts, and a path encircling the pole.

The convergence of the integrals in Equations (20) and (21) is assured, even in the presence of the Hankel function, when ( )Im 0λ ≤ as ρ →∞ . Convergence is also assured for

( )1/22 21Im 0k λ− < , (23)

and for

( )1/22 2

1 1 10k k jk

λλ

→′ ′′− → + ,

10k ′ > , (24)

10k ′′ ≤ .

If

( )1

2 2 21 z zk k j kλ ′ ′′− = + , (25)

then

1 1z

z

k kkkλ λ′ ′′ ′ ′′−′′ =

′ (26)

and

( )1/222 2 2 2

21 11 12z

k kk k kλ λ λ λ ′ ′′ ′ ′′ − − + ′ ′ ′′ ′ ′′= + −

1/22 2 2 21 1 0

2k k λ λ ′ ′′ ′ ′′− − + + >

(27)

where

( ) ( )Re Imj jλ λ λ λ λ′ ′′= + = + . (28)

The positive sign is chosen for zk ′ since 0zk ′ > (from Equa-tions (24), (25), and (27)). On the path of integration, 0zk ′′ < , and if 1 0k ′′ ≠ , convergence would be assured even if 0λ′′ = . If

Figure 3a. The closed contour of integration as imple mented by Sommerfeld in the lower complex λ plane.

Figure 3b. The actual location of the pole in the lower complex λ plane.

Figure 3c. The contribution of the pole generating the sur-face wave is excluded when the branch-cut contour is cho-sen vertically (Kahan and Eckart [22], Baños [6, pp. 55]).



medium 1 is lossless, 1 0k ′′ = , and then from Equation (26), 0λ′ > when 0λ′′ > , and 0λ′ < when 0λ′′ < . The fi rst condi-

tion, 0λ′′ > , confl icts with the convergence requirement for the

Hankel function, ( ) ( )20H λρ . This problem can be avoided if medium 1 is assumed to be lossy (i.e., 1 0k ′′ ≠ ), and the “lossless case” then is assumed to be the limiting form of the expression

1 0k ′′→ . Since ( ) ( )20H λρ can be integrated through the origin

( )0λ = even if 0λ′′ = , the path 2C of Figure 2 can be modifi ed to the path 1C of Figure 3a, follow ing the real axis from −∞ to +∞ . Now, Equation (20) can be rewritten as

1

2 222 2

1sV

C

kg j

k

λ

λ

−= −

−∫

( ) ( ) ( )2 2 20 12 2 2 2

1 2

exp j k z zd

k k

λρ λλ λ

ε λ λ

′Η − − +

− + −

(29)

where the contour of integration, 1C , is shown in Figure 3a, along with the location of the branch points at 1k and 2k , and their associated branch cuts, together with the pole of Equa-tion (29). The presence of the free term, λ , will nullify the singularity of the Hankel function at 0λ = . The integral from ‒∞ to 0 goes slightly below the negative real axis, as the Hankel function has a branch cut along that line. sVg is now a spectrum of plane waves traveling away from the ground plane, with the vertical component of the propagation constant given as

2 21k λ− . The integral in Equation (29) also con tains double-

valued functions 2 21,2k λ− . The proper sheets of the double-valued functions are those on which the radia tion condition,

2 212 2 0|j k zsV sV z

gj k g e

zλλ

∞− −=−∞

∂ − − = ∂ , (30)

is satisfi ed. This can occur only if 2 21Im 0k λ − >

, i.e.,

0sVg → as ρ →∞ for a fi xed z , and 0sVg → as z →∞ for a fi xed ρ .

3. Sommerfeld’s Original Formulation Had No Error in Sign, But...

Historically, in 1909, Sommerfeld computed the integral along the positive real axis of Figure 2. He did this by fi rst applying the Cauchy principal-integral method to close the contour by a large semicircle at infi nity, given by the semi-circular contour with indentations of Figure 3a, lying in the third and the fourth quadrants. The result was the integral given

by the contours of Figure 3a. As seen in his book [4], this closed contour of integration is equivalent to two integrals around the branch cuts associated with the branch points at 1k+ and 2k+ , and a contour integration around the pole Pλ λ= , where

1 22 21 2

Pk k

k kλ =

+. The other branch points, 1k− and 2k− , and

the pole located at 1 22 21 2

k k

k k−

+, are of no concern, as they are

located in the upper half-plane where the contour is not closed, as seen in Figure 3a. Sommerfeld then evaluated the residue at the pole, and showed that it has the form of a surface wave (this wave is defi ned in Appendix 1). Using Equation (21), Sommerfeld then showed that

( ) ( )(2)2 2 22 101 2 2

2 12 2 2 2

1 2

H exp2

P Ppolez

P P

k z kjP

k k

k k

λ ρ λπ

λ λ

− −

Π = − +

− −

,

(31)

where 1polezΠ is part of the solution from the pole contribution.

Next, a large-argument approximation was made for the Hankel function by following the path of integration of Fig ure 3a. For large values of ρ , the asymptotic representation for the Hankel function was used:

( ) ( ) [ ]Limit202 exp 4j j

λρλρ λρ π

πλρ→ ∞Η → − + ,

(32)

resulting in

( )2 2 22 11 2 2

2 12 2 2 2

1 2

exp2 P Ppolez

P

P P

k j z kP

j k k

k k

λ ρ λπλ ρ

λ λ

− − −

Π = +

− −

.

(33)

As Sommerfeld, in his book, then pointed out: this for mula bears all the marks of surface waves [a true surface wave is a slow wave, and the fi elds become concentrated on the interface as the frequency increases]:

It was the main point of the author’s work of 1909 to show that the surface wave fi elds are automati-cally contained in the wave complex. This fact has of course, not changed. What has changed is the weight which we attached to it. At that time it seemed conceivable to explain the overcoming of the Earth’s curvature by radio signals with the help of the character of the surface waves; however we know now that it is due to the ionosphere. In any case, the recurrent discussion in the literature on the

reality of the Zenneck waves seems immaterial to us.

Sommerfeld revisited the problem in 1926, redeveloped the same equations by using a different approach, and referred to the saddle-point method of Ott [20] “which he says is best represented by the one contour integral that goes near the pole and the second saddle point at 2k ” [20]. At this point, it is important to note that Sommerfeld never referred to an error in the sign in his original work.

In a companion paper [21], we discussed the various prob lems associated with the original Sommerfeld formula-tion, and what were the controversies. Here, we summarize our conclusions from [21].

The main problem with Sommerfeld’s 1909 paper was that he overlooked the actual location of the surface-wave pole. The actual location of the pole is not as shown in Fig ure 3a, taken from Sommerfeld’s work, but it is located as shown in Figure 3b. For example, the specifi c locations for the pole for different dielectric constants are given by 2.5 5r jε = − ; ( ) 10.95 0.07p j kλ = −

9 5r jε = − ; ( ) 10.96 0.02p j kλ = −

15 5r jε = − ; ( ) 10.97 0.001p j kλ = −

81 5r jε = − ; ( ) 10.99 0.0004p j kλ = −

These values indicate that the location of the pole is as shown in Figure 3b, and not as originally given by Sommerfeld in Figure 3a. As pointed out by Kahan and Eckart [22],

Sommerfeld did not notice while computing his asymptotic development of the branch cut integral that this contains besides the space wave, the sur-face wave with a negative sign and so cancels the residue of the pole, and that therefore, the path taken primitively by Sommerfeld is the correct one and not the one proposed by Epstein [23].

This is clear in Figure 3c when one changes the path of the branch cut. Furthermore, Kahan and Eckart [22] pointed out that the pole should not come into the picture, as the singular-ity does not meet the radiation condition, and it appears only through an inadvertency in Sommerfeld’s calculations. It was also illustrated by Baños [6, pp. 55-61] that the pole is not located on the right plane of the branch cut. This will be much clearer when the saddle-point method is applied, as the saddle-point path never crosses the pole, and so the surface-wave contribution as envisaged by Sommerfeld never arises. This was the difference in the solutions of Sommerfeld and Weyl [24], as also pointed out by Baños [6, pp. 55-61].

In summary, the real cause of the source of disagree ments among various authors that Sommerfeld made an error in the sign in his 1909 paper has no basis, as the error in the sign is a myth. We will further discuss this in a companion paper [21].

The most succinct and clear explanation of this contro-versy is available in Schelkunoff [25], which interestingly is not referenced in any of the papers. In his book, Schelkunoff [25, p. 430] categorically states “that the denominator equa tion in Equation (16) can have no roots, the integrals can have no poles, and there are no surface waves. This conclusion is contrary to that reached by early writers on the subject.” Schelkunoff then proves his point by substituting the solution into the equation for the pole. When we substitute

1 22 2

1 2P

k k

k kλ = ±

+ in the equation

2 2 2 2 2 22 1 1 2 0P Pk k k kλ λ− + − = ,

it then becomes clear that

2 2 2 2 2 22 1 1 2k k k kλ λ− + − =

2 2 2 2

2 2 2 21 2 1 22 1 1 22 2 2 2

1 2 1 2

k k k kk k k kk k k k

− + −+ +

2 4 2 42 1 1 2 0k k k k= − + − ≠ ,

so there does not appear to be any surface wave in the fi nal solution, a conclusion that everybody agrees with, and what Schelkunoff pointed out about 70 years ago [25]!

4. Properties of the Integration Path Related to the Saddle-Point Method of Integration

We next are going to follow Ott’s formulation [20], which is more complete than that of Weyl’s [24], as Ott deals with the presence of the pole near the saddle-point path of integration. The integral in Equation (29) can be simplifi ed by making the following substitutions:

1 sinkλ β= , (34)

2 sinRρ θ= , (35)

2 cosz z R θ′+ = . (36)

The interpretation of the angle θ is shown in Figure 1. Hence, the application of Equations (32), (34), (35), and (36) to Equation (29) yields



1

1/2 21

22

sin2 sinsin cos sin

sVkgR

ε ββπ θ ε β ε βΓ

−≈

+ −∫

( ){ }1 2exp 4 cosj k R dπ β θ β− − − , (37)

where 1Γ is a path in the complex β plane, as shown in Fig-ure 4.

There is one obvious weakness in the arguments pre sented to derive Equation (37), namely, that there are points on the

path for which the argument of the Hankel function, ( ) ( )20H λρ, used in Equation (37) through Equation (32) is not large and may even be zero, so that the asymptotic expansion for large arguments cannot be used. However, as argued by Brekhovskikh [9], the arguments will be rigorous if the large-argument approximation is used only after the path of integra tion has been changed to the path of steepest descent, 0Γ , of Figure 4. The result will then be the same.

Assuming medium 1 to be lossless, then 1k in the trans-formation of Equation (34), 1 sinkλ β= , implies, for a com-plex β ( )jβ β β′ ′′= + , that the complex value of λ is

( )1 sin cosh cos sinhj k jλ λ β β β β′ ′′ ′ ′′ ′ ′′+ = + . (38)

Equivalently,

1 sin coshkλ β β′ ′ ′′= , (39)

1 cos sinhkλ β β′′ ′ ′′= . (40)

Hence, the mapping in Equation (34), 1 sinkλ β= , transforms the quadrants of the λ plane in parallel strips of width 2π radians, and the path of integration from λ′ = −∞ to ∞ is transformed to the path 1Γ , where ( )1Im sin 0kλ β′′ = = , as

shown in Figure 4. The requirement ( )2 21Im 0k λ− <

amounts to ( )1Im cos 0k β < on the path of integration, or, for 1k real,

sin sinh 0β β′ ′′ > , (41)

as ( )cos cos cosh sin sinhj jβ β β β β β′ ′′ ′ ′′ ′ ′′+ = − . The script U in Figure 4 denotes the strips of the β plane on which the above inequality, Equation (41), is satisfi ed (the upper Riemann sheet). The other strips are denoted by L (the lower Riemann sheet). The path 1Γ then totally lies on U . The loca tion of the branch points at 1kλ = ± in the λ plane are now transformed into

1sin 1Bβ = ±

in the β plane, and are situated at 2π± ,

3 2π± , and so on. The branch cuts along ( )2 21Im 0k λ− =

are now transformed into ( )1Im cos 0k β = , and begin at the branch points / 2.β π= ±

Since ( ) ( )1 1Im cos Im sin 2 0k kβ π β= ± = , these branch cuts will run parallel to the path 1Γ , ( )1Im sin 0k β λ′′= = , but shifted by 2π± from the origin along the real axis. The transformation 1 sinkλ β= has thus transformed the upper and lower sheets associated with the branch points 1k± into one sheet, where certain strips on the sheet belong to the pre vious upper (U ) and lower ( L ) Riemann sheet on the λ plane.

The remaining branch points, 2kλ = ± , are transformed into

2sin Bβ ε= ± ,

which has solutions

2ln 1B j jβ ε ε = ± ± − . (42)

The branch cuts 2 22Im 0k λ − =

are transformed into

2Im sin 0ε β − = . In the β plane, there are then two

Riemann sheets, connected along the branch cuts

( )1/22Im sin 0ε β − = of the branch points

2Bβ .

Finally, the poles in the λ plane are now given by 2cos sin 0P Pε β ε β+ − = . Hence,

Figure 4. The complex β plane, showing possible branch points, branch cuts, poles, and the path of steepest descent for an imperfect ground plane with the material parame ter

( )21 jε ε ′= − and 1ε > .

sin

1Pεβε

= ±+

(43)

with

1cos1P

βε

= −+

, (44)

since ( )1Im cos 0k β < . The possible locations for the poles can be approximated by

1sin 1 cos1 2 2P Pε πβ βε ε

± = ≈ − ≈ + ,

which results in

12Pπβ

ε

≈ ± . (45)

For the parameters of a highly conducting ground,

( )21r jε ε ′= − , the locations of the poles and the branch cuts are pictorially depicted in Figure 4 (not to scale). Out of the possible locations of the branch points and poles, 32B ,

42B , 2P

, and 3P are situated on the upper Riemann sheet of the branch points

2Bβ , on which ( )1Im cos 0k β < . It is also important to note that none of the poles ( 2P , 3P ) are situated between the original path of the integration, 1Γ , and the path of the steepest descent, 0Γ . However, when the path of steep est descent, 0Γ , lies in close proximity of the pole 2P , special precautions must be taken in the evaluation of the integral of Equation (37), as carried out by Ott [20]. The pole 1P is of no concern, since it lies on the second Riemann sheet of the branch point

2Bβ , on

which ( )2Im sin 0ε β− > . The pres ence of the branch point 32B should ordinarily be taken into account when ε is close to

unity, when deforming the path 1Γ into the path of steepest descent, 0Γ . Often, the contribu tion along the borders of the branch cut would be a quickly decreasing exponential that could be neglected in comparison to the contribution from the saddle-point integration. It is just important to note that when the saddle-point path crosses the branch cut, one would get a lateral wave (as defi ned in Appendix 1), which we do not discuss in this paper [9; 26, p. 508].

In summary, the saddle-point method clearly shows that the pole contribution is in no way contained in the fi nal solu-tion, and Sommerfeld simply overlooked the actual location of the pole when he transformed the contour of the branch cuts from Figure 3a to Figure 3c. When the branch-cut contour is modifi ed for the true location of the pole, illustrated in Fig-ure 3b, the contribution from the surface-wave-pole term would then have cancelled, as illustrated in Figure 3c. This was pointed out by Kahan and Eckart [22].

5. Analysis of the Refl ected Field

Application of Equations (32), (34), (35), and (36) to Equation (29) yields

1

214

22

sin2 sinsin cos sin

jsV

kg eR

π ε ββπ θ ε β ε β

−

Γ

−≈

+ −∫

( )1 2exp cosjk R dβ θ β− − , (46)

where 1Γ is a path in the complex β plane as shown in Fig-ure 4. The path of steepest descent never crosses any of the poles. The contributions along the borders of the second branch cut associated with the branch point k2 – particularly for low values of the dielectric constant, ε – are not necessary, as they would be quickly decreasing exponentials [7] that can be neglected in comparison to the contribution from the sad dle-point integration.

Hence, by application of the method of steepest descent [Equations (65) and (70), as explained in Appendix 2] to Equation (46), for 2θ π< (i.e., when the pole is not near the saddle point), one obtains

[ ] 21 222

2exp sin

cos sinsV

jk Rg

Rε θ

ε θ ε θ

− −≈

+ −

( ) ( ) ( )( )

2 2

221 2 2 2

1 2 1 cos 3 cos112

sin cos sinjk R

ε ε ε ε ε θ θ

ε θ ε θ ε θ

− − + − − − + −

( ) ( )( )

2 2

2 222 2

1 cos sin 2 sin 14sinsin cos sin

ε ε θ ε θ ε θ

θε θ ε θ ε θ

− − + + − − + −

.

(47)

Hence, 1reflectedzΠ of Equation (20) can now be written as

( ) 21 21 22

exp cos sin

cos sinreflectedz

jk RP

Rε θ ε θ

ε θ ε θ

− − −Π ≈ + −

( ) ( ) ( )( )

2 2

33/21 2 2 2

1 2 1 cos 3 cos1

sin cos sinjk R


ε θ ε θ ε θ

− − + − +

− + −

( ) ( )( )

2 2

33/22 2

1 cos 2 sin sin

sin cos sin


ε θ ε θ ε θ

− + − + − + −

2

2 2

sin

2sin cos sin

ε θ

θ ε θ ε θ

− −

+ −

.

(48)



1

1/2 21

22

sin2 sinsin cos sin

sVkgR

ε ββπ θ ε β ε βΓ

−≈

+ −∫

( ){ }1 2exp 4 cosj k R dπ β θ β− − − , (37)

where 1Γ is a path in the complex β plane, as shown in Fig-ure 4.

There is one obvious weakness in the arguments pre sented to derive Equation (37), namely, that there are points on the

path for which the argument of the Hankel function, ( ) ( )20H λρ, used in Equation (37) through Equation (32) is not large and may even be zero, so that the asymptotic expansion for large arguments cannot be used. However, as argued by Brekhovskikh [9], the arguments will be rigorous if the large-argument approximation is used only after the path of integra tion has been changed to the path of steepest descent, 0Γ , of Figure 4. The result will then be the same.

Assuming medium 1 to be lossless, then 1k in the trans-formation of Equation (34), 1 sinkλ β= , implies, for a com-plex β ( )jβ β β′ ′′= + , that the complex value of λ is

( )1 sin cosh cos sinhj k jλ λ β β β β′ ′′ ′ ′′ ′ ′′+ = + . (38)

Equivalently,

1 sin coshkλ β β′ ′ ′′= , (39)

1 cos sinhkλ β β′′ ′ ′′= . (40)

Hence, the mapping in Equation (34), 1 sinkλ β= , transforms the quadrants of the λ plane in parallel strips of width 2π radians, and the path of integration from λ′ = −∞ to ∞ is transformed to the path 1Γ , where ( )1Im sin 0kλ β′′ = = , as

shown in Figure 4. The requirement ( )2 21Im 0k λ− <

amounts to ( )1Im cos 0k β < on the path of integration, or, for 1k real,

sin sinh 0β β′ ′′ > , (41)

as ( )cos cos cosh sin sinhj jβ β β β β β′ ′′ ′ ′′ ′ ′′+ = − . The script U in Figure 4 denotes the strips of the β plane on which the above inequality, Equation (41), is satisfi ed (the upper Riemann sheet). The other strips are denoted by L (the lower Riemann sheet). The path 1Γ then totally lies on U . The loca tion of the branch points at 1kλ = ± in the λ plane are now transformed into

1sin 1Bβ = ±

in the β plane, and are situated at 2π± ,

3 2π± , and so on. The branch cuts along ( )2 21Im 0k λ− =

are now transformed into ( )1Im cos 0k β = , and begin at the branch points / 2.β π= ±

Since ( ) ( )1 1Im cos Im sin 2 0k kβ π β= ± = , these branch cuts will run parallel to the path 1Γ , ( )1Im sin 0k β λ′′= = , but shifted by 2π± from the origin along the real axis. The transformation 1 sinkλ β= has thus transformed the upper and lower sheets associated with the branch points 1k± into one sheet, where certain strips on the sheet belong to the pre vious upper (U ) and lower ( L ) Riemann sheet on the λ plane.

The remaining branch points, 2kλ = ± , are transformed into

2sin Bβ ε= ± ,

which has solutions

2ln 1B j jβ ε ε = ± ± − . (42)

The branch cuts 2 22Im 0k λ − =

are transformed into

2Im sin 0ε β − = . In the β plane, there are then two

Riemann sheets, connected along the branch cuts

( )1/22Im sin 0ε β − = of the branch points

2Bβ .

Finally, the poles in the λ plane are now given by 2cos sin 0P Pε β ε β+ − = . Hence,

Figure 4. The complex β plane, showing possible branch points, branch cuts, poles, and the path of steepest descent for an imperfect ground plane with the material parame ter

( )21 jε ε ′= − and 1ε > .

sin

1Pεβε

= ±+

(43)

with

1cos1P

βε

= −+

, (44)

since ( )1Im cos 0k β < . The possible locations for the poles can be approximated by

1sin 1 cos1 2 2P Pε πβ βε ε

± = ≈ − ≈ + ,

which results in

12Pπβ

ε

≈ ± . (45)

For the parameters of a highly conducting ground,

( )21r jε ε ′= − , the locations of the poles and the branch cuts are pictorially depicted in Figure 4 (not to scale). Out of the possible locations of the branch points and poles, 32B ,

42B , 2P

, and 3P are situated on the upper Riemann sheet of the branch points

2Bβ , on which ( )1Im cos 0k β < . It is also important to note that none of the poles ( 2P , 3P ) are situated between the original path of the integration, 1Γ , and the path of the steepest descent, 0Γ . However, when the path of steep est descent, 0Γ , lies in close proximity of the pole 2P , special precautions must be taken in the evaluation of the integral of Equation (37), as carried out by Ott [20]. The pole 1P is of no concern, since it lies on the second Riemann sheet of the branch point

2Bβ , on

which ( )2Im sin 0ε β− > . The pres ence of the branch point 32B should ordinarily be taken into account when ε is close to

unity, when deforming the path 1Γ into the path of steepest descent, 0Γ . Often, the contribu tion along the borders of the branch cut would be a quickly decreasing exponential that could be neglected in comparison to the contribution from the saddle-point integration. It is just important to note that when the saddle-point path crosses the branch cut, one would get a lateral wave (as defi ned in Appendix 1), which we do not discuss in this paper [9; 26, p. 508].

In summary, the saddle-point method clearly shows that the pole contribution is in no way contained in the fi nal solu-tion, and Sommerfeld simply overlooked the actual location of the pole when he transformed the contour of the branch cuts from Figure 3a to Figure 3c. When the branch-cut contour is modifi ed for the true location of the pole, illustrated in Fig-ure 3b, the contribution from the surface-wave-pole term would then have cancelled, as illustrated in Figure 3c. This was pointed out by Kahan and Eckart [22].

5. Analysis of the Refl ected Field

Application of Equations (32), (34), (35), and (36) to Equation (29) yields

1

214

22

sin2 sinsin cos sin

jsV

kg eR

π ε ββπ θ ε β ε β

−

Γ

−≈

+ −∫

( )1 2exp cosjk R dβ θ β− − , (46)

where 1Γ is a path in the complex β plane as shown in Fig-ure 4. The path of steepest descent never crosses any of the poles. The contributions along the borders of the second branch cut associated with the branch point k2 – particularly for low values of the dielectric constant, ε – are not necessary, as they would be quickly decreasing exponentials [7] that can be neglected in comparison to the contribution from the sad dle-point integration.

Hence, by application of the method of steepest descent [Equations (65) and (70), as explained in Appendix 2] to Equation (46), for 2θ π< (i.e., when the pole is not near the saddle point), one obtains

[ ] 21 222

2exp sin

cos sinsV

jk Rg

Rε θ

ε θ ε θ

− −≈

+ −

( ) ( ) ( )( )

2 2

221 2 2 2

1 2 1 cos 3 cos112

sin cos sinjk R


ε θ ε θ ε θ

− − + − − − + −

( ) ( )( )

2 2

2 222 2

1 cos sin 2 sin 14sinsin cos sin


θε θ ε θ ε θ

− − + + − − + −

.

(47)

Hence, 1reflectedzΠ of Equation (20) can now be written as

( ) 21 21 22

exp cos sin

cos sinreflectedz

jk RP

Rε θ ε θ

ε θ ε θ

− − −Π ≈ + −

( ) ( ) ( )( )

2 2

33/21 2 2 2

1 2 1 cos 3 cos1

sin cos sinjk R


ε θ ε θ ε θ

− − + − +

− + −

( ) ( )( )

2 2

33/22 2

1 cos 2 sin sin

sin cos sin


ε θ ε θ ε θ

− + − + − + −

2

2 2

sin

2sin cos sin

ε θ

θ ε θ ε θ

− −

+ −

.

(48)



The fi rst term of Equation (48) represents a spherical wave originating from the image, and can be rewritten as

( )1 21

2

expreflectedTMz

j k RP

R−

Π ≈ Γ , (49)

where TMΓ can be recognized as the TM refl ection coeffi cient associated with the spherical wave [7, 10]. This is given by

2

2

cos sin

cos sinTM

ε θ ε θ

ε θ ε θ

− −Γ =

+ −. (50)

The name refl ection-coeffi cient method is derived from Equa-tion (49), since 1

reflectedzΠ is now obtained as the refl ection

coeffi cient times the potential from the image of the source. The method represents a good approximation, as long as the fi elds are computed far away from the ground plane, and also away from the source dipole to ensure 2θ π< . This implies that the use of the refl ection coeffi cient in the computations of the refl ected fi elds are not valid near the ground, where 2θ π≈ [10, 12].

The total Hertz potential in medium 1, when the conduc-tivity of the relative permittivity of the lower medium is large, i.e., 1ε > , is given by

( ) ( )1 1 1 21

1 2

exp expz

jk R jk RP

R R− −Π ≈ +

3

1 2

cos 1 2 1 ...cos 1 cos 1jk R

ε θ εε θ ε θ

− + + + +

.

(51)

Note that when ε → ∞ , 1zΠ of Equation (51) goes properly into the form of a source plus an image term due to a vertical electric dipole located above a perfectly conducting ground plane.

However, when 2θ π≈ , this becomes

1zΠ ≈

( ) ( ) ( )1 1 1 2 1 221 2 1 2

exp exp 2 exp ...jk R jk R

P jk RR R jk R

ε − −− + − +

(52)

It is now important to recognize from Equation (52) that the sum of the fi rst two terms may be smaller than the third term. As a matter of fact, when both the transmitter and the receiver are near the ground, i.e.,

1 2R R ρ= ≈ , (53a)

0z z′≈ ≈ , (53b)

then we observe that the fi elds will be solely determined by the second- and higher-order terms of Equation (52). Also, there is no surface-wave term in the expression, and the domi nant term behaves as 21 R . The reason for this poor conver gence in the vicinity of 2θ π≈ is that the effect of the pole at 2π becomes important. Also, it is important to point out that the saddle-point path does not cross any of the poles. We will address this point in a companion paper [21]. The bottom line is that since it is the higher-order terms that are responsible for the calculation of the fi elds along the interface in Equa tion (52), we need to carry out a different asymptotic expan sion, starting using Equation (21), and not Equation (20). This is the topic of the next section.

6. Fields Near the Interface

In order to solve for the total fi elds near the interface [7], a modifi ed saddle-point method, as explained in Appendix 3, is applied to take into account the effect of the pole Pβ near the saddle point. In the expressions for both sVg and sVG in Equations (20) and (21), there is a pole, Pβ , which is seen from [27]

( ) ( )2

22

sin cos1 1sin sin1cos sin P P

ε β ε ββ β β βεε β ε β

− −=

+ − −+ −

,

(54)

where 2cos sin 0P Pε β ε β+ − = , with sin

1Pεβ

ε= ±

+

and 1cos1P

βε

= −+

. Applying Equations (54) and (108) to

Equation (21), we obtain

1

1/21

2

2 sinexp

4 sinsVkG jR

βπεπ θ

Γ

= −

∫

( )1 22

exp cos cos

cos sin

jk Rd

β θ ββ

ε β ε β

− −

+ −

21

22

4 cos sin cos1 1cos

1

k jRπ θ ε θ ε θε

εθε

− −≈

−−+

( )21 2exp erfc

cos sin11 1

jk R W jW

θ ε θε ε

− −

+ ++ +

,

(55)

where

2 21 2 2sin 2

PW jk Rθ β− = −

1 2

cos sin11 1

jk R θ ε θε ε

= − + −

+ + , (56)

and W was called the numerical distance by Sommerfeld [1]. If 1ε > and 2θ π≈ , and if W is very small, then we have

( )2exp erfc 1W jW − ≈ . Under this assumption, applying Equation (36) to Equation (55), one gets a simplifi ed expres sion for

[ ] ( )1 1 2 22 22

exp1

sVz zk jG jk R

R Rπ ε

ε

′+≈ − −

−

( ) [ ]1 21 1.5

2

exp2

z z jk Rk j

Rπ

′+ −≈ − . (57)

Equation (57) thus illustrates that when 2θ π≈ , the domi nant

term of the potential 1 1 52

1z

R ⋅Π ∝ , and therefore the lead ing

term for the fi elds will also vary as 1 51ρ ⋅

, if ( )z z′+ is small

compared to ρ in Equation (18). It is interesting to observe that Equation (57) is not a function of the ground parameters. The path-loss exponent factor in mobile urban cellular communication should thus be three near the ground and the refl ection-coeffi cient method is not applicable, under those circumstances. This should approximately hold for any types of ground parameters, such as urban, suburban, or even lakes and oceans. One possible reason for this is given by Stratton [3]: the refl ection coeffi cient is approximately 1+ for a perfectly conducting ground when the fi elds are observed far from the ground, and it transforms to 1− when the fi elds are observed near the ground, when 2θ π≈ . This particular varia tion of the fi eld near an imperfect ground will be verifi ed by experimental data and by a more-accurate numerical analy sis in the next sections.

However, as W becomes large, then

( )2 21exp erfc 1

2jW jW

W Wπ− − ≈ +

for W →∞ and 34

arg W π< , (58)

and for 1ε > , 2 1 22jk RWε

−≈ . Under this condition,

[ ] ( )1 2 2

1 222 exp 1sV

z zG jk R

jk RRεε

′+ ≈ − −

. (59)

The total Hertz potential in medium 1, which is valid near the interface for 1ε > and 2θ π≈ , thus becomes

( ) ( )

( ) ( )

( ) ( )

( ) ( )

1 1 1 2

1 2

1 21 1.5

21

1 1 1 2

1 2

1 22

1 22

exp exp

exp 2 , 1

exp exp

exp 2 1 , 1

z

jk R jk RP

R R

jk Rj k z z W

R

jk R jk RP

R R

jk Rz z W

jk RR

π

εε

− −−

− ′− +

The above simplifi ed expressions illustrate that a Norton surface wave decays asymptotically as 21 R , and this applies only in the far-fi eld region, where 1W > , as the fi rst two terms cancel in the second expression. It is also interesting to note that the third term for 1W > provides the so-called height-gain for the transmitting and receiving antennas. How ever, this height-gain again only applies to the far-fi eld regions. In the intermediate region, the fi elds decay as approximately 1.51 ρ . This represents a two-dimensional radiation fi eld often associated with a surface wave, as we shall see in a companion paper [21], which will discuss the physical characteristics of the fi elds that arise in a mobile cel lular wireless communication system. Observe also that for 1W < , the above expression is independent of the ground parameters. This will be confi rmed in the next section using a more-accurate numerical analysis.

In summary, Sommerfeld characterizes W as the numeri-cal distance. When the large-argument approximation is invoked for W, then the fi elds decay as 21 R . Interestingly, this is one of the confusing parts in all the discussions, as Sommerfeld stated:

...for small values of the numerical distance, the spatial-wave type predominates in the expression for the reception intensity; in this case the ground peculiarities have no marked infl uence and we can make computations using an infi nite ground con-ductivity without introducing great errors. For lar-ger values of W the rivalry between the space and the surface waves are apparent.

However, our interests in a cellular wireless communication system are for small to intermediate values of W, for which very little information is available. Our observations for the path-loss exponent in a cellular wireless communication is that the value is three for moderate distances from the base-station antenna. In the fringe region (i.e., further away from the base-station antenna) the value is four, which will be further illus-trated by a more-accurate numerical analysis and experimental results! In addition, in this region, the ground parameters have little effect, as seen by Equation (57).



At this point, it is important to point out the novelty of our solution, which is not available in the popular literature, because for the intermediate region, we have used two differ-ent procedures that deviated from the classical formulations. First, we used the second form of the Green’s function, as shown in Equation (21), to observe the fi elds near the inter face. Second, we used a different saddle-point method of inte gration in handling the pole near the saddle point in evaluating the integrals of Equation (21), which was outlined by Clemmow [27] and also used by Hill and Wait [13].

If one applies the modifi ed saddle-point method of evalu-ating the integral – as explained in Appendix 3 – to the Green’s function given by Equation (20), one then obtains

sVg ≈

( )( )( )( ) ( )

2 2

1

2 2

sin sin cos2exp

41 cos 1 1 2sin

2P

kjR

ε θ ε θ ε θππ θ β

ε θ ε

− − −− +

− − +

( )( )

1

1 2exp cos

sin2

P

jk Rd

β θβ

β βΓ

− − −∫

2 21

22

4 sin cos sincos 1 1 1

k jRπ ε θ ε θ ε θ

θ ε ε− − −

≈− + −

( )21 2exp erfc

cos sin11 1

jk R W jW

θ ε θε ε

− −

+ ++ +

. (60a)

For small values of W, and when the fi elds are desired close to the interface, then we also require 2θ π≈ . In this case, for

1ε > we obtain

[ ]1 212

exp21sV

jk Rk jgRπ

ε

−≈ −

+. (60b)

By incorporating Equation (60b) into Equation (20), it is seen that the space-wave term dominates, and the additional contri-bution of the surface-wave term, given by Equation (60b) and as predicted by Sommerfeld, is small.

However, when W is large, then

[ ]1 22 1 2

2exp1sV

jk Rg

R jk Rε−

≈ −

. (60c)

Substituting this expression into Equation (20), it is seen that the dominant terms for the space waves cancel each other, and the Hertz potential is given by the higher-order terms. There-fore,

[ ]1 2

1 21 2

2 expz

P jk R

jk R

ε −Π ≈ . (61)

A similar asymptotic form was previously obtained, as seen in Equation (52). That is why we expanded the Hertz potential in a different form, given by Equation (21), which cancelled the space waves and provided the dominant ground-wave term. The rationale for doing this was explained by Stratton, as the refl ection coeffi cient is 1+ for a perfect ground when the fi elds are evaluated far from the interface, but it then trans forms to

1− when the fi elds are evaluated near the interface. This second form was also originally used by Sommerfeld.

In short, there are two unique features of this presentation as it differs from other researchers’ work. First, there is the use of Equation (21) in the modifi ed saddle-point method to cal culate the fi elds in the regions both near and far from the base-station antenna. Second, there is the use of a different mathe matical form when applying the modifi ed saddle-point method when there is a pole near the saddle point, as explained in Appendix 3. Almost all researchers factored out the pole term and then had a remainder term that did not contain the pole, and applied the saddle-point method to that expression. We decided to follow a different route, which was outlined by Clemmow [27] and is presented in Appendix 3. That is why our expressions are different than, for example, those of Tyras [7] and Collin [18], who used the conventional method of extracting the pole, even though they made use of Equa tion (21). It is also true that for small values of the numerical distance W, the variation of the fi elds is quite different than from the results when the value of W is large.

However, in all fairness, there have been so many approximations

electromagnetic macro modeling of propagation in mobile ... · for the fi elds of the ground wave,...

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