Download Lecture 2: Wave Equations

Lecture 2: Wave Equations, Green’s Function, Huygens’ Principle
and Fresnel Zones
2.1. Wave Equation Presentation
Physically, EM-wave propagation phenomena can be described by use of both the
scalar and vector wave equation presentations. In the case of an isotropic
inhomogeneous medium one can present Maxwell’s equations in the following form,
using their time harmonic presentations (1.7):
   1 (r)  E(r)   2 (r)E(r)  ij(r)     1 (r)M(r)
(2.1)
   1 (r)  H(r)   2 (r)H(r)  iM(r)     1 (r) j(r)
Because most problems of wave propagation above the terrain, including built-up
environments, reduce to propagation in a homogeneous, source-free isotropic
medium, this system can be easily simplified from system (1.31) by taking into
account the relations (1.17a)-(1.17c) with (r)  , (r)  , (r)   , that is,
    E(r )   2 E(r)  0
(2.2)
    H(r)   2 H(r )  0
Because both equations are symmetric, one can use one of them, namely that for E ,
and by introducing the vector relation     E  (  E)   2 E and taking into
account that  E  0 , finally obtain
 2 E(r)  k 2 E(r)  0 ,
(2.3)
where k 2   2  . It can be shown that all other electromagnetic vectors satisfy as
well the same wave equation as (2.1) (see, particularly, the equations (1.34) for
Hertzian vectors in a source-free medium).
In special cases of a homogeneous, source-free, isotropic medium, the three
dimensional wave equation reduces to a set of scalar wave equation. This is because in
Cartesian coordinates, E(r)  E x x 0  E y y 0  E z z0 , where x 0 , y 0 , z 0 are unit vectors
in the directions of the x, y, z coordinates, respectively. Hence, the equation (2.3)
consists of three scalar equations such as
 2  ( r )  k 2 ( r )  0 ,
1
(2.4)
where (r ) can be either E x , E y , or E z . This statement is not true in cylindrical or
spherical coordinate systems. The problems of independent solution of each scalar
wave equation, such as (2.4) is the subject next lecture.
2.2. Boundary Conditions
Equations (2.4) describe all the propagation phenomena within an infinite
inhomogeneous isotropic medium. But if we consider two inhomogeneous finite or
semi-finite regions, we need to introduce boundary conditions at the interface between
these two regions in order to solve one of the two equations (2.2). In this case the
procedure to solve the vector wave equation is as follows.
As a first step this equation is solved separately for each region. Then, in the
second step, by patching the solution together via boundary conditions, we obtain the
solution for two neighboring regions. It can be easily shown that the boundary
conditions follow from one of the two vector wave equations (2.2). To do so, we
integrate the first equation of (2.2) within a small region in the interface of the two
inhomogeneous semi-finite or finite regions, as presented in Fig. 2.1.
Fig. 2.1
Then using Stokes’s theorem for the surface integral of a curl and using the same
integration over surface S for both equations (2.2), we finally obtain after
straightforward derivations and taking the limit   0 (see Fig. 2.1) respectively for
the magnetic-field component
n  H1  n  H 2  j S
2
(2.5)
and for electric-field component
n  E1  n  E 2  M S
where M S and j S
(2.6)
is a magnetic and electric current sheet at the interface,
respectively. Equation (2.5) states that the discontinuity in the tangential component
of the magnetic field is proportional to the electric current sheet j S . This is the first
boundary condition for solving any one vector electromagnetic equation from (2.6).
Equation (2.6) states that the discontinuity in the tangential component of the electric
field is proportional to the magnetic current sheet M S . This is the second boundary
condition for (2.1).
Both boundary conditions (2.5) and (2.6) can be simplified for the case of
wave propagation above a flat intersection. In the case considered, the first boundary
condition (2.5) for an isotropic non-magnetized (  =1) source-free ( M S  0 , j S  0 )
sub-soil medium reduces to
H1n  H2 n
and
(2.7)
H1  H2 
Both conditions are valid in the case of finite conductivity of each medium, which is
satisfied within the air-ground surface. The first condition in (2.7) states, that the
normal components of the magnetic field of the wave are continuous at the interface
of air-conductor intersection surface. The second condition in (2.7) states that the
tangential component of magnetic field is also continuous at the interface of the airground surface.
As for the second boundary condition (2.6), it also can be simplified for the
interface of the air-conductor intersection as
n  E1  n  E 2
or
(2.8)
E1  E 2 
Condition (2.8) states that the tangential components of the electric field of an EMwave are continuous at the interface of the air-conductor intersection surface. One
may notice that conditions (2.5) and (2.6) are more general than those described by
(2.7) and (2.8), and satisfy various kinds of isotropic inhomogeneous media that
consist of both electric and magnetic sources.
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Example 1: Two homogeneous linear isotropic magnetic materials have an
interface at x=0, as shown in Fig 1E.
Fig. 1E.
At the interface, there is a surface current with density j s  20u y  A / m  . The relative
permeability,  r1  2 , and the magnetic field strength, H1  15u x  10u y  25u z , are
given in the region x<0, and  r 2  5 in the region x>0.
Find: The magnetic field strength H 2 in the region x>0.
Solution
1) The magnetic field can be separated according to (2.7) on two components
with independent boundary conditions. Thus, for the normal component, we
write
 r1 0 H1n   r 2  0 H 2n  H 2 x 
 r1
2
H 1 x   H 1x  6 A / m
r2
5
2) For the tangential component, we write
H 2  H1  j s  H 2 z  H1z  j y  25  20  45 A / m
3) Since there is no change in the y-component of the magnetic field, we write
H 2  6u x  10u y  45u z A / m
2.3. Wave Equation Solutions
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To analysis different kinds of wave equation solutions, let us consider free space, as a
simple example of an infinite isotropic homogeneous source-free medium, the vector
wave equation of which can be presented in very simple form (2.2)-(2.3) for one of
the component of the EM field, or by use of the Hertzian vector (r ) :
 2 (r)  k 2 (r)  0
(2.9)
where, once more, the wavenumber k    (in many practical cases of terrain
propagation  =1 with great accuracy, and one can rewrite it as k    ).
2.3.1. Plane Waves in Homogeneous Medium
For plane waves in a Cartesian coordinate system each of the equation (2.3) or (2.9)
can be rewritten in scalar form (2.4) for any Cartesian component of vectors E(r ) ,
H(r) or (r ) . Usually, in the literature there is other form of presentation of
equation (2.4) by introducing instead wavenumber k, the phase velocity
v ph  v 

c
. In this case (2.4) can be rewritten as:

k

 2 (r ) 
2
(r )  0
v2
(2.10)
Wave equations (2.4) or (2.10) have the well-known solution [1-3]
(r)  expik  r
(2.11)
The waves that satisfy scalar equation (2.10) and are determined by expression (2.11)
are called plane waves. Wave vector k denotes here the direction of propagation of
the plane wave in free space (see Fig. 1.1, Lecture 1). If one considers the plane wave
that propagates in any direction, say along the x-axis, then the fundamental solution of
(2.10) is
( x)  A expikx  B exp ikx
(2.12)
This solution describes the waves, propagating in the positive direction (with the sign
“+” in the exponent) and in the negative direction (with the sign “-” in the exponent)
respectively along the x-axis with phase velocity v ph 
ideal free space.
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c
which equals c in an

But one can note that EM fields have a vector character and satisfy Maxwell’s
equations (1.7) or wave equations such as (2.3), (2.4). Thus one can find the field
vectors in the following form
E(r )  e E E 0 expik  r 
or
(2.13)
H(r )  e H H0 expik  r 
where e E and e H are the constant unit vectors, i.e., | e E | | e H |  1 ; E 0 and H 0 are the
complex amplitudes, which are constant in space and time. From conditions in free
space without sources
 E  0
H  0
and
it follows that
eE  k  0
and
eH k  0
(2.14)
which denote that E and H are perpendicular to direction of wave propagation k .
Moreover, because in free space the first Maxwell equation (1.1a) or (1.7a) reduces to
  E  iB  0,
B  H  H
(2.15)
we can finally obtain from (2.43) for a plane wave (2.41) in free space (with   1 ):
i(k  e E ) E 0  ke H B0  expik  r   0
(2.16)
Equations (2.16) have solutions:
eH 
k  eE
,
k
B0   E 0
(2.17)
Hence, vectors e E , e H and k form the system of orthogonal vectors, where vectors
E and B oscillate in phase and their ratio is constant (see Fig. 2.3). The wave, which
is described by relations (2.13) and (2.17), is a transverse wave propagating in the k direction.
2.3.2. Poynting Vector and Poynting Theorem
This theorem is the simple law of EM-wave energy conservation. It is known from
electrostatics and magnetostatics, that the work of the electric field to move a single
charge q is equal q  v  E , where v is the vector of the charge velocity. The same
work of the magnetic field for this charge is equal to zero, because the magnetic field
direction is perpendicular to the velocity vector [1-3]. For a continuous distribution of
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charges and currents in a medium, the total work of the EM field in the volume V in
unit time is equal [1-3]:
 j  Edv
(2.18a)
V
This expression determines the velocity of the decrease in the field energy within the
volume V.
Let us now obtain the law of energy conservation, using Maxwell’s equations
(1.1). We shall substitute the current density j in (2.10a) using (1.1b)
D 

 j  Edv   E    H  E  t dv
V
(2.18b)
V
Taking into account the vector equality [4]
  E  H  H    E  E    H
one can easily rewrite (2.18) as:
D

 j  Edv      (E  H)  E  t
V
 H
V
B 
dv
t 
(2.19)
If we now present the density of total field energy according to [1-3] as
W  E  D  B  H ,
(2.20)
then (2.11) will be rewritten in the following form:
 W

  j  Edv   
   (E  H)dv
t

V
V 
(2.21)
Because expression (2.21) is written for any volume V, it can be presented in
differential form:
W
   S = -j  E
t
(2.22)
Equation (2.22) is the equation of EM-field energy conservation, or the equation of
continuity. It can be easily shown that the vector S  E  H in the brackets on the right
hand side of (2.22) has the dimension of watt/ m 2 , which is that of power density.
From (2.22) is clear that it may be associated with the direction of power flow.
The vector that determines the power flow of EM field is called the Poynting vector.
Equation (2.21) is the integral Poynting theorem and equation (2.22) is its vector
presentation.
7
Using the time harmonic presentation of Maxwell’s equations, one can convert
(2.13) to the time harmonic form. In fact, if we now introduce instead of the
derivation


the term i, and present the operation of averaging < E  H >, as
t

1
Re E  H * , taking into account Gauss’s theorem for the term   (E  H* ) , we
2
finally obtain from (2.21) the Poynting theorem presented in time harmonic form:
 ds  (E  H
*
)   dv (H *  B  E  D * )   dvE  j
S
V
(2.23)
V
or accounting for (2.20), we can present the total energy of the electromagnetic field
as
Wtotal  WE  WH 
E 2

2
H 2
(2.24)
2
Then, the Eq. (2.22) can be rewritten as

E
 E  H ds   t   2
H 2 
dv   E 2 dv

2
V 
V



 

S

total power flow
leaving the region
2
storage e / m energy
(2.25)
heating
Example 2: The field vectors in free space are given by the following
u E
4
expressions: E  10 cos(t  z )u x (V/m) and H  z
( A / m) . The frequency
3
Z0
f=500 MHz, Z 0  120 ()
Find: Poynting vector.
Solution
1) In phasor notations, the fields are expressed as
E  10e
4 
j   z
3 
ux
and H  10
4 
j   z
3 
4 
j   z
3 
e
e
uy 
120
12
uy
2) The Poynting vector is
S av (r ) 
1
10
ReE(r )  H * (r ) 
u z  0.133u z (W / m 2 )
2
2  12
8
Example 3:
In Fig. 2E a capacitor of radius a and of width b, which is
connected with the source of voltage V(t) via the corresponding electrical circuit, is
given.
Fig. 2E.
Find: the power flowing via the area of the radial edge of a capacitor using Poynting
theorem, if the density of the total magnetic field equals zero. Show that this power
equals the rate of total energy changes.
Solution
According to conditions of the problem, the total energy equals only the energy of the
electric field, i.e.,

 E 2 
 E 2 
 V  
  a 2 b
Wtotal  WE  WH  WE  



 2 
 2 
According to Poynting's theorem (2.25), since
wH 
H 2
2

 0 and heating is not
accounting, we finally get:

dWtotal dWE
d  E 2 
dE
  a 2 b    a 2 b E

 

dt  2 
dt
dt
dt





On the other hand, according to 2nd Maxwell equation (see Lecture 1):
 B

   0 dl    j 
L
S
D 
 E 
ds    
ds
t 

t


S
or
9
(E1)
  dE
dt
2aH   a 2
 H
a dE
2 dt
Then, the Poynting's vector power density equals:
PS  E  H   S   E 
a dE
2 dt
 (2ab)   (a 2 b) E
dE
dt
(E2)
Comparing (E1) and (E2), we obtain well-known result:
PS  
dWtotal
dt
(E3)
2.3.3. Wave Polarization
The vector of electric field in the plane wave as described by formula (2.13) is
directed along unit vector e E . To obtain the more general case of wave polarization
we need an additional linear polarized wave independent of the first one. It can be
easily shown that two linear independent solutions, which satisfy wave equation
(2.12) or (2.13) can be presented in the following form
E1  e1 E1 expik  r
(2.26)
E 2  e 2 E 2 expik  r
The magnetic field components of the EM wave satisfy, according to (2.13) in free
space (  =1), the following relations:
Bj  
k Ej
k
, B j  H j , j=1, 2
(2.27)
Here amplitudes E 1 ( H1 ) and E 2 ( H 2 ) are the complex values, which enable us to
introduce the phase difference between the two components of the EM wave. Thus the
common solution for the plane EM-wave propagated along vector k , can be
presented as a linear combination of E 1 and E 2 :
E(r )  e1 E1  e 2 E 2  expik  r 
(2.28)
If E 1 and E 2 have the same phase, then solution (2.20) describes the linear polarized
wave with polarization vector directed to the e 1 -axis at angle
E 
  tan 1  2 
 E1 
and with amplitude
10
(2.29a)

E  E12  E 22

1/ 2
(2.29b)
as presented in Fig. 2.2.
If E 1 and E 2 have different phases, then the EM wave (2.28) is elliptically
polarized. If E1  E 2 and phase difference equal

, then the elliptically polarized
2
wave becomes a circularly polarized wave.
Fig. 2.2
In this case solution (2.28) can be rewritten as
E(r )  E 0 e1  e 2  expik  r 
(2.30)
The sign “+” corresponds to anticlockwise rotation (sometimes called the wave with
left-hand circular polarization). The sign “-” corresponds the waves with right-hand
circular polarization (see Fig. 2.3).
Fig. 2.3
11
Then two waves with circular polarization can be considered as a basic system for
describing the common case of polarized waves. Let us introduce in the common case
the orthogonal complex unit vectors:
e 
1
e1  ie 2 
2
(2.31)
Then, the common presentation of a polarized wave (2.28) by the use of linearly
polarized waves, now, by the use of two circularly polarized waves (2.30) and
expression (2.31), can be rewritten as
E(r )  e  E   e  E   expik  r 
(2.32)
where E  and E  are the complex amplitudes of two circularly polarized waves with
opposide directions of rotation. If their modules are different, but their phases are
equal, then expression (2.32) describes, as above, an elliptically polarized wave with
main elliptical axes directed along e 1 and e 2 . The ratio of these semi-axes equals of
(1-q)/(1+q), where q 
E
. If the complex amplitudes have different phases, so that
E
E
 q  expi 
E
then the ellipses’ axes for E - vector are rotated by angle
(2.33)

. In Fig. 2.4 the common
2
case of an elliptical polarized EM wave is presented.
Fig. 2.4
At each spatial point the vector E (the same applies to the vectors H or B ) describes
ellipses, as shown in Fig. 2.4. For the case q  1 , we once more return to the case of
linearly polarized wave.
12
2.3.4. Cylindrical and Spherical Waves in Free Space
As mentioned in the literature [1-3, 5-7], to obtain the common vector presentation of
cylindrical and spherical waves is a very complicated problem, which can be reduced
for the case of an isotropic homogeneous source-free medium (with properties which
limit to those in free space, but with   1 ) to the simple scalar form, as was done
above for the plane wave in the Cartesian coordinate system.
The scalar wave equation in the cylindrical coordinate system , , z can be written
as:
1  

1 
2



 k 2   (r)  0

2
2
2
z
     

(2.34)
The above partial differential equation can be solved by separation of variables, and
its one-dimensional (along the z-axis) solution can be presented in the following form
[4]
 (r )  n () expin  ik z z
(2.35)
where n is an integer since the wave field has to be 2 periodic in  . Then, by
substituting (2.35) in (2.34), we reduce it to an ordinary differential equation with the
full derivative (d/dx, x is variable) presentation [4]:
 1 d d n2

  2  k r2  n ()  0

  d d 


where k r  k 2  k z2

1/ 2
(2.36)
. One may notice that (2.28) is the Bessel equation [10] with
two lineary independent solutions. Its general solution is a linear superposition of the
following four spectral functions [4]: Bessel function, J n ( k r ); Neumann function,
N n ( k r ); Hankel functions of first order, Hn(1) ( k r ) , and second order, Hn( 2 ) ( k r ),
respectively. Since only two of these four special functions are independent, they are
lineary related each other, that is,


J n (k   ) 
1 (1)
H n (k   )  H n( 2) (k   ) ,
2
N n (k   ) 
1
H n(1) (k   )  H n( 2) (k   ) ,
2i


13
(2.37a)
H n(1) (k   )  J n (k   )  iN n (k   ),
or
(2.37b)
H n( 2) (k   )
 J n (k   )  iN n (k   ).
Full information about the properties of these special functions can be obtained from
the special books [4, 10], which describe all mathematical functions. For our purposes
the exponential approximation of those functions is very important.
For example, the general representation of a cylindrical wave (2.27) can be
reduced in a very simple exponential form:
 (r ) ~
 n

2

 n
exp i
 i  exp i  ()  ik z z  ik r  
k r 
4

 2
 
(2.38)
Here  is the arc length in the  direction, and n /  can be thought of as the
component of vector k , if one compares the cylindrical wave presentation (2.38) with
that for a plane wave (2.11). Consequently, expression (2.30) looks like a plane wave
propagating mainly in the direction k  k z zˆ  k r ̂ , when    .
We now consider a spherical wave presentation in free space. In the spherical
coordinate system r, ,  , the scalar wave equation is [4]:
 1  2 

1


1
2
r

sin


 k 2   (r )  0
 2
2
2
2
2
 r sin  
 r r r r sin  

(2.39)
Following [3], we present the solution of this equation in the form:
 (r)  (r , ) expim
(2.40)
The general equation (2.31) can be further simplified by the separation of variables by
letting
(r , )  bn ( kr ) Pnm (cos )
(2.41)
where Pnm (cos ) is the associate Legendre polynomial satisfying the equation
 1 d
d 
m2  m
sin 
 n(n  1) 

 Pn (cos )  0
d 
sin 2  
 sin  d
(2.42)
Analogously, bn ( kr ) satisfies the equation
n(n  1) 
 1 d 2 d
r
 k2 
 2
 bn ( kr )  0
 r dr dr
r2 
14
(2.43)
Equation (2.43) is just the spherical Bessel equation, and b n ( kr ) is either the
spherical Bessel function, jn ( kr ) , spherical Neumann function, n n ( kr ) , or the
(2)
spherical Hankel functions, h (1)
n ( kr ) and h n ( kr ) [4, 10].
As is shown in [4, 10], the spherical special functions can be approximated by
the spherical functions proportional to ~
expikr
r
. If so, one can represent spherical
wave, as a plane one when    .
In an unbounded homogeneous medium using any kind of source, it is a
difficult problem to obtain a strict solution of the wave equation, which describes EM
wave propagation in such a medium. Usually, to obtain a particular solution of a wave
equation one can assume that the source is a point with respect to volume metric
dimensions around this source. In the literature [1-3, 6-9] to determine the criterion of
a point source requires that the linear dimensions of the source, l must be smaller
than the wavelength in the considered medium, that is, l <<

. In this case the so
called Green’s function, as a solution of the wave equation for a point source, can be
introduced. Moreover, if any real antenna can be represented as a general real source
by a linear superposition of point sources, one can obtain a general solution for the
wave equation with such a source by using the solution of the wave equation for a
point source, in the other words, by use of Green’s function as a point source function.
This result is also connected with the topic of linearity of the wave equation in the
considered medium. Below, we will examine the boundary-value problems both for
scalar and vector waves by employing the Green’s function presentation.
2.4. Green’s Function Presentation
Scalar presentation of Green’s function. First of all we examine the Green’s function
presentation for the scalar wave equation. In this case let us construct the solution of a
scalar wave equation in any volume V of free space having an arbitrary source s( r ).
Such a solution can be written in the same form, as (2.10), but with a source in its
right-hand side:
 2 ( r)  k 2 ( r)  s( r)
15
(2.44)
First, we will introduce the same equation for the Gereen’s function, but with a point
source in its right-hand side:
 2 G( r, r')  k 2 G( r, r')  ( r  r')
(2.45)
The given functions G( r, r') and ( r) can be easily found from the principle of
linear superposition, since G( r, r') , as was mentioned above, is the solution of (2.44)
with a point source in the right-hand side. In fact, one can notice that an arbitrary
source s(r ) is just
s(r)   dr' s(r')(r  r')
(2.46)
which actually a linear superposition of point sources in mathematical terms.
Consequently, the solution of (2.45) is just
(r )    dr ' G (r , r ' ) s(r ' )
(2.47)
V
which is the linear superposition of the solution of (2.37).
To find the solution of (2.45) for free space, or more correctly, for an
unbounded, homogeneous medium, one can solve it in spherical coordinates with the
origin at vector r' . In this case (2.45) reduces to
 2 G( r)  k 2 G( r)  ( r)
(2.48)
But due to the spherical symmetry of a point source, G( r) must also be spherically
symmetric. Then, for r  0, the homogeneous, spherically symmetric solution of
(2.48) is given by
G ( r)  A
expikr
r
B
expikr
(2.49)
r
Since sources are absent at infinity, a physically correct solution of (2.48) can be
presented as
G ( r)  A
expikr
(2.49a)
r
To determine the unknown constant A, we substitute (2.41a) into (2.39) and integrate
it over a small volume about the origin to yield
 dv    
V
A expikr
r
  dv  k 2
V
A expikr
r
 1.
(2.50)
Note that the second integral in (2.50) vanishes when V  0 , because dv=4r 2  dr.
16
The first integral in (2.50) can be converted into a surface integral using
Gauss’s theorem to obtain
 2 d expikr 
A
 1
4 r

dr
r

 r 0
or
(2.51)
A
1
4
As was mentioned above, the solution of (2.45) must depend only on | r - r' | .
Therefore, Green’s function must be presented, as a solution of (2.45), in the
following form:
G( r, r')  G( r - r') 
expik | r - r' |
| r - r' |
(2.52)
Moreover, it can be seen that G( r, r')  G( r', r) from reciprocity irrespective of the
shape of volume V [3]. This fact and formula (2.52) imply that Green’s function is
translationally invariant for unbounded, homogeneous media. Consequently, a general
solution of inhomogeneous equation (2.44) by using (2.47) can be finally presented as:
(r )    dr '
V
expik | r  r '|
| r - r' |
s(r ' )
(2.53)
2.5. Huygens’ Principle – Mathematical Description
Huygens’ principle, which is named by the Danish researcher Christian Huygens,
shows how a wave field on the surface S determines the wave field off the surface S
(Fig. 2.5a) or, inversely, inside the area bounded by the surface S (Fig. 2.5b). In the
other words, each point on the surface S can be interpreted as a source of a spherical
wave, which can be observed at any point A, either in the outside space with volume
V, if a source O is inside it (Fig. 2.5a), or inside the bounded area S, if a source O is
outside the surface S (Fig. 2.5b).
17
Above we indicated that any scalar wave in an unbounded source-free
homogeneous isotropic medium can be described by the homogeneous equation (2.10)
(or (2.36) without the right-hand side).
If we now multiply (2.10) by G(r, r') and (2.45) by (r ) , subtracting the resulting
equations and integrating over a volume V containing vector r' , we have
 drG(r, r' )
2

(r )  (r ) 2 G (r , r ')  (r ' )
(2.54)
V
Taking into account the following relations [4]
G 2    2 G    G  G 
we can rewrite (2.54) by use of the so-called Green’s theorem. Green’s theorem or, as
it is sometimes called, the second Green formula [2] states the equivalence of volume
integral (2.54) with the surface one, i.e.,


 drG(r, r') (r)  (r) G(r, r')   dsG(r, r') n  (r)
2
2
V
S
G(r, r') 
n 
(2.55)
It is well known [4], that for any two scalar functions f and g there is some relation
between them: f g  n  f
g
. Taking into account this relation and Green’s theorem,
n
based on Gauss’ divergence law, one can rewrite the left-hand side of (2.46) as:
(r' )   dsn  G(r, r' )(r)  (r)G(r , r' )
(2.56)
S
Equation (2.56) is the mathematical expression of the statement that once (r ) and
n  ( r) is known on surface S, then ( r' ) away from S can be found.
18
Fig. 2.5
From general equation (2.56) one can obtain two different situations at the bounded
surface S. In fact, if G(r, r') satisfies equation (2.10) or (2.44) (without its right-hand
side) with the boundary conditions n  G( r, r' )  0 for r S , then expression (2.56)
becomes
(r ' )   dsG (r , r ' )n  (r ).
S
19
(2.57a)
On the other hand, if now G( r, r') has only to satisfy equation (2.10) or (2.44)
(without its right-hand side) for both r and r' in volume V, and no boundary
condition has yet been imposed on G( r, r') , then, in the case of G( r, r')  0 for
r S , (2.48) becomes
(r ' )    ds(r )n  G (r , r ' ).
(2.57b)
S
Equations (2.56), (2.57a), and (2.57b) are various forms of Huygens’ principle
depending on the definition of Green’s function G( r, r') on the bounded surface S.
For example, equations (2.57a) and (2.57b) state that only n  ( r) or ( r) need be
known, respectively, on the surface S in order to determine wave function (r' ) .
2.6. Huygens’ Principle – Geometrical Description
To show the well-known fact that propagation of rays can be explained by a concept
of electromagnetic wave propagation, let us now formulate Huygens’ principle,
presented above, for free space without obstacles or discontinuities, i.e., for an
unbounded homogeneous medium. This case was an early description of what actually
happens with wave energy when it simply travels in free space in straight-ahead
manner. Here, in simple terms, the principle suggests that the energy from each point
propagates in all forward directions to form many elementary spherical wavefronts,
which Huygens called wavelets. The envelope of these wavelets forms the new
wavefront. In other words, each point on a wavefront acts as the source of secondary
elementary spherical waves described by Green’s function G( r, r') . These waves
combine to produce a new wavefront in the direction of wave propagation. With great
20
accuracy each wavefront can be represented by the plane which is normal to wave
vector k (see Fig. 2.6, line AA’, as a starting wave position).
Spherical elementary waves originate from every point on AA’ to form a new
wavefront BB’ which is drawn tangential to all elementary waves with equal radii. As
can be seen from the illustration of Huygens’ principle presented in Fig. 2.6a, the
secondary waves originating from points along AA’ do not have a uniform amplitude
in all directions.
In fact, if  represents the angle between the direction to any point C on the
elementary sphere (see Fig. 2.6b) and the normal to the wavefront (or parallel to k ),
then the amplitude of the secondary wave in a given direction is proportional to
(1  cos ) . If so, the amplitude in the k -direction is proportional to ~ (1  cos 0)  2 .
In any other direction the amplitude is less than 2. In particular, the amplitude
of any elementary wave in the backward direction is ~ (1  cos )  0 , that is, the
waves not propagate backward. The waves propagate forward along straight lines
normal to their wavefronts. Moreover, the consideration of elementary waves
originating from all points on AA’ leads to the expressions for the field at any point
on BB’ in the same form, the solution of which shows that the field at any point on
BB’ is exactly the same as that at the nearest point on AA’.
The phase difference between the oscillations at these neighbouring points of
lines AA’ and BB’ depends on the distance between them, d, and is therefore
proportional to ~ kd  2d /  . If d   , all points at the fronts AA’ and BB’ oscillate
in phase; if d   / 2 , all points oscillate in anti-phase etc.
21
'
A
'
B
'
C
A
B
C
Old wavefront
New
wavefront
Current
wavefront
Fig. 2.6a: Geometry of Huygens principle in free space.
Fig. 2.6b
22
Hence, from Huygens’ principle in the particular case of unbounded free space
follows the phenomenon of straight line wave propagation (that is, along the traces
where is the minimum of loss of energy, according to Fermat's principle), as light rays
in optics. Moreover, this also follows from Fermat principle, according to which rays
propagate in free space along the lines where the integral of energy is minimal, that is,
along the straight lines. This concept will be used for the case of optical rays'
reflection and diffraction from intersections by use the same vector and scalar wave
equations described in Lecture 2.
2.7. Fresnel-Zone Concept for Free Space
The existence of Fresnel zones also follows from Huygens’ principle not only in
obstructive conditions for both terminals, transmitter and receiver, when any obstacles
are placed around them and diffraction phenomenon is predominant.
In the case of free unbounded space, let us once more return to the integral
form (2.55) of EM-field presentation at any point r between the observer and the
source of radiation, using the Green’s function source presentation. In the case of free
space, instead of the virtual sources at any boundary limit of the space volume V,
within which the real source exists, we will introduce virtual sources to describe the
EM field in each point of the wavefront in space along the wave propagation path. A
main result, which follows from Huygens’ principle is that at any point of a sourcefree unbounded medium, the total field is a superposition of elementary spherical
waves, which are radiated by virtual sources in space and reach to the observation
point along straight paths.
Thus, let us consider that the radiation source is placed in free space at point A
and the receiver is at point B, as shown in Fig. 2.7. We also consider an imaginary
23
plane with area S normal to the line-of-sight path at any point between A and B, which
passes across the point O at the line AB (see Fig. 2.7).
Now, if we “work” with infinite volume V, Green’s theorem (2.55) can be
represented for any vector of the EM wave, namely, the Hertzian vector, as
(R )  
(R ') expik | R  R '|
ds
n
| R  R '|
(2.58)
where | R  R '|  r is the distance from any point in the imaginary plane S and
observer at point B.
Fig. 2.7
If the initial radiation source can be assumed to be a point source with Green’s
r
r
e ikr1
function G ~
, then for any point in the plane S, because 1   0 , we finally
n
r1
r1
obtain:

'
 r0 exp ik r1  r1
1 1
(R ) ~
  ik 
2   r1
r1r1'
 r1
 ds
All the mentioned distances are presented schematically in Fig. 2.7.
24
(2.59)
We will talk about the so-called wave zone or far zone between the plane S and
two terminals A and B, i.e., when
kr1  1 ,
kr  1
(2.60)
In this case, the first term in brackets within integral (2.59) is less than the second one,
and can be neglected. Moreover, in the process of integration the variables r and r1 are
changed. Therefore, because the inequality (2.60) is valid for line-of-sight propagation
links in free space, relatively small changes of variable r cause fast oscillations of the

product ~ exp ik r1  r1'
 . On the other hand, this fact leads to fast changes of sign
both for real and imaginary parts of integrand in (2.59). At the same time, other
products in the integrand of (2.59) are changed more weakly with relatively small
deviations of r and r1 . In this case, the known method of stationary phase is usually
used to derive such an integral, containing both slow and fast terms inside the
integrand [4, 7, 10]. We will not present here all the complicated analyses and
derivations of the integral in (2.59), only its final form for the observed point B, as
( B) ~ 
 
ik
exp ik r0  r0'
2 r0 r0'



 dxdy expi k2  r1  r1  x

0
'
0
2



(2.61)
 k  1 1 
exp i   '  y 2 
 2  r0 r0  
For each integral in the two-dimensional integral of (2.61) one can use the well known
integral presentation [4, 10]

 expi d 
2

Therefore,
25
i


( B) ~ 
 
ik
exp ik r0  r0'
'
2 r0 r0
 k  1i
1
  '
2  r0 r0 

expikd 
d
(2.62)
where d  r0  r0' is the distance between the source (point A) and the observer (point
B). Hence, as is shown from expression (2.62), if at the plane S the source A creates a
field ~
e ikr1
, then at the observed point B, the virtual dipoles, uniformly distributed at
r1
e ikd
S, will create a field ~
, the same as for the direct wave from A to B. This is the
d
main content of Huygens’ principle. Additional analysis of integral (2.51), extended
above S for the farthest wave zones


'
ik r0 exp ik r1  r1
( B) ~ 
ds
2  r1
r1r1'
(2.63)
shows that the plane S can be split into the concentric circles (hoops) of arbitrary
radius. It is apparent that any wave, which has propagated from A to B via point Ci ,
i  1, 2,... , on any of these hoops has traversed a longer path than AOB (namely,
A C i B>AOB at Fig. 2.7). While passing from one hoop to another, the real and
imaginary parts of the integrand in (2.63) change its sign. The boundaries of these
hoops are determined by the condition:

 
k r1  r1'  r0  r0'
  n 2 ,
n  1, 2,...
(2.64)
The physical meaning of these hoops for wave propagation is that if the virtual
sources of the elementary waves lie within the first hoop, they send to observer
radiation with the same phase for each elementary wave. Sources from two
neighboring hoops send respective radiation, which extinguish each other. Some
elementary analysis of integral (2.63) shows that a no-vanishing result exists only
26
from the first, central hoop. The hoops are usually called the Fresnel zones [1-3, 5-9].
Let us derive the width of these zones b . For the first Fresnel zone, assuming
x 2  y 2  b 2 , and using expressions (2.63) and (2.64) for n=1, we have:
k  1 1 2

2
  ' x  y 
2  r0 r0 
2


(2.65)
and
b1 
x
2

 y2 
r0 r0'
 1 1
~ R
  ' 
k  r0 r0 
(r0  r0' )
(2.66)
where R is the minimal range from each r0 and r0' .
Hence, the width of the first Fresnel zone is larger than wavelength, i.e.,
b1 ~ R   . For greater zones (with number n>1), for which the width b is
smaller than the distance to each zone center (simply, the radius of each circle b  bn ),
one can easily obtain after differentiation of (2.66) the following equation:
 1 1 
kbb  '  
 r0 r0  2
(2.67)
and
b ~
 R b12
~
 b1
2k b
b
(2.68)
Here R is, once more, the minimal range from each r0 and r0' .
Thus the width of the hoops with n>1 decreases with an increase of radius of
each zone bn . At the same time it can be shown that the radius of each Fresnel zone of
any specific number of the family of zones can be expressed in terms of zone numbers
n and the distance between both points A and B and the imaginary plane S as [1-3, 9]
27
b  bn 
nr0 r0'
(r0  r0' )
(2.69)
from which, introducing in (2.69) n=1, one can immediately obtain expression (2.58)
the radius of the first Fresnel zone. As follows from (2.68) and (2.69) the width of the
Fresnel zones b decreases with increasing zone number n. At the same time, the
area of these zones is not dependent on zone number n, that is,
2 bb ~

R
2
(2.70)
It is clear that the radii of the individual hoops depend on the location of the imaginary
plane with respect to points A and B. The radii are largest midway between points A
and B and become smaller as the points are approached. Moreover, as follows from
(2.64), the family of hoops have a specific property: the path length from A and B via
each circle is n
AOB=

longer than the direct path AOB, i.e., for n=1 (first zone) ACB2


, so the excess path length for the innermost circle is . Other zones will
2
2
have an excess proportional to

with parameter of proportionality n=2, 3, 4, ....
2
The loci of the points for which the excess A C i B - AOB= n

defines a family
2
of ellipsoids, the radii of which are described by expression (2.69). But in free space
without any obstacles, as we showed above mathematically, only the first ellipsoid is
valid and presents the first Fresnel zone which passes through both points, transmitter
(T) and receiver (R), as illustrated in Fig. 2.8.
28
Fig. 2.8
This is why, despite the fact that in free space reflection and diffraction phenomena
are not observed, and no effect of interference between neighboring zones exists to
describe the loss-less phenomenon of wave propagation, the concept of Fresnel zones
is also used. This approach allows us to obtain the first hoop’s width in line-of-sight
propagation conditions and then to estimate through formula (2.58), by use of the
“working” frequency for the respective radio link, the range R of wave propagation in
conditions of direct visibility between any receiver and transmitter.
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