>1<
Dan Censor
Ben Gurion University of the Negev
Department of Electrical and Computer Engineering
Beer Sheva, Israel 84105 censor@ee.bgu.ac.il
Abstract
—This study attempts to summarize and systematize the mathematical tools used in free-space electromagnetic wave scattering problems. The elements of
Relativistic Electrodynamics relevant to the subject are reviewed. Then, various two- and three-dimensional representations for waves scattered by objects at rest are summarized. Finally the two aspects are combined into a package of various representations for the scattered wave by moving objects and the associated scattering amplitude.
1. Free-Space Relativistic Electrodynamics
2. Spatiotemporal Fields and Plane Waves
3. Phase Invariance and Minkowski Four-Vectors
4. Surface Integral Representations
5. Plane Wave Integral Representations
6. Special Functions Representations
7. Differential Operators Representations
8. Applications to Relativistic Scattering
9. Concluding Remarks
References
1. FREE-SPACE RELATIVISTIC ELECTRODYNAMICS
The sourceless free-space (vacuum) Maxwell’s equations can be written as [1, 2]
 r
 t
H ,
 r
 t
E (1) where
   
0
,

are the free-space (vacuum) parameters and are associated
0 with the speed of light c

(
 
0
)

1/ 2
, and with the free space impedance for plane waves that
  t
Z r

(
 
0 0
)
1/ 2
0,
  t r
. Applying the divergence operator
  r
0 implies equation. In (1) all fields, e.g.,
 r
E

( )
0,
 r
0
in (1) and assuming
, the remaining Maxwell
are dependent on the spatiotemporal coordinates grouped as a Minkowski [3, 4, 2] four-space quadruplet in the form
R

( , ict ) which so far is only notational, without any further content.
(2)

>2<
Einstein’s Special Relativity theory [5-8, 2], or its Relativistic
Electrodynamics component with which we are presently dealing, is based on two postulates: The kinematical postulate asserts that the speed of light c in free space is identical in all inertial (i.e., non-accelerated) frames of reference. With some additional considerations regarding the symmetry of space, Einstein [5] derived the
Lorentz transformation r

U r v t ), t
  
( t
  c
2
/ )

(1

2 )

1/ 2 ,
   v , (3)
U (

1) vv v
 v / v relating the spatiotemporal coordinates in two inertial systems, where the origin of the primed system velocity v
 
, as observed from the unprimed system

, is moving with a
. In (3) the tilde denotes dyadics,
~
I is the idemfactor (or unit) dyadic.
Using the Minkowski notation, the Lorentz transformation (3) can be symbolized in the form R
   
. The role of U is to multiply the component along the velocity by

. It is easy to show that (3) can be solved to yield the inverse transformation R
  
, whose form is obtained from (3) by exchanging primed and unprimed coordinates, and replacing v by v
   v .
Using the chain-rule of calculus, one finds the Lorentz transformation for space and time differential operators, which can be grouped into the four-space gradient operator quadruplet

R

(
 r
,
 i c
 t
)
 r

U (
 r
 v
 t c
2
/ ),
 t

   t
 r
) which can be symbolized as

R

  
. The associated inverse transformation by

R
  
also exists.
(4)
For the dynamical part, Einstein postulated the form-invariance or functional-invariance of Maxwell’s equations in all inertial systems, i.e., in

'
We now have, similarly to (1)
 r

E
 t

H

,
 r


H
   t

E
 where in (5) the fields, e.g., E
 
( ), R
   ict

) spatiotemporal coordinates in

' .
are dependent on the
By exploiting (4) in (5), and comparing to (1), Einstein [1] derived the transformation formulas for the fields, which in the present case take the form
E
  
(
  
), H
  
(
  
)
V
 
I (1

)
(5)
(6)
The role of V in (6) is to multiply the components perpendicular to the velocity by

.

>3<
This summarizes the elements of relativistic electrodynamics in so far as we need it for the subsequent discussion.
2. SPATIOTEMPORAL FIELDS AND PLANE WAVES
A basic solution of the Maxwell’s equations (1) is a transversal plane electromagnetic wave
E

  
ˆ
ˆ e
0 h
0

 e i

, K

( , i
  k

,
0
/ h
0

Z (7) where in (7) E H E k H k 0,
ˆ ˆ ˆ
, defining a right-handed triad of mutually perpendicular vectors. We also introduce a new Minkowski quadruplet
K for the spectral components—the wave vector k and the (angular) frequency

.
The multiplication between Minkowski quadruplets is defined as
  k i
  r ict )
 t (8)
Inasmuch as Maxwell’s equations (1) are linear, a superposition (integral) of plane waves (7)
E
 e
0
  d
4
K

C
( d
4
K ) e i

 dk dk dk d x y z i
 c


E g ( )
H

 (9) is also a solution, where C is the integration path, and provided that in (9) e
0
/ e
0

Z
 k
 c , and similarly to a single plane wave,
ˆ
E
 ˆ
H
 k ,
ˆ
E
 ˆ
H
 ˆ
E
ˆ ˆ
H
 
0 . Note carefully that the constraint

/ k
 c , which is the dispersion relation for the present case, will cause the four-fold integral (9) to collapse to a three-fold integral. In some of the representations shown below this was automatically taken into account and is not discussed. For example, in the
Sommerfeld integral for cylindrical functions

/ k
 c is understood, and the integral is only on the direction of propagation of the plane waves [1].
Application of the transformation (6) for the fields to (7) yields
E V E v
ˆ ˆ ˆ   
(
  ˆ ˆ
)
 
H V H
 v
ˆ ˆ ˆ
HZ
 c
2
/( ))
 
(
  ˆ ˆ
)
 
W V I
 ˆ ˆ
)
(10)
In (10) the new dyadic W applies to both fields, which is very convenient. Thus we obtain in (7), (9)

>4<



( ) 
 e
0
C

( d
4
K ) e i

W

 g
E
H
K


(11)
Scrutinizing (11) it is clear that we have derived the fields in

' , expressed in terms of R , the coordinates native to frame of reference

, including the description of the integration path C , sometimes in the form ( ) , i.e., depending on the spatial variables in

. We can of course substitute the Lorentz transformation R
   
(3) and derive (11) in terms of
 
coordinates R

, but unless this is followed by a numerical simulation, very little is gained by it. For example, the limits of the integral will now become space and time dependent
C (
  
) , in a complicated way that we do not know to deal with analytically.
Had we started with fields in
 
, i.e., from (cf. (9)
E
H

 e h
0
0

C


( d
4
K

) e i
 
 g


E
(

)
H
( K

)


(12) we would find the analog of (11)
E R

)
 (

)


 e h
0
0

  
C


( d
4
K

) e i K R
W


(
 ˆ ˆ 
I )
W




E
(

)
 g

H
( K

) 
(13) once again defining the field in one frame of reference in terms of the coordinates of another frame. This is typical to the present class of problems, and is often overlooked until one tries to solve some concrete problem.
3. PHASE INVARIANCE AND MINKOWSKI FOUR-VECTORS
The plane waves in (9), (11), or (12), (13) possess the same phase

, or K R

, respectively, but thus far we have no way of relating phases of plane waves in different inertial systems. Einstein [5] tacitly assumes that a plane wave in one frame of reference transforms into a plane wave in another frame, and when we compare the phase at a spatiotemporal location R with the phase at the corresponding R
   
, the measured phases are identical. This is the principle of phase invariance, or phase conservation, i.e.,
K R K R

(14)
Accordingly, substitution of (3) in (14) yields a transformation commonly referred to as the relativistic Doppler Effect [2, 4, 5-8] k

( v
 c
2
/ ),
   v k ) (15)

>5< and the inverse transformation, similar to the one applying to (3), follows.
This brings us to the subject of Minkowski four-vectors [2-4], which up to this point were used only as a compact notation. Rather than going into this subject in detail, only a few remarks will be made here, to put the subject in proper perspective. An arbitrary quadruplet Q is a proper Minkowski four-vector if and only if
Q R Q R (16) where in (16) R is a-priori considered as a Minkowski four-vector, i.e.
2 2 c t

R R
 r r
2 c t

2
(17)
By substituting the Lorentz transformation (3) into (17) it is verified that (17) is identically satisfied. Therefore the assumption of R being a Minkowski four-vector is tantamount to postulating the Lorentz transformation, and vice-versa .
Once a quadruplet Q has been established as a Minkowski four-vector, it can be used to test any new quadruplet P . In general, any four-vector multiplied by itself is an invariant
Q Q Q Q (18) and this can be used as a definition of a four-vector in the statement: the length of a four-vector in the Minkowski space is a scalar invariant under rotation. However, it must be remembered that this whole mathematical edifice rests on the original definition (17), i.e., when the physics comes into the game, the Lorentz transformation is already assumed here.
The use of Minkowski’s four-vectors provides a very convenient technique for dealing with various aspects of the Special Relativity theory. However, for the purpose of dealing with the present subject of relativistic free-space scattering problems, the important aspect to note is that Minkowski’s theory, in spite of its mathematical content and compact and elegant notation, is not an essential part of
Einstein’s theory. In other words, anything that we can do using the Minkowski four-space formalism, we can also do without it. For example, using the Minkowski theory, once K is defined as in (7) and declared as a Minkowski four-vector, (14) follows without stipulating the phase invariance principle as an extra postulate, and
(15) can be derived. Conversely, if (15) is postulated, (14) will follow as a consequence.
The upshot of this section’s discussion is that when a plane wave (7) is stated in

, we are able to recast it as a plane wave as observed in
 
, in terms of native
 
coordinates
E
H
 
  

K
  k
 i
 
0
 
0
 

 e i
 
/ ),

/

W




ˆ
H e h
0
0


,
0
 
0 e i


Z
(19)

>6< where the spectral components are related by (15) and the field amplitude follow
(10).
4. SURFACE INTEGRAL REPRESENTATIONS
The mathematical tools are mainly based on [1] and Twersky’s articles [9-12], which should be consulted for earlier references. In the following sections 4-7 an overview of various representations of the scattered field in the frame of reference
 
of the object at rest are given. This encumbers the notation with a profusion of primes, but to avoid confusion subsequently, no shortcuts are used. Note also that the factor

0
 i t
is carried along by the fields, and is therefore included below.
With z as the cylindrical axis, the two-dimensional problem is analyzed.
The excitation wave is given by (19), with
ˆ   ˆ
. Incorporating the relevant Green function
G ( r
  ρ 
)
 iH k
0
( | r ρ 
|) / 4 (20) satisfying (
 r
2

 k

2 ) ( r
  ρ 
)
  
( r   ρ 
) , the Helmholtz wave equation, where in
(20) H denotes the Hankel function of order zero of the first kind. This goes
0 together with the time factor e
 i t
in ensuring outgoing scattered waves. Applying
Green’s theorem to the scattered wave E
 sc
 ˆ
E
 sc
and G , (20), in the region external to the scatterer, the solution of the Helmholtz wave equation
(
 r
2

 k

2
) E
 sc

0 for the scattered wave is given by the surface integral representation
E sc

1
4 i
S


[ H
0

ρ 
E
 sc

E
 sc

ρ 
H
0
]
 d S
  
H
0
, E
 sc

H
0

( | r
 ρ 
|), E
 sc

E sc
( , ), d
S   d ( )
 n dS
(21) with n

in (21) denoting the outward unit normal, and the integral is compacted by the expression in braces.
For k | r
 ρ 
| 1, r
  
, H
0

H k
0 r
 ρ 
|) in (21) becomes
H k
0
( | r
 ρ 
|) (
 
)
 ik r ρ
, (
 
)

(2 /

)
1/ 2 i k r e (22)
Substituting (22) into the surface integral (21) yields (note a sign typo in [9])
E sc
( , ) H k r g r t g r t

 e
 ik
ˆ r ρ
, E sc
( , )

(23) where in (23) the scattering amplitude g
 
is a function of the angle defined by the unit vector ˆ

, and the primed symbol g

has been chosen to indicate that this is the pertinent function observed in
 
.

>7<
For three-dimensional vector waves [11, 12], (20) is replaced by the dyadic
Green function
(
  ρ 
)
    r

/ k

2 ) ( | r  ρ 
I
 
ρ 
/ k

2
) k h k r
 ρ   

 h
(1)
0
  ie i

/

(24) satisfying the Helmholtz wave equation (
 r

  r

  k

2
I )

G ( r
  ρ 
)
  
(
  
) .
In the region external to the scatterer the solution of the Helmholtz wave equation (
 r

  r

  k

2 )
  sc

0 for the scattered wave is given by the surface integral [12]
E
   sc r
 k

4
 i
S


[(

ρ 

E
 sc
 d S
  h

) ( d S
 
E
 sc
) (

ρ 
 h

)]



,
 sc h
  h ( k | r
 ρ 
|)

(
  ρ 
4
) , k

 i
E  sc
 E ρ   d
S   d
S ρ  n

For k | r
 ρ 
| 1, r
  
, ( | r
 ρ 
|) in (24) becomes
(25)
( | r
 ρ 
|) (
 
)
 ik
(26) and operating on (26) with ( I
  
ρ 
/ k

2
) yields (
 ˆ ˆ
) which is the part of the r

, ,i.e., perpendicular to the direction of propagation in the far field. Hence (25) reduces to the analog of (23)
E
   sc
( g r
ˆ t g r t


(

) e
 ik
, E
 sc
ρ t

(27)
4. PLANE WAVE INTEGRAL REPRESENTATIONS
Plane wave integrals have been discussed in Section 2 above. Scattered fields can be represented in the object’s rest frame

' in terms of plane wave integrals, derived from surface integral representations, by exploiting Sommerfeld-type integrals for the special functions involved.
Consider the Sommerfeld integral representation [1]
H k r
0
 
)

1


  
/ 2 i
/ 2 i e
 
  d
 
, C

 cos
 with the limits (28) needed for the convergence of the integral. Defining a new angle
   
, d
   d
 
renders (28) as
(28)
H k r
0
 
)

1

 / 2 i e ik p r d

, k
/ 2 i
ˆ k r C (29)

>8< and multiplying (29) by e
 i t
clearly displays the plane wave integral for this case.
Now use (29) to recast H in (20), (21) as
0
H k P
0
 
)

1

   
  
/ 2 i
/ 2 i e
 
  d
P
 r
ρ 
|,
P    
, P C
 
  
1

 e ik p r ρ ) d
 
(30)
Substituting (30) in (21) and interchanging order of integration leads to
E
   sc r
 
H
0
, E
 sc


 e ik p r ρ ) d
 
, E sc
ρ t t
 d
  
1


(31)


 e ik p r
 e
 ik
, E sc
ρ e ik g p t d
 g  ˆ g

For more detail regarding the integration limits and the domain of validity of the representation (31) see Twersky [9] and cited references there. Suffice it to say that the surface integral representation (31) applies to any surface enclosing the scatterer, while the plane wave representation (31) is at least valid for r
    max
, i.e., outside the circle circumscribing the scatterer, which in the special case of a circular cylinder is the surface itself.
Similarly to (28), the spherical Hankel function (
 
)
 h (1)
0
( k r ) is recast as an integral [1, 11]
(
 
)

 
 
0
    d
/ 2 i e
 
  S d
 
  
  
0
/ 2 i e   S d
 
(32)

1
2
 where in the second expression (32) the Weyl limits are introduced. In the form (32) this is a trivial modification, but it serves to show that the integration here is extended over the unit sphere. Similarly to (29), we recast cos
 
in terms of the scalar product of two unit vectors
C
 
   
)
 r
ˆ   
)

S S C
  

C C
 
(33)
Accordingly, we have to redefine the paths in the complex planes

,

, so that the integrals converge. One simple way of doing it is to redefine for each ˆ

the r

becomes always the polar axis.
Consequently the Weyl limits can be used. Equivalently, we could choose a fixed polar axis and change the limits to achieve two appropriate Sommerfeld-type paths.
For example, if the limits on
  are chosen similarly to (30), from
/ 2 i to / 2 i , then in the integral e iS S C
vanishes at the limits, and taking
 
from
  
0 to upper limit while e iS S C
/ 2 i makes e iC C
 vanish at the
remains oscillatory. See also Noether [13].
The analog of (29) will therefore be

>9<

(
  
1
2

 ik p r e d

 ik p r e d
  p


    d
  
  
0
/ 2 i e ik p r
S d
 
(34)
The analog of (30) is
(
  
1
2

 e ik ˆ ( ρ ) d


(35)
By substituting (34) in (24), (25) we obtain, similarly to (31)
E
   sc r



,
 sc
 
( I
   r

/ k

2 h k r
 ρ 
|),
 sc
 

  ( I
   r
 r
 k

2
/ )
1
2

 e ik p r e
 ik d


, E
 sc p
 

1
2


(

) e ik p r e
 ik d


, E
 sc p
 

1
2

 e ik p r

(

) e
 ik
, E
 sc
ρ t
 d
1
2

 e ik g p  r

Once again, the plane wave integral representation (36) applies at least for
   max
, i.e., outside the circumscribing sphere, which in the special case of a
 spherical scatterer is the surface itself.
6. SPECIAL FUNCTIONS REPRESENTATIONS
One way of deriving the special functions representations is by starting from the plane wave integrals. In (23), (31) expand g

in a Fourier series
(36)
 
0
 i
     m m im
  and interchange in (31) integration and summation. This yields
E
   sc r
 
0
 i
    m m
 m m
  im
  i a H ( ) , m
 
, ... ,

(37)
(38) valid at least for r
    max
.
The three-dimensional case is discussed by Twersky [12]. Accordingly we start with (27), (36) and expand g

as a series in vector spherical harmonics

>10< g r t
 e e
0
  i
 t  nm
 m
C r n
  nm
 m
B r n
  nm
 m
P r n
  nm
 m
C r n
 r r

Y n m r
 ˆ 
B m n

( θ  
 
/ S
 
ψ 
 
) Y n m r 
B m n r
 r
 r

Y n m ˆ  r
ˆ 
P n m r
  r

Y n m
( r
ˆ 
),

 
Y n m
C r
 m n


(
ψ imY n
  m
  r
ˆ 
/ S
  n

θ 
 
) Y n m r 
 m
  n , ... , n
(39) where in (39) Y n m
are spherical harmonics and c
 nm
, b
 nm
, p
 nm
are coefficients.
Substitution of the spherical harmonics m ˆ
B m n
ˆ
P n m
( ) in (36) yields for each type a corresponding vector spherical wave according to
1

2

 i
 n
M e ik p r
 m
C p n

B nm
( k ), i n

1 Ν nm m n
( ), P n m ˆ 
( k ), i n

1 L nm
 d

( k )


M nm
( k )

J
1 m
C r n
ˆ  Ν nm
( k )

J
2 n
 
J
3
L nm
( k )

J
4 m
P r n
 
J
5 m
B r n

J
1

(
 
), J
2

(
 
) /(
 n k r )
J
3
 
[
  n
(
 
)] / k r , J
4
J
5
 h k r n
 
) /(
 
),
 n
 

1/ (

1)
[ ( )] n
Finally the special functions representation takes the form n

(40)
E
   sc r
 
0
     nm i n

M nm
( k
   nm
 i
Ν nm
( k r
) b nm
 i
L nm
( k r
) p nm

(41)
Nota bene, in (39)-(41) we have included for completeness p
 
0 , the coefficients associated with longitudinal waves which are irrelevant to the problem of free space scattering involving solely transversal waves.
7. DIFFERENTIAL OPERATORS REPRESENTATIONS
The differential operator representations lead to inverse distance power series, a point of extreme importance for scattering by moving objects, because an object at rest in
 
is observed as moving in

, with time dependence distance from the observer.
We start with the two-dimensional asymptotic series for Hankel functions
[1, 9] m i H m
( k r ) (
  


 m
2  m
2
  2  m
2
(1 4 )(9 4 )...((2 1) 4 )...
( 8
 
)
 
!
(42)
(
 
,
 m
2
),
 
0, 1, 2, ...

>11<
Substituting (42) in (38) and noting that (
  2
) im
  
0 facilitates the interchange of summation and differentiation, yielding
E
   sc r

(
 
,
 2
 
) g


(
 
)
 g
  g

2

 
 g


9 g

40
2
  g

128(
 
)
2
16
4
  g

 


(43) g
  g r
ˆ t
 g
 t
 z
ˆ 
0
 i
    a e m m im
 
An exact (as opposed to asymptotic) operational form has been derived
[10], based on the Lommel polynomials [14]. Thus
E sc
( , )

O k r e
,
 2
 
) g e
 t
 o
(
 
,
 2
 
) g o
 g e o
  g
 t
 g

(
 
O e

H Q
0 e

H Q
1 e
, O o

H Q
0 o

H Q
1 o
Q e
2
 
2(2
2
  k r )
2
4)

2
 
(
2
 
2
4) (
 
4!(2
 
)
2

4
16)
 
(44)
Q e

2
 2
 k r

 

Q o
/ i
 
2
 
2
 
3!(2
 
)
4)
3
2

2
 
(
2
 
2
4) (
 
16)
2
5!(2 )
5
 
2

2
 
1

(
 
9)(
 
3!(2 )
3
1)
2

(
 
25)(
  2
9) (
 
5!(2 )
5
1)
2
 
Q / i 1 o
 
(
 
1)
2(2
 
)
2
2

(
 
4!(2
2
1) (
  k r )
4
9)
2
 
For the three-dimensional case we have a dyadic operator [12]
E
  sc r t

O ( , )
ˆ
O

(
 
)




0


D D I D I

( D (

1)

I )
  
1
D
 
2

(

2 )
 n

(
 
I

( D
 
1) n I )
D
 ˆ 
( D

2)
ˆ   ˆ 
2 S


1


 
( S
 
θ 
)
 
 
ψ   ˆ 
( D

S


2

)





2


2
  
ˆ   
  r ˆ   ˆ 
( D

S


2

)
ˆ   ˆ 
2


2 

ˆ   ˆ 
2 S


1


 

( /(2 k r
 
D

S


2 2

[
 
S


( S


)]

(45) where in (45)

 
( S
 
ˆ 
)
 ˆ t
 
 
[ S
 
is understood. This terminates the discussion on the various representations of the scattered field in the object’s rest frame  
.
8. APPLICATIONS TO RELATIVISTIC SCATTERING

>12<
The key to free space relativistic scattering is given in (13), which tells us how to transform a plane wave back from
 
to

. Accordingly we can start with the far field approximations (23), (27) and write
E r sc
 
(
 
) W
ˆ r

 g r t
E r sc
   z
ˆ
E sc r
W


 
H k r ) W
ˆ r
 g r
(
 ˆ ˆ 
I ) t

(
 
) W


 z g r t
(46) v z 0 , i.e., the velocity is perpendicular to the cylindrical axis, and the three-dimensional, two-dimensional, cases are considered, respectively. Note that in (46) we obtain the field E sc observed in

, in terms of the coordinates R
 substitute from the Lorentz transformation R

, native to
 

 
. Afterwards one can
, (3), in order to express the scattered field as E sc
( ) in terms of native coordinates of

.
Similarly, the plane wave representations (31), (36), yield
E r sc
  
1

 e ik g p t d

,
 z
ˆ g

E r sc
  
1
2

 e ik g p

 g

W


 
,


(
 ˆ ˆ 
I )
(47) for the two- and three-dimensional cases, respectively.
Thus we have derived an expression g r

W r

 g r
ˆ
W


(
 ˆ ˆ 
I ) (48) relating the scattering amplitudes relevant to the scattered fields in the two systems

.
The results (47) can be derived from (31), (36) by multiplying both sides by a differential operator dyadic. Inasmuch as the integration and differentiation operators affect different variables, they are interchangeable, therefore we have
W
 r



E
   sc r

1
2


W

 r
 e ik p r  g p



1
2

 e ik p r
W
 g p
1
2

 e ik p r g p E r sc
 
W
 r



E
   sc r

W

 r

 z
ˆ
E
   sc r

1

 e ik p r
W

 r

    ˆ ik
I ) V I i
   r
 g p )
I / k

) d
  
E sc r
  for the two- and three-dimensional cases, respectively.
This remarkable result supplements (48) in the form
E sc r

W

 r


E sc
( , )
(49)
(50)

>13<
In (27), for large distances ( ) can be interpreted as a quasi plane wave
(
 
)
  ie /
    ie ik /
    ie ik / (51) with a slowly varying function 1/ k r in (51). Hence only for this special case (50) becomes
E sc r

W

 r


E sc
( , )
 r

     sc r (52)
Mathematically, there is no difference between g and g

, both are functions depending on angles only. Therefore similarly to (37), (39) g r
  
W


 z g r

W

  z
ˆ   i
    m
 im
   g r

W


 
0


W

 g r i
   
 i
    nm
 nm m
C r n
 m
C r
 c nm
  nm

B m n

B m n r
  nm
( ) b nm

 m
P r n

P n m r p nm p nm


  i
    m a e im
 
(53) where in (53) a
 m
and c
 nm
, b
 nm
, p
 nm
are new coefficients, obtained by recasting the pertinent functions in series. For additional material on how such representations are derived, see e.g., [15].
By substitution of (53) in (47), (49), and considering the formal similarity to
(31), (36), it becomes clear that the scattered fields in

, given by
E sc r

W

 r


E sc
( , ) are also solutions of
(
 r
2

 k

2
) E sc

0,
 r

  r


E sc
 k
 equations, respectively, in terms of
2
E
  sc

0 , the corresponding Helmholtz wave
native coordinates.
It follows that (21), (23), and the corresponding (25), (27), can be represented by the surface integrals
E sc
( , )
 
H
0
, E sc
ρ

( , )


, sc
( , )

(54) for the three- and two-dimensional cases, respectively.
The differential operators representations follow too. Similarly to (43)-(45) we have
E r sc
  
(
 
,
 2
  g
  t
E r sc
   e
(
 
,
 2
 
) g e
    o
(
 
,
 2
 
) g o

E r sc
 t

O ( , )
ˆ
(55) for the corresponding three cases, respectively.
9. CONCLUDING REMARKS

>14<
The mathematical tools used in free-space electromagnetic waves scattering have been summarized and sytematized. The elements of Relativistic Electrodynamics relevant to the subject are revisited. Then various two- and three-dimensional representations for waves scattered by objects at rest are summarized. The mathematical tools are chiefly given by Stratton’s comprehensive book [1] and the monumental articles by Twersky [9-12].
The two aspects are combined into a package of various representations for the scattered wave by moving objects and the associated scattering amplitude.
Results are shown for four types of representations of the solution: The surface integral representations, the plane wave representations, the special function representations, and finally the differential operator representations. Only the surface integral applies to arbitrary surfaces, all the way down to the object’s surface. The rest have to satisfy stronger conditions. Essentially, these representations are valid outside the circumscribing cylinder or sphere enclosing the object, for the two- and three-dimensional cases, respectively.
New dyadic differential operators have been introduced which facilitate relations between scattered waves and their corresponding scattering amplitudes in various inertial frames of reference.
REFERENCES
1. J.A. Stratton, Electromagnetic Theory , McGraw-Hill, 1941.
2. D. Censor, “Application-oriented relativistic electrodynamics (2)”,
PIER,Progress In Electromagnetics Research , Vol. 29 , 107–168, 2000.
3. H. Minkowski, “Die Grundgleichungen für die elektromagnetische Vorgänge in bewegten Körpern”, Göttinger Nachrichten , 53-116, 1908.
4. D. Censor, “Relativistic electrodynamics: various postulates and ratiocinations”,
PIER--Progress In Electromagnetic Research, Vol. 52, pp.301-320, 2005.
5. A. Einstein, “Zur Elektrodynamik bewegter Körper”, Ann. Phys. (Lpz.) , Vol. 17 ,
891-921, 1905; English translation: “On the Electrodynamics of moving bodies”, The Principle of Relativity , Dover .
6. W. Pauli, Theory of Relativity , Pergamon, 1958, also Dover Publications.
7. J. Van Bladel, Relativity and Engineering , Springer, 1984.
8. J. A. Kong, Electromagnetic Wave Theory , Wiley, 1986.
9. V. Twersky, "Scattering of waves by two objects", pp. 361-389, Electromagnetic
Waves , Ed. R.E. Langer, Proc. Symp. in Univ. of Wisconsin, Madison, April
10-12, 1961, The University of Wisconsin Press, 1962.
10. J.E. Burke, D. Censor, and V. Twersky, "Exact inverse-separation series for multiple scattering in two dimensions", The Journal of Acoustical Society of
America, Vol. 37, pp. 5-13, 1965.
11. V. Twersky, "Multiple scattering by arbitrary configurations in three dimensions",
Journal of Mathematical Physics, Vol. 3, pp. 83-91, 1962.
12. V. Twersky, "Multiple scattering of electromagnetic waves by arbitrary configurations", Journal of Mathematical Physics, Vol. 8, pp. 589-610, 1967.
13. F. Noether, "Spreading of electric waves along the earth", p. 167, in Theory of
Functions , R. Rothe, F. Ollendorf, and K. Pohlhausen, eds., Dover, 1933.
14. G.N. Watson, A Treatise on the Theory of Bessel Functions , Cambridge University

>15<
15. D. Censor, "Non-relativistic scattering in the presence moving objects: the Mie problem for a moving sphere", PIER-Progress In Electromagnetic Research,
Editor J.A. Kong, Vol. 46, pp. 1-32, 2004.