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EM WAVE SCATTERING BY OBJECTS MOVING
ON BOWDITCH-LISSAJOUS TRAJECTORIES
Dan Censor
Department of Electrical and Computer Engineering,
Ben–Gurion University of the Negev
84105 Beer–Sheva, Israel
censor@ee.bgu.ac.il
Abstract—A method for analyzing scattering of electromagnetic waves by objects
performing complex periodic and quasi-periodic motion on Bowditch-Lissajous
trajectories is presented. The method is based on the previously introduced Quasi
Lorentz Transformation, facilitating the approximate analysis of scattering in the
presence of varying velocity. In the present class of problems the method is
specialized to time-dependent motion. A special case of scattering by cylinders is
analyzed. The resulting spectrum is shown to be discrete, with sidebands determined
by the frequencies of initial carrier incident wave and the mechanical motion.
1. Introduction
Scattering of electromagnetic waves by objects performing complex motion is of
interest both theoretically and for applied scientific and engineering purposes.
Monitoring motion by means of wave scattering facilitates remote sensing of
properties of objects and constituent media.
This class of problems includes quasi-periodic motion along BowditchLissajous trajectories. Salient examples are orbiting devices abounding in machinery
and aviation. For comprehensive mathematical discussions and relevant references to
Lissajous’ original work [1], see e.g., [2], [3]. Presently motion along the principal
Cartesian coordinates is investigated. Investigations of scattering by nonuniformly
moving objects are found in the literature, see Van Bladel [4] for a comprehensive
review of early and contemporary work.
Consider a 3D Cartesian family of parametric curves defining the velocity of a
point periodically moving along the axes
v(t )  xˆ v0 x cos  x  yˆ v0 y cos  y  zˆ v0 z cos  z ,  i  i t   i , i  x, y, z
(1)
where 0  i  2 . The corresponding spatial location is derived by integrating
v  dt ρ(t ) . Ignoring constants of integration yields
ρ(t )  xˆ Ax sin  x  xˆ Ay sin  y  xˆ Az sin  z , v0i  i Ai
(2)
In our special example both velocity and location are boxed in rectangular-cuboid
regions in their respective Cartesian spaces. For rational quotients i , trajectories are
closed, otherwise quasi-periodic trajectories are encountered, eventually scanning all
points within the confining cuboid. Assuming time-harmonic incident plane waves,
we investigate the scattered wave from objects moving along such trajectories.
The electromagnetic model used below is the Quasi Lorentz Transformation
(QLT), Presently the QLT for time-dependent velocity is defined in terms of
differentials, relating spatiotemporal coordinates in two relatively moving frames of
reference (e.g., see [5, 6])
dr  dr  v(t )dt
(3)
2
dt   dt  c v(t )  dr
(4)
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valid to the First Order in v / c (FO) and therefore prescribing in (2) i Ai much
smaller than c , the speed of light in free space.
Ignoring higher order terms, the Inverse Transformations (IT) of (3), (4), are
approximated to FO as
(3*)
dr  dr  v(t )dt 
2
(4*)
dt  dt   c v(t )  dr
Henceforth the asterisk notation for IT will be understood even without explicitly
writing out the expressions. Formally, all we have to do is exchange primed and
unprimed quantities and replace v by v . Note that v (t )  v (t ) are already FO
expressions.
For constant velocity v , integration of (3), (4) with zero constants of
integration leads to
(5)
r  r  vt
2
t  t  c v  r
(6)
recognized as the FO velocity approximation of the exact SR (Special Relativity)
Lorentz Transformation (LT) [7]. The corresponding IT (5*), (6*), follow. It is noted
that both (3), (4), and (5), (6), lead to the same FO SR law for addition of velocities,
which motivated the introduction of the QLT for varying velocities in the first place.
Substituting (5), (6), in the (purely mathematical) chain rule of calculus
 r  ( r r)   r  ( r t ) t
(7)
 t  ( t t ) t   ( t r)   r
(8)
yields for the differential operators related to (5), (6)
 r   r  c 2 v t 
(9)
 t   t   v   r
(10)
respectively, where  r symbolizes the Nabla operator, and applied to a vector, creates
a dyadic. Note that (10*) coincides with the well known “material” or “moving”
derivative used in continuum mechanics. Furthermore, taking c   renders (9),
(10), as a Galilean transformation with  r   r .
Integrating the differential relations (3), (4) yields the global relations
t
r  r   v( t )dt
(11)
(12)
t   t  c 2 v(t )  r  t  c 2 | v(t ) | 
where t denotes the dummy integration variable and  is a coordinate in the
direction of the velocity at a given time t . Differentiating (11), (12), with respect to
t ,  , respectively, using the Leibnitz rule of differentiating integrals yields
dr / dt  dr / dt  v(t )
(13)
2
dt  / d  dt / d  c | v(t ) |
(14)
and multiplying (13), (14), by dt , d  , respectively retrieves (3), (4).
For constant v , substituting (9), (10) into the Maxwell Equations (ME)
 r  E   t B,  r  H   t D,  r  D  0,  r  B  0
(15)
and collecting terms yields the ME in another reference frame as
 r  E   tB,  r  H   t D,  r  D  0,  r  B  0
(16)
subject to the FO Field Transformations (FT)
E  E  v  B, B  B  v  E / c2 , D  D  v  H / c 2 , H  H  v  D (17)
For variable v (t ) we substitute (11), (12), into (7), (8), to derive (cf. (9), (10))
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 r   r  c 2 v(t ) t 
(18)
(19)
 t   t   v(t )   r
Now upon substitution of (18), (19) into the ME (15), we encounter terms like
(20)
v(t ) t  E   t ( v(t )  E)  ( t ( v(t ))  E
Because of the additional term  t  ( v(t )) we cannot extend (17) to v (t ) . However, it
is noted that that field time derivatives as in  t   E involve the wave frequencies, say
 , while  t ( v(t ) involves the mechanical frequencies i , as in (1), (2). Provided
i is much smaller than  , the term  t  ( v(t )) (20) is negligible and the field
transformations (17) are valid for v (t ) as well.
2. Plane Waves
Plane waves are characterized by constant vector amplitudes obeying the pertinent
FT, in our case (17). Space and time variations are delegated to the appropriate phase
exponentials ei  ei  , hence the invariance of the phase follows by definition
 (r, t )   (r, t )
(21)
although in general the phase is not form-invariant.
The initial incident plane wave is given in the “laboratory” unprimed
reference system
E  zˆ E0ei , H  yˆ H 0ei , E0 / H 0  ( 0 /  0 )1/2
(22)
  kx  t  kr cos   t ,  / k  c  ( 0 0 )1/2
propagating in free space (vacuum) in direction x̂ , with the E -field polarized along
the cylindrical axis ẑ . The constitutive relations in free space are
D   0E, B  0 H, D   0E, B  0 H
(23)
By substitution of (23) in (17), the FT reduce to two equations
E  E  0 v(t )  H, H  H   0 v(t )  E
(24)
where v is chosen in the xy -plane according to (1) with v0 z  0 . Accordingly
E  zˆ (1   0 x cos  x ) E0 ei ,  0 x  v0 x / c
(25)
H  yˆ H0ei (1  0 x cos  x )  xˆ H0ei 0 y cos  y , 0 y  v0 y / c
For arbitrary direction of propagation  (22) is replaced by
E  zˆ E0ei , H  yˆ cos  H 0ei  xˆ sin  H 0ei
  k  r  t  k x x  k y y  t , k x  k cos  , k y  k sin 
According to (24), replace (25), (26), by
E  zˆ E0ei (1  0 x cos  cos  x  0 y sin  cos  y )
i
(26)
(27)
(28)
i
(29)
H  yˆ H0e (cos   0 x cos  x )  xˆ H0e (sin   0 y cos  y )
The phase for arbitrary direction of propagation is found from (21), (27),
(11*), (12*)
   k  r   x sin  x   y sin  y  t   (xˆ Cx cos  x  yˆ C y cos  y )  r
(30)
 i  ki v0i / i  Ai ki , Ci  c 2v0i  k 0i , i  x, y
where  i is of the order Ai /  , the mechanical amplitude normalized to the
wavelength, which is finite but not necessarily small, and Ci is FO in the velocity. it
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follows that for distances satisfying r  much smaller than Ai / 0i , (30) can be
approximated by
   k  r   x sin  x   y sin  y  t , r  Ai /  0i
(31)
3. Scattering by Circular Cylinders
For the incident wave (22) we have in (30)  y  0 . For simplicity we also assume
presently  i  0 . Furthermore, we assume the approximation (31) to hold, hence
(25), (31) reduce to
E  zˆ PE0 ei  , P  1   0 x cos  x t ,    k  r   x sin  x t   t 
(32)
Recasting (32) in terms of a Bessel-Fourier series (e.g., see [8]) yields
E  zˆ PE0ei   zˆ PE0eikrit n J neinxt   zˆ PE0n J neikrin t
(33)
n    n x , n  nn
,
J

J
(

)

n
n
x
Furthermore, by recasting P  1   0 x (ei xt   e  i xt  ) / 2 and rearranging the series, we
obtain
E  zˆ E0  n eik rin t  J n , J n  ( J n   0 x ( J n 1  J n 1 ) / 2)  (1   0 x n /  x ) J n (34)
In (33), (34), we obtain a discrete spectrum of frequencies n , with sidebands
separated by the mechanical frequency  x . The reason for the absence of  y
is due to the specific choice of the direction of the incident wave (22). For scattering
by a circular cylinder of radius a we now encounter the classical canonical problem
for each discrete spectral component n . Without delving into specific scattering
problems (e.g., see [9]), it is assumed that the scattered wave must satisfy the
boundary conditions, and the wave equation, for each individual excitation frequency,
and is therefore represented in the exterior domain r  a as
Es  zˆ E0  nm am ,n K m ,n e  in t  ,  nm   n  m , K m ,n  J ni m H m ,n eim 
(35)
H m ,n  H m (kn r ), kn  n / c
where H m  H m(1) are the first kind Hankel functions, and am ,n denote the scattering
coefficients for each circular mode and frequency m, n , respectively.
Using the Sommerfeld integral representation (e.g., see [8]) for the cylindrical
functions, (35) is recast as a superposition of plane waves
Es  zˆ E0  n J n 1 
  ( /2) i i 
n
 ( /2)  i
e g n ( )d 
zˆ E0  n (2 / i kn r )1/2 J n eikn rin t g n ( )
(36)
 n  k n  r   n t   kn r  cos(    )  n t , g n ( )   m am ,n eim 
The far field in (36) asymptotically becomes an outgoing cylindrical wave governed
for each frequency by the scattering amplitude gn ( ) .
In the integrand (36) we have plane waves propagating in complex directions
  , with appropriate amplitudes g n ( ) . Substituting (24*) in (36) leads to (28*),
(30*) for each plane wave in (36), becoming the scattered wave measured in the
initially (unprimed) reference frame, but still expressed in terms of the primed system
of reference coordinates
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Es  zˆ E0  n J n 1 
  ( /2) i i 
n
  ( /2)  i
e g n ( )d , g n ( )   m am,n eim 
g n ( )  g n ( )(1   0 x cos  cos  xt    0 y sin   cos  y t )
(37)
It is noted that  y featuring in (37) results from (24*), although it does not feature in
(32), due to the special direction   0 presently chosen for the incident wave.
Expressing cos  , sin   , in terms of exponentials and rearranging the series,
we find
Es  zˆ E0 n J n 1 
  ( /2) i i 
n
  ( /2)  i
e g n ( )d , g n ( )   m Am,n eim 
Am,n  am,n  cos  x t 0 x (am 1,n  am 1,n ) / 2  i cos  y t 0 y (am 1,n  am 1,n ) / 2
(38)
where the new coefficients Am ,n are still time dependent, but this has no effect on the
Sommerfeld integral representation. Therefore, by inspection of (35), (36), we recast
(38) as
(39)
Es  zˆ E0nm Am,n Km,n ein t
The time dependent Am ,n contribute to the spectrum. Rearranging indices in
(39) it is recast as
i t 
 i t 
E s  zˆ E0  nm e in t  ( Lm ,n  iK m ,n (e y  e y )  0 y (am 1,n  am 1,n ) / 4)
(40)
Lm ,n  K m ,n am ,n   0 x ( K m ,n 1am 1,n 1  K m ,n 1am 1,n 1  K m ,n 1am 1,n 1  K m ,n 1am 1,n 1 ) / 4
A choice of an obliquely incident wave (27) instead of (22) would have resulted in
more symmetrical expressions with respect to  x ,  y , but considerably complicating
the mathematical manipulations.
In order to express E s exclusively in terms of unprimed coordinates one has
to substitute (11), (12) into (39), (40) using the velocity (1) relevant to the present
case
r  r  (xˆ Ax sin  xt  xˆ Ay sin  y t )
(41)
t   t  c2 (xˆ v0 x cos xt  yˆ v0 y cos  yt )  r
(42)
At distances large compared to the motional amplitudes Ax , Ay we approximate
r   r , and     , and for all FO terms in  we approximate t   t , yielding in (40)
Es  zˆ E0  nm [e
 in t ikn x0 x cos  xt ikn y 0 y cos  y t
(43)
i t
 i t
( Lm,n  iK m,n (e y  e y ) 0 y (am1,n  am1,n ) / 4)]
Once again exponentials in (43) are expressed in terms of Bessel-Fourier series [8]
 i
t
i t
 i t
Es  zˆ E0  nmpq S p ,q ,n e n , p ,q ( Lm,n  iK m ,n (e y  e y ) 0 y (am1,n  am1,n ) / 4)
(44)
S p ,q ,n  i p  q J p (kn x 0 x ) J q (kn y 0 y ), n , p ,q  n  p x  q y
Next cos  y t in Am ,n is expressed in terms of exponentials and series indices
adjusted, finally becoming
 i
t
Es  zˆ E0  nmpq Qn ,m, p ,q e n , p ,q
Qn ,m, p ,q  S p ,q ,n Lm,n  iK m,n ( S p ,q 1,n  S p ,q 1,n ) 0 y (am1,n  am1,n ) / 4
(45)
clearly showing the spectral content of the scattered wave.
Subject to the approximations stated above, (45) concludes the derivation of
the scattered wave for the present case.
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4. Discussion and Concluding Remarks
A general discussion of scattering of EM waves by objects moving periodically and
quasi periodically on Bowditch-Lissajous trajectories is presented, and the
implementation to a relatively simple problem is analyzed.
SR is restricted to LT involving constant velocities. One thus encounters the
problem of employing an adequate theory. Rather than assuming Galilean physics,
here an attempt is made to approximate the LT for varying time dependent velocity
with the QLT which in the limiting case of constant velocity becomes a LT to the FO
in the velocity.
The feasibility of implementing the model is demonstrated. It is shown that
the scattered wave spectrum is discrete, involving the initial carrier frequency, and
sidebands at frequencies which are harmonics of the mechanical motion frequencies.
The information provided by the scattered wave facilitates the remote sensing
of the motion of vibrating and orbiting objects, as often encountered in engineering
and applied science.
5. References
1. J. A. Lissajous, "Mémoire sur l'étude optique des mouvements vibratoires,"
Annales de Chimie et de Physique 3e série 51, pp. 140-231, 1849.
2. A. Deprit, “The Lissajous Transformation, I. Basics”, Celestial Mechanics and
Dynamical Astronomy, Vol. 51, pp. 201-225, 1991.
3. A. Deprit and A. Elipe, “The Lissajous Transformation, II. Normalization”,
Celestial Mechanics and Dynamical Astronomy, Vol. 51, pp. 227-250, 1991.
4. J. Van Bladel, Relativity and Engineering, Springer, 1984.
5. D. Censor, “Non-relativistic scattering: pulsating interfaces” Progress In
Electromagnetics Research, PIER 54, 263–281, 2005.
6. D. Censor, “The Need for a First-Order Quasi Lorentz Transformation”,
Proceedings of the 2nd International Conference AMiTaNS’10, Sozopol,
Bulgaria, June 21-26, 2010, AIP Conference Proceedings, Vol. 1301,
pp. 3-13, 2010.
7. 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.
8. J.A. Stratton, Electromagnetic Theory, McGraw-Hill, 1941.
9. D. Censor, “The mathematical elements of relativistic free-space scattering”,
JEMWA—Journal of Electromagnetic Waves and Applications,
Vol. 19, pp. 907-923, 2005.