EM WAVE SCATTERING BY OBJECTS MOVING ON BOWDITCH-LISSAJOUS TRAJECTORIES

Dan Censor Department of Electrical and Computer Engineering,

Ben–Gurion University of the Negevy g84105 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.

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Introduction Consider a 3D Cartesian family of parametric curvesConsider a 3D Cartesian family of parametric curves defining the velocity of a point periodically moving along the axesthe axes

0 0 0ˆ ˆ ˆ( ) cos cos cosx x y y z zt v v v

t i x y z

v x y z

, , ,i i it i x y z where 0 2i . The corresponding spatial location is d i d b i i ( )d I i fderived by integrating ( )td tv ρ . Ignoring constants of integration yields

ˆ ˆ ˆ( ) i i iA A A A0ˆ ˆ ˆ( ) sin sin sin ,x x y y z z i i it A A A v A ρ x x x

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a = 1, b = 2 (1:2) a = 3, b = 2 (3:2)

a = 3 b = 4 (3:4) a = 5 b = 4 (5:4)

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a 3, b 4 (3:4) a 5, b 4 (5:4)

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Quasi Lorentz Transformation for time-dependent velocity is d fi d i t f diff ti l l ti ti t ldefined in terms of differentials, relating spatiotemporal coordinates in two relatively moving frames of reference

( )d d t dt r r v 2 ( )dt dt c t d v r( ) ,d d t dtr r v ( )dt dt c t d v rvalid to the First Order /v c Ignoring higher order terms, the Inverse Transformations are approximated to FO aspp

( ) ,d d t dt r r v 2 ( )dt dt c t d v r For constant velocity v , integration with zero constants of integration leads to

,t r r v 2t t c v r ,t r r v 2t t c v r

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( ) ,d d t dt r r v 2 ( )dt dt c t d v r Integrating yields the global relationsIntegrating yields the global relations

( ) ,t

t dt r r v 2 2( ) | ( ) |t t c t t c t v r v

where is a coordinate in the direction of the velocity at a given time t . Differentiating with respect to ,t , respectively, using the Leibnitz rule / / ( ),d dt d dt t r r v 2/ / | ( ) |dt d dt d c t v multiply by ,dt d , respectively to retrieve QLT.

Similarly ( )t

t dt

r r v 2 ( )t t c t v rSimilarly ( ) ,t dt r r v ( )t t c t v r

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Substituting ,t r r v 2t t c v rinto chain rule of calculus

( ) ( ) ,tt r r r rr ( ) ( )t t t tt rr( ) ( ) ,tr r r r ( ) ( )t t t t r

yields for the differential operators 2c v v,tc r r v t t rv

Similarly ( ) ,t

t dt

r r v 2 ( )t t c t v ry ( ) , ( )

yields for the differential operators 2 ( ) ( ) 2 ( ) ,tc t r r v ( )t t t rv

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For const.v , substituting 2 ,tc r r v t t rv

i t th ME 0 0 E B H D D Binto the ME , , 0, 0t t r r r rE B H D D Band collecting terms yields ME in another reference frame as

, , 0, 0 E B H D D B, , 0, 0t t r r r rE B H D D Bsubject to the FO Field Transformations (FT)

2 2, / , / ,c c E E v B B B v E D D v H H H v D *For variable ( )tv 2 ( ) ,tc t

r r v ( )t t t rv we encounter terms like ( ) ( ( ) ) ( ( ( ))t t tt t t v E v E v E field time derivatives t E involve wave frequencies, , while

( ( )t t v involves the mechanical frequencies i Provided i the terms ( ( ))t t v are negligible and * are valid for ( )tv as well.

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Plane Waves Constant vector amplitudes. Space-time variations given by h ti l i i ( ) ( )t t phase exponentials i ie e , ( , ) ( , )t t r r .

in general not form-invariant. Initial incident plane wave in unprimed reference systemInitial incident plane wave in unprimed reference system

1/20 0 0 0 0 0

1/2

ˆ ˆ, , / ( / )

cos / ( )

i iE e H e E H

kx t kr t k c

E z H y

0 0cos , / ( )kx t kr t k c propagating in direction x , with the E -field z polarized. In vacuum 0 0 0 0, , , D E B H D E B H0 0 0 0, , , FT reduce to 0 0( ) , ( )t t E E v H H H v E

ˆ ˆ(1 cos ) /iE e v c E z v z0 0 0 0(1 cos ) , / ,x x x xE e v c E z v z

0 0 0 0 0 0ˆ ˆ(1 cos ) cos , /i ix x y y y yH e H e v c H y x

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A bit di ti f tiArbitrary direction of propagation 0 0 0ˆ ˆ ˆ, cos sin

cos sin

i i iE e H e H et k x k y t k k k k

E z H y xk r

, cos , sinx y x yt k x k y t k k k k k r

Accordingly 0 0 0ˆ (1 cos cos sin cos )iE e E z 0 0 0(1 cos cos sin cos )x x y yE e E z

0 0 0 0ˆ ˆ(cos cos ) (sin cos )i ix x y yH e H e H y x

The phase for arbitrary direction of propagationThe phase for arbitrary direction of propagation

20 0 0

ˆ ˆsin sin ( cos cos )

/x x y y x x y yt C C

k v A k C c v k i x y

k r x y r

0 0 0/ , , ,i i i i i i i i ik v A k C c v k i x y

i is of the order /iA , finite but not small, iC is velocity FO. For distances satisfying 0/i ir A approximatelyy g 0i i pp y

0sin sin , /x x y y i it r A k r

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Scattering by Circular Cylinders Incident wave 0y . Assume 0i . Assume 0/i ir A y i 0i i

0 0ˆ , 1 cos , sinix x x xPE e P t t t E z k r

Recasting in Bessel-Fourier seriesRecasting in Bessel Fourier series 0 0 0ˆ ˆ ˆ

( )

x nin t i i ti i i tn n n n

n

PE e PE e J e PE J e

n J J

k rk rE z z z

, , ( )n x n n n n xn J J Recast 01 ( ) / 2x xi t i t

xP e e and rearrange the series ˆ i i tE J k rE 0

0 1 1 0( ( ) / 2) (1 / )

ni i tn n

n n x n n x x n

E e JJ J J J n J

k rE z

For circular cylinders of radius a we have the classical scattering problem for each discrete spectral component n .

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In the exterior domain r a For each frequency n

0ˆ ,ni t m ims nm m n m n m n n m nE a K e K J i H e E z 0 , , , ,

,

,( ), /

s nm m n m n m n n m n

m n m n n nH H k r k c

d t th tt i ffi i t,m na denote the scattering coefficients.Recast as a superposition of plane waves S f ld i l iSommerfeld integral representation

( /2)1

0 ( /2)ˆ ( )n

i is n n ni

E J e g d

E z 0 ( /2)

,cos( ) , ( )

s n n ni

imn n n n m m nk r t g a e

The far field becomes an outgoing wave governed by the scattering amplitude ( )ng .

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For plane waves in the integrand, propagating in complex , have scattered waves measured in the unprimed frame, , p ,but still expressed in primed coordinates

( /2)1

0ˆ ( ) , ( )ni i imE J e g d g a e

E z 0 ,( /2)

0 0

( ) , ( )

( ) ( )(1 cos cos sin cos )

s n n n n m m ni

n n x x y y

E J e g d g a e

g g t t

E z

Recast cos , sin in exponentials and rearrange series( /2)

10ˆ ( ) , ( )n

i iE J e g d g E z 0 ( /2)

, , , 0 1, 1,

( ) , ( )

, cos ( ) / 2

s n n n ni

imm m n m n m n x x m n m n

E J e g d g

A e A a t a a

E z

0 1, 1,cos ( ) / 2y y m n m ni t a a

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The ,m nA are still time dependent, but this has no effect on Sommerfeld’s integral representation.on Sommerfeld s integral representation. Therefore, by inspection we recast sE as

ˆ ni tE A K e E z 0 , ,n

s nm m n m nE A K e E zTime dependent ,m nA contribute to the spectrum. Rearranging indices sE is recast as

0ˆ (ni ts nm m nE e L E z 0 ,

, 0 1, 1,

(

( ) ( ) / 4)y y

s nm m n

i t i tm n y m n m niK e e a a

, , , 0 , 1 1, 1 , 1 1, 1

1 1 1 1 1 1

() / 4

m n m n m n x m n m n m n m n

m n m n m n m n

L K a K a K aK a K a

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, 1 1, 1 , 1 1, 1)m n m n m n m n

To express sE in terms of unprimed coordinates substitute ˆ ˆ( sin sin )x x y yA t A t r r x x 2

0 0ˆ ˆ( cos cos )t t c v t v t x y r0 0( cos cos )x x y yt t c v t v t x y rAt distances large compared to ,x yA A , approximate r r ,and and for FO terms in approximate t t and , and for FO terms in approximate t t

0 0cos cos0ˆ [ n n x x n y yi t ik x t ik y t

s nmi t i t

E e

E z

, , 0 1, 1,( ( ) ( ) / 4)]y yi t i tm n m n y m n m nL iK e e a a

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Once again express exponentials in Bessel-Fourier series , ,ˆ ( ( ) ( ) / 4)n p q y yi t i t i tE S e L iK e e a a E z , ,

0 , , , , 0 1, 1,

, , 0 0 , ,

( ( ) ( ) / 4)

( ) ( ),

n p q y ys nmpq p q n m n m n y m n m n

p qp q n p n x q n y n p q n x y

E S e L iK e e a a

S i J k x J k y p q

E z

Next cos t in A is expressed in terms of exponentialsNext cos yt in ,m nA is expressed in terms of exponentials and series indices adjusted, finally becoming

, ,ˆ n p qi tE Q e E z 0 , , ,

, , , , , , , , 1, , 1, 0 1, 1,( ) ( ) / 4s nmpq n m p q

n m p q p q n m n m n p q n p q n y m n m n

E Q e

Q S L iK S S a a

E z

clearly showing the spectral content of the scattered waveclearly showing the spectral content of the scattered wave.QED

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Discussion and Concluding Remarks A general discussion of scattering of EM waves by objects moving g g y j gperiodically and quasi periodically on Bowditch-Lissajous trajectories is presented, and the implementation to a relatively simple problem is analyzed. y 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 g p y , p ppfor 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 y p gshown that the scattered wave spectrum is discrete, involving the initial carrier frequency, and sidebands at frequencies which are harmonics of the mechanical motion frequencies.q 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.

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g g pp

Questions time

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Lissajous lasso curves

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Lissajous frosting on toaster strudel

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THIS IS ALL, FOLKS, THANK YOU

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