1

NON-RELATIVISTIC SCATTERING BY TIME-VARYING BODIES AND MEDIA

Dan CensorBen-Gurion University of the Negev

Department of Electrical and Computer EngineeringBeer Sheva, Israel 84105

Download:http://www.ee.bgu.ac.il/~censor

Files: varying.pdf, varying.zip, varying-paper.pdf

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1 .Introduction2 .First Order Lorentz Transformation

3 .Uniformly Moving Plane Interface4 .Uniformly Moving Half Space

5 .Oscillating Plane Interface6 .Oscillating Half Space Medium

7 .Boundary-Value Problem: Oscillating Cylinder8 .Derivation of the Scattered Field

9 .Boundary-Value Problem: Oscillating Cylindrical Medium

10 .Concluding RemarksReferences

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1. Introduction First order Lorentz trx.

2 2

( , ), ( , ),

/ /

ˆ / , / ,

ict ict

t t

t t c t t c

v v c v

R r R R R R r R R R

r r v r r v

v r v r

v v v

0

0

2 2

Variable velocity quasi Lorentz trx.

( ) , ( )

( ) , ( ) /

0

dt d d dt

t t c d dt dt d c

R

R

R

R

r

r r v R r r v R

v R r v R r

v

4

Phase and uniqueness

0 0 0

( )

( )

2

( ) ( ) ( )

, , ( , / )itc

d d d

i c

R R R

RR R R

R r

R R R K R R

K k

, , 1,2,3,4ji

j i

KK

R R i j

, ( ) 0 R K R

( ) 0

( ) ( ) 0t

r

r

k R

k R R

Fresnel drag effect and Doppler effect

2 2

[ ] [ ]

( ) / ( ) /

( ) ( [ ])

c c

K K K K K K

k k v R k k v R

v R k v R R k

5

Phase invariance

0 0

( ) ( ) ( ) ( ) , [ ]

( ) ( )

d d

d d

d dt d dt

R R

R RR K R R R K R R R R R

K R R K R R

k r k r

Lorentz force Field trx.

2

2

( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) /

( ) ( ) ( ) ( ) /

( ) ( ) ( ) ( )

c

c

E R E R v R B R

B R B R v R E R

D R D R v R H R

H R H R v R D R

Boundary conditions ( ) (1) (2) ( ) (1) (2)

(1) (1) (1) (1) (1) (1)

ˆ ˆ( ) 0, ( ) 0

,

b beff eff

eff eff

n E E n H H

E E v B H H v D

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3. Uniformly Moving Plane Interface Excitation plane wave

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

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

ˆ ˆ, , / ( / )

, / ( ) 1/

ex exi iex ex ex ex ex ex

ex ex ex ex ex ph

E e H e E H

k z t k v

E x H y

Equation of motion Tz z vt

Time signal at local coordinate origin (1) (1) (1)

0 0| , (1 ), /Tex ex z exT exT ex pht v v

2 (1) (1)

(1) (1) (1) (1) (1)

(1) (1) (1) 2

, / (1 )

(1 ) /(1 ) (1 ( 1))

/ , ( / )

exT exT exT exT ex ex ex

exT exT

exT exT ph ph

k Z t k k v c k A

q A q A

q v A v c

7

Effective phase velocity (1) (1) (1) (1)

,

(1) (1) (1)

/ (1 ) /(1 )

(1 ( 1))

eff ex exT exT ph

ph

v k v A

v A

Excitation field at boundary

(1)

ˆ ˆ,

/ / 1

exT exTi iexT exT exT exT

exT ex exT ex

E e H e

E E H H

E x H y

Scattered (Reflected) wave (1)

(1)

ˆ ˆ, , /

, / 1/

sc sci isc sc sc sc sc sc

sc sc sc sc sc ph

E e H e E H

k z t k v

E x H y

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Time signal at local coordinate origin (1)

0 0

(1) (1) (1)

| , (1 )

/ / (1 ) /(1 ) 1 2

Tsc sc z scT scT sc exT

sc ex sc ex

t

k k

Scattered field at boundary (1)

(1)

2

(1) (1) (1) (1),

(1) (1) (1),

ˆ ˆ, /

/ / 1

, /

(1 ) / (1 ( 1))

/ (1 ( 1))

scT scTi iscT scT scT scT

scT sc scT sc

scT scT exT scT sc sc

sc exT eff sc exT

eff sc exT scT ph

E e E e

E E H H

k Z t k k v c

k A v q A

v k v A

E x H y

Field in internal domain (2) (2) 1/ 2 (2)

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

ˆ ˆ, , / ( / )

, / ( ) 1/

in ini iin in in in in in

in T exT exT ph

E e H e E H

z t v

E x H y

At the boundary Tz Z

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Solving the boundary-condition problem Perfectly conducting interface

(1) (1) (1) (1)

(1) (1)

(1 ) (1 )(1) (1)

( ) ( ) (1)

( ) (1)

0

(1 ) (1 ) 0

/ [ (1 2 )]

(1 2 ), ,

ex sc

ex sc sc ex

ex sc

exT scT

iK A iK Aex sc

i K K i K K Asc ex

i K Ksc sc ex ex

E e E e

E E e e

e K k Z K k Z

E E

Arbitrary media

(1) (1) (1) (1)

(1) (1) (1) (1)

(1 ) (1 )(1) (1)

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

, = ,

(1 ) (1 )

(1 ) (1 ) /

ex sc

ex sc

exT scT inT exT scT inT

iK A iK A iex sc in

iK A iK A iex sc in

Z

E e E e E e

E e E e E e

E E E H H H

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4. Uniformly Moving Half Space-The Fizeau experiment Internal field in medium at rest, , Tz vt z z

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

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

ˆ ˆ, , / ( / )

, / ( ) 1/

in ini iin in in in in in

in in in ph

E e H e E H

t v

E x H y

Phase at boundary's origin 0z (2) (2) (2)

0 0| (1 ), /in in z ex in pht t v v

Phase at boundary z Z 2 (2) (2)

(2) (2) (2),

(2) (2) (2) (2) (2) 2,

, / (1 )

(1 ) / /

(1 ( 1)), ( / )

inT T ex T in

in ph ex eff in

eff in ph ph

Z t v c A

A v v

v v A A v c

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Boundary conditions: ( ) (2) (1) ( ) (2) (1)

(2) (2) (2) (2) (2) (2)

ˆ ˆ( ) 0, ( ) 0

,

b beff eff

eff eff

n E E n H H

E E v B H H v D

Internal field at boundary

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

ˆ ˆ,

/ / 1 , / ( / )

inT inTi iinT inT inT inT

inT in inT in inT inT

E e H e

E E H H E H

E x H y

Solving the boundary value problem

( 2) ( 2)

( 2) ( 2)

(2) (1 )

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

, =

(1 )

(1 ) / ,

ex ex

ex ex

exT scT inT exT scT inT

iK iK i Aex sc in

iK iK i Aex sc in

E e E e E e

E e E e E e Z

E E E H H H

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5. OSCILLATING PLANE INTERFACE

0 , sinT t tz z z S S t

0 0 0( ) / , , cost tv t dz dt v C v z C t

Incident phase at 0Tz 0

0 0 0

0

| ,

, ( ),

ex n

T

i i tex ex z ex t ex n n

nn ex n n ex n n

k z S t e I e

n I J k z

Incident phase at boundary

(1) (1), 0

(1) (1) (1)0 0

/ (1 ( 1) )

/ , /

n nT n

nT n eff ex n t

n n ph ph

k Z t

k v k A C

k v v v

Incident field at boundary (1)

(1)0 0

ˆ ˆ ˆ, /

(1 ) ( ) n

exT exT exT exT exT

iexT t ex n n ex

E H E

E C E I k z e

E x H y y

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Reshuffle indices (1) (1)0

1 1

1

(1 ( 1) ) (1)0

(1)0

(1)0 ;

(1); 0 1 1 1 1

(1 )

(1 ( ))

( ( ))

( (

n t n

n n

n n n n n

n n

iK A C i texT ex n n t

iK i t i t i tex n n n

iK i t i t i t i tex n n n n ex n

iK iKex n ex n n n n n

E E I e C

E I e B e e

E I e e B e e E e

E E I e B I e B I

1

(1)

))

( (1 ) 1) / 2,

niK

n n n n

e

B iK A K k Z

Internal field at boundary (2)

(2) (2) 1/ 2 (2);

ˆ ˆ ˆ, /

, / ( ) 1/n T n

in in in in in

z i tin n in n n n ph

E H E

E E e v

E x H y y

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Scattered wave ; ;(1)

;

(1) (1) 1/ 2 (1); ; ;

ˆ ˆ ˆ, / ,

, / ( ) 1/

sc scik z i tsc sc sc sc sc sc sc

sc ex sc sc ph

E H E E E e

k v

E x H y y

...expressed at 0Tz ; 0 ;

;

( )0 ; ; ; 0

; ; 0

(1); ; 0 0 0

| ( )

( )

( ) ,

sc t sc ex

T

sc

ik z S i t i t i tsc z sc sc sc

i tsc sc

sc ex sc ex

E E e E e J k z e

E e J k z

k z k z

Constraint n , i.e., a Kronecker delta function ;n

0 ; ; ; ; 0 ;| , ( ), n

T

i tsc z n sc n sc n sc n sc n scE e E E E J k z

Include amplitude effect (1)0/ 1scT sc tE E C

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(1) (1)0

1 1

(1)

(1 ( 1) ) (1); 0

(1); 0

(1); 0 ;

ˆ ˆ ˆ, /

(1 )

(1 ( ))

( ( ))

n t n

n n

n n n n n

scT scT scT scT scT

iK A C i tscT n sc n t

iK i t i t i tn sc n n

iK i t i t i t i tn sc n n n scT

E H E

E E e C

E e B e e

E e e B e e e E

E x H y y

1 1(1); ; 0 1 ; 1 1 ; 1

(1)

( )

( (1 ) 1) / 2

n n n

n

iK iK iKscT n sc n n sc n n sc n

n n

E E e B E e B E e

B iK A

6. Oscillating Half Space Medium

0 tz S , 0 0 0( ) / , tv t d dt v C v

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Excitation wave ˆ ˆ, , exT exTi i

exT exT exT exT exT ex exE e H e k Z t E x H y

Scattered wave ;(1)

;

(1) (1) 1/ 2 (1);

ˆ ˆ ˆ, / ,

/ ( ) 1/

sc n nk Z i tsc sc sc sc sc sc n sc n

sc n n ph

E H E E E e

k v

E x H y y

Internal wave (2)

;

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

ˆ ˆ ˆ, / ,

, / ( ) 1/

i i tin in in in in in in

ex ph

E H E E E e

v

E x H y y

... at 0z 0

0 ; ; ;

(2) (2) (2) (2)0 0 0 0 0 0

|

( ), , / , /

,

t ni S i t i t i ti tin z in in in

ex ph ph

n ex

E E e E e J e E e J

J J v v v

n n

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Finally

( 2) ( 2)0

1

(2)

(2)0

(1 ( 1) ) (2); 0

(2); 0

(2); 0

ˆ ˆ ˆ, /

/ / (1 )

(1 )

(1 ( ))

( (

n t n

n

n n

inT inT inT inT inT

inT in inT in t

i A C i tinT in t

i t i t i tin

i t i t iin

E H E

E E H H C

E E e C

E e B e e

E e B e e

E x H y y

1

;

(2); ;

(2); ; 0 ; ; 1 ; 1 ; ; 1 ; 1

))

, , ( (1 ) 1) / 2

( )

n n

n

t i tin

iin in n n n

in in in in

e E

E E e J Z B i A

E E E B E B

;

; ; ; ;, in ni tinT n inT n inT n in nE e E E E

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7. Boundary-Value Problem: Oscillating Cylinder Incident field at Tr

(1)

; ; ;

(1); 0 1 1; 1 1;

(1)0 1 1 1; 1 1 1 1; 1

(1)0 1 1 1; 1 1 1

ˆ ˆ ˆ, /

,

[ ( ) / 2

( )

(

n T

exT exT exT exT exT

i t immexT n ex n ex n m ex nm

ex nm ex n nm n n m n n m

n n n m n n n m

n n n m n n

E H E

E E e E i E e

E E I L I L I L

i B I L B I L

i B I L B I L

E x H y y

1; 1

(1)0

)]

(1 ) / 4, ( ), L ( )

n m

n n n n ex nm m nB ik A I J k z J k

Internal field at Tr

; ; ; ;

(2) (2) 1/ 2 (2);

ˆ , , ( )

/ ( ) 1/

n Ti t imminT inT inT n in n in n m in nm m in n

in n n ph

E E E e E i E J k e

k v

E x

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Wave in arbitrary direction (1)

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

ˆ ˆˆ ˆ ˆ, / ,

C

, / ( ) 1/

i

z y

ex ph

E H E E E e

t k r t k z k y t

k v

E x H k x k x

k r

...evaluated at 0T r 0

0

0 0 0 0 0

0

| , | C

( C ) , ,

T T

n

it

i i tn ex

E E E e k z S t

e J k z e n n

r r

...constraint n is a ;n , prescribing

0 0, ( C )ni tn n n nE E e E E J k z

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Phase at Tr

(1) (1)0

(1) (1)0

(1)

ˆ

, (1 ( 1) ),

( 1)

ˆ C , , /

T T

T

T

nT nT T n nTy nTz n

nTy ny nTz nz t nz n

nT nR n t

nR n T n n n n n n n ph

t k S k C t

k k k k A C k k C

K A C C C

t K t K k k v

k r

k r

Include factor (1)01 CtC prescribed by the boundary conditions

(1)0

(1) (1) (1)0 0

(1) (1)0

C (1)0

ˆ , (1 C )

(1 C )(1 ( 1) )

(1 ( )), =C ( ( 1) 1) / 2

(

nT n

nR

T

nR

T

n T

i i tT T T n n t n nT

in n t n t

i i t i tn n n n n

iK

nT n

E E E e C E e

E e C iK A C C C

E e B e e B iK A C

E e E e

E x

1 1C C

; 1 ; 1 ; 1 ; 1)n nT TiK iK

n n n nB E e B E

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Construct outgoing cylindrical functions 1

1

C C(1)10 ; 1 ; 1

( / 2)C(1)0 ; 1 ; 1 ( / 2)

ˆ [

] ,

n nn T T

Tn T

T

iK iKi tscT n n n n

iiK

n n i

e e E e B E

e B E d

E x

Use Fourier series imn m nmE a e

1

1

C C(1)10 ; 1 1;

C(1) (1)0 ; 1 1;

C C(1)10 1; 1

ˆ [

] , ( ( 1) 1) / 4

ˆ [ ]

n nn T T

n T

T

nn T T

iK iKi tscT nm nm n n m

iK imn n m n n

iK iKi t imnm nm n m

e e a e B a

e B a e d U iK A C

e e a a U e e d

E x

x

Express in terms of Hankel functions

(1)0 1; 1 ;

ˆ [

], ( )

n Ti t immscT nm nm nm

n m m nm m n

e i e a M

a U M M H K

E x

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Reshuffle indices related to Time

(1) (1)0 1; 1 1 ; 1 0 1; 1 1 ; 1

(1)0 1; 1 ;

ˆ ,

/ 4

n Ti t immscT nm nm nm nm nm

n m m n m m

n m m

e i e F F a M

a V M a V M

a M

E x

Solve boundary-conditions problem 0 |exT scT inT E E E , ˆ ( ) | 0T exT scT inT r H H H

8. Derivation of the Scattered Field 2

0 0, , /T t T T tz z z S y y t t v zC c

Do not include (1)0(1 C )tC also replace nK by n n TK k r

(1) (1)0 1; 1 1 ; 1 0 1; 1 1 ; 1

ˆ , ( ), n T Ti t immsc nm nm nm m n nm nm nm

n m m n m m

e i e F M H K F a M

a V M a V M

E x

Only good for positions near the object, for short times

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Re-define coefficients im

n m nmE a e

Use the Twersky differential operator representation in inverse powers of the distance

2 2 4

2

2 2 2 2

0

1/ 2

ˆ ( )

1 4 9 40 16( ) 1 ( )

8 128

(1 4 )(9 4 ) ([2 1] 4 )( )

( 8 ) !

( ) 2 /( )

n T

T

T

n

i tsc n n

n nn n

nn

iKn n

e D E

D E H Ei K K

H Ei K

H H K i K e

E x

Ahhhh... what a relief....

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1 .Introduction2 .First Order Lorentz Transformation

3 .Uniformly Moving Plane Interface4 .Uniformly Moving Half Space

5 .Oscillating Plane Interface6 .Oscillating Half Space Medium

7 .Boundary-Value Problem: Oscillating Cylinder8 .Derivation of the Scattered Field

9 .Boundary-Value Problem: Oscillating Cylindrical Medium

10 .Concluding RemarksReferences

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THIS IS ALL, FOLKS, THANK YOU