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1 NON-RELATIVISTIC SCATTERING BY TIME-VARYING BODIES AND MEDIA Dan Censor Ben-Gurion University of...
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Transcript of 1 NON-RELATIVISTIC SCATTERING BY TIME-VARYING BODIES AND MEDIA Dan Censor Ben-Gurion University of...
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
3
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
6
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
8
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
9
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
12
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
13
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
14
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
15
(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
16
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
17
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
18
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
19
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
20
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
21
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
22
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
23
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....
24
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