EM WAVE PROPAGATION AND SCATTERING IN

SPATIOTEMPORALLY VARYING MOVING MEDIA

The Exponential Model

Dan Censor

Department of Electrical and Computer Engineering, Ben–Gurion University of the Negev, 84105 Beer–Sheva, Israel

censor@ee.bgu.ac.il

Keywords: Electromagnetic Wave Propagation; Electromagnetic Wave Scattering; Special Relativity; Moving Media.

Abstract: An approximate method for analyzing EM wave propagation and scattering in the presence of temporally

and spatially varying media is investigated. The method is quasi-relativistic in the sense that for constant

velocity it reduces to Einstein’s Special Relativity theory to the first order in the normalized speed

/vc

The present exponential model was previously used for temporally invariant velocity only. The motion must

be irrotational and the characteristic wavelength and period scales of the mechanical motion must be much

larger compared to those of the EM field ones. For simple periodic motion it is shown that the EM field is

modulated by the motion, and a spectrum of discrete sidebands is created, with frequencies separated by the

mechanical frequencies. The results suggest new approaches to the celebrated Fizeau experiment. Rather

than using an interferometer setup as in the traditional experiment, the equivalent phase velocity in a

periodically moving medium can be deduced from the measured. Simple examples are computed: the effect

of the motion on an initially plane harmonic wave, and scattering by perfectly conducting and refractive

planes and cylinders.

1 INTRODUCTION AND

ABBREVIATIONS

Scattering of EM waves in the presence of moving

media and scatterers is of interest for theoretical and

engineering applications, see (Van Bladel, 1984) for

a comprehensive introduction to the relevant

literature. Einstein’s SR; Minkowski, 1908;

Sommerfeld, 1964; Pauli, 1958) facilitates the

analysis for problems involving constant velocities.

Historically this is related to the FE and the

associated Fresnel drag phenomenon (Einstein,

1905; Pauli, 1958). Heuristic approximations are

required for varying velocities, and it stands to

reason that they will adequately apply to cases

involving the normalized speed

/vc

to the FO only.

Historically, the present exponential model

seems to have originated with Collier and Tai

(1965), and later considered for general temporally

invariant velocities (Nathan and Censor, 1968;

Censor, 1969a, 1972).

1.1 Glossary of Abbreviations

BC=Boundary Condition/s

AKA=Also Known As

BFS= Bessel-Fourier Series

EM=Electromagnetic

EX=Exponential Model/s

FE=Fizeau Experiment/s

FO=First Order in

/vc

FT=Field Transformation/s

GT=Galilean Transformation/s

IT=Inverse Transformation/s

LT=Lorentz Transformation/s

MCR= Minkowski Constitutive Relations

ME=Maxwell Equation/s

MM=Mechanical Medium

RE=Relativistic Electrodynamics

RF=Reference Frame/s

SC=Scattering Coefficient/s

SP=Scattering Problem/s

SR=Special Relativity

ZO=Zero Order in

/vc

Censor D.

EM WAVE PROPAGATION AND SCATTERING IN SPATIOTEMPORALLY VARYING MOVING MEDIAThe Exponential Model.

DOI: 10.5220/0004785400620068

In Proceedings of the Second International Conference on Telecommunications and Remote Sensing (ICTRS 2013), pages 62-68

ISBN: 978-989-8565-57-0

2 FIRST ORDER RELATIVISTIC

ELECTRODYNAMICS

Concepts and notation are introduced via a short

recapitulation of the FO RE. The source-free ME in

a RF

′

are

0, 0

∂∂∂∂

∂∂

′′′′

′′

′′′′

×=− × =

′′

⋅= ⋅=

EBHD

(1)

Fields are functions of native coordinates, e.g.,

(,)t

′′′

Er . According to SR, in an inertial RF

have the form-invariant ME, i.e., (1) without

apostrophes

0, 0

∂

∂∂ ∂

∂∂

×=− × =

⋅= ⋅=

EBHD

(2)

where the fields are functions of native coordinates,

e.g.,

(,)tEr .

The FO LT and its IT are given by

,tt t c

−

′′

=− =− ⋅rrv vr

(3)

,tt t c

−

′′ ′ ′

=+ =+ ⋅rr v vr

)

respectively. Henceforth the underline 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

. Effecting the limit

c →∞

in (3),

), yields the GT

′

Substituting (3) in the chain rule of calculus

() ()

tttt

∂

∂∂∂∂

∂

∂∂ ∂ ∂

′

′′

=⋅+

′′

=+⋅

rr rr

(4)

and its IT (4

) leads to the FO differential LT and its

associated IT

tt t

∂

∂∂∂∂∂

−

′′

=+ =+⋅

rr r

(5)

tt t

∂

∂∂∂∂∂

−

′′′′

=− =−⋅

rr r

)

respectively. Note that the second formula (5) is in

fact the “material derivative” or the “moving

derivate” as referred to in continuum mechanics. In

the limit

c →∞

the first equations in (5), (5)

become the GT

∂

′

, as usually used in

continuum mechanics.

Substituting (5

) into (1) and regrouping terms

yields (2), subject to the FO FT

−

′′

=+× =− ×

′′

+× =−×

E EvBB B vE

D D vHH H vD

(6)

and similarly for corresponding IT FT

−

′

′′ ′

=−× =+ ×

′

′′′

−× =+×

E E vBB B vE

DD vHHH vD

(6)

Assuming in

′

Γ simple linear constitutive

relations

′

′′ ′

=DEB H (7)

and substituting from (6) into (7) leads to the FO

MCR (Minkowski, 1908)

/( )

+× = +×

−

×=−×

DvH EvB

BvE HvD

(8)

Although applicable to constant

only, when

arbitrarily stipulated to hold for constant local

, it

provides the basis for many scattering problems

involving rotating spheres and cylinders (see Van

Bladel, 1984, pp. 392-3, for relevant articles by D.

De Zutter and others).

3 FIRST ORDER RELATIVISTIC

ELECTRODYNAMICS,

VARYING VELOCITIES

Inasmuch as SR deals with constant

only, there

exists no exact transition to varying velocity.

Consequently an heuristically extension of the above

FO model must be stipulated, e.g., by generalizing

(5), (5

) to

(,) , (,)

tt t

ct t

∂

∂∂∂∂∂

−

′′

+=+⋅

rr r

vr vr

(9)

(,) , (,)

tt t

ct t

∂

∂∂∂∂∂

−

′

′′ ′

′′ ′′

−=−⋅

rr r

vr vr

(9)

The correspondence makes (9), (9

), plausible.

Note that terms involving the velocity are already of

FO, hence ZO coordinates can be dropped or added,

e.g.,

(,) (,)tt

′

vr vr . Once again note that (9)

tallies with the material derivative concept.

It is easily seen that the form invariance of the

ME subject to (9), (9

) is not preserved here, since by

substitution of (9

) into the ME (1) we encounter

terms like

()(), (,)

tt t

∂

∂∂

′′ ′

′′

=×− ×=vE vE vEvvr (10)

where the term

(( ( , ) )

∂

′

′′

×vr E

obviates the

extension of the FT (6) to varying velocity.

EM Wave Propagation and Scattering in Spatiotemporally Varying Moving Media

However, it is noted that field time derivatives as in

∂

′

×E involve wave frequencies, say

, while

(,)

∂

′

′′

involves MM frequencies. Similarly,

space derivatives of

(,)t

′′

vr , are characterized by a

wave number

, while differentiating the velocity

involves

, the MM wave number.

We conclude that when the length and time

scales characterizing the velocity are larger than the

corresponding parameters of the fields, the FT (6),

and with them the form invariance of the ME (1),

(2), can be assumed. This also implies that the MCR

(8) are valid subject to the present restrictions.

4 THE EXPONENTIAL MODEL

Previously the EX (Collier and Tai, 1965; Nathan

and Censor, 1968; Censor, 1969a, 1972) was based

on the stipulation that (6) remains valid for local

time independent

, although ()vr is spatially

varying. Since the MCR (8) are already of ZO, they

can be recast in a simplifies form

( , )( ), 1/tC c C

−−

=+× = −×

=−=

DEΛ HB H Λ E

Λ vr

(11)

Substituting (11) into (2) yields the relevant FO

ME for moving media (Tai, 1964; Nathan and

Censor, 1968)

∂

μ∂ ∂ ε∂ ∂

∂

ε∂ ∂ μ∂ ∂

×=

−+× ⋅+⋅×=

=+× ⋅−⋅×=

H Λ EEΛ H

E Λ HHΛ E

(12)

Note that by interchanging

↔↔−HE

(12), we switch between the two equations. For

irrotational

(,)tvr we have 0

∂

×=

Λ , entailing a

conservative field, associated with the scalar

potential

, d

∂

ΦΦ= ⋅

∫

ΛΛl

(13)

therefore the path integral (13) depends on limits

only. Accordingly (12) can be recast as

*,*

*0,*0,*

∂μ∂∂ε∂

∂∂∂∂∂

×=− × =

⋅= = = − ×

rrrr

EHHE

EH Λ

(14)

Incorporating the time scales argument as in

(10), whereby the velocity’s time-derivative is

neglected, solutions of (12), (14), can be constructed

in the form

,0,

∂∂

∂

ΦΦ

=⋅ ×= =Φ

∫

EEHH

Λ l ΛΛ

(15)

The operator exponential is understood as a

symbolic Taylor series

1...

∂

≡+Φ +

. The ZO

fields

,EH satisfy the ME

1111

0, 0

∂

μ∂ ∂ ε∂

∂∂

×=− × =

⋅= ⋅=

EHHE

(16)

Inasmuch as the operator

∂

acts on the ZO

fields, for time harmonic fields possessing the factor

−

with frequency

, we identify

∂

ω∂ ∂ ω

↔

−=+×

Λ (17)

Thus the EX is a perturbation scheme whereby

we start with well-known solutions of the ME in

media at rest (16), and with the exponential operator

(15) as a factor, a FO solution of the ME in moving

media is created. Of course, BC, where applicable,

must be taken on the complete fields (15). For

simplicity, incompressible media are considered

here, therefore

satisfies the Laplace equation

∂

Φ=

Once (11)-(16) are accepted as our working

formalism, everything takes place in the

“laboratory” unprimed RF.

5 PLANE WAVE PROPAGATION

IN OSCILLATING MEDIA

Consider a plane harmonic wave satisfying (16)

111 1 11 11 1

ˆˆ

Ee He t

θθ

===⋅−EE HH kr

(18)

with mutually perpendicular

11 1

,,kEH, launched

into the moving medium. The medium time-

dependent velocity is given according to (11), (13),

() cos ,ttd

ΩΦ= ⋅ = ⋅

∫

ΛΛ Λl Λ r

(19)

It follows that the solution of the ME (15) is given

Second International Conference on Telecommunications and Remote Sensing

11 1 1

110 110

ˆˆ

cos cos

Ee He

θθ

θθω θω ξ

=− ⋅ Ω=−Λ Ω

EE HH

Λ r

(20)

where

is the coordinate in the direction of Λ .

Note that in (19), (20),

can assume any value,

therefore

⋅Λ r

is not necessarily small, in spite of

Λ being FO. This is a consequence of choosing a

time-dependent velocity as in (19) (cf. (23) below).

Recasting

in terms of a BFS (e.g., see Stratton,

1969) yields

,()( )

iit

nn n n

nnn

eFe FiJ

νω

⋅−

=∞

=−∞

=Σ = − Λ

=−ΩΣ=Σ

(21)

revealing the spectral structure the plane wave

assumes in the moving medium, with the initial

carrier frequency

for 0n = and additional

discrete sidebands

for integers

. Throughout

denotes the non-singular Bessel function of order

q . Thus (20) can be recast as

ˆˆ

iit iit

nn n n

nnn

Ee He

EE HH F

⋅− ⋅−

=Σ = Σ

kr kr

EE HH

(22)

The time periodic velocity (19) can be

generalized

to a MM space and time harmonic plane

velocity wave

(,) cos( ),| | 2 /

=⋅−Ω=Λ r ΛΚr Κ (23)

with

denoting the MM wavelength. For

longitudinal compression waves

,ΚΛ are parallel,

hence we have

(cos( ))

sin( ) 0

∂∂

×= × ⋅−Ω

=× ⋅−Ω=

ΛΛΚr

ΛΚ Κr

(24)

as prescribed for (13). Furthermore (15) prescribes

000

cos( )

ˆˆ

(/)sin( ), ,

ξξ

Φ= ⋅ −Ω ⋅ =

ΛΚ−Ω=

=Λ Κ Κ −Ω =Κ =Λ

∫

ΛΚrr

ΚξΛ ξ

(25)

with

defining the coordinate in the direction of

,ΚΛ. Instead of (20) we now have

11 1 1

11 0

ˆˆ

,(/)sin( )

Ee He

θθ

ξξ

θθω ξ

=−ΦΦ=ΛΚ Κ−Ω

EE HH

(26)

For

→ , i.e., for

→∞, the problem reduces

to (19)-(22).

The analog of (21) is now

11 1

(/),

nn nn n

eGe

GJ t

nn n

ωθ ν

νω

=Σ

=ΛΚ=⋅−

−=−Κ =−Ω

κ r

κ k Κ k ξ

(27)

It is noted that even though the Bessel functions

argument (27) is of FO, it involves the ratio of the

MM and EM wavelengths

Κ= , which is

not necessarily small and must be assessed for each

concrete case. The analog of (22) is now

ˆˆ

,/ /

nn n n

nn nn n n

Ee He

tE E H H G

θθ

θν

=Σ =Σ

⋅− = =

EE HH

κ r

(28)

In (27), (28), in addition to the temporal

spectrum

, we have a discrete spatial spectrum of

κ . Due to the vector character of

κ , when

k is

not parallel to

, each spectral component

propagates in a slightly different direction,

possessing a different phase velocity according to

/| | ( )/| |

nnn

Cnn

=−Ω −κ k Κ (29)

6 IMPORT FOR NEW FIZEAU-

TYPE EXPERIMENTS

Doppler Effect frequency shifts are usually

associated with moving sources or boundaries. It is

therefore of interest to note, as shown in (28) that

wave-fronts in moving media can also create a

spectrum, without involving moving material

boundaries. As far as this author is aware, this

phenomenon was not documented before in the

present EX context. In a sense, it is akin to some

acousto-optics experiments involving interaction of

sound and EM waves, but rather than having

constitutive parameters modulated by sound, here

medium velocity is involved. The present results

might suggest new approaches to the celebrated FE.

The classical FE (e.g., Van Bladel, 1984, p.

120ff.) measures the EM wave effective phase

velocity

eff

in a column of a moving medium

(water in the original FE), characterized by

C in the

rest RF. The results tally with the SR velocity

addition formula (Pauli, 1958). Consider (18)-(22),

or (23)-(28), with

→ , for parallel velocity and

propagation directions, and

0Ω= . From

(20) or

(26) we then find effective values

EM Wave Propagation and Scattering in Spatiotemporally Varying Moving Media

110 1 10

eff

kk C

ωω

=−Λ= −Λ

(30)

//1/( )

v(1 1/ )

eff eff eff

CkcnC

CC C n

−

===−Λ

≈+ Λ=+ −

(31)

with index of refraction

/ncC=

in the rest RF.

This is the basis for the classical FE. Essentially,

with

C known, the quest is for the value of

order to compute

eff

or vice-versa.

Exploiting the present theory, rather than

using an interferometer setup and measuring the

displacement of diffraction fringes, as done in the

traditional FE,

Λ can be found from measurements

of spectral components. Thus by measuring the

amplitude of waves, and solving for relevant

arguments of

G in (22), (28), respectively, the

value of

Λ can be extracted. Using (30), (31), the

effective parameters can be computed. One can

envision a medium set into periodic motion as in

(19) or (23). The EM wave propagated through the

medium will display a spectrum of discrete

frequencies

, (21), (27). In both cases the

cumbersome interferometer setup involving a

moving water column is obviated. This also solves

the problem of the irregular flow at the source and

sink regions where the fluid is injected and drained,

as in the classical FE. Better resolution (AKA

selectivity) of sidebands can be attained by

electronically down-shifting frequencies after

detection (AKA mixing, or heterodyning) employed

in radio communications techniques.

7 SCATTERING PROBLEMS

SP for the EX with time-independent

have been

discussed before (Collier and Tai, 1965; Censor,

1969; Censor, 1972).

As in the FE and other cases (Censor, 1969b),

fluid-dynamics continuity problems of the medium

flow in the presence of the scatterer are arbitrarily

ignored, assuming that the flow is maintained as if

the material scatterer has no effect. Otherwise

complicated problems ensue that cannot be tackled

with the analytical tools employed here. Realistic

flows have been considered in (Censor, 1972).

7.1 Scattering by Plane Interfaces

As the simplest example for a SP consider a

perfectly conducting plane at

0x = , with a

perpendicularly incident wave as in (26) with

replacing

111 11

ˆˆ

,/ /

Ee He

kx t E H

θθ

ωω μεη

==−

−−Φ = =

Ez H y

(32)

the ratio

defining the medium impedance in the

region

The reflected wave

E must satisfy the BC

rix

−EE. The BC prescribe identical time

variation for all waves at the boundary

0x = , hence

111

ˆˆ

Ee He

kx t

ρρ

θωω

=− − − Φ

Ez Hy

(33)

with

−

denoting the reflection coefficient.

Consider next a refractive medium in the region

0x > , with frequency dependent rest RF

constitutive parameters

()

, ()

Correspondingly the RF phase velocity, impedance,

are

() 1/C

με

= , () /

νμε

= , respectively. In

the region

the parameters

()

remain dependent on the excitation frequency

only, in order to satisfy the ME subject to the EX,

(15).

The EX solutions (32), (33), are recast in

spectral components as in (28)

ˆˆ

ni ni

inn i nn

ni n n

Ee He

θκν

=Σ =−Σ

=−

Ez H y

(34)

ˆˆ

nr nr

rnnn rnnn

nr n n

Ee He

θθ

ρρ

θκν

=Σ =Σ

=− −

Ez Hy

(35)

and the transmitted wave is given by

ˆˆ

,/,()

/, /

(), ()

nt nt

tnn t nn nnn

nt n n n n n n n

nnnn nn n

nnnnn

Ee He E E

xt CCC

EH H H

θθ

θκνκν ν

ητηη

μμνττν

=Σ =−Σ =

=− = =

Ez H y

(36)

The solution for

, are given by the familiar

formulas

()/(),2/()

nn n n nn

ηηηητ ηηη

−+= + (37)

7.2 Scattering by Circular Cylinders

For the SP of a perfectly conducting circular

cylinder of radius

, the incident excitation plane

wave is once again given by (32). Leaving the EX

Second International Conference on Telecommunications and Remote Sensing

factor intact and recasting the ZO solution

ik x

BFS yields, (15)-(17), in cylindrical coordinates

,1 1

(/)

(),

im i t

ii mm

im i t

mm m km

eEeLe

iE e e

LiJkr L

ωω

ψω

∂ωμ

∂

−Φ −Φ

−

−Φ

−

==Σ

=×

=Σ

==×

EEz

(38)

Accordingly we construct the scattered wave as

(1)

(/)

(),

im i t

smmm

im i t

smmm

mm m km

Ee a M e

iE e a e

iH kr L

ψω

∂

−Φ

−

−Φ

−

=Σ

==×

(39)

with

(1)

denoting the first kind Hankel functions.

On application of the BC

rira=

=−EE, the EX

factor cancels and we find

(1)

()/ ()

mm m

aJkrHkr=−

(40)

the familiar SC of the ZO problem.

For a material cylinder we start with (38), (39),

and recast

1 x

−Φ

in BFS

(/)sin( )

(cos )

,,,

,()()

ixt

in r t

in t ip p

np n np np p

eGe

Ge P e P i J n r

−ΛΚ Κ−Ω

−Κ −Ω

=Σ

=Σ = − Κ

(41)

Hence

()

1,, ,,

()

1,,,,

,, , ,, ,

(/)

ipm i t

inpmnpm

ipm i t

inpmnpm

npm n np m npm n np m

EQe

iE e

QGPL GP

ψν

+−

=Σ

(42)

Similarly, (39) becomes

()

1,, ,, ,,

()

1,,,,,,

,, , ,, ,

(/)

ipm i t

s npm npm npm

ipm i t

snpmnpmnpm

npm n np m npm n np m

E aSe

iE a e

SGPM GP

ψν

+−

=Σ

(43)

In the cylinder’s interior

fields are obtained as

a superposition of regular cylindrical waves of

modes

, at frequencies

, satisfying the ME (16)

with rest RF parameters

(),()

νεν

1, , ,

1,,,

,,,

(/)

(),

iu i t

tnununu

iu i t

tnununu

nu u n nu nu

EbTe

iE b e

TiJr T

ψν

κ∂

−

=Σ

==×

(44)

On application of BC

0| , ( )|

ist ra i s tra

−= ⋅ + −EEE ψ HHH (45)

prescribing the fields continuity on the interface, the

orthogonality of angular modes

prescribes

nonzero coefficients

≠

for

upm≠+

. Hence

(44) can be recast to include the constraint

1,, , ,

1,, ,,

()

(/) ( )

iu i t

tnpm nunu

iu i t

tnpm nunu

EpmubTe

iE p m ub e

ψν

ηδ

−

=Σ +−

=Σ+−

(46)

where

denotes the Discrete Kronecker Delta

Function. For each spectral component

, and

angular mode

the BC lead to an infinite set of

equations, which can only solved if properly

truncated.

Consider (41)-(46) for the case of a monopole

0upm

. This only works for thin cylinders,

hence

must be properly truncated, otherwise

higher multipole terms must be included.

Accordingly

1,0,0,00

(/) ,

innnnnn

EQe Q Q GPL

iE e GP

−

=Σ = =

=Σ ==

HQQQL

(47)

and similarly

,0,0 ,0 0 ,0,0 ,0 0

(/)

snnn

nn nn nn nn

EaSe

iE a e

SS GPM GP

−

=Σ

== ==

SS M

(48)

,0 0 ,0

(/)

(),

tnnn

nn n n n n

EbTe

iE b e

TT J r T

κ∂

−

=Σ

== ==×

TT z

(49)

From (45) and (47)-(49), we get explicit

equations for the SC

(1)

01 0 1 0

(1)

01 0 1 0

10 0

() () ( )0

() () (/)()0

/[ ( / ) ( )]

nnn

Jka aH ka bJ a

Jka aH ka b J r

bbJ Jna

ηη κ

+−=

′′ ′

−=

=ΛΚΚ

(50)

EM Wave Propagation and Scattering in Spatiotemporally Varying Moving Media

with the prime denoting differentiation with respect

to the argument. In form (50) is similar to the

classical SP, but including the present velocity

effects, therefore solving for the coefficients

is straightforward.

8 SUMMARY AND

CONCLUDING REMARKS

The advent of SR (Einstein, 1905) facilitated the

analysis of SP involving moving objects and media.

However, SR is founded on the concept of inertial

RF moving at constant

A multitude of scientific and engineering

problem involve varying velocities. Heuristic

models that in the case of constant

merge into

exact SR are not unique. Presently the Quasi Lorentz

Transformation (Censor, 2005, 2010) (9) is

employed. Subject to the constraint of MM and EM

space and time scaling, the FO ME and FT (1), (2),

(6), apply to varying

(,)tvr .

The EX, originating with Collier and Tai (1965)

provides FO SR solutions to ME in moving media.

The method is generalized here to time-dependent

irrotational velocity fields. Previously (Collier and

Tai, 1965) only time-independent velocity systems

have been considered.

In periodically moving media the solution for the

ZO case of plane waves displays discrete sideband

spectra. This provides new approaches to the FE.

Unlike the original FE, employing interference

experiments, the present results suggest

measurements based on analysis of the spectra

created by periodical mechanical flows or waves.

Canonical SP examples are given for scattering

by plane interfaces and by circular cylinders, in the

presence of periodically moving embedding media.

It is shown that opaque objects, like the perfectly

conducting interfaces above, yield the classical SC

for media at rest, involving only the excitation

frequency

. On the other hand, refractive

scatterers are excited by the frequencies created by

the MM motion, (37), (43), displaying SC depending

on the sideband frequencies.

The results suggest new methods for remote

sensing the material parameters of objects that are

not directly accessible. To further investigate the

present model, more canonical SP will have to be

investigated, with various MM motional modes.

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