The Theory of Low Frequency Physics Revisited George Venkov - - PowerPoint PPT Presentation

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The Theory of Low Frequency Physics Revisited

George Venkov Department of Applied Mathematics and Informatics, Technical University of Sofia, 1756 Sofia, Bulgaria <gvenkov@tu-sofia.bg> Martin W. McCall Department of Physics, The Blackett Laboratory Imperial College London, London SW7 2AZ, UK <m.mccall@imperial.ac.uk> Dan Censor Department of Electrical and Computer Engineering, Ben–Gurion University of the Negev Beer–Sheva, Israel, 84105 <censor@ee.bgu.ac.il>

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The Theory of Low Frequency Physics Revisited

Download the present presentation from: http://www.ee.bgu.ac.il/~censor/presentations-directory/dani-low- frequency.ppt OR http://www.ee.bgu.ac.il/~censor/presentations-directory/dani-low- frequency-ppt.pdf Download a reprint of the paper, which appeared in JEMWA—Journal of ElectroMagenetic Waves and Applications, Vol. 21, pp. 229-249, 2007 at: http://www.ee.bgu.ac.il/~censor/low-frequency-paper.pdf

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SUMMARY: INTRODUCTION OLD LOW-FREQUENCY THEORY (RED FRAME) CONSISTENT MAXWELL SYSTEMS HELMHOLTZ EQUATION AND PLANE WAVES PLANE-WAVE (SOMMERFELD) INTEGRALS LOW-FREQUENCY THEORY DIFFRACTION (KIRCHHOFF) INTEGRALS ANOTHER EXAMPLE: ACOUSTICS ANOTHER EXAMPLE: ELASTODYNAMICS EM SCATTERING FROM A CYLINDER CONCLUDING REMARKS

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INTRODUCTION Space-Time Source-Free Maxwell Equations , ( , ), ( , )

t t

t t ∂ μ∂ ∂ ε∂ ∂ ∂ × = − × = ⋅ = ⋅ = = =

r r r r

E H H E E H E E r H H r Time-Harmonic Maxwell Equations

t

i ∂ ω ⇔ − ( ) ( ), ( ) ( ) ( ) ( ) ( , ) ( ) , ( , ) ( )

i t i t

i i t e t e

ω ω

∂ ωμ ∂ ωε ∂ ∂

− −

× = × = − ⋅ = ⋅ = = =

r r r r

E r H r H r E r E r H r E r E r H r H r

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Taylor series:

1 2 2

( ) {1 ( /1!) ( / 2!) ( / !) } ( ) | ( / !) ( ) | [( / !] ( ),

p p x n n n n n x n x

f x d d p d f n d f n d f x x x

ξ ξ ξ ξ ξ ξ

ξ ξ

= ∞ ∞ = = =

= + Δ + Δ +⋅⋅⋅+ Δ +⋅⋅⋅ = Σ Δ = Σ Δ Δ = − Sympolic 3D Taylor expansion, ∂r gradient operator on r : ( ) ( ) [ / !] ( ),

n n n

f e f n f

⋅∂ ∞ =

= = Σ ⋅∂ = −

r

Δ r

r r Δ r Δ r r Plane wave: ( / !) ( ) / !

i i i n n n n n

f e f n e f i e n

⋅ ⋅ ⋅ ∞ ∞ = =

= Σ ⋅∂ = Σ ⋅

k r k r k r r

Δ k Δ For = r : ˆ ˆ ( ) ( ) / !, / , /

i n n n

f e f ik n k k c ω ω με

⋅ ∞ =

= Σ ⋅ = = =

k r

k r k k

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Low-Frequency Scattering series: Incident waves: ˆ ˆ ( ) ( ) ( ) / ! ˆ ˆ ( ) ( ) ( ) / !

n n i i n n n i i n

e ik n h ik n

∞ = ∞ =

= Σ ⋅ = Σ ⋅ E r e k r H r h k r Scattered waves: ( ) ( ) ( ) / ! ( ) ( ) ( ) / !

n n n n n n

e ik n h ik n

∞ = ∞ =

= Σ = Σ E r E r H r H r Substitute into Maxwell Equations ( ) ( ) ( ) ( ) ( ) ( ) / ( ) ( ) 0, / / i ikZ i ik Z Z e h ∂ ωμ ∂ ωε ∂ ∂ μ ε × = = × = − = − ⋅ = ⋅ = = =

r r r r

E r H r H r H r E r E r E r H r

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OLD LOW-FREQUENCY THEORY (RED FRAME) Stevenson, A.F., “Solution of Electromagnetic Scattering Problems as Power Series in the Ratio Dimension of Scatterer/Wavelength”, J. Appl. Phys., Vol. 24, 1134-1141, (1953). Asvestas, J.S. and Kleinman,R.E., “Low-Frequency Scattering by Perfectly Conducting Obstacles’, J. Math. Phys., Vol. 12, 795-811, (1971). Dassios, G. and Kleinman, R., Low Frequency Scattering, Oxford Mathematical Monographs, Clarendon Press, (2000).

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Substituting into the divergence equations:

( ) ( ) ( )/ ! 0 ( ) ( ) ( )/ ! 0

n n n n n n

e ik n h ik n ∂ ∂ ∂ ∂

∞ = ∞ =

⋅ = Σ ⋅ = ⋅ = Σ ⋅ =

r r r r

E r E r H r H r

These are power-series, therefore each term vanishes individually and we get:

( ) 0, ( )

n n

∂ ∂ ⋅ = ⋅ =

r r

E r H r

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Substituting into the rotor equations: ( ) ( ) ( ) / ! ( ) ( ) ( ) / ! ( ) ( ) ( ) / ! ( ) / ( / ) ( ) ( ) / !

n n n n n n n n n n n n

e ik n ikZ ikZh ik n h ik n ik Z ik e Z ik n ∂ ∂ ∂ ∂

∞ = ∞ = ∞ = ∞ =

× = Σ × = = Σ × = Σ × = − = − Σ

r r r r

E r E r H r H r H r H r E r E r Choosing equal powers of ik in the power-series:

1 1

( ) ( ) ( ) ( ), /

n n n n

n n e h Z ∂ ∂

− −

× = × = − =

r r

E r H r H r E r

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What is a power-series? Given a series with variable x ( )

n n n

f x x a

∞ =

= Σ To compute the coefficients

n

a we need variable x :

2 3 1 2 3 1 1 2 2

+ ( ) | ( ) | , ( ) | ( ) / 2!| ( ) ( )/ ! | ,

n n n n n x x x x x x x n x x n

a xa x a x a x a f x d f x a d f x f x x a d f x n a a d f x a

∞ = = = = = =

= + + + + ⋅⋅⋅ = = Σ = = = =

Power series proper cannot be defined with respect to constant parameters

/ ik i c ω =

is a constant parameter!

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POWER-SERIES METHOD Choosing equal powers

• f ik in the power-series:

1 1

( ) ( ) ( ) ( ), /

n n n n

n n e h Z ∂ ∂

− −

× = × = − =

r r

E r H r H r E r

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CONSISTENT MAXWELL SYSTEMS Helmholtz wave equation:

2 2 2 2

, , ( ) ( ) i i k ∂ ωμ ∂ ωε ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ × = × = − ⋅ = ⋅ = × × = ⋅ − = − + =

r r r r r r r r r r r

E H H E E H E E E E E First consistent Maxwell system:

2 2

( ) 0, , k i ∂ ωμ ∂ ∂ ∂ + = × = ⋅ = ⋅ =

r r r r

E E H E H Second consistent Maxwell system:

2 2

( ) 0, , k i ∂ ωε ∂ ∂ ∂ + = × = − ⋅ = ⋅ =

r r r r

H H E E H

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HELMHOLTZ EQUATION AND PLANE WAVES A plane wave and its Taylor expansion ( ˆ k = k x =constant): ˆ ( ) ( ) ( ) / ! ˆ ( ) ( ) ( ) / !,

i n n n ikx n n n

e ik n e ik x n k

⋅ ∞ = ∞ =

= = Σ ⋅ = = Σ =

k r

E r e e k r E r e e k x Helmholtz equation applied to plane wave:

2 2 2 2 2 2 2 2 2 2

ˆ ( ) ( ) ( )( ) / ! ( ) ( ) ( ) / ! ( ) ( ( 1) ) / !

i n n n ikx n n x n x n n n n

k e ik k n d k e ik d k x n ik n n x k x n

⋅ ∞ = ∞ = ∞ − =

∂ + = Σ ∂ + ⋅ = + = Σ + = Σ − + =

k r r r

e e k r NOT satisfied IDENTICALLY term by term

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Equating powers of x PERMITTED, because x =variable! Only the whole series satisfies IDENTICALLY:

2 2 2 2 2 2 2 2 2 2

( ) ( ) ( ) / ! ( ) [ ( 1) ]/ ! ( ) ( 1) / ! ( ) / ! ( ) ( 2)( 1) /( 2)! ( ) / !

ikx n n x n x n n n n n n n n n n n n n n n n

d k e ik d k x n ik n n x k x n ik n n x n ik x n ik n n x n ik x n

∞ = ∞ − = ∞ − ∞ + = = ∞ + = ∞ + =

+ = Σ + = Σ − + = Σ − − Σ = Σ + + + −Σ =

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A “trivial” recurrence relation:

2 2 2 2 2 2

( ) ( ) ( ) / !, ( ) ( ) ( ) ( ) ( ( ) ( )) / ! ( ) ( ( 1) ( ) ( )) / !

ikx n n n n n n x n x n n n n n n

f x e ik f x n f x x d k f x ik d f x k f x n ik n n f x k f x n

∞ = ∞ = ∞ = −

= = Σ = + = Σ + = Σ − + =

2 2 2 2 1

( ) ( 1) ( ) 0, ( ) 1, ( )

x n n x x

d f x n n f x n d f x n d f x

−

= − = = = =

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PLANE-WAVE (SOMMERFELD) INTEGRALS Plane wave integral solves Helmholtz equation:

ˆ ˆ /2 1 ˆ / 2 / 2 1 ˆ 2

ˆ ( ) ( ) , sin

ik C i C i i C

e e d d d d d S d S

β π π β π β π α π α α π β π α

β β α α

⋅ = − ∞ =− + ∞ = = − ∞ =− =

= Ω Ω = Ω = =

∫ ∫ ∫ ∫ ∫ ∫

k r k k k

E r g k Complex contour C , complex

R I

i = + k k k , real k

2

2

R I R R I I R I R I

i k i = + = ⋅ = ⋅ − ⋅ + ⋅ ⋅ = k k k k k k k k k k k k k

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Taylor expansions and partial waves:

ˆ ˆ ˆ

ˆ ˆ ˆ ( ) ( ) ( ) ( ) ( ) / ! ˆ ˆ ( ) ( ) / !, ( ) ( )( )

i n n n C C n n n n n C

e e d e ik n d e ik n d

⋅ ∞ = ∞ =

= Ω = Σ ⋅ Ω = Σ = ⋅ Ω

∫ ∫ ∫

k r k k k

E r g k g k k r E r E r g k k r Helmholtz equation solutions:

2 2 2 2 ˆ 2 2 ˆ 2 2

ˆ ( ) ( ) ( ) ( ) ˆ ˆ ( ) ( ) ( )( ) / ! ( ) ( ) ( ) / !

i C n n n C n n n

k k e e d e ik k n d e ik k n

⋅ ∞ = ∞ =

∂ + = ∂ + Ω = Σ ∂ + ⋅ Ω = Σ ∂ + =

∫ ∫

k r r r k r k r

E r g k g k k r E r

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Recurrence relation:

2 2 2 ˆ ˆ 2 2 ˆ ˆ 2

ˆ ˆ ˆ ˆ ( ) ( )( ) ( ) ( ) ˆ ˆ ˆ ˆ ( ) ( 1)( ) ( 1) ( )( ) ( 1) ( )

n n n C C n n C C n

d d n n d n n d n n

− − −

∂ = ∂ ⋅ Ω = ∂ ⋅ Ω = − ⋅ Ω = − ⋅ Ω −

∫ ∫ ∫ ∫

r r r k k k k

E r g k k r g k k r g k k r g k k r E r Vector Laplace equations:

2 2 2 2 1

( ) ( 1) ( ) 0, ( ) 1, ( )

n n

n n n n

−

∂ = − = ∂ = = ∂ =

r r r

E r E r E r E r

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LOW-FREQUENCY THEORY Low-Frequency. First consistent Maxwell system:

2 2 2 2

( ) 0, , ( ) ( 1) ( ), ( ) ( ) 0, 0, ( ) ( ) ( ) / !

n n n n n n n n n

k i n n ik e ik n ∂ ωμ ∂ ∂ ∂ ∂ ∂

− ∞ =

∂ + = × = ⋅ = ⋅ = ∂ = − × = ⋅ = ⋅ = = Σ

r r r r r r r r

E E H E H E r E r E r H r E H E r E r Low-Frequency. Second consistent Maxwell system:

2 2 2 2

( ) 0, , ( ) ( 1) ( ), ( ) ( ) 0, 0, ( ) ( ) ( ) / !

n n n n n n n n n

k i n n ik h ik n ∂ ωε ∂ ∂ ∂ ∂ ∂

− ∞ =

∂ + = × = − ⋅ = ⋅ = ∂ = − × = − ⋅ = ⋅ = = Σ

r r r r r r r r

H H E E H H r H r H r E r E H H r H r

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DIFFRACTION (KIRCHHOFF) INTEGRALS Scalar longitudinal (e.g., Acoustics) Surface-Integral solution:

2 2 2 2 | |

( ) ( ) 0, ( ) ( ) ( ) ( ) / 4 | | ( ) [ ( ) ( ) ( ) ( )]

ik S

k p k G G e p p G G p d δ π ∂ ∂

−

∂ + = ∂ + − = − − − = − = − − − ⋅

∫

ρ ρ r ρ ρ ρ

ρ r ρ ρ r r ρ r ρ r ρ r ρ r ρ ρ S

• Vector transversal (e.g., Electromagnetics):

2 2 2 2 2

( ) ( ) 0, ( ) ( ) ( ) ( ) ( ) [( ) ( ) ( ) ( )] ˆ ( ), ( ) ( ), ( ) ( ) ( / ) ( )

T T T S T T T

k k d d d d dS k G ∂ δ ∂ ∂ ∂ ∂ ∂ ∂ + = ⋅ = ∂ + − = − − = × ⋅ × − × ⋅ × = = = ⋅ = − = − = + −

∫

ρ ρ ρ ρ ρ ρ ρ ρ

F ρ F ρ G r ρ I ρ r F r S F G F S G F F ρ S S ρ n ρ G G r ρ G ρ r I r ρ

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Vector longitudinal (e.g., Acoustics, Elastodynamics):

2 2 2 2 2

( ) ( ) 0, ( ) ( ) ( ) ( ) ( ) [( )( ( ) ) ( ( ))( )] ˆ ( ), ( ) ( ), ( ) ( ) ( ) /

L L L S L L L L

k k d d d d dS G k ∂ δ ∂ ∂ ∂ ∂ ∂ ∂ + = × = ∂ + − = − − = ⋅ ⋅ − ⋅ ⋅ = = = × = = − = − = − −

∫

ρ ρ ρ ρ ρ ρ ρ ρ

F ρ F ρ G r ρ I ρ r F r G F ρ S F ρ G S F F ρ S S ρ n ρ G G G ρ r G r ρ ρ r

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Corresponding Surface-Integral solutions for partial waves: Scalar longitudinal waves: ( ) [ ( ) ( ) ( ) ( )]

n n n S

p p G G p d ∂ ∂ = − − − ⋅

∫

ρ ρ

r ρ r ρ r ρ ρ S

• Vector transversal waves:

( ) [( ) ( ) ( ) ( )]

n n T n T S

d d ∂ ∂ = × ⋅ × − × ⋅ ×

∫

ρ ρ

F r S F G F S G

• Vector longitudinal waves:

( ) [( )( ( ) ) ( ( ))( )]

n L n n L S

d d ∂ ∂ = ⋅ ⋅ − ⋅ ⋅

∫

ρ ρ

F r G F ρ S F ρ G S

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ANOTHER EXAMPLE: ACOUSTICS Acoustics basic equations: ( ) ( ) 0, ( ) ( ) 0,

t t

p p γ∂ ∂ ρ∂ ∂ ∂ + ⋅ = + = × =

r r r

r v r v r r v First consistent system:

2 2 2 2

( ) ( ) 0, ( ) ( ) 0, , ( ) k p p i k ∂ ωρ ω γρ ∂ ∂ + = − = = × =

r r r

r r v r v r

Second consistent system:

2 2

( ) 0, k i p ∂ ωγ ∂ + = ⋅ − =

r r

v v

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First consistent system, series and recurrence relation:

2 2

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

n n n n n

p p ik p n p n n p

∞ = −

= Σ ∂ = −

r

r r r r Derivation of acoustic velocity: ( ) ( ) ( ) ( ) / ! ( ) ( ) ( ) / ! ( ) ( ), /

n n n n n n n n

i p p ik p n v ik n p v p i ωρ ∂ ∂ ∂ ωρ

∞ = ∞ =

= = Σ = Σ = =

r r r

v r r r v r v r v r r

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Second consistent system, series and recurrence relation:

2 2

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

n n n n n

v ik n n n

∞ = −

= Σ ∂ = −

r

v r v r v r v r Derivation of acoustic pressure: ( ) ( ) ( ) ( ) / ! ( ) ( ) ( ) / ! ( ), /

n n n n n n n n

i p v ik n p p ik p n p p v i ωγ ∂ ∂ ∂ ωγ

∞ = ∞ =

= ⋅ = Σ ⋅ = Σ = ⋅ =

r r r

r v r v r r r v r

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ANOTHER EXAMPLE: ELASTODYNAMICS Elastodynamics basic equations: , ( ) [( ) ( ) ]

t T

i i i ∂ ωρ ∂ ω ω λ ∂ μ ∂ ∂ ⋅ = − = = − − = ⋅ + +

r r r r

τ v v w w τ v I v v

• Separation to pressure and shear:

, 0, , ( ) [( ) ( ) ] , [( ) ( ) ]

p s p s T p p p p p p T s s s s s

i i i i ∂ ∂ ∂ ωρ ω λ ∂ μ ∂ ∂ ∂ ωρ ω μ ∂ ∂ = + × = ⋅ = ⋅ = − − = ⋅ + + ⋅ = − − = +

r r r r r r r r r

v v v v v τ v τ v I v v τ v τ v v

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First consistent system:

2 2 2 2 2 2 2 2 2 2 2 2

( ) 0, ( ) 2 ( ) / , ( 2 ) / ( ) 0, [( ) ( ) ] / , /

p p p p p p p p T s s s s s s s s

k i k c c k i k c c ∂ λ ∂ μ ∂ ω ω λ μ ρ ∂ μ ∂ ∂ ω ω λ ρ + = ⋅ + + = = = + + = + + = = =

r r r r r r

v v I v τ v v v τ

• Second consistent system:

2 2 2 2

( ) 0, ( ) 0,

p p p p s s s s

k i k i ∂ ∂ ωρ ∂ ∂ ωρ + = ⋅ = − + = ⋅ = −

r r r r

τ τ v τ τ v

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First consistent system, series and recurrence relations:

2 0, , , , 2 2 0, s , , , 2

( ) / !, ( 1) ( ) / !, ( 1)

n p p n p p n p n p n n s n s s n s n s n

v ik n n n v ik n n n

∞ = − ∞ = −

= Σ ∂ = − = Σ ∂ = −

r r

v v v v v v v v Second consistent system, series and recurrence relations:

2 0, , , , 2 2 0, s , , , 2

( ) / !, ( 1) ( ) / !, ( 1)

n p p n p p n p n p n n s n s s n s n s n

v ik n n n v ik n n n

∞ = − ∞ = −

= Σ ∂ = − = Σ ∂ = −

r r

τ τ τ τ τ τ τ τ

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Surface integrals for longitudinal dyadics:

2

( ) [( )( ( ) ) ( ( ))( )] ( ) ( ) /

L L S L L

d d G k ∂ ∂ ∂ ∂ = ⋅ ⋅ − ⋅ ⋅ = − = − −

∫

ρ ρ ρ ρ

F r G F ρ S F ρ G S G G ρ r ρ r

• Surface integrals for transversal dyadics:

2

( ) [( ) ( ( )) ( ) ( ( ))] ˆ , ( / ) ( )

T T T T S T

d d d dS k G ∂ ∂ ∂ ∂ = × ⋅ × − × ⋅ × = = + −

∫

ρ ρ r r

F r G S F ρ S G F ρ S n G I r ρ

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EM SCATTERING FROM A CYLINDER Perfectly conducting infinitely long circular-cylinder radius r a = . TM case. Incident and scattered fields are denoted by

cos ,

ˆ ˆ ˆ ( ) ( ) ( cos ) / ! ˆ ˆ ˆ ˆ ˆ ( ) ( ) ( ) / !, cos

i ikr n n i n n s n s n

e e e e e ik r n e ik E n

ψ

ψ ψ

⋅ ∞ = ∞ =

= = = Σ = Σ = ⋅ = ⋅

k r

E r z z z E r z r x r k r Boundary-condition ˆ ( ( ) ( )) 0 |

i s r a =

× + = r E r E r

Ei Es ψ

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, 0 , 0 2 1 , 0

ln ˆ ( ) ln ˆ ˆ ( ) ( ln ) ln

s s s r r

r a r d rd r a κ ∂ κ κ κ

−

⎛ ⎞ = − , ⋅ = ⎜ ⎟ ⎝ ⎠ ⎛ ⎞ ∂ = − = ⎜ ⎟ ⎝ ⎠

r r

E r z E z z E r

z Ei Es

The low-frequency solution for the scattered wave Monopole logarithmic solution is the only solution satisfying implied Laplace’s equation and appropriate radiation condition

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The field

, 0 s

H is given by

, 0 , 0 , 0 , 0

ˆ ( ) ( ) /[ ln ]

s s s s

ikr a ∂ κ ∂ ∂ = × = ⋅ = , × =

r r r

H r E r ψ H H c.f. the ‘n’ theory yields

, 0

ln ˆ ( ) ln

s

r a κ κ ⎛ ⎞ = − ⎜ ⎟ ⎝ ⎠ H r ψ

EsHs

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The analytic solution for the scattered wave

(1) (1)

ˆ ( ) ( )

m m im s m m m m m m

e i a H kr e H J iN

ψ =∞ =−∞

= Σ = + E r z For 1 , << kr ka

ln ( ) ( ) ln r a ka H kr a ⎛ ⎞ ≈ −⎜ ⎟ ⎝ ⎠

c.f.

, 0 , 1

ln ln ˆ ˆ ( ) ln ln

s r a fixed

r r a a

κ κ κ

κ κ

<<

⎛ ⎞ ⎛ ⎞ = − → − ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ E r z z

EsHs

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CONCLUDING REMARKS Low-Frequency wave theory is important in physics, occurring for example in the Rayleigh scattering theory for the blue color of the sky in daytime, and the red color at sunset. It was therefore recognized that a concise mathematical theory is very desirable. Starting with Stevenson, 1953, the old theory assumed that the fields can be expanded in power-series in ik , in spite of the fact that ik =constant. This led to a different set of equations, which we call for short the “n” algorithm, for the partial fields in electromagnetics and elastodynamics. Unknowingly, because they started with the Helmholtz equation, previous theorists derived for the scalar acoustics case the correct equations, with “n(n-1)” recurrence relations. The inconsistency with electromagnetics was unfortunately missed. The present study unifies the three wave physics models: acoustics, electromagnetics, elastodynamics, subject to a common “n(n-1)” algorithm.

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THIS IS ALL, FOLKS, THANK YOU