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

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> 1

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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 9

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 10

INTRODUCTION Space-Time Source-Free Maxwell Equations ∂ × = − μ∂ ∂ × = ε∂ E H H E , r r t t ∂ ⋅ = ∂ ⋅ = E H 0 r r = = E E r H H r ( , ), ( , ) t t ∂ ⇔ − ω Time-Harmonic Maxwell Equations i t ∂ × = ωμ ∂ × = − ωε E r H r H r E r ( ) ( ), ( ) ( ) i i r r ∂ ⋅ = ∂ ⋅ = E r H r ( ) ( ) 0 r r − ω − ω = = E r E r H r H r i t i t ( , ) ( ) , ( , ) ( ) t e t e 11

Taylor series: = + Δ + Δ +⋅⋅⋅+ Δ +⋅⋅⋅ ξ 1 2 2 p p ( ) {1 ( /1!) ( / 2!) ( / !) } ( ) | f x d d p d f ξ ξ ξ ξ = x 0 = Σ ∞ Δ ξ = Σ ∞ Δ Δ = − n n n n ( / !) ( ) | [( / !] ( ), n d f n d f x x x = ξ ξ = = 0 0 0 0 n x n x 0 0 ∂ r gradient operator on r : Sympolic 3D Taylor expansion, 0 0 ⋅∂ Δ = = Σ ∞ ⋅∂ = − r r Δ r Δ r r n n r ( ) ( ) [ / !] ( ), f e f n f 0 = r 0 0 0 0 n 0 Plane wave: ⋅ ⋅ ⋅ = Σ ∞ ⋅∂ = Σ ∞ ⋅ k r k r k r Δ i k Δ i i n n n ( / !) ( ) / ! f e f n e f i e n 0 0 = = r 0 0 0 0 0 n n 0 = r For 0 : 0 ⋅ ∞ ˆ ˆ = Σ ⋅ = = ω = ω με k r k r k k i n n ( ) ( ) / !, / , / f e f ik n k k c = 0 0 0 n 12

Low-Frequency Scattering series: Incident waves: ∞ ˆ = Σ ⋅ E r e k r n n ˆ ( ) ( ) ( ) / ! e ik n = 0 0 i i n = ˆ Σ ∞ ˆ ⋅ H r h k r n n ( ) ( ) ( ) / ! h ik n = i i 0 n 0 Scattered waves: ∞ = Σ E r E r n ( ) ( ) ( ) / ! e ik n = 0 n 0 n ∞ = Σ H r H r n ( ) ( ) ( ) / ! h ik n = 0 n 0 n Substitute into Maxwell Equations ∂ × = ωμ = E r H r H r ( ) ( ) ( ) i ikZ r ∂ × = − ωε = − H r E r E r ( ) ( ) ( ) / i ik Z r ∂ ⋅ = ∂ ⋅ = = μ ε = E r H r ( ) ( ) 0, / / Z e h r r 0 0 13

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). 14

Substituting into the divergence equations: ∞ ∂ ⋅ = Σ ∂ ⋅ = E r E r n ( ) ( ) ( )/ ! 0 e ik n = r r 0 0 n n ∂ ⋅ = Σ ∞ ∂ ⋅ = H r H r n ( ) ( ) ( )/ ! 0 h ik n = r r 0 0 n n These are power-series, therefore each term vanishes individually and we get: ∂ ⋅ = ∂ ⋅ = E r H r ( ) 0, ( ) 0 r r n n 15

Substituting into the rotor equations: ∂ × = Σ ∞ ∂ × E r E r n ( ) ( ) ( ) / ! e ik n = r r 0 0 n n ∞ = = Σ H r H r n ( ) ( ) ( ) / ! ikZ ikZh ik n = 0 0 n n ∂ × = Σ ∞ ∂ × H r H r n ( ) ( ) ( ) / ! h ik n = r r 0 0 n n ∞ = − = − Σ E r E r n ( ) / ( / ) ( ) ( ) / ! ik Z ik e Z ik n = 0 0 n n Choosing equal powers of ik in the power-series: ∂ × = E r H r ( ) ( ) n − r 1 n n ∂ × = − = H r E r ( ) ( ), / n e h Z − r 1 0 0 n n 16

What is a power-series? Given a series with variable x = Σ ∞ n ( ) f x x a = 0 n n To compute the coefficients a we need variable x : n = Σ ∞ = + + + + ⋅⋅⋅ 2 3 n n ( ) + f x x a a xa x a x a x a = 0 0 1 2 3 n n n = = = 0 1 ( ) | ( ) | , ( ) | f x d f x a d f x a = = = 0 0 0 0 1 x x x x x = = 2 n ( ) / 2!| , ( )/ ! | d f x a d f x n a = = 0 2 0 x x x x n Power series proper cannot be defined with respect to constant parameters = ω is a constant parameter! / ik i c 17

POWER-SERIES METHOD Choosing equal powers of ik in the power-series: ∂ × = E r H r ( ) ( ) n − r 1 n n ∂ × = − = H r E r ( ) ( ), / n e h Z − r 1 0 0 n n 18

CONSISTENT MAXWELL SYSTEMS Helmholtz wave equation: ∂ × = ωμ ∂ × = − ωε ∂ ⋅ = ∂ ⋅ = E H H E E H , , 0 i i r r r r ∂ × ∂ × = ∂ ∂ ⋅ − ∂ = − ∂ E E 2 E 2 E ( ) r r r r r r ∂ + = 2 2 E ( ) 0 k r First consistent Maxwell system: ∂ + = ∂ × = ωμ ∂ ⋅ = ∂ ⋅ = 2 2 E E H E H ( ) 0, , 0 k i r r r r Second consistent Maxwell system: ∂ + = ∂ × = − ωε ∂ ⋅ = ∂ ⋅ = 2 2 H H E E H ( ) 0, , 0 k i r r r r 19

HELMHOLTZ EQUATION AND PLANE WAVES = k x =constant): ˆ A plane wave and its Taylor expansion ( k = ⋅ = Σ ∞ ˆ ⋅ k r E r e e k r i n n ( ) ( ) ( ) / ! e ik n = 0 0 0 n ∞ = = Σ = E r e e k x ikx n n ˆ ( ) ( ) ( ) / !, e ik x n k = 0 0 0 n Helmholtz equation applied to plane wave: ∂ + ⋅ = Σ ∞ ∂ + ˆ ⋅ = k r 2 2 e e 2 2 k r i n n ( ) ( ) ( )( ) / ! 0 k e ik k n = r r 0 0 0 n ∞ + = Σ + 2 2 2 2 ikx n n ( ) ( ) ( ) / ! d k e ik d k x n = 0 x n x = Σ ∞ − − + = 2 2 n n n ( ) ( ( 1) ) / ! 0 ik n n x k x n = 0 n NOT satisfied IDENTICALLY term by term 20

Equating powers of x PERMITTED, because x =variable! Only the whole series satisfies IDENTICALLY: ∞ + = Σ + 2 2 2 2 ikx n n ( ) ( ) ( ) / ! d k e ik d k x n = 0 x n x ∞ − = Σ − + 2 2 n n n ( ) [ ( 1) ]/ ! ik n n x k x n = 0 n ∞ − ∞ + = Σ − − Σ n n 2 n 2 n ( ) ( 1) / ! ( ) / ! ik n n x n ik x n = = n 0 n 0 = Σ ∞ + + + + 2 n n ( ) ( 2)( 1) /( 2)! ik n n x n = 0 n −Σ ∞ + = 2 n n ( ) / ! 0 ik x n = 0 n 21

A “trivial” recurrence relation: = = Σ ∞ = ikx n n ( ) ( ) ( ) / !, ( ) f x e ik f x n f x x = 0 n n n ∞ + = Σ + 2 2 2 2 n ( ) ( ) ( ) ( ( ) ( )) / ! d k f x ik d f x k f x n = 0 x n x n n ∞ = Σ − + = 2 n ( ) ( ( 1) ( ) ( )) / ! 0 ik n n f x k f x n = − 0 2 n n n = − 2 ( ) ( 1) ( ) d f x n n f x − 2 x n n = = 2 0, ( ) 0 n d f x 0 x = = 2 1, ( ) 0 n d f x 1 x 22

PLANE-WAVE (SOMMERFELD) INTEGRALS Plane wave integral solves Helmholtz equation: ∫ ˆ ⋅ ˆ = Ω k r E r g k ik ( ) ( ) e e d ˆ 0 k C β π = − ∞ /2 ∫ ∫ i Ω = β 1 d d π ˆ k β =− π + ∞ / 2 C i β π = α π = − ∞ / 2 ∫ ∫ ∫ i Ω = β α = α 1 , sin d d S d S α α π ˆ k 2 β =− π α = 0 C = + k k k , real k Complex contour C , complex i R I = + k k k i R I = ⋅ = ⋅ − ⋅ + ⋅ 2 k k k k k k k k 2 k i R R I I R I ⋅ = k k 0 R I 23

Taylor expansions and partial waves: ∫ ∫ ⋅ ˆ ˆ ∞ ˆ = Ω = Σ ⋅ Ω k r E r g k g k k r i n n ( ) ( ) ( ) ( ) ( ) / ! e e d e ik n d = ˆ ˆ 0 0 0 k n k C C ∫ ∞ ˆ ˆ = Σ = ⋅ Ω E r E r g k k r n n ( ) ( ) / !, ( ) ( )( ) e ik n d = ˆ 0 0 n n n k C Helmholtz equation solutions: ∫ ⋅ ∂ + = ∂ + ˆ Ω k r 2 2 E r 2 2 g k i ( ) ( ) ( ) ( ) k k e e d r r ˆ 0 k C ∫ ∞ = Σ ∂ + ˆ ˆ ⋅ Ω 2 2 g k k r n n ( ) ( ) ( )( ) / ! e ik k n d = r ˆ 0 0 n k C = Σ ∞ ∂ + = 2 2 E r n ( ) ( ) ( ) / ! 0 e ik k n = r 0 0 n n 24