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MMET 2002, Kiev, Ukraine, September 11, 2002

THE WIENER-HOPF FACTORIZATION METHOD FOR THE DIFFRACTION BY WEDGES HAVING ARBITRARY

APERTURE ANGLE

V. G.Daniele

Dipartimento di Elettronica Politecnico di Torino

C.so Duca degli Abruzzi 24, 10129 Torino (Italy)

email:daniele@polito.it

Acknowledgement

This work was supported by the “Mario Boella Institute on the Information and Telecommunication Advanced Technologies (ISMB)”and by the Italian Ministry of Education, University and Scientific Research (MIUR) under grants MM093227718

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AIM OF THIS WORK

-To solve diffraction problems involving wedges having arbitrary aperture angles with the classical Wiener-Hopf Technique

-To compare the Wiener-Hopf factorization method with the Sommerfeld-Malyuzhinets method

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OUTLINE

-Introduction and some previous results

-The Wiener-Hopf Equations for the Maliyuzhinets problem

-Factorization of the matrix kernel relevant to the Maliyuzhinets problem

- Equivalence of the Wiener-Hopf and the Maliyuzhinets solution

- Comments

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-Introduction and some previous results (MMET 2000, ICEAA 2001)

In this author’s opinion the Wiener-Hopf technique is the more general and powerful analytical method to solve diffraction problems in presence of geometrical discontinuities. However, as far as wedge problems are concerned, there is the belief that, except for particular aperture angles, these problem cannot be faced by Wiener-Hopf techniques. The more important motivation of his work was to introduce the W-H technique for dealing with wedges of arbitrary angles.

An example: PEC wedge (E-polarization).

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particular case: Φ=π

ß ß)()(

2mAV −=− ++ ηξ

ωµ)()(

2ηηξ

ωµ−=− ++ AV

22 ηξ −= k 22 ηξ −= k

η=m

dxexEV xjz∫

∞

+ =0

)0,()( ηη [ ] ρρρ ρρρ deHHmA mj∫

∞ −+ Φ−−Φ=−

0),(),()(

V+(η) (classical) plus function

A+(-m) generalized minus function , A+(−η) classical minus function

Φ−+Φ−= sincos 22 ηη km (Φ=π)

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particular case: Φ=π

Generalized Factorization Classic Factorization)(22 ηηξ gk =−=

Kernel to factorize in classic way:

)()()( 22 ηηηξ +−=−== gg gmgkg )()()( 22 ηηηηξ +−=−== cc ggkg

Kernel to factorize in generalized way:

Φ−=− 2

]cos[cos)( k

mArc

kmg g

π

Φ

−

−=+

2

]cos[sin

)(22

kArc

k

kg g η

π

ηη

)(22 ηηξ gk =−=

ηη +=− kgc )(

ηη −=+ kgc )(

η=mΦ−+Φ−= sincos 22 ηη km

Scalar kernel!

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Solution of the Generalized W-H equation: )()(2

mAV −=− ++ ηξωµ

V+ HhL = -w m2

k SinA ArcCosA- h

kEn

Eè k2- h2

R

CosA -2 p +2 ArcCosA hkE

n E -CosA 2 jon ERemarks:

- The generalized W-H solution coincides exactly with that obtainable by the Sommerfeld (Malyuzhinets) solution

- The particular case Φ=π (n=2) yields to the same solution obtained by the classic W-H technique for the half-plane

In the following:

R is a constant related to the incident field,

n=2Φ/π

By generalized factorization and decomposition of the involved functions (details reported in ICEAA -2001 ) one gets:

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Comments:

The W-H formulation yields generalized W-H equations that involve plus functions in the h-plane and minus functions in the m-planes. As it happens for classical W-H equations, the solution of the generalized W-H equations requires the concepts of additive and multiplicative decompositions. The theory of the (generalized) decomposition has been developed in the previous work (MMET 2000 and ICEAA 2001) .

New result presented in this Conference:

A mapping h=h(a) has been found for the wedges problems that reduces the generalized W-H equations in the h-plane to classical W-H equations in the a-plane

To give an indication of this reduction is efficient for solving wedge problems with the W-H technique in the following the Maliuzhinets problem will be considered.

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Malyuzhinets Problem

),(),( Φ=Φ ρρ ρHZE az

),(),( Φ−−=Φ− ρρ ρHZE bz

The generalized W-H equations are:

)()()( mYXG −+Φ =ηη

where:

)0,()0,(

)(ηη

η+

++ =

IV

X),(),(

)(Φ−−Φ−

=+

+− mI

mImY

Φ+Φ−== sincos)( ξηηmm 22 ηξ −= k

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( ) ( )

( ) ( )

++

++−

=Φ

ξµωξξ

ξµωξξ

η

bbbb

aaaa

nnZnnZ

nnZnnZG )(

Φ−Φ−== sincos)( ηξηnnba

ba Zn

,,

ωµ=

)()()( mYXG −+Φ =ηηThe equation

is not a classical Wiener-Hopf equation since is regular

in and not

)(mY−

0]Im[ ≤m 0]Im[ ≤η

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]][arccos[cos[)(k

kη

πηηη −

Φ−==

Remarkable result (one year of work).

The mapping :

reduces the equation: )()()( mYXG −+Φ =ηη

to the classical Wiener –Hopf equation planethein −η

)()())( ηηη −+Φ = YXGUsed Notations:

))(()( ηηη FF =

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)()()( ηηη gd YYY −−− +=

o

og RY

ηηη

−=⇒ − )(

)cos( oo k ϕπ

ηΦ

−=

To solve the W-H equation we separately consider the geometrical optic contribution:

oϕ

}cos( ojko

iz eEE ϕϕρ −=

Plane wave excitation with angle

0)sin(

)sin2o

oao

o

o EZZ

jR

ϕ

ϕπ

π−Φ+

ΦΦ

=

where: and by supposing the face in the shadow region

Φ−=ϕ

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o

oo

RGGX

ηηηηη

−= −

−Φ−+Φ+ )())())( 11

))(η+ΦGandG )(η−Φ

))()())( ηηη +Φ−ΦΦ = GGG

Wiener-Hopf solution:

where:

are the factorized matrices of ))(ηΦG

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Algebraic manipulations yield:

( )( )( )

−

−++

−=

+−

Φ 1001

)(0

01

11

)(0

01

1111

2)(

ηξµω

ηξµωξ

ηb

annnnZZ

nZZG

baba

ba

)()(

ηξ

µωη

−

= faµω

ηξη

)()( −= fb

)()(2

1)(ba

aa

ZZnZnn

f−

+−=η

The factorization of is reduced to that of: ))(ηΦG

1

1b

a

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planetheink −=−= − ηηξηξηξ )()(22

)(ηξ−follows from the factorization of

Standard procedures of scalar factorization yield:

2)(

ηηξ +=−k

)())((

)(ηξηηξ

ηξ−

+ =

Factorization of the matrix:

1

1b

a

)(21

)(2 ηηξ +=− k)(2 ηξ

ωµ

−

=ba

1

1b

a

Since a/b is a rational function of η

is a Daniele-Khrapkov matrix

:

It follows that

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The factorization of Daniele-Khrapkov matrices is reduced to scalar additive and multiplicative decomposition by a well known method:

See for example:

A.Büyükaksoy and A.H Serbest (1993), Matrix Wiener-Hopf Factorization Methods and Applications to Some Diffraction Problems,ch.6 in “Analytical and Numerical Methods in Electromagnetic Wave Theory”, M. Hashimoto, M.Idemen and O.A. Tretyakov Editors, Science House Co.,Ltd., Tokio,

Final results:

=

−+−+−

−+−

−+

++ )()(

2cosh)()(

2sinh

)(

)()(2

sinh)(

)()(2

cosh

)(1

1

ηξηµω

ηξηµω

µω

ηξ

ηξηµω

ηξ

µωηξη

µω

η

tt

tt

gb

a

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•

−

−

−

−

=

−

−−

− ]log[21

cosh]log[21

sinh)(

]log[21

sinh)(

]log[21

cosh)(

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a

b

a

b

a

b

a

b

ZZ

ZZ

ZZ

ZZ

gb

a

µω

ηξηξ

µω

η

•

−−−−−

−−−

−−

)()(2

cosh)()(2

sinh)(

)()(2

sinh)(

)()(2

cosh

ηξηµω

ηξηµω

µωηξ

ηξηµω

ηξµω

ηξηµω

tt

tt

where: )()(]log[)(

1)( ηη

ηξµωη +−

−

+=++

= ttnnnn

ta

b

( ))()(

4)( 2 ηηη +−=

++−

−= ggn

nnn

nnZZZZ

g ba

ba

ba

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In particular it is required the factorization of the functions:

+−

+

+=

+

n

nn

n

nn

n

nn bababa ,,,

and the decomposition of the function:

+−−−

−

++

+

++

=

=++

=

]log[)(

1]log[

)(1

]log[)(

1)(

a

b

a

b

a

b

nnnn

nnnn

nnnn

t

ηξηξ

ηξηµω

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Standard decomposition equations based on the Cauchy formula yield:

duuku

u

kn

nn baba

ηπ

ϑ

π +Φ=

+∫

∞

+cosh

sinh]]sinh[

sinarctan[

0

,,(3)

duuk

u

uu

knnnn ab

a

b

ηπ

ϑ

π

ϑπηξ +

Φ−

Φ=

++

∫∞

+− cosh2

sinh]

]sinh[

sinarctan[]

]sinh[

sinarctan[

2]log[

)(1

0

bababa kkn ,,,~

cossin ϑϑ ==

(4)

where:

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Comment:

The Wiener-Hopf solution require the evaluation of the integrals (3) and (4).

This evaluation do not requires the introduction of special functions since the integrals are very easy to evaluate numerically

COMPARISON WITH THE SOMMERFELD-MALYUZHINETS SOLUTION.

It is convenient to introduce the complex variables defined by:

wk cos−=η wwπΦ

=

It follows:

wk cos−=η

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In the plane w a cumbersome analytical procedure shows that the integrals (3) and (4) can be expressed in terms of the Malyuzhinets function : )(wΦΨ

)~

()~

()~

()~

( ,,,,

cos

,babababa

wk

ba wwwwnnn

ϑϑϑϑη

−Φ−Ψ+Φ−Ψ−Φ+Ψ+Φ+Ψ=

+ΦΦΦΦ

−=+

+Φ−Ψ−Φ−Ψ+Φ+Ψ−Φ+Ψ

+Φ+Ψ−Φ+Ψ+Φ−Ψ−Φ−Ψ

Φ

=

++

ΦΦΦΦ

ΦΦΦΦ

−=+− )~

()~

()~

()~

(

)~

()~

()~

()~

(log

]2

sin[

1]log[

)(cos aabb

aabb

wka

b

wwww

wwwwwnn

nnkϑϑϑϑ

ϑϑϑϑπηξ

η

These expressions allows to verify that the obtained Wiener-Hopf solution agrees completely with the Malyuzhinets solution.

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CONCLUSIONS AND COMMENTS

-This paper shows that the wedge problems solved with the Sommerfeld-Malyuzhintes method can be successfully solved also by the classical Wiener-Hopf technique.

- Even though it seems that the two methods have the same capability for solving wedge problems, it should be observed however that the extension of the W-H technique also to arbitrary angular regions, definitively establishes this last method as the most general and powerful method for solving field problems involving geometricaldiscontinuities. The young researchers interested to diffraction problems should invest major time for studying this fascinating technique.

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-The Sommerfeld- Maliuyzhinets approach requires the solution of difference equations. These equations do not formulate a closed mathematical problem and their solutions must satisfy additive conditions . Beside the difficulty to find general solutions of the difference equations there is the difficulty to select (if it is possible) the solution satisfying the additive regularity conditions.

-The Wiener-Hopf technique involves closed mathematical problems constituted by the decomposition-factorization of functions.

To this author's opinion the superiority of Wiener-Hopf with respectSommerfeld-Malhyuzinets is more manifest when no closed solutions are possible and approximate techniques become necessary. In fact factorizations obtained with Pade’ or rational approximants face these problems better than approximate solutions of difference equations . (see next presentation: V.G.Daniele-R. Zich: Approximate factorizations for the kernels involved in the scattering by a wedge at skew incidence).