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THE WIENER-HOPF SOLUTION OF THE PENETRABLE WEDGE

PROBLEM V.G.Daniele

Dipartimento di Elettronica

Politecnico di Torino C.so Duca degli Abruzzi 24

10129 Torino (Italy) also at Istituto Superiore Mario Boella

Torino (Italy) daniele@polito.it

Website: http://personal.delen.polito.it/vito.daniele/

Rapporto ELT-2009-1 OCTOBER 2009

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Remarks: -In this Report there are present errors and omissions. - After 2009, a part of this Document has been improved and now it is published on four papers: Daniele (2010,2011a,2011b,2011c)). 1. Introduction The first rigorous studies of waves in presence of geometric discontinuities are due to Poincare' (1892) and Sommerfeld (1896). These two authors considered the diffraction by a half-plane immersed in the free space. The generalization of the half-plane problem leads to the wedge problem. It constitutes an important and challenging subject of applied mathematics in the last century. There is a vast number of papers that face the wedge problem. They concern many disciplines such as electromagnetism, acoustics, hydrodynamics, fracture mechanics and so on. The wedge can be classified into impenetrable and penetrable. If an impenetrable wedge is immersed in a homogenous isotropic medium, in the diffraction problem only one wave number is involved. The wedge problems with only one wave number are relatively simple to be studied. However in general we must face different wave numbers and moreover, as far as the penetrable wedge is concerned, there is always an additional complication, due to the presence of two or more different aperture angles. The Wiener-Hopf (W-H) formulation for impenetrable wedge problems has been exhaustively considered in the past by Daniele (2001a,2003a,2004b) and by Daniele & Lombardi (2006) so this Report only deals with the penetrable case or more precisely with the penetrable wedge problem that consists in evaluating the electromagnetic field when a plane wave propagating in the free space with permittivity oε ε= and permeability oµ , is skew incident to a wedge having a dielectric permittivity 1 r oε ε ε= and 1 r oµ µ µ= (Fig.1). In particular the penetrable wedge will be called dielectric wedge when 1 oµ µ= . The more interesting attempt to solve the dielectric wedge problem was that of Radlow (1964). This author obtained the solution relevant to the diffraction by the right-angled dielectric wedge. Unfortunately, it has been ascertained that this solution is wrong by Kraut & Lehaman (1969). The interest in the Radlow method comes from the fact that it is founded on an a general technique based on the solution of multidimensional Wiener-Hopf equations. However in these cases the factorization problem needs function-theoretic techniques employing two complex variables, and the difficulty to work on a two-dimensional complex plane easily introduces errors (Albani, 2007). The difficulty of solving the dielectric wedge problem has suggested many techniques of solution. For instance sophisticated extensions of the geometrical optics ( Morgan & Rawlins, 2007; Kim, 2007) and other advanced approaches have been proposed (Zavadskii, 1966; Aleksandrova & Khiznyak,1975; Lewin & Sreenivasiah, 1979; Bates,1985; Wojcik ,1995). Numerical solutions for this problem have been provided by Balling 1973a,b; Vasil'ev & Solodukhov, 1974; Wu & Tsai,1977; Stratis & Anantha and Taflove, 1997. Many of the above solutions do not consider the dielectric wedge problem as a canonical geometry. This is unsatisfactory since it is only the solution of the canonical problem that provides a complete and rigorous theory for evaluating the

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several different components that are present in the wedge field: reflected and refracted plane waves, surface waves , lateral waves, diffraction coefficients. In the author’s opinion, the more significant results obtained for the penetrable wedge geometries arise from the reduction of the problems to integral equations (IE) both one-dimensional and two-dimensional. These equations have been formulated both in the space domain and in the spectral domain (Vasil'ev & Solodukhov, 1974; Rawlins, 1977,1999; Berntsen, 1983; Kim & Ra and Shin, 1991a,b; Marx, 1993; Budaev, 1995; Crosille & Lebeau, 1999; Fujii, 1994; Osipov,1999; Buldyrev & Lyalinov, 2001; Gautesen, 2001; Daniele & Graglia,2001; Tyzhnenko, 2002; Daniele, 2005; Salem, Kamel & Osipov, 2006). Several techniques were used to find a solution and some of them are based on regularization methods . In the spatial domain the classical formulation of the dielectric wedge problem in terms of IE was obtained by (Vasil'ev & Solodukhov, 1974). These authors provided an extension of the Fredholm IE method of the scattering problem by finite bodies ( Muller ,1969) by introducing the concept of non uniform current (Ufimtsev,1962). However in these works the considered geometry is substantially not a canonical one. In fact, the considered wedge presents a finite length and moreover, to satisfy Liapunov’s condition on the contour of integration of the IE, the corners have been assumed as rounded edges with finite curvature radii. The formulation in the spectral domain has been considered by many authors. Some of these works are very important and produced significant contributions to find the solution of the penetrable wedge canonical problem. In particular in the Monograph (Crosille & Lebeau, 1999) the problem is formulated in terms of singular integral equations in the Fourier domain; these equations were successfully solved by using the Garlekin collocation method. Alternatively the Monograph (Budaev, 1995]) uses the very popular Sommerfeld-Malyuzhinets (S-M) representations (Osipov & Norris,1999; Buldyrev & Lyalinov, 2001). The difference equations that arise from this formulation are originally reduced to singular integral equations and successively the regularization method reduces them to Fredholm equations. In the spectral formulations the fields are defined through Laplace-Fourier, Sommerfeld -Malyuzhinets or Kontorovich-Lebedev representations. It must be remarked that even though the three different representations are intimately related (Malyuzhintes 1958a,b; Daniele,2003b), they conceptually yield three very different mathematical problems: the function factorization in the case of Laplace-Fourier domain, the solution of difference equations in the case of Sommerfeld -Malyuzhinets domain, and the solution of algebraic linear equations (that in the more general cases have the form of IE) in the case of Kontorovich-Lebedev domain. The above three techniques present advantages and disadvantages and, when exact solutions are possible, the choice of one of them depends on the personal preferences of the single author. In most cases the wedge problems are not amenable to exact solutions; in these circumstances we can always resort to IE formulations. Starting from the above representations, the deduction of the IE can be more or less straightforward and also different difficulties can be encountered. This paper concerns the W-H formulation of the dielectric wedge problem. This formulation represents the unknowns with one dimensional Laplace transforms, but conceptually differs from the formulation used by Crosille & Lebeau, 1999. In effect the integral equations deduced by the above authors immediately derive from the

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classical convolutional IE in the space domain (Bowman & Senior and Uslenghi, 1969), when the unknowns are rephrased simply in the spectral domain. Conversely in the W-H formulation the introduction of new unknown function is fundamental, that is the value, in its complementary support, of the integral present in the spatial IE. By introducing this complementary function, in the Fourier domain we have algebraic equations. The basic problem becomes the factorization of a scalar or matrix kernel. It is only in the framework of approximate factorization technique that we resort to the introduction of IE approaches. The IE factorization is very convenient because it involves Fredholm IE of second kind (Daniele, 2004a). This is not surprising, because the WH equations can be reduced to Hilbert-Riemann problem and the standard regularization of the involved singular IE is discussed in Muskhelishvili,1953; Gahkov,1966; Vekua 1967. The W-H formulation can be directly accomplished in the spectral Fourier domain (Jones method). For the dielectric wedge problem it yields generalized W-H equations. (GWHE) that are not amenable to exact solution. According to the above considerations, in this work we get approximate factorization by reducing the solution of the GWHE to the solution of coupled Fredholm equations of second kind. The Fredholm IE obtained in this paper present a form that is not suitable for a direct solution with standard methods. In fact they consist of two coupled equations that are defined in two different complex planes α and β . To get classical Fredholm equations, this author introduced three successive steps: new complex planes u and v , an analytical continuation process and the application of the classical Cauchy formula. Despite the apparent complexity of this procedure the final form of the Fredholm equations is very simple. In particular the involved kernels present a well suited behavior that allows very quickly to obtain stable numerical solutions, by using simple quadrature scheme. The proposed quadrature scheme for the solution of the Fredholm integral equation introduces artificial singularities that derive from the approximation of exact integrals with finite sums. Consequently the approximate spectrum on the two faces of the wedge is accurate within certain limitations. When the evaluated spectrum is not sufficient to get the field on the faces and in general in every point of the physical space, instead of solving numerically the Fredholm IE with techniques different from quadrature schemes, a process of numerical continuation is suggested. The analytical continuation of numerical results is an old and yet persistent problem of applied mathematics that can be solved by resorting to recursive equations. In this paper these equations have been deduced directly from the generalized W-H equations by introducing the angular complex plane w and by taking into account the properties of the W-H unknowns in this plane. Besides enlarging the validity regions of the numerical solutions of the integral equations, the recursive equations provide the exact complete evaluation of the geometric optics field. For this task, analytical exact expressions are available that do not require the solution of the Fredholm equations and of course are in accord with the laws of geometric optics. This circumstance is to be expected since it is well known in the literature (Budaev, 1995; Crosille & Lebeau, 1999; Buldyrev & Lyalinov, 2001) . One of the drawbacks of the quadrature scheme, is that the integration intervals are assumed finite, whence the near edge field , according to the Watson lemma, could be inaccurate. In order to have sufficient high spectrum for accurately evaluating the near

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field, we should resort to other solution techniques different from quadrature schemes. For instance assuming a Neumann-Gebengauer representation of the field in the spatial domain and taking into account that the Laplace transforms of the Bessel functions are known, we can solve the GWHE by using the moment method in the spectral domain. Another method could be to resort to an iterative solution of the IE starting from the solution of the isorefractive wedge problem that is known. However the consideration of the best techniques to get the near field is beyond the scope of this work. In this report we will limit ourselves to some comments about the utilization of the results obtained with the quadrature method. This report is organized as follows: Section 2 deduces the GWHE of the dielectric wedge at normal incidence. After introducing useful mappings, in section 3 the GWHE are reduced to Fredholm equations of second kind. In this section we also consider the solution of some particular wedge geometries amenable to known exact solutions. It allows to ascertain the validity of our IE formulation. Section 4 presents the numerical solution of the Fredholm equations and provides representations of the unknowns in appropriate regions of the spectral domains. The efficiency, convergence, and validation of the obtained approximate solutions are illustrated in Section 5 through comparison between approximate and exact solutions of the problems solved in closed form in section 3. Since the spectral solutions are analytical functions, relative errors are presented in vertical and horizontal lines of the w -plane. Section 6 addresses the problem of enlarging the regularity strips of the representations obtained with the numerical solutions of the Fredholm IE. Section 7 deals with the evaluation of the electromagnetic field in the spatial domain. In particular in this section it has been shown that the geometrical optics contribution can be obtained without the necessity of solving the Fredholm IE equations. Finally a discussion on the difficulties encountered in the evaluation of the spatial near field is presented in this section too. Sections 8 12÷ report the formulation of the problem in the case of oblique incidence. In particular section 9 concerns with the formulation of the integral equations, section 10 with the evaluation of the starting spectra, section 11 with the analytical continuation and section 12 with the far fields. Appendix A succinctly deduces the Fredholm equations for every value of the incidence angle oϕ , starting from the GWHE. Appendix B deals with an application of the Cauchy integral that is fundamental for the aims of this work. Appendix C discusses the characteristics of the kernels and in particular their properties of compact operators. Appendix D presents the rotating waves representations of the spectra, Appendix E reports an discussion on the recursive equations that allow the analytical continuations, Appendix F reports the MATHEMATICA Code Program for the deduction of the diffraction coefficients in the general case. Appendix G reports the plots of the Diffraction coefficients for several values of the aperture angle Φ , the incident angle oϕ , the skewness angle β and the electromagnetic parameters rε and rµ of the penetrable wedge. Appendix H reports the complete MATHEMATICA code program that produces the plots of the diffraction coefficients reported in Appendix G. To conclude this introduction we remark that an important advantage of the GWHE formulation (as well as of the spectral method used by Crosille & Lebeau, 1999;) is that it can be extended to solve wedge problems involving anisotropic or bianisotropic

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media (Daniele, 2004b). Apparently this extension is not possible in the framework of the SM formulation, since the Sommerfeld representations only occur in media where the Helmholtz wave equation holds. For these general problems, even though new conceptually difficulties are not present, the GWHE yield Fredholm IE that are very complicated to solve. 2. Generalized W-H equations of the problem For simplicity’s sake, at first only the case of a plane wave incident in a plane perpendicular to the edge (normal incidence) is reported. Figure 1 illustrates the problem of the diffraction by a plane wave on a dielectric wedge having relative permittivity rε immersed in the free space with permittivity oε and permeability oµ .

Fig.1 : The dielectric wedge problem We only consider time harmonic electromagnetic fields with a time dependence specified by the factor tje ω which is omitted. For illustrative purposes, in this section the incident field is constituted by the E- polarized plane wave having the following longitudinal components:

cos( }o ojkiz oE E e ρ ϕ ϕ−= (1)

where oϕ is the angle of the direction of the plane wave and o o ok ω µ ε= . The general case of incidence will be considered in the sections 8,9,10 and 11. The two different angular regions in fig. 1 are defined by: Free space: 0 ϕ< < Φ

Dielectric wedge : ϕ πΦ < <

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The Wiener-Hopf technique for wedge problems is based on the introduction of the following Laplace transforms:

ρϕρϕη ρη deEV jzz ∫

∞

+ =0

),(),( , ρϕρϕη ρηρρ deHI j∫

∞

+ =0

),(),( (2)

where the subscript + indicates plus functions, i.e. functions having convergence half-planes that are upper half-planes in the η -plane. We also define minus functions ( )F η− the functions having convergence half-planes that are lower half-planes in the η -plane. To avoid the presence of singularities on the real axis η , the propagation constant ok will be assumed with negative (vanishing) imaginary part also in the presence of lossless media. In the following the Laplace transforms will be called the spectra. For the spectra relevant to the axial directions 0ϕ = and ϕ π= , and to the two faces ϕ = Φ and ϕ = −Φ , the following functional equations hold (Daniele,2001a, 2003a, 2004b) :

( ) ( ) ( ) ( )o o o a o aV I n V m I mξ η ωµ η ωµ+ + + +− = − − − − (3) ( ) ( ) ( ) ( )o o o b o bV I n V m I mξ η ωµ η ωµ+ + + ++ = − − + − (4)

1 1 1 1( ) ( ) ( ) ( )o a o aV I n V m I mπ πξ η ω µ η ωµ+ + + ++ = − − + − (5)

1 1 1 1( ) ( ) ( ) ( )o b o bV I n V m I mπ πξ η ω µ η ωµ+ + + +− = − − − − (6) where:

, ( ) ( , )a b zV m V m+ +− = − ±Φ , , ( ) ( , )a bI m I mρ+ +− = − ±Φ , ( ) ( ,0)o zV Vη η+ += , ( ) ( ,0)oI Iρη η+ += , ( ) ( , )zV Vπ η η π+ += , ( ) ( , )I Iπ ρη η π+ +=

and 1 πΦ = −Φ , 2 2( ) okξ ξ η η= = − , 2 21 1( ) r okξ ξ η ε η= = − ,

( ) cos sinm m η η ξ= = − Φ + Φ , ( ) cos sinn n η ξ η= = − Φ − Φ ,

1 1 1 1 1( ) cos sinm m η η ξ= = − Φ + Φ , 1 1 1 1 1( ) cos sinn n η ξ η= = − Φ − Φ

The branches of ξ and 1ξ are defined by: (0) okξ = , 1 1(0) r ok kξ ε= = . In the algebraic equations (3)-(6), the unknowns are plus functions in the η -plane and minus functions in the m -plane (equations (3) and (4)) or minus functions in the 1m -plane (equations (5) and (6)). The fundamental idea in the W-H technique is the separation for each of the above equations, in two members that are respectively plus and minus functions. To get it, the factorization of a function is an important tool. In particular, in order to simplify the form of the previous equations, we introduce the generalized factorization of the scalars (Daniele, 2001a, 2003a,2004b) 1 1, , ,n nξ ξ :

1 1 1 1 1 1, , ,n n n n n nξ ξ ξ ξ ξ ξ− + − + − + − += = = = where the subscript + means plus function in the η -plane and the subscript − means minus function in the m -plane or 1m -plane. The explicit expressions of the factorized functions can be obtained as described in (Daniele, 2003a, 2004b) and will be reported later. Simple algebraic manipulations on the equations (3)-(6) yield the following GWHE of the problem:

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1 1 2( ) ( ) ( )Y X m X mnξη −

+ + ++

= − − − (7)

12 1 1 2 1

1

( ) ( ) ( )Y X m X mnξη −

+ + ++

= − + −! ! (8)

3 3 4( ) ( ) ( )Y X m X mnξη −

+ + ++

= − − − (9)

14 3 1 4 1

1

( ) ( ) ( )Y X m X mnξη −

+ + ++

= − + −! ! (10)

where:

1 ( ) 2 ( )oY Vnξη η+

+ ++

= , 12

1

( ) 2 ( )zY Vn πξη η+

+ ++

= ,

[ ]1 ( ) ( ) ( )a bnX m V m V mξ−

+ + +−

− = − − + − , [ ]11 1 1 1

1

( ) ( ) ( )a bnX m V m V mξ

−+ + +

−

− = − − + −! ,

[ ]2 2( ) ( ) ( )oa bX m I m I mωµ

ξ+ + +−

− = − − − , [ ]2 1 1 121

( ) ( ) ( )oa bX m I m I mωµ

ξ+ + +−

− = − − −! ,

32( ) ( )o

oo

Y In kωµη η+ ++

= , 41 1

2( ) ( )oY In k πω µη η+ ++

= −

[ ]3 ( ) ( ) ( )a bo

nX m V m V mk−

+ + +− = − − − , [ ]13 1 1 1

1

( ) ( ) ( )a bnX m V m V mk−

+ + +− = − − −!

[ ]4 ( ) ( ) ( )oa b

o

X m I m I mkωµ

ξ+ + +−

− = − − + − , [ ]4 1 1 11 1

( ) ( ) ( )oa bX m I m I m

kωµξ+ + +

−

− = − − + −!

Notice that the two uncoupled systems (7), ( 8) and (9), (10) have the same form so in the following all the theoretical considerations refer only to the system (7), ( 8). 3. Reduction of the generalized W-H equations to Fredholm equations. System (7), (8) is simple , however this author was able to obtain a closed form solution of it only for very special cases. In particular, as far as penetrable wedges are concerned, these special cases are: the wedge with 1rε = , the wedge with 1 / 2πΦ = Φ = and, after a slight modification of the equations (7-10), the right isorefractive wedge (Daniele & Uslenghi,2000). In this report, to get the solution in the general case we will reduce the system (7),(8) to a Fredholm IE of second kind. To simplify this reduction, we introduce the mapping

cos arccos( )okkαη

πΦ = − −

for the the equation (7) and the mapping

11

1

cos arccos( )kkβη

π Φ

= − −

for the equation (8) . The importance of these mappings is

related to the fact that, in the complex planes α and β , we are dealing with classical W-H equations (Daniele,2001a, 2003a,

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2004b). In fact, taking into account that in these planes 1 1, , and n nξ ξ− − + + have the forms:

2okn α

+−

= , 11 2

kn β+

−= , nn

n−+

= , 11

1

nnn−

+

= ,

2ok αξ−+

= , 1 2ok βξ −+

= , ξξξ+−

= , 11

1

ξξξ+

−

= ,

we get:

1 1 2( ) ( ) ( )o

o

kY X X

kα

α α αα+ − −

+= −

− (11)

12 1 2

1

( ) ( ) ( )k

Y X Xk

ββ β β

β+ − −

+= +

−!!! !!! !!! (12)

where : 1 1( ) ( )Y Yα η+ += , 1 1( ) ( )X X mα− += − , 2 2( ) ( )X X mα− += − ,

2 2( ) ( )Y Yβ η+ +=!!! , 1 1 1( ) ( )X X mβ− += −!!! ! , 2 2 1( ) ( )X X mβ− += −!!! ! . To obtain approximate solutions of the equations (11) and (12) we will use the Fredholm factorization described in (Daniele,2004a; Daniele & Lombardi, 2006, 2007). By using the standard procedure indicated in the Appendix A , we get:

2

1 2

'( ')

'1( ) ( ) '2 '

, Im( ) 0( )

o o

o oo

o

oo

o

k kX

k kkX X d

jk

T

α αα

α ααα α α

π α αα

αα α

−∞

− − −∞

+ +− − −+ − + =

−−

= >−

∫ (13)

1 12

1 111 2

1

'( ')

'1( ) ( ) ' 02 '

k kX

k kkX X d

jk

β ββ

β βββ β β

π β ββ

−∞

− − −∞

+ +− − −+ + − =

−− ∫!!!

!!! !!! (14)

where: cos( )O o ok πα ϕ= −Φ

, 4o oT j E π= −

Φ

The equations (13) and (14) eliminate the plus functions 1 ( )Y α+ , 2 ( )Y β+

!!! and present several advantages. For instance, even though the integration lines are not finite, the kernels involved in the integrals are compact kernels Daniele,2004a. Besides, the singularities in the α -plane (β -plane) defined by the integral are only the branches

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points okα = ± ( 1kβ = ± ). It means that since the poles present in 2 ( ')X α− ( 2 ( ')X β−!!! ) do

not occur in these integrals (Daniele,2004a), the evaluation of geometrical optics does not require the solution of the integral equations (see sect.7). Finally, we experienced that the accuracy of the obtained numerical solutions considerably increased if one deforms the contour path constituted by the real axis of planeα − ( planeβ − ) into the straight line ( )α βλ λ that joins the points 1( )ojk jk− − and 1( )ojk jk (Daniele,2004a; Daniele & Lombardi, 2006). Unfortunately the above coupled equations have a form that is not suitable for a direct solution with standard known methods. In fact they are defined in two different complex planesα and β . To get classical Fredholm equations, in the following we will introduce three successive steps: new complex planes u and v , an analytical continuation process to eq.(14) and the application of the classical Cauchy formula. Despite the apparent complexity of this procedure, we get a final form that is simple. In particular the new kernels present a well suited behavior that allows to obtain stable numerical solutions very quickly, by using simple quadrature scheme. For facilitating function-theoretic manipulations that may be obscure in the ,α β -planes, we also introduce the angular complex variables 1,ow w defined by :

1 1cos , coso ok w k wα β= − = − (15) or

1 11

cos , coso ok w k wπ πα β= − = −Φ Φ

(16)

where : 1 11

,o ow w w wπ π= =Φ Φ

In order to have more simple notations, in the following we will suppress the subscript o in the variables w , w : ow w≡ , ow w≡ In the plane ,w w and 1 1,w w , the plus functions are all even functions and the following representations hold (Daniele, 2003a, 2004b):

1 1cos cosok w k wη = − = − , sinok wξ = − , 1 1 1sink wξ = − , cos( )om k w= +Φ ,

1 1 1 1cos( )m k w= +Φ , sin( )on k w= +Φ , 1 1 1 1sin( )n k w= +Φ , (17)

11 1cos , cos ,

2 2ow wn k n k+ += = 1

1 1sin , sin2 2ow wk kξ ξ− −= = ,

Besides, the straight lines and α βλ λ are represented by the vertical lines:

1,2 2

w j u w j vπ π= − + = − + where u and v are real.

Algebraic manipulations of (13) and (14) yield:

11

( )'

1 2 2'12( ) tan ( ) ( ') '

2 1 ( sinh cos )

u uo

u uu

o

ju e e j TP u P u P u duj ee j k j u

π

ππ ϕ

∞

+−∞

− + +− − = −

+− −Φ

∫ (18)

( )1

'

1 2 2'12( ) ( ) tan ( ) ( ') ' 0

2 1

v v

Q v vv

jv e e je v Q v Q v Q v dvj ee j

π

π∞

+−∞

− + += + + =

+− ∫ (19)

where:

1 1( ) ( cos )2oP u X k juπ

π+Φ = − − + +Φ

2 2( ) ( cos )2oP u X k juπ

π+Φ = − − + +Φ

(20) 1

1 1 1 1( ) ( cos )2

Q v X k jvππ+Φ = − − + +Φ

! , 12 2 1 1( ) ( cos )

2Q v X k jvπ

π+Φ = − − + +Φ

!

We observe that an analytical continuation of the integral in equation (19) makes 1( ) 0Qe v = valid for every value of v also outside the real axis. Hence it is convenient

to assume values of ( )v v u= such as :

11 1cos cos

2 2ok jv k juπ ππ πΦ Φ − + +Φ = − + +Φ

. It yields:

11

( 2 )cos2( ) 2arccos

2 r

j ujv v u

ππ π

ε

+ Φ

= = Φ − Φ

(21)

For these values of ( )v u we get:

, , 1( ) ( )a b a bV m V m+ +− = − , , , 1( ) ( )a b a bI m I m+ +− = − ;

taking into account the definitions of ,i iX X+ +! (see equations (7), (8), it yields:

1

1

1 1 1

( ( ) )2cosh( )sin[ ]

( ) ( ( )) ( )( )

2cosh( ( ))sin[ ]

jv uu

Q v Q v u P uju

v u

π

ππ

π

− + Φ− +Φ

= =− + Φ

− +Φ

(22)

2

1

2 2 22

1sin [ ( )]2 2( ) ( ( )) ( )1sin [ ( ( ))]

2 2o

Z juQ v Q v u P u

Z jv u

π

π

− += =

− + (23)

12

where 1 1

o r

ZZ ε

= is the ratio of the wedge impedance 11

oZ µε

= and the free space

impedance oo

o

Z µε

= .

Rewriting the equation (18) and taking into account the representations (22) and (23) in (19) we get the system:

( )'

1 2 2'

12( ) tan ( ) ( ') '2 1 ( sinh cos )

u uo

u uu

o

ju Te e jP u P u P u duj ee j k j u

π

ππ ϕ

∞

+−∞

− + +− − = −

+− −Φ

∫ (24)

( )

21 11 1 2

2

( ) '

2( ) '( )

( ) 1cosh( )sin[ ] ( ) ( ) sin [ ( )] ( )2 2 2 2tan 12cosh( ( ))sin[ ] sin [ ( ( ))]

2 2 21 ( ') ' 0

1

o

v u v

v u vv u

jv uu P u jv u Z ju P u

juv u Z jv u

e e j Q v dvj ee j

π ππ

ππ

π∞

+−∞

Φ Φ+ − + − ++

Φ Φ+ − +

++ =

+− ∫

(25)

Now we are facing two coupled equations that are defined in the same plane u . However, equation (23) cannot be used for substituting 2( ')Q v in the integral of equation (25) since 'ν must be real. For this task we use an alternative representation that derives from the Cauchy formula. By referring to using Appendix B , we get:

2

221

22 1

1

( )1 sin[ ( ) ] ( )sin [ ( )]

22 212 sin [ ( )] cos[ ( ) ] cos[ ( ) ]2 2 2 2o r

Q v

ju P uZ judu

Z jv ju jv

πππ

π π ππ επ π

∞

−∞

=Φ + +Φ− +Φ

= −Φ Φ− + + +Φ − + +Φ

∫(26)

The same procedure applied to the system (9), (10) yields the following equations for the unknowns 3( )P u and 4( )P u

( )'

3 4 4'

12( ) tan ( ) ( ') '2 1 ( sinh cos )

u uo

u uu

o

ju Ue e jP u P u P u duj ee j k j u

π

ππ ϕ

∞

+−∞

− + +− − = −

+− −Φ

∫ (27)

( )

1 13 1 4

( ) '

4( ) '( )

1 ( ) 1cos[ ( )]sin[ ] ( ) ( ) sin[ ( )] ( )2 2 2 2 2 2tan1 12cos[ ( ( ))]sin[ ] sin[ ( ( ))]

2 2 2 2 21 ( ') ' 0

1

o

v u v

v u vv u

jv uju P u jv u Z ju P u

jujv u Z jv u

e e j Q v dvj ee j

π π ππ

π ππ

π∞

+−∞

Φ Φ− + + − + − ++ +

Φ Φ− + + − +

++ =

+− ∫

(28)

where:

13

41

4 21

1

1 sin[ ( ) ] ( )sin[ ( )]22 2( ) 12 sin[ ( )] cos[ ( ) ] cos[ ( ) ]

2 2 2 2o r

ju P uZ juQ v du

Z jv ju jv

πππ

π π ππ επ π

∞

−∞

Φ+ +Φ− +Φ

= −Φ Φ− + + +Φ − + +Φ

∫

3 3( ) ( cos )2oP u X k juπ

π+Φ = − − + +Φ

, 4 4( ) ( cos )2oP u X k juπ

π+Φ = − − + +Φ

,

4 sin( )2o o oU j E π π ϕ= −

Φ Φ.

We verified that the equations (24-28) are exactly satisfied in the very few cases amenable of analytical solution. This check is important since it shows the correctness of the deductions of the equations (24-28) starting from the functional equations (3-6), The cases we have considered are the wedge having 1rε = , 1ok k= , 1oZ Z= (absence of the wedge in free space), the wedge having rε = ∞ (perfectly electric conducting

(PEC) wedge) and the plane wedge that is the dielectric wedge where 1 2π

Φ = Φ = . In

the following we report the exact solutions of these three cases but, for brevity’s sake, we will show neither their deduction starting from the GWHE, nor the cumbersome algebraic manipulations that verify that these exact solutions satisfy the IE. Some exacts solutions of the equations (3-6) a) The case of absence of the wedge: 1rε = ( π ϕ π− ≤ ≤ ):

( )( cos , )

cos coso

z oo o

EV k w jk w

ϕϕ ϕ+ − = −

− − , π ϕ π− ≤ ≤

( )

( )sin

( cos , )cos cos

o oo

o o o

EI k w j

k Z wρ

ϕ ϕϕ

ϕ ϕ+

−− = −

− − , π ϕ π− ≤ ≤

b) The case of the PEC wedge: rε = ∞ :

( cos , )

cos cos csc( )sin2 2 2

( ) ( )sin sin sin sin2 2 2 2

z o

oo

o oo

V k wwE w

jw wk

ϕπϕ πϕ ππ

π ϕ πϕ π ϕ πϕ

+ − =

Φ Φ Φ= −− + Φ + − + Φ Φ Φ Φ

, ϕ−Φ ≤ ≤ Φ

14

( cos , )

( ) ( )cos sin sin 2sin2 2 2 2

( ) ( )2 sin sin sin sin2 2 2 2

o

o oo

o oo o

I k w

w wEj

w wk Z

ρ ϕ

πϕ π ϕ π ϕ πϕπ

π ϕ πϕ π ϕ πϕ

+ − =

− + − + − Φ Φ Φ Φ = −− + Φ + − + Φ Φ Φ Φ

, ϕ−Φ ≤ ≤ Φ

c) The case of the plane dielectric wedge: 1 2π

Φ = Φ = , rε arbitrary:

( ) ( )

( cos , )

cos cos cos cos

z o

o o

o o o o

V k wE Ej j

k w k w

ϕ

ϕ ϕ ϕ ϕ

+ − =Γ

= − −− − + +

, 02πϕ< ≤

( )( )

( )( )

( cos , )

sin sincos cos cos cos

o o o o

o o o o o o

I k w

E Ej j

k Z w k Z w

ρ ϕ

ϕ ϕ ϕ ϕϕ ϕ ϕ ϕ

+ − =

− Γ += − +

− − + +

, 02πϕ< ≤

( )1 11 1

( cos , )cos cos

oz

t

T EV k w jk w

ϕϕ ϕ+ − = −

− − ,

2π ϕ π≤ ≤

( )

( )1 11 1 1

sin( cos , )

cos coso t

t

T EI k w j

k Z wρ

ϕ ϕϕ

ϕ ϕ+

−− = −

− − ,

2π ϕ π≤ ≤

where: 1arcsin sin( )t o

r

ϕ ϕε

=

1

1

cos cos

cos cos

o

t o

o

t o

Z Z

Z Zϕ ϕ

ϕ ϕ

−Γ =

+, 1T = + Γ

The elimination of 1( )P u in the equations (24),(25) and that of 3( )P u in the equations (27),(28) yields two decoupled scalar Fredholm integral equations of second kind in the unknowns 2( )P u and 4 ( )P u . For selected values of the aperture angle of the wedge (for example for right wedges) the kernel of these equations is a simple function of and 'u u . However, these equations are not reported here since, in order to obtain numerical solutions, it is better to solve the original systems constituted by the equations (24-28). For illustrative purposes in this section we have considered

15

/ 2oϕ < Φ that implies Im( ) 0oα > . The case / 2oϕ > Φ requires a slight modification of known terms of the Fredholm equations that is reported in Appendix A.

4. Spectra of Wiener-Hopf unknowns.

In order to solve Fredholm equations of second kind, efficient approximate methods that are well known in literature are available. In particular we obtained very accurate numerical results by using a simple quadrature scheme in the integrals present in the two systems constituted by the equations (24-28). Before proceeding this way it is interesting to observe that considering the normalized values / okη η= , / om m k= ,

1 1 / om m k= , the normalized unknowns ( ) ( )i o iY k Yη η= , , ,( ) ( )a b o a bV m k V m+ +− = − ,

, ,( ) ( )a b o a bI m k I m+ +− = − , , 1 , 1( ) ( )a b o a bV m k V m+ +− = − , , 1 , 1( ) ( )a b o a bI m k I m+ +− = − , satisfy the generalized W-H equations of our problem. This means that if we obtain a solution

( )F η assuming a given arbitrary value of k , the functions ( ) ( )oo o

k kF Fk k

η η=

represents the solution when the propagation constant assumes the different value ok . At the same time if ( )f ρ is the inverse Laplace transform relevant to k , the function

( )okfkρ is the inverse Laplace transform relevant to ok . This fact allows us to assume a

suitable value of k for the numerical solutions for the IE. In the following we will assume 1k j= − . Besides, in order to simplify the numerical procedure, we will

consider only acute wedges that yield aperture angle 2π

Φ ≥ .

A simple quadrature scheme for the integrals defined in the equations (24)-(28) yields the sampled equations:

( )/

1 2 2/

2( ) tan ( ) ( )2 1

, 0, 1,..,( sinh cos )

h r h iA h

h r h ih ri A h

o

o

jh r h e e jP h r P h r P h ij ee j

T Arhk j h r

π

π

π ϕ

+=−

− + +− − =

+−

= − = ± ±−

Φ

∑ (29)

( )1 1 1

11 1

21 11 1 2

2

/( )1

2 1( )( )/

( ) 1cosh( )sin[ ] ( ) ( ) sin [ ( )] ( )2 2 2 2tan 12cosh( ( ))sin[ ] sin [ ( ( ))]

2 2 2

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

o

A h h iv h r

v h r h iv h ri A h

jv h rh r P h r jv h r Z jh r P h r

jh rv h r Z jv h r

h e e j AQ h i rj e he j

π ππ

ππ

π +=−

Φ Φ+ − + − ++ +

Φ Φ+ − +

++ = = ± ±

+− ∑

(30)

16

2 1

2/ 2

12

2 1 1/1 1

1

1

( )1 sin[ ( ) ] ( )sin [ ( )]

22 2 ,2 sin [ )] cos[ ( ) ] cos[ ( ) ]

4 2 2 2

0, 1,...,

A h

i A hor

Q h r

jh i P hijh ih Zh rZ j jh i jh r

Arh

πππ

π π ππ επ π

=−

=Φ

+ +Φ− +Φ= −

Φ Φ− + + +Φ − + +Φ

= ± ±

∑ (31)

where ,A h and 1 1,A h are truncations and step parameters for the integrals in 'u and 'v . We observe that the as 1,A A →∞ and 1, 0h h → , the numerical solution converges to the exact solution (Kantorovich & Krylov, 1964). Consequently 1,h h have to be chosen as small as possible and 1,A A have to be chosen as large as possible. The kernel

1

u

v ue je +

++

presents a well suited behavior that allows to obtain stable numerical solutions

very quickly. For instance by assuming 1 10A A= = , 1 0.05h h= = in all our numerical simulations, we get stable solutions that provide very accurate values of the samples

( )rP h i and ( )rQ h i ( 1,..,4r = ). In fact, when a comparison with exact results was possible, we ascertained relative errors inferior to 2 -3 % . From the samples 2,4 ( )P hi and 2,4 ( )Q hi and taking into account the equations (7)-(10), (18),(19), after algebraic manipulations we get:

/

2/

( cos ,0)

sin( )( ), ( )

4sin 2 (cos cos )

z o

A ho

ai A ho

o

V k w

wk h j TM w hi P h ik w j k w

ππ

π ππ ϕ

+

=−

− =

Φ Φ= − − + − Φ − Φ Φ

∑ (32)

/

4/

( cos ,0)

cos( )2 ( ), ( )

2 2 (cos cos )

o

A ho

ai A ho o

o

I k w

wk h j UM w hi P h ik Z j k w

ρ

ππ

π ππ ϕ

+

=−

− =

Φ Φ= − + − Φ − Φ Φ

∑ (33)

1 1

1 1

1 /1 11

1 1 1 2/1 1

sin( )( cos , ) ( ), ( )

4sin 2

A h

z ai A ho

wk h jV k w M w hi Q hik w j

πππ

π+=−

ΦΦ− = − + Φ ∑ (34)

1 1

1 1

1 /1 11

1 1 1 4/1 1

cos( )2( cos , ) ( ), ( )2 2

A h

ai A ho

wk h jI k w M w hi Q hik Z jρ

πππ

π+=−

ΦΦ− = − + Φ ∑ (35)

where :

17

( )'

'1( , ')

1

u u

a u uu

e e jM u uj ee jπ +

+=

+−

Since in the representations (32),(33) the intervals of integrals are truncated, we can have not small relative errors for high values of Im[ ]w . These errors do not influence the far field since it requires the accurate evaluation of the spectra only for real values of w (see section 7). Instead, they are conceptually important for the near field, since Meixner conditions require Im[ ]w →∞ . However in practice we can ignore these conditions and in the η plane a band / 10 Im[ ] 3ok wη ≈ → ≈ could be sufficient also for an acceptable evaluation of the near field (see section 7). Similar considerations apply for the representations (34) and (35). In the following a strip Re[ ]a w b≤ ≤ of the complex plane w , where the relative error evaluated in the rectangle ( Re[ ]a w b≤ ≤ , 3 Im[ ] 3w− ≤ ≤ is sufficiently small, is called a “regularity strip” of the given representation. Sufficiently small means that in the points located in the interior of a regularity strip (but not too near the boundary of it), the relative error is less than 2 -3 % . The regularity strips of the representations (32)-(35)

are provided starting from the vertical lines Re[ ] (or Re[ ] )2 2

w wπ Φ= − = − and

11 1Re[ ] (or Re[ ] )

2 2w wπ Φ

= − = − until we cross the artificial singularities due to the

approximation of exact integrals with finite sums. We observe these artificial singularities are poles 1 0u hie + + = of ( , )aM u hi and in the plane w and 1w the nearest ones are located respectively in:

32

w j h iπΦ

= − Φ − and 11 1

32

w j h iπΦ

= − Φ − , / ,....., /i A h A h= − .

For small values of h these poles are densely distributed and consequently we have a vertical barrier that strongly limits the accuracy for of the representations (32)-(35) for

3Re[ ]2

w < − Φ and 1 13Re[ ]2

w < − Φ . However there are not other singularities in these

representations in the strips 3 Re[ ] 02

w− Φ < ≤ and 1 13 Re[ ] 02

w− Φ < ≤ . In fact the poles

/ 2u j wπ= → = −Φ and 1 1/ 2v j wπ= → = −Φ that are present in ( , ')aM u u and ( , ')aM v v do not occur since they arisen from the definitions of 2,4 ( )P u and 2,4 ( )Q u .

Whence the strips 3 Re[ ] 02

w− Φ < ≤ and 1 13 Re[ ] 02

w− Φ < ≤ are regularity strips of

( cos ,0)z oV k w+ − , ( cos ,0)z oI k w+ − and of 1 1( cos , )zV k w π+ − , 1 1( cos , )zI k w π+ − respectively. Let’s indicate with ˆ ( ) ( cos ,0)oV w V k w+ += − , ˆ ( ) ( cos ,0)oI w I k wρ+ += − ,

1 1 1ˆ ( ) ( cos , )zV w V k wπ π+ += − and 1 1 1

ˆ ( ) ( cos , )I w I k wπ ρ π+ += − the analytical continuations of the representations (32-35). Taking into account that these functions are even

18

functions of w and 1w (Daniele,2003a, 2004b) , we get the following expressions in the

strips 30 Re[ ]2

w≤ < Φ and 1 130 Re[ ]2

w≤ < Φ respectively:

ˆ ˆ ˆ ˆ( ) ( ), ( ) ( ) V w V w I w I w+ + + += − = − (36)

ˆ ˆ ˆ ˆ( ) ( ), ( ) ( ) V w V w I w I wπ π π π+ + + += − = − (37) In the following we define as the starting spectra of ˆ ( )V w+ , ˆ ( )I w+ and 1

ˆ ( )V wπ + ,

1ˆ ( )I wπ + the spectra given by the equations (32)-(37)). The starting spectra respectively

have the following regularity strips

3 3Re[ ]2 2

w− Φ < < Φ and 1 1 13 3Re[ ]2 2

w− Φ < < Φ (38)

To obtain the other W-H unknowns: the spectra on the faces ,ϕ ϕ= Φ = −Φ , we rewrite the GWHE equations (7)-(10) in the w and 1w planes by taking into account the expressions (17). We get:

ˆ2 sin ( )ˆ ˆ ˆ ˆ( ) ( ) sin( ) ( ) ( )

o

a b o a b

k w V w

I w I w k w V w V wωµ+

+ + + +

− =

= − +Φ − +Φ − +Φ +Φ + +Φ (39)

ˆ ˆ ˆ ˆ ˆ2 ( ) ( ) ( ) sin( ) ( ) ( )a b o a bI w I w I w k w V w V wωµ ωµ+ + + + + = +Φ + +Φ + +Φ +Φ − +Φ (40)

1 1 1

1 1 1 1 1 1 1 1 1 1 1

ˆ2 sin ( )

( ) ( ) sin( ) ( ) ( )a b a b

k w V w

I w I w k w V w V wπ

ωµ+

+ + + +

− =

= +Φ − +Φ − +Φ +Φ + +Φ " " " " (41)

1

1 1 1 1 1 1 1 1 1 1 1

ˆ2 ( )

( ) ( ) sin( ) ( ) ( )a b a b

I w

I w I w k w V w V wπωµ

ωµ+

+ + + +

=

= +Φ + +Φ + +Φ +Φ − +Φ " " " " (42)

where:

,

, 1 1 1

ˆ ( ) ( cos( ), )

( ) ( cos( ), )a b z o

a b z

V w V k w

V w V k w+ +

+ +

= − ±Φ

= − ±Φ" , ,

, 1 1 1

ˆ ( ) ( cos( ), )

( ) ( cos( ), )a b o

a b

I w I k w

I w I k wρ

ρ

+ +

+ +

= − ±Φ

= − ±Φ"

Taking into account the definitions of α , β and the equations (15)-(16), we have

1 1cos( ) cos( )ok w k w= . It yields :

, 1 ,ˆ( ) ( )a b a bV w V w+ +=

", , 1 ,

ˆ( ) ( )a b a bI w I w+ +="

(43)

19

Consequently, if we substitute w with w −Φ in equations (39),(40) and 1w with

1 1w −Φ in equations (41),(42) , we get a system of four equations in the four unknowns

,ˆ ( )a bV w+ , ,

ˆ ( )a bI w+ that yields the solutions:

1 1 1

1 1

1 1 1 1 1

1 1

ˆ ˆ( ) ( )ˆ ( )sin( ) sin( )

ˆ ˆsin( ) ( ) sin( ) ( )sin( ) sin( )

oa

o

o

o

Z Z I w I wV w

Z w Z w

Z w V w Z w V wZ w Z w

π

π

+ ++

+ +

−Φ − −Φ = ++

−Φ −Φ + −Φ −Φ+

+

(44)

1 1 1 1

1 1

1 1 1 1 1

1 1

ˆ ˆsin ( ) sin ( )ˆ ( )sin( ) sin( )

ˆ ˆsin sin( ) ( ) sin sin( ) ( )sin( ) sin( )

oa

o

o

w Z I w w Z I wI wZ w Z w

w w V w w w V wZ w Z w

π

π

+ ++

+ +

−Φ + −Φ= +

+

−Φ −Φ − −Φ −Φ+

+

(45)

1 1 1

1 1

1 1 1 1 1

1 1

ˆ ˆ( ) ( )ˆ ( )sin( ) sin( )

ˆ ˆsin( ) ( ) sin( ) ( )sin( ) sin( )

ob

o

o

o

Z Z I w I wV w

Z w Z w

Z w V w Z w V wZ w Z w

π

π

+ ++

+ +

− −Φ + −Φ = ++

−Φ −Φ + −Φ −Φ+

+

(46)

1 1 1 1

1 1

1 1 1 1 1

1 1

ˆ ˆsin ( ) sin ( )ˆ ( )sin( ) sin( )ˆ ˆsin sin( ) ( ) sin sin( ) ( )

sin( ) sin( )

ob

o

o

w Z I w w Z I wI wZ w Z w

w w V w w w V wZ w Z w

π

π

+ ++

+ +

−Φ + −Φ= +

+

− −Φ −Φ + −Φ −Φ+

+

(47)

where 1 1cos( ) cos( )ok w k w= or: 1

1 1( ), ( )w g w w g w−= = with:

cos( ) arccosr

wg wε

=

, 1

1 1( ) arccos cosrg w wε− = (48)

The properties of the function ( )g w and its inverse 11( )g w− are well known

(Budaev,1995; Buldyrev & Lyalinov, 2001). For instance they are odd functions of w and 1w . The branch points of the above functions are respectively

( )arcosh rj nε π± + , 1arccosr

nπε

± + , with n arbitrary integer. In the following we

assume as branch lines the segments that join the two branch points defined by the same n . For real values of w and 1w , the real parts of these functions are plotted in Figure 2 and 3.

20

Fig.2 . The function Re[ ( )]g w for real values of w ( 10rε = )

Fig.3 The function 1

1Re[ ( )]g w− , for real values of 1w ( 10rε = ) Taking into account the regularity strips of the starting spectra and that, for real values of

rε , it is 1(Re[ ( )])g g w w− = , we get that the starting representations of ,ˆ ( )a bV w+ , ,

ˆ ( )a bI w+ have the regularity strip:

1 11 1

5 5 5 5-min , Re[ ( )] Re[ ] min , Re[ ( )]2 2 2 2

g w g− − Φ Φ < < Φ Φ (49)

The numerous numerical simulations have confirmed the previous statement. The inversion (Bromwich integral) of a generic Laplace transform ( )F η+ requires the values of this function in its convergence region. By studying the mapping cosok wη = − we can ascertain that the convergence region of ( ) ( cos )oF F k wη+ += − is mapped in the strip Re[ ]wπ π− ≤ ≤ (Daniele, 2004;Daniele&Lombardi,2006.) In the following we define adequate, a representation whose regularity strip comprises the points π± . For instance the starting spectra ˆ ( )V w+ , ˆ ( )I w+ are adequate provided that 2 / 3πΦ > .

21

Concerning the face spectra given by (44)-(47), we observe that they are adequate

provided that 15( )2

g w < Φ for every value 0 Re[ ]w π≤ ≤ . Looking at fig. 2 we deduce

that the inequality 11 1 5arccos arccos

2r r

πε ε

−= − < Φ

holds. We have three cases:

15 3(or )2 5

π πΦ > Φ < , 14 (or )

5 5π πΦ < Φ > and 3 4

5 5π π< Φ <

In the first case the above inequality is satisfied for every value of 1rε > . In the second

case , since 1arccos2r

πε

<

, it is never satisfied. In the third case we must have

21

15cos2

rε >Φ

. For instance for 23πΦ = , we have adequate representations for the

starting face spectra, provided that 1.33rε > . Instead for 34πΦ = it occurs if

6.83rε > . The above limits are not mandatory when the values rε are very high. In fact in this

case the transmitted spectra 1ˆ ( )V wπ + and 1

ˆ ( )I wπ + are negligible and the regularity strip

for the starting face spectra is: 5 5Re[ ]2 2

w− Φ < < Φ (see fig. 7).

The above considerations show that for not sufficiently large values of rε or small aperture angles, the starting face spectra could be inadequate. To overcome this inconvenient, we could solve the Fredholm IE with techniques different from quadrature schemes. However in this report we suggest to resort to the process of analytical continuation presented in the section 6. In particular, due to the evenness of the spectra, the validity of a given representation in the interval Re[ ] 0B w− ≤ ≤ can be extended to the interval 0 Re[ ]w B≤ ≤ , by changing w with w− .

5. Validation of the numerical results

In all our numerous numerical simulations we verified that the functions ˆ ( )kV w+ ,

ˆ ( )k I w+ , 1ˆ ( )kV wπ + , 1

ˆ ( )k I wπ + , ,ˆ ( )a bk V w+ , ,

ˆ ( )a bk I w+ are independent on the value of the assumed k . As we said we checked analytically the validity of the integral equations in three problems amenable of exact solution. Concerning with the numerical simulations, to avoid troubles, the absence of the wedge has been modeled with a wedge of arbitrary aperture having 1.01rε = or 1 0.001r jε = − . However this very small modification always introduces the branch points okη = (for the free space) and 1kη = (for the wedge) in the numerical spectra. These singularities that are not present in the exact case

22

1rε = , in the 1,w w − plane, yield the poles 1,w w π= . The presence of these poles modifies the limits of the regularity strips (38) that become:

3 3min( , ) Re[ ] min( , )2 2

wπ π− Φ < < Φ and 1 1 13 3min( , ) Re[ ] min( , )2 2

wπ π− Φ < < Φ (50)

respectively. The above considerations do not apply for the other two problems. In particular the PEC wedge can be modeled without any trouble with a very high value of

rε (also real). For instance in the following the PEC wedge was numerically modeled with the wedge having 10000rε = . We increased the values of the integration parameters A and 1/ h in order to get stable numerical solutions. In all the worked case, we ascertained that choosing A=10 and h=0.05 is sufficient for this purpose. Beside all the simulations have confirmed the validity of the limits of the regularity strips (38) , (49) and (50) (when 1rε ≈ ) . We remark that even though it is possible to improve the accuracy of the obtained results by using more sophisticated quadrature schemes, the simple scheme adopted in this report is accurate enough for engineering applications We performed dozens of numerical simulations. To save space we will report only some of them. Since in the validity strips the numerical spectra and the exact ones are always indistinguishable , we will report only the plots of the relative differences.

fig.4 The relative error for real values of w relevant to the plane case 1 2

πΦ = Φ = ,

6oπϕ = , 2rε = .

In Fig. 4 , plots (1) and (2) respectively report the relative error versus real values of w , of the starting spectra ˆ ( )V w+ and ˆ ( )V wπ + in their regularity strips:

(1) ˆ ˆ( ) ( )( ) ˆ ( )

nV w V we wV w

+ +

+

−= , (2)

ˆ ˆ( ) ( )( ) ˆ ( )nV w V we w

V wπ π

π

+ +

+

−=

The subscript n refers to numerical evaluations. Notice the explosion of the relative

errors in the limits 13 32 2

w = ± Φ = ± Φ of the regularity strip.

23

Fig.5 The relative error for complex values of w relevant to the plane case 1 2

πΦ = Φ = ,

6oπϕ = ,

2rε = . In Fig. 5 , plots (2) and (3) report the relative error versus imaginary values of w ju= of the starting spectra ˆ ( )V w+ and ˆ ( )V wπ + . Plots (1) and (4) instead refer to values of w

that are complex: 4

w juπ= − + : (1)

ˆ ˆ( ) ( )4 4( )

ˆ ( )4

nV ju V jue u

V ju

π π

π+ +

+

− + − − +=

− +, (2)

ˆ ˆ( ) ( )( ) ˆ ( )nV ju V jue u

V ju+ +

+

−= (3)

ˆ ˆ( ) ( )( ) ˆ ( )nV ju V jue u

V juπ π

π

+ +

+

−= ,

(4) ˆ ˆ( ) ( )

4 4( )ˆ ( )

4

nV ju V jue u

V ju

π π

π

π π

π+ +

+

− + − − +=

− +

The subscript n refers to numerical evaluations. Notice that the errors are less than 0.03 provided that Im[ ] 3u < Fig. 6 compares the exact spectra on the faces a and b of a PEC wedge ( rε = ∞ ) with the numerical spectra in the same faces of a dielectric wedge where 10000rε = . This comparison has the aim of showing both the accuracy of the numerical solution of the equations (24)-(28) and the validity of modeling a PEC with high values of rε . Taking

into account that ˆ ˆ( ) ( ) 0ea ebV w V w+ += = where the subscript e means the exact evaluation of the spectrum for rε = ∞ , the plots reports the quantities:

(1) ˆ ˆ( ) ( )( ) ˆ ( )ea a

a

I w I we wI w

+ +

+

−= , (2)

ˆ ( )( ) ˆ ( )a

o a

V we wZ I w

+

+

= , (3) ˆ ( )( ) ˆ ( )b

o b

V we wZ I w

+

+

=

(4) ˆ ˆ( ) ( )( ) ˆ ( )eb b

b

I w I we wI w

+ +

+

−=

24

Fig.6 Comparison for real values of w of the spectra on the faces between the PEC wedge and the wedge with 10000rε = .

34π

Φ = , 3oπϕ = .

Fig. 7 differs from the fig.6 since now we assume: 56π

Φ = , 8oπϕ =

Fig.7 Comparison for real values of w of the spectra on the faces between the PEC wedge and the wedge

with 10000rε = . 56π

Φ = , 8oπϕ = .

We remark that even though in fig.7 we considered a small aperture angle ( 1 6π

Φ = ), the

comparison is very good also in this case since 10000rε = implies that the spectra

25

ˆ ( ( ))V g wπ + and ˆ ( ( ))I g wπ + are negligible (order of 510− for 1oE = ). As we said, at the

end of the previous section it yields the validity strip 5 5Re[ ]2 2

w− Φ < < Φ .

6. Enlargement of the regularity strips As will be seen in section 7, the point w π= − represents an important point where ˆ ( )V w+ and ˆ ( )I w+ must be evaluated very accurately. For instance if 2

3πΦ < , we need

an enlargement of the regularity strip (38) because it does not include w π= − . But also

if 23πΦ > the enlargement of the strip is recommended. In fact, taking into account the

equations (24) and (25), ˆ ( )V w+ and ˆ ( )I w+ should have the branch line constituted by the

vertical segment [ arcosh , arcosh ]r rj jπ ε π ε− − − + . Whence, even though the starting spectra are numerically accurate also in w π= − (generally the branch line contributions are negligible, see also sect.7) , the representations (32) and (33) are not conceptually acceptable since they do not present the above branch line. For 1

ˆ ( )V wπ + and

1ˆ ( )I wπ + , the same considerations apply. In particular taking into account that 1 2

πΦ <

the starting spectra 1ˆ ( )V wπ + , 1

ˆ ( )I wπ + are never adequate.

To enlarge the regularity strips of the longitudinal spectra ˆ ( )V w+ , ˆ ( )I w+ and 1ˆ ( )V wπ + ,

1ˆ ( )I wπ + we start from the equations (44)-(47) and put w w→ − . Taking into account

that all the involved plus functions are even functions, we can represent

, ,ˆ ˆ( ) ( )a b a bV w V w+ +− = , , ,

ˆ ˆ( ) ( )a b a bI w I w+ +− = with two alternative expressions. Comparing

these two expressions and taking into account that 1 ( )w g w= and 11( )w g w−= are odd

functions we get four recursive equations for the spectra ˆ ( )V w+ , ˆ ( )I w+ , ˆ ( )V wπ + , ˆ ( )I wπ + . These equations are:

[ ][ ]

[ ]

1

1

11

1

sin( 2 ) sin( ) sin( ( ))ˆ ˆ( ) ( 2 )sin( ) sin( ) sin( ( ))

2 sin( )sin( ( ) ) ˆ ( ( ) )sin( ) sin( ) sin( ( ))

o

o

o

o

w Z w Z g wV w V w

w Z w Z g wZ w g w V g w

w Z w Z g w π

+ +

+

− Φ −Φ − −Φ= − Φ +

−Φ + −Φ

−Φ −Φ −Φ+ −Φ −Φ

−Φ + −Φ

(51)

1

1

11

1

sin( ) sin( ( ))ˆ ˆ( ) ( 2 )sin( ) sin( ( ))

2 sin( ) ˆ ( ( ) )sin( ) sin( ( ))

o

o

o

Z w Z g wI w I wZ w Z g w

Z w I g wZ w Z g w π

+ +

+

−Φ − −Φ= − − Φ +

−Φ + −Φ−Φ

+ −Φ −Φ−Φ + −Φ

(52)

26

111 1 1 1 1

1 1 111 1 1 1 1 1

11 1 1 1 11 1

1 111 1 1 1 1 1

2 sin( )sin( ( ) )ˆ ˆ( ) ( ( ) )sin( ) sin( ( ) sin( )

sin( ( )) sin( )sin( 2 ) ˆ ( 2 )sin( ) sin( ( )) sin( )

o

o

o

Z w g wV w V g ww Z g w Z w

Z g w Z ww V ww Z g w Z w

π

π

−−

+ +−

−

+−

−Φ −Φ −Φ= −Φ −Φ +

−Φ + −Φ −Φ − −Φ− Φ − − Φ −Φ + −Φ

(53)

11 11 1 11

1 1 1 1 11

1 1 1 1 11 11

1 1 1 1 1

2 sin( )ˆ ˆ( ) ( ( ) )sin( ( )) sin( )

sin( ( )) sin( ) ˆ ( 2 )sin( ( )) sin( )

o

o

o

o

Z wI w I g wZ g w Z w

Z g w Z w I wZ g w Z w

π

π

−+ +−

−

+−

−Φ= −Φ −Φ +

−Φ + −Φ

−Φ − −Φ+ − Φ

−Φ + −Φ

(54)

The above equations provide an analytical continuation of the starting spectra. In particular it is important to enlarge the regularity strip of the starting spectra 1

ˆ ( )V wπ + and

1ˆ ( )I wπ + . In fact, it is the second limit of (38) that is responsible for the inadequacy of

the representations of the face spectra for small values of rε and/or of 1Φ . Taking into account the regularity strip of the starting spectra, the right limit of regularity strip of the equations (53) and (54) is the minimum of the values of 1w satisfying the inequalities:

1 17Re[ ]2

w < Φ

11 1

5Re[ ( )]2

g w− −Φ < Φ

Since we have numerically ascertained that the second inequality is always satisfied, it follows that the equations (53) and (54) represent transmitted spectra 1

ˆ ( )V wπ + and

1ˆ ( )I wπ + that are adequate, provided that 1

27πΦ > .

Regarding the spectra on the faces, we observe that after the enlargement of the regularity strips of the starting spectra 1

ˆ ( )V wπ + and 1ˆ ( )I wπ + , the same reasons considered

at the end of the section 4 yield the following conclusions:

If 79πΦ < , the representations (44)-(47) where 1

ˆ ( )V wπ + and 1ˆ ( )I wπ + are given by the

recursive equations (53) and (54) are always adequate. If 89πΦ > and the spectra

1ˆ ( )V wπ + and 1

ˆ ( )I wπ + are not negligible, they are never adequate; finally in the interval 7 89 9π π< Φ < they are adequate provided that

21

19cos2

rε >Φ

. For instance for

45πΦ = it must be 1.11rε > . For 5

6πΦ = it must be 2rε > .

27

The drawback of the enlargement of the regularity strip is that we are faced with expressions of the spectra that are not uniform. Beside these representations could be equivalent in some sub interval. In this case numerical experiments show that the analytical continuations are more accurate than the starting representations. For instance for , 1

ˆ ( )V wπ + , 1ˆ ( )I wπ + we could use:

- the starting representation for 1 1 11 1Re[ ]2 2

w− Φ ≤ ≤ Φ

- the equation (53) for 1 1 11 7Re[ ]2 2

wΦ ≤ < Φ

- the equation (53) with 1w substituted with 1w− , for 1 1 17 1Re[ ]2 2

w− Φ ≤ < − Φ

After we have enlarged the regularity strips of the transmitted spectra 1ˆ ( )V wπ + , 1

ˆ ( )I wπ + ,

we can enlarge the regularity strips of spectra of ˆ ( )V w+ , ˆ ( )I w+ using the equations (51)

and (52). We can repeat the above reasoning and if evaluate 1ˆ ( )V wπ + , 1

ˆ ( )I wπ + , through

the equations (53) and (54) we can enlarge the regularity strip of the ˆ ( )V w+ , ˆ ( )I w+ . It can

be shown that in the enlarged strip, the spectra ˆ ( )V w+ , ˆ ( )I w+ are adequate for every

value of rε , provided that 89πΦ < .

Using iteratively the equations (51)-(54) we can further enlarge the regularity strips. However for values of 1rε ≈ and/or very small 1Φ , this procedure can require several iterations. To overcome this drawback it should be more convenient to solve the Fredholm IE with other methods. For instance for value of 1rε ≈ we can adopt an iterative solution of the IE that starts from the exact values for 1rε = . Instead, for very small values of the complementary angle 1Φ perhaps it is better to model the dielectric wedge as a thick half-plane (Senior, & Volakis, 1995) and resort to the solution of this problem that is well known (Volakis & Senior, 1987). Using a similar procedure we can directly enlarge the regularity strips of the face spectra. We get:

11

1 11 1 1 1

( )

{[ ( 2 ) ( ( ( ) 2 ) sin( 2 ) ( 2 )

sin( ( ) 2 ) ( ( ( ) 2 ))}[ sin( ) sin( ( ))]

a

o o b b o b

b o

V w

k Z I w I g g w k w V w

k g w V g g w k w k g w

+

−+ + +

− −+

=

= − Φ − − Φ + − Φ − Φ + + − Φ − Φ +

(55)

( )

11 1

1 11 1 1

( )

{ ( ( ( ) 2 )sin( ) [ ( 2 )sin( ( )) sin( ( ))sin( 2 )

( 2 ) sin( )sin( ( ) 2 ) ( ( ( ) 2 )]}[ sin( ) sin( ( )) ]

a

o o b o b

b b o o

I w

k Z I g g w w k Z I w g w g w w

V w w g w V g g w Z k w k g w

+

−+ +

− −+ +

=

= − Φ + − Φ + − Φ ⋅

⋅ − Φ − − Φ − Φ +(56)

11

1 11 1 1 1

( )

{[ ( 2 ) ( ( ( ) 2 ) sin( 2 )

( 2 ) sin( ( ) 2 ) ( ( ( ) 2 ))}[ sin( ) sin( ( ))]

b

o o a a o

a a o

V w

k Z I w I g g w k w

V w k g w V g g w k w k g w

+

−+ +

− −+ +

=

= − − Φ + − Φ + − Φ ⋅ ⋅ − Φ + − Φ − Φ +

(57)

28

( )

11

11 1

1 1 1

( ) { ( ( ( ) 2 )sin( )[ ( 2 )sin( ( )) sin( ( ))sin( 2 ) ( 2 )

sin( )sin( ( ) 2 ) ( ( ( ) 2 )]}[ sin( ) sin( ( )) ]

b o o a

o a a

a o o

I w k Z I g g w wk Z I w g w g w w V w

w g w V g g w Z k w k g w

−+ +

+ +

− −+

= − Φ ++ − Φ − − Φ − Φ +

+ − Φ − Φ +

(58)

The equations (51)-(54) and (55)-(58) that have been deduced from the GWHE by working in the w-plane ( w is the parameter used in the SM representation) resemble the difference equations obtained by (Budaev 1995) (see also Buldyrev & Lyalinov. 2001) However in our work we only use these equations for providing analytical continuations of the starting representations. Moreover we use plus Laplace transforms instead of Sommerfeld functions. In this way there is the advantage of having even functions of w or 1w . For this reason we always deal with regularity strips that are symmetric with respect to the imaginary lines Re[ ] 0w = or 1Re[ ] 0w = ; in particular in order to increase the negative limit of the regularity strip, it is enough to increase the positive one. We also observe that to get the numerical continuation of the approximate solution of the IE, the use of the plane w is not mandatory. In effect Crosille & Lebeau, (1999) obtain recursive equations similar to (55)-(58) by working directly in the η -plane. 7. Field evaluation in every point of the space The above considered spectra only evaluate the W-H unknowns. Whence the problem we now are facing is the knowledge of the electromagnetic field in every point of the physical space. There are two methods to accomplish this task. The first method is based on the use of equivalent currents. These currents are the tangential fields on the faces and their spectra have been evaluated and discussed in the previous sections (see equations (44)-(47) and (55)-(58)). This method is very general and can be used for arbitrary wedge problems. It has been used in Crosille & Lebeau, 1999. (see also Ciarkowski & Boersma and. Mittra, 1984) For the sake of brevity the application of this method is beyond the aims of this report. In this report we will use the second method that is based on the use of suitable Sommerfeld representations. To make the evaluation symmetric in the free space and in the wedge, it is convenient to introduce the angular coordinate defined by 1ϕ π ϕ= − , in the interior of the wedge. By this way we have: Free space: 0 ϕ< < Φ

Dielectric wedge : 1 10 ϕ< ≤ Φ In the following we will consider only equations for observation direction in the free space (no subscript or subscript o). For observation directions in the dielectric wedge it is enough to substitute ϕ with 1ϕ , w with 1w , Φ with 1Φ , ˆ ˆ( ), ( )V w I w+ + with ˆ ˆ( ), ( )V w I wπ π+ + , ok with 1k and oZ with 1Z . Given the axial spectra ˆ ( )V w+ , ˆ ( )I w+ , the

following representations of the spectra ˆ ( , ) ( cos , )oV w V k wϕ ϕ+ += − ˆ ( , ) ( cos , )oI w I k wϕ ϕ+ += − hold (Daniele, 2003b):

29

( ) ( )

ˆ ( , )ˆ ˆ ˆ ˆ( ) ( ) sin ( ) sin ( )

2sino

V w

Z I w I w w V w w V w

w

ϕ

ϕ ϕ ϕ ϕ ϕ ϕ+

+ + + +

=

− − + + − − + + + = (59)

0 ϕ< < Φ

( ) ( )

ˆ ( , )ˆ ˆ ˆ ˆ( ) ( ) sin ( ) sin ( )

2o

o

I w

Z I w I w w V w w V w

Z

ϕ

ϕ ϕ ϕ ϕ ϕ ϕ+

+ + + +

=

− + + + − − − + + = (60)

Provided that the regularity strip of the spectra ˆ ( )V w+ , ˆ ( )I w+ is Re[ ]B w B− ≤ < , it follows that the regularity strip of the above representations is :

Re[ ]B w Bϕ ϕ− + ≤ < − . Given the spectra of ˆ ( , )V w ϕ+ , the exact electric field is given by the inverse Laplace transforms:

1( , ) ( , )2 r

jz zB

E V e dηρρ ϕ η ϕ ηπ

−+= ∫ , (61)

where rB is the Bromwich contour that encloses the singularities of ( , )zV η ϕ+ that are all located in an lower half-plane. In the following the points η of rB satisfy: Im[ ] okη < − . The wedge field comprises many kinds of waves, the representations (59) and (60) and their analytical continuations enable to separate these different components. For instance the poles of ˆ ( )V w ϕ+ ± and ˆ ( )I w ϕ+ ± produce the geometrical optics contributions, the zeroes of 1 sin( ) sin( ( ))oZ w Z g w+ produce the surface waves, the branch lines of ( )g w produce the lateral waves. For the representations in the interior of the wedge dielectric, similar considerations apply. A full evaluation of the field of the dielectric wedge problem is cumbersome and it will not be considered here. However it is interesting to recall briefly the Sommerfeld theory that provides the wedge field starting from the spectra in the w plane− . In particular we will report a slight modification of this theory that also allows to deal with near fields. By introducing in the eq. (61) the substitution cosok wη = − , we get:

cos( )

( )

1 ˆ( , ) ( , ) sin( )2

o

r

jk wz B

E V w e w dwρ

λρ ϕ ϕ

π+

+= − ∫ (62)

30

The line ( )rBλ is the image of rB in the w plane− . We can choose rB so that ( )rBλ

does not cut the branch lines present ˆ ( , )V w ϕ+ . These branch lines arise from the presence of the function ( )g w and taking into account the dependence on ϕ of the spectrum, they

are defined by the segments ( ) ( )[ arcosh , arcosh ]r rj n j nε π ϕ ε π ϕ− + ± + ± .

Whence ( )rBλ will be assumed as indicated in fig.8, where the branch points apw and

anw are defined by: ( )arcoshap rw j ε ϕ= + , ( )arcoshan rw j ε ϕ= − .

In order to evaluate the integral (62) we introduce the saddle points of the function cosojk wρ and the steepest descent path (SDP ) relevant to them. Both the saddle points

and the SDP do not depend on ϕ . The fig.8 reports the saddle point w π= − and the SDP relevant to this saddle point. This line is defined by:

2 2 2( ) ( )w w gd w j wπ= − + + where 2( )gd w is the gudermann function and 2w is the abscissa on the imaginary axis. On this SDP we get cos (1 ( ))o ojk w jk j h wρ ρ= − + where ( )h w is a continuous real function that assumes infinite negative values at the end points of the SDP (located in

32

jπ− − ∞ and 1 )2

jπ− + ∞ and has only a maximum in w π= − .

Fig. 8 The integration lines in the w -plane relevant to the evaluation of the electromagnetic field in the

free space in the case4πϕ = and 2rε = . In this case the vertical segments that constitute the branch

lines are crossed by the SDP relevant to saddle point w π= − .

31

For illustrative purposes, in the following we operate by assuming the values 2rε = and

4πϕ = of fig.8. Different values of rε and ϕ could modify the effects of the presence

of the branch lines. By warping the line ( )rBλ on the SDP we get:

( )

cos( ) cos( )

cos( ( ))( )

1 ˆ( , ) ( , ) sin( )2

ˆ ˆ( , ) sin( ) ( , ) sin( )2 2

ˆRes[ ( , )] sin( ( ))

o o

o o

p n

o i

i

jk k h wz o zSDP

jk w jk wo ob b

jk wo w w i

i

E k e V w e w dw

k kV w e w dw V w e w dw

jk V w e w

ρ ρ

ρ ρ

ρ ϕϕ

ρ ϕ ϕπ

ϕ ϕπ π

ϕ ϕ

−+

+ ++ +

++ =

= − +

− − +

−

∫

∫ ∫∑

(63)

where ,p nb are the line indicated in fig. 8 that enclose the half branch line relevant to the

branch points ( )arcoshbp rw j ε π ϕ= − − + and ( )arcoshbn rw j ε π ϕ= − − − located

in the region R that is the region between the two sides of the SDP and ( )rBλ .

The residues are to be evaluated in the poles ( )iw ϕ of ˆ ( , )V w ϕ+ that are in the region R. Similar considerations apply to the magnetic field. Concerning the evaluation of the electromagnetic field in the interior of the wedge we work in the 1w -plane and have a similar situation except that the branch lines of 1( )g w− are horizontal lines defined by:

1 1{ arccos Re[ ] arccos , Im[ ] 0}r r

n w n wπ ϕ π ϕε ε

− + ± ≤ ≤ + ± = where 0, 1,....n = ± .

In fig. 8 the branch lines cut the SDP. However in general, provided that 0ϕ ≠ , there is not discontinuity of the contribution of the SDP for the far field since the spectra are required only on the saddle point w π= − . Conversely the contribution on the SDP is always discontinuous when evaluate the field for 0ϕ = . To get the continuity we must add the integrals on pb and nb . Fortunately in the considered examples the above discontinuity is very small and it can be ignored (see figures (9-11). An important remark concerns the contributions of the poles. In fact we will later show that this contribution is known without the necessity of solving the IE. Taking into account these considerations, fig. 8 shows that the spectrum to be known is only that on the SDP. Moreover if we limit ourselves to the evaluation of the far field we only need the value of the spectrum in the saddle point w π= − . For the not uniform far field evaluation we get ( 1ok ρ >> ):

( )/ 4( )1 1ˆ ( , ) sin( ) ( , )2 2

o j kk h wjkz o oSDP

e V w e w dw E e Dk

ρ πρρ ϕ ϕ ϕπ π ρ

− +−+ =∫ (64)

where the diffraction coefficients ( , )oD ϕ ϕ is expressed by

32

( , )( , )oo

vDjEπ ϕϕ ϕ −

=

with ˆ( , ) ( , )sinzv w V w wϕ ϕ+= When a pole ( )iw ϕ is near the saddle point sw π= − we must use the uniform equation:

( )

( )

/ 4 2

1 ˆ ( , ) sin( )2

1 ( , ) ( )(2 cos )2 2

ok h wjkzSDP

j k iKP

o

e V w e w dw

v we F kk jE

ρρ

ρ π

ϕπ

π ϕ ϕρπ ρ

−+

− +

=

−=

∫ (65)

where ( )KPF α is the Kouyoumjian- Pathak transition function defined in Senior&Volakis (1995) . By changing ϕ , ( )iw ϕ could cross the SDP and equation (65) has a discontinuity that is compensated by a discontinuity in the plane waves contribution present in (63). The poles present in the region R provide the surface waves and the plane waves that constitute the optical geometrics contribution. Their number depends on the incident angle oϕ and observation angle ϕ . Whence we can have several different situations that can be completely described in specific problems. For instance given ϕ , for certain ranges of oϕ we can have only incident plane waves, incident plus reflected plane wave, incident plus transmitted plane waves and so on. Besides also the poles relevant to the plane waves could be complex for the presence of total reflections inside the wedge. For the elaboration of the spectrum ˆ ( )I w+ , the same remarks apply and, after we have

substituted ϕ with 1ϕ , w with 1w , Φ with 1Φ , ˆ ˆ( ), ( )V w I w+ + with ˆ ˆ( ), ( )V w I wπ π+ + , ok with 1k and oZ with 1Z , to the elaboration of the spectra in the interior of the wedge. In the following we will provide some further details about the three items:

a) Diffraction coefficients. b) Optical geometrical contribution c) Near field consideration

a) Diffraction coefficients Looking at the equation (64) that provides the diffraction coefficients in the free space, we need to evaluate ˆ ˆ( ) ( )V Vπ ϕ π ϕ+ +− ± = ∓ and ˆ ˆ( ) ( )I Iπ ϕ π ϕ+ +− ± = ∓ for every value of ϕ−Φ < < Φ . Let’s consider first the range 0 ϕ< < Φ . We begin by using the starting spectra for evaluating ˆ ( )V π ϕ+ − and ˆ ( )I π ϕ+ − . These spectra are not sufficient

for ˆ ( )V π ϕ+ + and ˆ ( )I π ϕ+ + and for these functions we resort to the analytical continuations (51), (52) by assuming the starting spectra in the second member of them. For negative values of ϕ , it is enough to change the roles of ˆ ( )V π ϕ+ − and ˆ ( )I π ϕ+ −

with that of ˆ ( )V π ϕ+ + and ˆ ( )I π ϕ+ + . The limitations of the above procedure can be determined through the study of the regularity strips (section 6); these strips concern

33

only values of Φ and rε and are defined by the inequalities: 3 / 2 0π − Φ ≤ and

15Re[ ( )]2

g π ϕ+ −Φ < Φ . Of course, in order to overcome these limitations, we can

always resort to the enlargement of the regularity strip by iteratively using the recursive equations (51)-(54). To the evaluation of the diffraction coefficient in the interior of the wedge, similar consideration apply. Fig. 9 reports some plots relevant to the diffractions coefficients expressed in dB for the

wedge with 23πΦ = . The condition 3 / 2 0π − Φ ≤ is satisfied. The condition

15Re[ ( )]2

g π ϕ+ −Φ < Φ is assured for every value of rε , provided that

0.7 1.45ϕ < Φ = . It is also assured in all the range ϕ <Φ , provided that 1.34rε > . The plots are singular in the direction 2 0.262oϕ ϕ π= Φ − − = that is relevant to the reflected waved from the face a and in the direction (2 ) 1.833oϕ ϕ π= − Φ + − = − that is relevant to the reflected waved from the face b.

Fig.9 The diffraction coefficients of a wedge with 23πΦ = ,

4oπϕ =

34

Fig.10 The diffraction coefficients of a wedge with 34πΦ = ,

8oπϕ =

Fig. 10 reports some plots relevant to the diffractions coefficients expressed in dB for the right wedge. For right wedges, the condition 3 / 2 0π − Φ ≤ is satisfied. The condition

15Re[ ( )]2

g π ϕ+ −Φ < Φ is assured in all the range ϕ <Φ , provided that 6.83rε > . It is

assured also for every value of rε , provided that 0.7 1.21ϕ < Φ = . The plots are singular in the direction 2 1.178oϕ ϕ π= Φ − − = that is relevant to the reflected waved from the face a and in the direction (2 ) 1.964oϕ ϕ π= − Φ + − = − that is relevant to the reflected waved from the face b.

Fig.11. The diffraction coefficient for a right wedge with 2 0.2r jε = − : (1) 9oπϕ = ,

(2) 0oϕ =

Fig. 11 compares two results of this report for the right wedge with the corresponding ones obtained by Vasil'ev & Solodukhov (1974, the plots 2 of fig.3). We assume

2 0.2r jε = − ; for this value of rε , the condition 15Re[ ( )]2

g π ϕ+ −Φ < Φ is assured

provided that 0.7 1.67ϕ < Φ = . The plot (2), reported in full line, considers the case 0oϕ = . The singularities of this plot are located in 2 1.571oϕ ϕ π= Φ − − = that is

relevant to the reflected waved from the face a and in the direction (2 ) 1.571oϕ ϕ π= − Φ + − = − that is relevant to the reflected waved from the face b. The

plot (1), reported in dashed line considers the case 9oπϕ = . The singularities of this plot

35

are located in 2 1.222oϕ ϕ π= Φ − − = that is relevant to the reflected waved from the face a and in the direction (2 ) 1.920oϕ ϕ π= − Φ + − = − that is relevant to the reflected waved from the face b. Taking into account the different definition of the diffraction

coefficient, in the range considered by Vasil'ev & Solodukhov ( 1.0473πϕ < = ) these

results completely agree. b) Optical geometrical contribution. Looking at equations (59) we observe that the poles ( )iw ϕ of ˆ ( , )V w ϕ+ arising from the

poles oiw of the axial spectra ˆ ( )V w+ and ˆ ( )I w+ : ( )i oiw wϕ ϕ= ±

Substituting in (63), we get: 0 ˆcos( ( )) cos( )o i o oi ijk w jk w jk ne e eρ ϕ ρ ϕ+ ± −= = ρi

where ˆ ˆ ˆcos sini oi oin x w y w= − ∓

It shows that the contributions of the poles in (60) represent plane waves having directions i oiwϕ = ∓ with unit vectors defined by ˆin . We remember that for a given value of ϕ , the poles ( )i oiw wϕ ϕ= ± to be considered are those located in the region R . It means that to evaluate the residues we must have representations valid in this region. For instance the starting spectra only provides the poles oiw located in the regularity strip (38). In general the adequate spectra are expressed by the recursive equations (51) and (52). These representations yield two type of pole: those arising from the spectra ˆ ( 2 )V w+ − Φ , ˆ ( 2 )I w+ − Φ , 1

ˆ ( ( ) )V g wπ + −Φ −Φ and 1ˆ ( ( ) )I g wπ + −Φ −Φ and those due to

zeroes of 1 sin( ) sin( ( ))oZ w Z g w−Φ + −Φ . The former are contributions of geometrical optics , the latter are surface waves. A very important observation on the integrals present in the IE is that they do not have the poles occurring in the unknowns of the integrand. This means that to get the poles and the relevant residues for all the spectra that are of interest in the wedge problem, we can ignore the values of these integrals; hence it is not necessary to solve the IE. For the field in the interior of the wedge, similar considerations apply. For instance let us consider the starting spectrum 1

ˆ ( )V wπ + . If we ignore the integral, there are not pole contributions in this spectrum. It should mean absence of geometrical transmitted field in the wedge. However this result is not absurd since the starting spectrum 1 1

ˆ ( , )V w ϕ+ is never adequate for all the value of 1ϕ . To get an adequate spectrum for the field transmitted inside the wedge, we must resort to the enlargement of the regularity strip as indicated in section 6. By using the recursive equations (53) and ignoring the integrals present in the starting spectra, we get:

36

( )

11 1 1

1 1 1 1 1

1 1 111 1 1

1 1 1 1 1 1 1

11 1 1

1 1 1 1 1

11 1 1

( )sin( )sin csc( )2ˆ ( , )

( )cos sin sin( ) sin( ( ))2 2

( )sin( )sin csc( )2

( )cos s2

og

oo o

o

o

g wjE Z w wV w

g wk Z w Z g w

g wjE Z w w

g wk

π ϕπ ϕϕ

π ϕ πϕ ϕ ϕ

π ϕπ ϕ

π ϕ

−

+ −−

−

−

+ −Φ+ −Φ

Φ= + + −ΦΦ + + −Φ + + −Φ Φ Φ

− −Φ− −ΦΦ

− −ΦΦ +Φ

( )11 1 1 1 1 1 1in sin( ) sin( ( ))

2o

oZ w Z g wπϕ ϕ ϕ− − −Φ + − −Φ Φ

The geometrical optics arises from the zeroes of 1

1 1 1( )cos sin2 2

og wπ ϕ πϕ− ± −Φ+

Φ Φ.

These poles are defined by:

11 1 1( )2 2 2

og wπ ϕ π πϕ− ± −Φ = ± − Φ Φ

or 1 1 1 ( )ow gϕ ϕ± −Φ = ± Φ − We must only consider the poles located in the region 1R . The number of them depends on the value of 1ϕ and oiϕ . For the sake of simplicity we suppose the presence of only a pole in this region. This pole is :

1ow = 1 1 ( )o otgϕ ϕ ϕ ϕ− +Φ + Φ − = − with: ( )ot ogϕ ϕ= Φ − Φ − . The evaluation of the residue in the above pole requires some algebraic manipulations.

Taking into account that 1

1 ( )sin[ ( )] sinrdg ug u u

duε

−− = , we get:

1 1 1

1 1

1

cos( ) cos( )1 1 1 1 1 1 1 1

cos( )

ˆ ˆRes[ ( , )] sin( ) Res[ ( , )] sin( )o ot

o o

ot

jk w jkgw o w o

jk

jk V w e w jk V w e w

T e

ρ ρ ϕ ϕ

ρ ϕ ϕ

ϕ ϕ+ + −+ +

+ −

− = − =

=

where:

1

1

2 sin( )sin( ) sin( )

o o

o o ot

E ZTZ Z

ϕϕ ϕ

Φ −=

Φ − + Φ −

The above expression represents the first transmitted waves in the wedge. Both the value of the direction otϕ of the transmitted wave and the transmission coefficients T are in accord with the laws of the geometrical optics. c) Near field

37

Using spectral formulations, generally the evaluation of near field is more difficult than that of the far field. Fortunately this evaluation is not very important in the practical application so usually it is not considered. However for impenetrable wedge problems amenable of closed form solutions, this study was accomplished by Osipov&Norris,1999. By using a very complete analysis the above authors were able to get the dominant terms of the near field. The difficulty to evaluate the near field is not present in the spatial IE formulation (Vasil'ev &Solodukhov, 1974, Marx,1993). In particular it has been ascertained that the Meixner conditions are satisfied (Budaev& Bogy, 2007). However the drawback of spatial domain IE is that we must assume wedges with finite length and moreover the corners are rounded edges with finite curvature radii. To remain in the framework of the method indicated in this report, we now indicate as to utilize the spectra in order to obtain the near field of a wedge. If 0ρ ≠ , the rapidly vanishing exponential ( )ok h we ρ present in the eq.(63) considerable limits the band of the significant spectrum. For instance if the strip

3 1Re[ ]2 2

wπ π− ≤ ≤ − , 3 Im[ ] 3w− ≤ ≤ is a regularity strip, numerical simulations show

that the near field can be accurately evaluated provided that ρ is greater than 1/10 of the wavelength. Conversely (0, )zE ϕ cannot be evaluated numerically since the integral on the SDP for

0ρ = requires the knowledge of the spectrum for infinity values of 2w . However this authors claims that the value (0, )zE ϕ coincides with that present in the isorefractive

wedge having the same 1ε of the dielectric wedge and 1o

r

µµε

= . This value is known

(Daniele & Uslenghi,2007). Even though a preliminary proof of this results has been shown for the right wedge (Daniele & Uslenghi,2000), at the moment the above statement has to be considered a conjecture. 8. The skew incidence case in presence of wedges with arbitrary rε and rµ In the presence of an incident plane wave with skew incidence:

zjjo

iz

ooo eeEE αϕϕρτ −−= }cos( zjjo

iz

ooo eeHH αϕϕρτ −−= )cos( (66) where oE and oH are known quantities, β is the angle between the incident direction in

and z , o o ok k ω µ ε= = , coso okα β= and 2 2sino ok kτ β α= = − , we have the following equations that relate the Laplace transforms of axial and tangential components of the electromagnetic field [Daniele,2003a; Daniele 2004b]

38

Fig.1 : The dielectric wedge problem (67)

2 2

( ,0) ( ,0) ( ,0) ( , ) ( , ) ( , )o o o oz z z z

mV I I n V m I m I mρ ρτ α η τ αξ η η ηωε ωε ωε ωε+ + + + + +− − = − − Φ − − Φ + − Φ

(68) 2 2

( ,0) ( ,0) ( ,0) ( , ) ( , ) ( , )o o o oz z z z

mI V V n I m V m V mρ ρτ α η τ αξ η η ηω µ ωµ ωµ ω µ+ + + + + ++ + = − − Φ + − Φ − − Φ

(69)

2 2

( ,0) ( ,0) ( ,0) ( , ) ( , ) ( , )o o o oz z z z

mV I I nV m I m I mρ ρτ α η τ αξ η η ηωε ωε ωε ωε+ + + + + ++ + = − − −Φ + − −Φ − − −Φ

(70)2 2

( ,0) ( ,0) ( ,0) ( , ) ( , ) ( , )o o o oz z z z

mI V V n I m V m V mρ ρτ α η τ αξ η η ηω µ ωµ ωµ ω µ+ + + + + +− − = − − −Φ − − −Φ + − −Φ

(71)

2 21 1 1 1 1 1

1 1 1 1 1 1 1 11 1 1 1

( , ) ( , ) ( , ) ( , ) ( , ) ( , )z z z zmV I I n V m I m I mρ ρ

τ α η τ αξ η π η π η πωε ωε ωε ωε+ + + + + +− − + − + − = − −Φ + − −Φ − − −Φ

(72)2 21 1 1 1 1 1

1 1 1 1 1 1 1 11 1 1 1

( , ) ( , ) ( , ) ( , ) ( , ) ( , )z z z zmI V V n I m V m V mρ ρ

τ α η τ αξ η π η π η πω µ ω µ ω µ ω µ+ + + + + +− − − − − − = − −Φ − − −Φ + − −Φ

(73)

2 21 1 1 1 1

1 1 1 1 1 1 1 11 1 1 1

( , ) ( , ) ( , ) ( , ) ( , ) ( , )z z z zmV I I n V m I m I mρ ρ

τ α η τ αξ η π η π η πωε ωε ωε ωε+ + + + + ++ + = − − Φ + − Φ − − Φ

39

(74)2 21 1 1 1 1

1 1 1 1 1 1 1 11 1 1 1

( , ) ( , ) ( , ) ( , ) ( , ) ( , )z z z zmI V V n I m V m V mρ ρ

τ α η τ αξ η π η π η πω µ ωµ ωµ ωµ+ + + + + +− + + = − Φ + − Φ − − Φ

where 1 1 1sin okα β α= = , 1 oη η= , 1 r oε ε ε= , 1 r oµ µ µ= 1 1 1 r r ok kω µ ε ε µ= =

2 2( ) oξ ξ η τ η= = − , 2 2o o okτ α= − , 2 2

1 1 1 e Okτ α λ τ= − = , 21 ( 1)csce r rλ ε µ β= + − 2 2

1 1 1 1( )ξ ξ η τ η= = − with the branch (0) oξ τ= ,

1 1(0)ξ τ= , ( ) cos sinm m η η ξ= = − Φ + Φ , ( ) cos sinn n η ξ η= = − Φ − Φ

1 1 1 1 1 1 1( ) cos sinm m η η ξ= = − Φ + Φ , 1 1 1 1 1 1 1( ) cos sinn n η ξ η= = − Φ − Φ Summing (67) , (69) and subtracting (71) by (73) we get:

2

2 ( ,0) ( ) ( ) ( )o oz vz i iz

mV n s m d m d mρτ αξ ηωε ωε+ + + += − − − − + −

21 1

1 1 1 1 11 1

2 ( , ) ( ) ( ) ( )oz vz i iz

mV n s m d m d mρτ αξ η πωε ωε+ + + +− = − − + − − −

Similar equations can be obtained by subtracting (67) , (69) and summing (71), (73) or by subtracting (72) , (74 ) and summing (68), (70) or by summing (72) , (74) and subtracting (68), (70) By using again the normalization considered in sect.2 ( notice that k must be substituted by oτ and 1k by 1τ )

1 1 1 1 1 1, , ,n n n n n nξ ξ ξ ξ ξ ξ− + − + − + − += = = = we get the four system ( 1,3,5,7i = ):

( ) ( ) ( 1)( ) ( ) ( )i i iY X m X mnξη −

+ + + ++

= − − − (75)

1( 1) ( ) 1 ( 1) 1

1

( ) ( ) ( )i i iY X m X mnξη −

+ + + + ++

= − + −! !

where: where:

( ) ( , ) ( , )ab b bs c a c a c+ + +− = − Φ + − −Φ ( ) ( , ) ( , )ab b bd c a c a c+ + +− = − Φ − − −Φ

a stands V , I ; b stands z , ρ ; c stands m , 1m ;

40

(76)

1 ( ) 2 ( ,0)zY Vnξη η+

+ ++

=

12

1

( ) 2 ( , )zY Vnξη η π+

+ ++

=

1 ( ) ( )vznX m s mξ−

+ +−

− = − −

2

2 21( ) ( ) ( )o o

i izmX m d m d mρ

τ αξ ωε ωε+ + +−

− = − − −

1

1 1 11

( ) ( )vznX m s mξ

−+ +

−

− = − −!

21 1

2 1 1 121 1 1

1( ) ( ) ( )oi iz

mX m d m d mρτ α

ξ ωε ωε+ + +−

− = − − −

!

2

32( ) ( ,0) ( ,0)o o

zo

Y I In ρ

τ α ηη η ηωε ωετ+ + +

+

= − −

21

41 11 1

2( ) ( , ) ( , )ozY I I

n ρα ητη η π η π

ωε ωετ+ + ++

= +

3 ( ) ( )vzo

nX m d mτ−

+ +− = − −

2

41( ) ( ) ( )o o

i izo

mX m s m s mρτ αωε ωετ ξ+ + +

−

− = − − −

13 1 1

1

( ) ( )vznX m d mτ−

+ +− = − −!

21 1

4 1 1 11 11 1

1( ) ( ) ( )oi iz

mX m s m s mρτ αωε ωετ ξ+ + +

−

− = − − −

!

5 ( ) 2 ( ,0)zY Inξη η+

+ ++

= −

16

1

( ) 2 ( , )zY Inξη η π+

+ ++

= −

41

5 ( ) ( )iznX m s mξ−

+ +−

− = −

2

6 21( ) ( ) ( )o o

v vzmX m d m d mρ

τ αξ ω µ ω µ+ + +−

− = − − −

1

5 1 11

( ) ( )iznX m s mξ

−+ +

−

− = −!

21 1

6 1 1 121 1 1

1( ) ( ) ( )ov vz

mX m d m d mρτ α

ξ ω µ ω µ+ + +−

− = − − −

!

2

72( ) ( ,0) ( ,0)o o

zo

Y V Vn ρ

τ α ηη η ηω µ ωµτ+ + +

+

= − +

21

81 11 1

2( ) ( , ) ( , )ozY V V

n ρτ α ηη η π η πω µ ω µτ+ + +

+

= +

7 ( ) ( )izo

nX m d mτ−

+ +− = −

2

81( ) ( ) ( )o o

v vzo

mX m s m s mρτ αω µ ω µτ ξ+ + +

−

− = − − −

17 1 1

1

( ) ( )iznX m d mτ−

+ +− = −!

211

8 1 1 11 11 1

1( ) ( ) ( )ov vz

mX m s m s mρατ

ω µ ω µτ ξ+ + +−

− = − − −

!

Applying the procedure indicate in section 3 (the only difference that k must be substituted by oτ , 1k by 1 e oτ λ τ= ), we get again the following integral equations: (77)

( 1)2 2 2 2

( ) ( 1)2 2

' ( ')'1( ) ( ) ' ( ),

2 '

o oi

o ooi i i

o

X

X X d nj

τ α τ α ατ α τ ατ αα α α α

π α ατ α

+ −∞

− + − −∞

+ + − − −+ + − =

−− ∫

42

( 1,3,5,7i = )

(78)

1 1 1 1( 1) 12 2 2 2

1 1 1 11 1( ) 1 ( 1) 1 12 2

1 11 1

' ( ')'1( ) ( ) ' 0

2 '

i

i i

X

X X dj

τ α τ α ατ α τ ατ αα α α

π α ατ α

+ −∞

− + − −∞

+ + − − −+ − + =

−−∫

!!!

!!! !!!

where ;

(79) cos( )o o oπα τ ϕ= −Φ

,

Notice that ,to avoid confusion with the variable β which defines the skewness angle of the incident plane wave, we used the new complex variable 1α .

(80)

10 201 2 2

30 403 2 2

50 605 2 2

7

4( ) [ , , ]

2 ( ) ( ) ( )

4 sin( )2( ) [ , , ]

2 ( ) ( )

4( ) [ , , ]

2 ( ) ( )

4( ) [ ,

2

ooi

oo o oo

oo

oo

o o oo

oo

oo o oo

o

j E R Rn If

j E R Rn If

j H R Rn If

jn If

πτ αα ϕ

α α α α α ατ απ π ϕ

τ αα ϕα α α α α ατ απ

τ αα ϕα α α α α ατ α

α ϕ

−Φ +Φ= < +− − −−

Φ +Φ Φ= < +− − −−

Φ +Φ= < +− − −−

−Φ= < 70 80

2 2

sin( )2 , ]

( ) ( )

oo

o

o o oo

H R Rπ π ϕ

τ αα α α α α ατ α

+Φ Φ +− − −−

,

0iR are defined in Appendix A. In particular we will rewrite equations (11) and (12) in the real coordinates u and v defined by :

(81)

1 1 1 1 1 11

cos cos cos2

cos cos cos2

o o ow w ju

w w jv

π πα τ τ τ

π πα τ τ τ

= − = − = − − + Φ = − = − = − − + Φ

And we get: (82) ( 1,3,5,7i = )

(83) 1 1/ 2( ) tan ( ) ( , ') ( ') ' 0

2i i ij vQ v Q v M v v Q v dvπ ∞

+ +−∞

− ++ + =∫

43

where:

(84) ( )'

'1( , ')

1

u u

u uu

e e jM u uj ee jπ +

+=

+−

( ) ( ( ))i in u n uα= and:

(85) ( ) ( cos )2i o i oP u X juπτ τ

π−Φ = − + +Φ

,

(86) 11 1( ) ( cos )

2i o iQ v X jvπτ τπ−Φ = − + +Φ

!

Again the first key step that allow to easily solve the equations (17) and (18) is to observe that (19) is valid for every value of v so that by imposing that:

(87) 11

( 2 )cos2( ) 2arccos

2 tr

j ujv v u

ππ π

ε

+ Φ

= = Φ − Φ

with 2

2 212 1 ( 1)csctr e r ro

τε λ ε µ βτ

= = = + −

we get: (88)

1

1

1

1

( ) ( )( ) ( )( ) ( )

( ) ( )

vz vz

vz vz

v v

v v

s m s md m d ms m s m

d m d mρ ρ

ρ ρ

+ +

+ +

+ +

+ +

− = −− = −− = −

− = −

,

1

1

1

1

( ) ( )( ) ( )( ) ( )

( ) ( )

iz iz

iz iz

i i

i i

s m s md m d ms m s m

d m d mρ ρ

ρ ρ

+ +

+ +

+ +

+ +

− = −− = −− = −

− = −

Or by putting: (89) ,

, , ,( ) ( )i vz i v zD u d mρ ρ+= − , ,

, , ,( ) ( )i vz i v zS v s mρ ρ+= −

,1 , , , 1( ) ( )i v

z i v zD v d mρ ρ+= − , ,1 , , , 1( ) ( )i v

z i v zS v s mρ ρ+= − we have: (90) , , , ,

1 , 1 , , ,( [ ]), ( [ ]) ( ), ( )i v i v i v i vz z z zD v u S v u D u S uρ ρ ρ ρ =

Looking at the expressions that relate ( ),iP u to ( )abs c+ − and ( )abd c+ − (see equations (85) and definitions reported in equations (75)), we are dealing with two groups of equations having the form:

44

1

7

2

8

( ) ( )( ) ( )

( )( ) ( )( )) ( )

vziz

a i

v

P u S uP u D u

h uP u D uP u S u

ρ

ρ

= ,

3

5

4

6

( ) ( )( ) ( )

( )( ) ( )( )) ( )

vziz

b i

v

P u D uP u S u

h uP u S uP u D u

ρ

ρ

=

where the matrices ( )ah u and ( )bh u are defined by:

11

22

32 33

41 44

0 0 00 0 0

( )0 0

0 0

a

aa

a a

a a

hh

h uh h

h h

= ,

11

22

32 33

41 44

0 0 00 0 0

( )0 0

0 0

b

bb

b b

b b

hh

h uh h

h h

=

11( 2 )2 ( )sin

2aj uh sech u ππ

+ Φ= − , 11 22b ah h= − , 22 11b ah h=

321 ( 2 )cos csc( ( 2 )) cos4 2b o

j uh Z j u πβ ππ

+ Φ= −

331sin csc( ( 2 ))4b oh Z j uβ π= − −

2

41

1 ( 2 )cos csc ( ( 2 )) cos4 2

bo

j uj uh

Z

πβ ππ

+ Φ−

= −

441 2 sin

sinhbo

jhZ j u

β=

+

221 ( 2 )( ( 2 ))sin4 2a

j uh sec j u πππ

+ Φ= −

232

1 ( 2 )cos csc ( ( 2 )) cos4 2a o

j uh Z j u πβ ππ

+ Φ= − −

332 sin

sinho

ajZh

j uβ

=+

41

1 ( 2 )cos csc( ( 2 ))cos4 2

ao

j uj uh

Z

πβ ππ

+ Φ−

=

44

1sin csc( ( 2 ))4

ao

j uh

Z

β π −= −

45

Similarly we have:

1 1

7 1

2 1

8 1

( ) ( )( ) ( )

( )( ) ( )( )) ( )

vziz

a i

v

Q v S vQ v D v

H vQ v D vQ v S v

ρ

ρ

= ,

3 1

5 1

4 1

6 1

( ) ( )( ) ( )

( )( ) ( )( )) ( )

vz

iz

b i

v

Q v D vQ v S v

H vQ v S vQ v D v

ρ

ρ

=

where the matrices , ( )a bH v have similar forms of , ( )a bh u and here are not reported. Taking into account (25) we get:

1 1

7 7

2 2

8 8

( ( )) ( )( ( )) ( )

( )( ( )) ( )( ( )) ( ))

ae

Q v u P uQ v u P u

h uQ v u P uQ v u P u

= ,

3 3

5 5

4 4

6 6

( ( )) ( )( ( )) ( )

( )( ( )) ( )( ( )) ( ))

be

Q v u P uQ v u P u

h uQ v u P uQ v u P u

=

where:

1( ) ( ( )) ( )ae a ah u H v u h u−= , 1( ) ( ( )) ( )be b bh u H v u h u−= The expressions of ( )aeh u and ( )beh u are the following:

11

22

32 33

41 44

0 0 00 0 0

( )0 0

0 0

ae

aeae

ae ae

ae ae

hh

h uh h

h h

= ,

11

22

32 33

41 44

0 0 00 0 0

( )0 0

0 0

be

bebe

be be

be be

hh

h uh h

h h

=

where:

oo

o

Z µε

= , 1r

or

Z Zµε

=

111

( 2 ) ( 2 ( ))cosh( )csc ( ( ))sin2 2ae

ju jv uh u sech v uπ ππ π

+ Φ + Φ=

122

1 ( 2 ) 1 ( 2 ( ))cos ( 2 )csc ( 2 ( ))sin4 2 4 2ae

ju jv uh ju sec jv uπ ππ ππ π

+ Φ + Φ= − −

46

32 1

21

1cos ( 2 )cos( )4

( 2 ) ( 2 ( )) ( 2 ) ( 2 ( ))[ cot cos csc ]csc2 2 2 4

ae

tr

h Z ju

ju jv u ju jv u

π β

π π π πεπ π π

= −

+ Φ + Φ + Φ −−

1

33( sinh )

( sinh ( ))tr

aeo

Z j uhZ j v uε +

=+

41

1

cos( )cosh( )( 2 ) ( 2 ( )) ( 2 ) ( 2 ( ))[ cot cos csc ]csc

2 2 2 42

ae

tr

o

h uju jv u ju jv u

Z

βπ π π πε

π π π

=+ Φ + Φ + Φ −−

44( 2 ( )) ( 2 )csc sin

4 4ae trjv u juh π πε − −

=

22 11be aeh h= , 11 22be aeh h= −

32 1 41be o aeh Z Z h= , 133 44be ae

o

Zh hZ

=

41 321

1be ae

o

h hZ Z

= −

44 331

obe ae

Zh hZ

=

The second key step is to observe that being in the integrals of (18) 'v real we can relate , , , ,

1 , 1 , , 1 ,( '), ( ') with ( '), ( ')i v i v i v i vz z z zD v S v D u S uρ ρ ρ ρ through the Cauchy formula

(Appendix B): (96) , , , ,

1 , 1 , 1 , 1 ,( ), ( ) ( , ) ( ), ( )i v i v i v i vz z z zD v S v T v u D u S u duρ ρ ρ ρ

∞

−∞ = ∫

with:

21

1

sin[ ( ) ]2( , )

2 cos[ ( ) ] cos[ ( ) ]2 2tr

juT v u

ju jv

ππ

π ππ επ π

Φ+ +ΦΦ

= −Φ Φ+ +Φ − + +Φ

It follows:

47

1 1

7 7

2 2

8 8

( ) ( )( ) ( )

( , )( ) ( )( ) ( ))

a e

Q v P uQ v P u

T v u duQ v P uQ v P u

∞

−∞= ∫ ,

3 3

5 5

4 4

6 6

( ) ( )( ) ( )

( , )( ) ( )( ) ( ))

be

Q v P uQ v P u

T v u duQ v P uQ v P u

∞

−∞= ∫

where:

1( , ) ( , ) ( ) ( )a e a aT v u T v u H v h u−= , 1( , ) ( , ) ( ) ( )be b bT v u T v u H v h u−= The expressions of ( , )a eT v u and ( , )beT v u are the following:

11

22

32 33

41 44

0 0 00 0 0

( , )0 0

0 0

ae

aea e

ae ae

ae ae

TT

T v uT T

T T

= ,

11

22

32 33

41 44

0 0 00 0 0

( , )0 0

0 0

be

bebe

be be

be be

TT

T v uT T

T T

=

observe that the above eight equations are two decoupled system of order four. The first system (system a) relate 1( ),P u 7 ( ),P u 1( ),P u 1( ),P u to ( )iQ u Taking into account the definition of ( )iP u ( )iQ u we have:

111

( 2 ) ( 2 )( , )cosh( )csc ( )sin2 2ae

ju jvT T v u u sech vπ ππ π

+ Φ + Φ=

122

1 ( 2 ) 1 ( 2 )( , )cos ( 2 )csc ( 2 )sin4 2 4 2ae

ju jvT T v u ju sec jvπ ππ ππ π

+ Φ + Φ= − −

32 1

21

1( , ) cos ( 2 )cos( )4

( 2 ) ( 2 ) ( 2 ) ( 2 )[ cot cos csc ]csc2 2 2 4

ae

tr

T T v u Z ju

ju jv ju jv

π β

π π π πεπ π π

= −

+ Φ + Φ + Φ −−

1

33( sinh )( , )

( sinh )tr

aeo

Z j uT T v uZ j vε +

=+

41

1

( , )cos( )cosh( )( 2 ) ( 2 ) ( 2 ) ( 2 )[ cot cos csc ]csc

2 2 2 42

ae

tr

o

T T v u uju jv ju jv

Z

βπ π π πε

π π π

=+ Φ + Φ + Φ −−

44( 2 ) ( 2 )( , ) csc sin

4 4ae trjv juT T v u π πε − −

=

48

22 11be aeT T= , 11 22be aeT T= −

32 1 41be o aeT Z Z T= , 133 44be ae

o

ZT TZ

=

41 321

1be ae

o

T TZ Z

= −

44 331

obe ae

ZT TZ

=

Taking into account the equations (90), (96) and the definitions (75) we get the two uncoupled systems of order four:

1 2 2 12( ) tan ( ) ( , ') ( ') ' ( )

2

juP u P u M u u P u du n u

π∞

−∞

− +− − =∫

7 8 8 72( ) tan ( ) ( , ') ( ') ' ( )

2

juP u P u M u u P u du n u

π∞

−∞

− +− − =∫

11 1 33 2 32 7 33 7

( )2( ) ( ) tan ( ) ( ) ( , ') ( ') ' ( , ') ( ') ' 0

2ae ae ae ae

jv uh u P u h u P u M u u P u du M u u P u du

π∞ ∞

−∞ −∞

− ++ + + =∫ ∫

22 7 44 8 41 1 44 8

( )2( ) ( ) tan ( ) ( ) ( , ') ( ') ' ( , ') ( ') ' 0

2ae ae ae ae

jv uh u P u h u P u M u u P u du M u u P u du

π∞ ∞

−∞ −∞

− ++ + + =∫ ∫

( , ') ( ( ), ') ( ', ') 'aei j aei jM u u M v u v T v u dv

∞

−∞= ∫ ,

, 1,2,3,4i j =

For instance as 2oϕΦ

<

49

1

3

5

7

4( )

sinh cos

4 sin( )2( )

sinh cos

4( )

sinh cos

4 sin( )2( )

sinh cos

oi

o

oo

o

o

o

oo

o

j En u

j u

j En u

j u

j Hn u

j u

j Hn u

j u

π

π ϕ

π ϕπ

π ϕ

π

π ϕ

π ϕπ

π ϕ

−Φ= −

−Φ

Φ Φ= −−

Φ

Φ= −−

Φ

−Φ Φ= −−

Φ

3 4 4 32( ) tan ( ) ( , ') ( ') ' ( )

2

juP u P u M u u P u du n u

π∞

−∞

− +− − =∫

5 6 6 52( ) tan ( ) ( , ') ( ') ' ( )

2

juP u P u M u u P u du n u

π∞

−∞

− +− − =∫

11 3 33 4 32 5 33 4

( )2( ) ( ) tan ( ) ( ) ( , ') ( ') ' ( , ') ( ') ' 0

2be be be be

jv uh u P u h u P u M u u P u du M u u P u du

π∞ ∞

−∞ −∞

− ++ + + =∫ ∫

22 5 44 6 41 3 44 6

( )2( ) ( ) tan ( ) ( ) ( , ') ( ') ' ( , ') ( ') ' 0

2be be be be

jv uh u P u h u P u M u u P u du M u u P u du

π∞ ∞

−∞ −∞

− ++ + + =∫ ∫

( , ') ( ( ), ') ( ', ') 'aeij aeijM u u M v u v T v u dv∞

−∞= ∫ ,

, 1,2,3,4i j =

9. Evaluations of the starting spectra on the axis z. From the equations (9), (17) and (18) we get the following representations of the plus functions

50

1,3,5,7i =

11( cos ) ( , ') ( ') ' ( )i o i io

Y w M u u P u du n uττ

∞

+−∞

− = + ∫

1 1 1 11( cos ) ( , ') ( ') 'i io

Y w M v v Q v dvττ

∞

+ +−∞− = ∫

Taking into account of the definitions of the ( )iY η , after many algebraic manipulations we get the following representations of the starting spectra :

1 71ˆ ( ) cos cot cot sin ( ) csc ( )2 2 2 o

w wV w w Y w Z Y wρπ πβ β+

= − + Φ Φ

11ˆ ( ) csc sin ( )4z

wV w w Y wπ+ = −

Φ

3 51ˆ ( ) cos csc ( ) cot cot sin ( )

2 2 2oo

w wI w Y w Z w Y wZρ

π πβ β+ = − + Φ Φ

51ˆ ( ) csc sin ( )4z

wI w w Y wπ+ =

Φ

1 11 1 2 1 8 1

1 1

1ˆ ( ) cos cot cot sin ( ) csc ( )2 2 2 o

e

w wV w w Y w Z Y wρππ πβ β

λ+

= − + Φ Φ

11 1 2 1

1

1ˆ ( ) csc sin ( )4z

wV w w Y wππ

+ = −Φ

1 11 4 1 1 6 1

1 1

1ˆ ( ) cos csc ( ) cot cot sin ( )2 2 2r o

o e

w wI w Y w Z w Y wZρπ

π πε β βλ+

= + Φ Φ

11 1 6 1

1

1ˆ ( ) csc sin ( )4z

wI w w Y wππ

+ =Φ

that have validity strips Re[ ]w−Φ ≤ ≤ Φ and 1 1 1Re[ ]w−Φ ≤ ≤ Φ . 10. Analytic Continuation For evaluating the fields on the faces , as we done in the case of normal incidence, we again rewriting the GWHE 1-4 in the w -plane and the GWHE 5-8 in the 1w − plane. Next we put w w→ −Φ and 1 1 1w w→ −Φ . Provided that 1 1cos( ) cos( )ow wτ τ=

or 1 1 1( ), ( )e ew g w w g w= = with:

cos( ) arccosee

wg wλ

= −

, ( )1 1 1( ) arccos coseg w wλ= − we have :

, 1 ,ˆ( ) ( )a b a bV w V wρ ρ ρ ρ+ + + +=

", , 1 ,

ˆ( ) ( )a b a bI w I wρ ρ ρ ρ+ + + +="

51

, 1 ,ˆ( ) ( )az bz az bzV w V w+ + + +=

", , 1 ,

ˆ( ) ( )az bz az bzI w I w+ + + +="

The above equations) provide an analytical element of the axial spectra in the strips

Re[ ]w−Φ < < Φ and 1 1 1Re[ ]w−Φ < < Φ , respectively. To get the global spectra we need a continuation of the analytical elements. There would be no problem if the analytical elements were known in exact form. Unfortunately the solution of the Fredholm equations are approximate and we must perform an analytical continuation of numerical data. To get the recursive equations in the planes w and 1w , first we must eliminate the unknowns iX + ( 1,..,8)i = in the GWHE equations (equations (7-10) of Daniele (2010)). For this task we again rewrite the GWHE in the w and 1w planes. Next if we substitute w with w± −Φ in equations (1-4) and 1w with 1 1w± −Φ in equations (5-8), by taking into account that 1 1cos( ) cos( )ow wτ τ=

or 1 1 1( ), ( )e ew g w w g w= = with:

cos( ) arccosee

wg wλ

= −

, ( )1 1 1( ) arccos coseg w wλ= − we have :

, 1 ,ˆ( ) ( )a b a bV w V wρ ρ ρ ρ+ + + +=

", , 1 ,

ˆ( ) ( )a b a bI w I wρ ρ ρ ρ+ + + +="

, 1 ,ˆ( ) ( )az bz az bzV w V w+ + + +=

", , 1 ,

ˆ( ) ( )az bz az bzI w I w+ + + +="

Using the above equations we get sixteen independent equations that contain the face spectra ,

ˆ ( )a bV w+ ± and ,ˆ ( )a bI w+ ± . The other unknowns are the axial spectra. All the plus

functions are always even functions (Appendix A of (Daniele (2011)), consequently we have four additional equations , ,

ˆ ˆ( ) ( )a b a bV w V w+ += − and , ,ˆ ˆ( ) ( )a b a bI w I w+ += − . By

eliminating the eight unknowns ,ˆ ( )a bV w+ ± and ,

ˆ ( )a bI w+ ± , we get four independent

recursive equations for the axial spectra ˆ ( )V w+ , ˆ ( )I w+ , 1ˆ ( )V wπ + , 1

ˆ ( )I wπ + . After cumbersome algebraic manipulations done by MATHEMATICA these equations can be rewritten as:

52

1

111 12

1

1

ˆ ( )ˆ ( )ˆ ( )ˆ ( )

ˆ ˆ( 2 ) ( ( ) )ˆ ˆ( 2 ) ( ( ) )

[ , ( )] [ , ( )]ˆ ˆ( 2 ) ( ( ) )ˆ ˆ( 2 ) ( ( ) )

z

z

t

z z tn t n t

z z t

t

V w

V w

I w

I w

V w V g w

V w V g wH w g w H w g w

I w I g w

I w I g w

ρ

ρ

ρ ρπ

π

π

ρ ρπ

+

+

+

+

+ +

+ +

+ +

+ +

=

+ Φ +Φ +Φ

+ Φ +Φ +Φ= +Φ +Φ + +Φ +Φ

+ Φ +Φ +Φ

+ Φ +Φ +Φ

1

1

1

1

1 1 1 1 1

1 1 1 1 121 1 1 1 1 1 22 1 1 1 1 1

1 1 1 1 1

1 1 1

ˆ ( )ˆ ( )ˆ ( )ˆ ( )

ˆ ˆ( ( ) ) ( 2 )ˆ ˆ( ( ) ) ( 2 )

[ ( ), ] [ ( ), ]ˆ ˆ( ( ) ) ( 2 )ˆ ˆ( ( ) ) (

z

z

t

z t zn t n t

z t z

t

V w

V w

I w

I w

V g w V w

V g w V wH g w w H g w w

I g w I w

I g w I

ρπ

π

π

ρπ

ρ ρπ

π

π

ρ ρπ

+

+

+

+

+ +

+ +

+ +

+ +

=

+Φ +Φ + Φ

+Φ +Φ + Φ= +Φ +Φ + +Φ +Φ

+Φ +Φ + Φ

+Φ +Φ 1 12 )w + Φ

where

11 1 12 11

21 1 22 1

( , ) ( , )( , )

( , ) ( , )n n

n n

H w w H w wH w w

H w w H w w=

have been evaluated by MATHEMATICA (see Appendix H) 11. Spectra for every direction ϕ Given the whole axial spectra we can obtaining the spectra for every direction ϕ by using the rotating waves (2003b). In the air we get the following expressions of the spectra for every direction ϕ−Φ ≤ ≤ Φ :

1 2( ) ( )( )sinz

o

v w v wV wwϕ

ϕ ϕτ+

+ + −=

−, 1 2( ) ( )( )

sinzo

i w i wI wwϕ

ϕ ϕτ+

+ + −=

−

53

where the rotating waves are expressed in terms of the axial spectra ˆ ( )zV w+ , ˆ ( )zI w+ by the following equations:

11 ˆ ˆ ˆ( ) sin ( cos cos ( ) sin ( ) sin( ) ( ))2 o o z o zv w k Z w I w Z I w w V wρβ β β+ + += − − +

1

ˆ ˆ ˆsin ( sin ( ) cos cos ( ) sin( ) ( ))( )

2o o z z

o

k Z w I w wV w V wi w

Zρβ β β+ + +− +

= −

2 1( ) ( )v w v w= − − , 2 1( ) ( )i w i w= − −

Similarly in the wedge ( 1 1 1ϕ−Φ ≤ ≤ Φ , 1ϕ π ϕ= − ) we get:

1 1 2 11

1 1

( ) ( )( )sin

d dz d

v w v wV wwϕ

ϕ ϕτ+

+ + −=

−, 1 1 2 1

11 1

( ) ( )( )sin

d dz d

i w i wI wwϕ

ϕ ϕτ+

+ + −=

−

where the rotating waves are expressed in terms of the axial spectra ˆ ( )zV wπ + , ˆ ( )zI wπ + by the following equations:

1 1 1 1 1 1 1 1 1 1 1 11 ˆ ˆ ˆ( ) sin ( cos cos ( ) sin ( ) sin( ) ( ))2d z zv w k Z w I w Z I w w V wπ ρπ πβ β π β π π+ + += − − − + − − + − − +

1 1 1 1 1 1 1 1 1 1

1 11

ˆ ˆ ˆsin ( sin ( ) cos( )cos ( ) sin ( ))( )

2z z

d

k Z w I w w V w V wi w

Zπ π ρπβ π β π β π+ + +− − + + − + + − +

= −

Taking into account the linearity we write the above equations in the form:

( ) ( ) ( )E Hz z o z oV w V w E V w Hϕ ϕ ϕ+ + += + ( ) ( ) ( )E H

z z o z oI w I w E I w Hϕ ϕ ϕ+ + += +

( ) ( ) ( )E Hz d z d o z d oV w V w E V w Hϕ ϕ ϕ+ + += + ( ) ( ) ( )E H

z d z d o z d oI w I w E I w Hϕ ϕ ϕ+ + += + 12. Far field According to the results of sect. 7 we get ( 1oτ ρ >> , ϕ−Φ ≤ ≤ Φ ):

( ) ( )/ 4 / 41 1( , ) ( , ) ( , )2 2

o oj jd j z EE j z EHz o o o o

o o

E e e D E e e D Hτ ρ π τ ρ πβ βρ ϕ ϕ ϕ ϕ ϕπτ ρ πτ ρ

− + − +− −= +

54

( ) ( )/ 4 / 41 1( , ) ( , ) ( , )2 2

o oj jd j z HE j z HHz o o o o

o o

H e e D E e e D Hτ ρ π τ ρ πβ βρ ϕ ϕ ϕ ϕ ϕπτ ρ πτ ρ

− + − +− −= +

where the external diffraction coefficients , ( , )A BoD ϕ ϕ (A,B=E,H) is expressed by

( ϕ < Φ ):

( )( , )

Eo zEE

o

VD

jϕτ π

ϕ ϕ +− −= ,

( )( , )

Ho zEH

o

VD

jϕτ π

ϕ ϕ +− −=

( )( , )

Eo zHE

o

ID

jϕτ π

ϕ ϕ +− −= ,

( )( , )

Ho zHH

o

ID

jϕτ π

ϕ ϕ +− −=

Similarly in the wedge ( 1 1τ ρ >> 1 1 1ϕ−Φ ≤ ≤ Φ , 1ϕ π ϕ= − ) we get:

( ) ( )1 1/ 4 / 4

1 1

1 1( , ) ( , ) ( , )2 2

j jd j z EE j z EHz d o o d o oE e e D E e e D Hτ ρ π τ ρ πβ βρ ϕ ϕ ϕ ϕ ϕ

πτ ρ πτ ρ− + − +− −= +

( ) ( )1 1/ 4 / 4

1 1

1 1( , ) ( , ) ( , )2 2

j jd j z HE j z HHz d o o d o oH e e D E e e D Hτ ρ π τ ρ πβ βρ ϕ ϕ ϕ ϕ ϕ

πτ ρ πτ ρ− + − +− −= +

where the internal diffraction coefficients ( , )EE

d oD ϕ ϕ is expressed by:

( )( , )

Eo z dEE

d o

VD

jϕτ π

ϕ ϕ +− −= , 1 ( )

( , )H

zEHd o

VD

jϕτ π

ϕ ϕ +− −=

( )( , )

Eo z dHE

d o

ID

jϕτ π

ϕ ϕ +− −= , 1 ( )

( , )Hz dHH

d o

ID

jϕτ π

ϕ ϕ +− −=

In the air, the components dEβ , dEϕ of the electrical field normal to the diffracted rays are (Senior&Volakis,1995):

sin

dd zEEβ β= − ,

sin

dd o zZ HEϕ β= − ,

Similarly in the dielectric , the components

1

dEβ , dEϕ of the electrical field normal to the diffracted rays are (Senior&Volakis,1995):

55

11sin

dd zEEβ β= − , 1

1sin

dd zZ HEϕ β= − ,

7. Conclusions This report formulates the dielectric wedge problem in terms of Generalized Wiener Hopf equations. Even though the GWHE of the dielectric wedge problem cannot be solved in closed form, an appropriate procedure reduces them to Fredholm equations of second kind. A simple quadrature scheme is sufficient to demonstrate good convergence of numerical results. An analytical continuation of the numerical results is discussed. It provides analytical representations that enable to analyze the different components ( geometrical optics and GTD contributions, surface rays, lateral waves) of the wedge field. Diffraction coefficients for some wedges are presented. Acknowledgement The author wishes to thank Prof. P.L.E. Uslenghi for the unceasing encouragements and for the many suggestions that made this work possible. References Albani, M, 2007. On Radlow’s quarter –plane diffraction solution. Radio Science .

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Vekua N.P.,1967. Systems of Singular Integral Equations, P. Noordhoff, Gronigen. Volakis, J.L. & T.B.A. Senior, 1987. Diffraction by a thin dielectric half plane. IEEE

Transactions on Antennas and Propagation . 35, pp.1483-1487. Wojcik, G.L, 1995. A formulation of self-similar dielectric wedge diffraction, Digest of

1995 IEEE Antennas and Propagation Int. Symp, pp.1068-1071. Wu T.K & L.L. Tsai.,1977. Scattering by a Dielectric Wedge: A numerical Solution.

IEEE Transactions on Antennas and Propagation . 25, pp.570-571 Zavadskii, Y.Yu., 1966. Certain diffraction problems in contiguous liquid and elastic

wedges. Soviet Physics-Acoustics. 12, pp.170-179.

59

Appendix A: The Fredholm integral equations of the problem The GWHE of the wedge penetrable consist in four uncoupled system having the form ( 1,3,5,7i = ) (A1)

( ) ( ) ( 1)( ) ( ) ( )i i iY X m X mnξη −

+ + + ++

= − − −

(A2) 1( 1) ( ) 1 ( 1) 1

1

( ) ( ) ( )i i iY X m X mnξη −

+ + + + ++

= − + −! !

In order to get classical Wiener Hopf equation we introduce the mapping

cos arccos( )oαη τ

π τΦ = − −

for the the equation (A1) and the mapping

1 11

1

cos arccos( )αη τπ τ

Φ= − −

for the equation (A2) . It yields:

(A3) ( ) ( ) ( 1)2 2( ) ( ) ( )o

i i i

o

Y X Xτ αα α ατ α

+ − + −+

= +−

(A4) 1 1

( 1) 1 ( ) 1 ( 1) 12 21 1

( ) ( ) ( )i i iY X Xτ αα α ατ α

+ + − + −+

= −−

!!! !!!

where ( ) ( )( ) ( )i iY Yα η+ += , ( 1) 1 ( 1)( ) ( )i iY Yα η+ + + +=

The not conventional part of a plus (minus) Laplace transform is defined by the part that presents singularities in the standard regularity half plane (in this case Im[ ] 0α ≤ ( 1Im[ ] 0α ≥ ). Generally in wedge problems, the non conventional singularities are poles arising from the plane waves present in the optical geometric contributions. The reduction of the W-H equations to Fredholm integral equations requires the plus and the minus unknowns to be conventional (Daniele 2004a). Concerning eq. (A4), since there are not sources in the interior of the dielectric wedge, this equation only involves conventional Laplace transforms and the standard procedure for reducing the W-H equations to Fredholm equation described in (Daniele,2004a; yields the equation (14) without any difficulty. Conversely in the equation (A3), at least one of the plus or the minus functions must be not conventional, otherwise the solution of the W-H equation would vanish. To evaluate the not conventional parts of the unknown Laplace transforms could be a difficult task for the presence in the wedge problem of several reflected, refracted and transmitted plane waves. However for the plus unknowns

60

( ) ( )( ) ( )i iY Yα η+ += , only the pole coso o oπα τ ϕ = − Φ

arising from the incident plane

wave can produce not conventional parts. To show it, we first evaluate the poles of the incident, reflected by the a- face and reflected by the b-face plane waves. For the plus

unknowns, in the planew − they are Yio ow π ϕ+ = −

Φ, ( 2 )Y

ao ow π ϕ+ = − − ΦΦ

and

( 2 )Ybo ow π ϕ+ = − + Φ

Φ respectively. These three poles produce the same point

coso o oπα τ ϕ = − Φ

but, taking into account Fig.13 of (Daniele 2004a), we observe that

the poles of the reflected waves do not occur in the source term of equation (11) because they are located in the improper sheet of the planeα − . Conversely the pole

coso o oπα τ ϕ = − Φ

of the incident field is in the proper sheet and, depending on the

direction oϕ , can be located in the upper Im[ ] 0α > or in lower Im[ ] 0α < half plane. In the first case the minus functions are conventional and the plus functions are not. In the second case the opposite holds. Taking this into account, let’s consider the source constituted by the plane wave (1) that is present in the free space. Taking into account that cos( )( , ) o oji

z oE E e τ ρ ϕ ϕρ ϕ −= , we have:

cos(0 }

0( ,0) o oji j o

z oo

jEV E e e dτ ρ ϕ ηρη ρη η

∞ −+ = =

−∫ (A1)

where: , cosO o oη τ ϕ= − , It yields:

(A5) i1 1 z+

( ) sin=Res[ ( )] 2 Res[V ( ,0)] 4( ) sin

o o

o oo

oo

T Y j Enη η

ξ η ϕη ηπη ϕ

++

+

= = − Φ

or in the planeα − :

(A6) 1 1

sin ( )sin= 4 4

sin( )sino

oo

o oo

o

dT T j E j Ed η

π ϕα ϕ π π

πη ϕϕ

− Φ = − = −Φ − Φ

Φ

Similarly we have: cos( )

( , ) ( cos( ) sin( ))o oj

io o o o o o o

o

eE E H Z kτ ρ ϕ ϕ

ρ ρ ϕ α ϕ ϕ ϕ ϕτ

−

= − − −

cos( )( , ) o ojiz oH H e τ ρ ϕ ϕρ ϕ −=

cos( )

( , ) ( cos( ) sin( ))o oj

io o o o o o o

o o

eH H Z E kZ

τ ρ ϕ ϕ

ρ ρ ϕ α ϕ ϕ ϕ ϕτ

−

= − + −

By repeating the same reasons that deduced 1T , after algebraic manipulations we yield:

61

(A7)

3 4 sin ( )2o oT j Eπ π ϕ = − Φ Φ

, 5 =4 oT j HπΦ

, 7 4 sin ( )2o oT j Hπ π ϕ = − Φ Φ

If we suppose 2oϕΦ

< , oα is located in the upper α −half plane, it follows that the

functions 1 ( )X α− , 2 ( )X α− , 3 ( )X α− and 4 ( )X α− are conventional in the α -plane and ( )iY α+ ( 1,3,5,7i = ) not are . By rewriting:

( ) ( )s ii i

o

TY Yα αα α+ += +−

, 1,3,5,7i = (A8)

the equation (A8) assumes the form that separates the conventional part from the not conventional one:

1 1 22 2( ) ( ) ( )s o o

o o

TY X Xτ αα α αα α τ α

+ − −

++ = +

− − (A7)

Applying the standard technique described in (Daniele,2004a); we yield the equation:

( 1)2 2 2 2

( 1)2 2

' ( ')'1( ) ( ) '

2 ' ( )

o oi

o oo ii i

oo

XTX X d

j

τ α τ α ατ α τ ατ αα α α

π α α α ατ α

+ −∞

− + − −∞

+ + − − −+ + = +

− −−∫

1,3,5,7i =

When 2oϕΦ

> the pole oα is located in the lower half-plane and the functions ( )iY α+

are conventional . Conversely the minus functions are not. In order to put in evidence their not conventional part, now there is the necessity to evaluate the poles of reflected field by the faces for the minus unknowns. In the planem − they are the points

cos( )ao o om k ϕ= − +Φ and cos( ) cos( 2 )bo o o o om k kϕ ϕ= − −Φ = − − Φ +Φ respectively; it

yields: Xao ow π ϕ− = −

Φ, ( 2 )X

bo ow π ϕ− = − + ΦΦ

. Again we observed that they produce the

same image cos( )O o ok πα ϕ= −Φ

in the planeα − . However for positive oϕ , only the

pole Oα due to the reflected wave by the face a is located in the proper sheet of the planeα − . So by rewriting:

11 1

22 2

( ) ( )

( ) ( )

s o

o

s o

o

RX X

RX X

α αα α

α αα α

− −

− −

= +−

= +−

,

33 3

44 4

( ) ( )

( ) ( )

s o

o

s o

o

RX X

RX X

α αα α

α αα α

− −

− −

= +−

= +−

(A8)

62

55 5

66 6

( ) ( )

( ) ( )

s o

o

s o

o

RX X

RX X

α αα α

α αα α

− −

− −

= +−

= +−

,

77 7

88 8

( ) ( )

( ) ( )

s o

o

s o

o

RX X

RX X

α αα α

α αα α

− −

− −

= +−

= +−

where the not conventional part io

o

Rα α−

presents the residues ioR that will be evaluated

later. By using again the standard technique described in (Daniele,2004a; we get the

following Fredholm equations ( 1,3,5,7i = ) that are valid for Im( ) 0oα < or 2oϕΦ

>

( 1)2 2 2 2

( 1)2 2

( 1)

2 2

' ( ')'1( ) ( ) '

2 '

,

Im( ) 0

o oi

o ooi i

o

i oio o

o oo

o

X

X X dj

RR

τ α τ α ατ α τ ατ αα α α

π α ατ α

τ αα α α ατ α

α

+ −∞

− + − −∞

+

+ + − − −+ + − =

−−

+= +

− −−

<

∫

(A9)

By invoking the continuity, in the case 2oϕΦ

= ( 0oα = ) we can deduce that the second

terms of (A9) become: ( 1) ( 1)

2 2

1 12 2

i o i oio o

o

R RR τ αα α ατ α

+ +++ +

− .

In order to get the residues ioR ( 1,3,5,7i = ) we need the first reflected plane wave and the first transmitted plane wave from the face a. For the longitudinal fields of these waves, we can write:

cos( )( , ) o rjrz orE E e τ ρ ϕ ϕρ ϕ −= , cos( )( , ) o rjr

z orH H e τ ρ ϕ ϕρ ϕ −=

1 cos( )( , ) tjtz otE E e τ ρ ϕ ϕρ ϕ −= , 1 cos( )( , ) tjt

z otH H e τ ρ ϕ ϕρ ϕ −= where the superscript r means reflected, the superscript t means transmitted, 2r oϕ ϕ= Φ −

and ( )t ogϕ ϕ= Φ + Φ − with cos( ) arccosr

wg wε

= −

.

Taking into account the equations that relates the transversal components to the longitudinal components :

63

we can force the four boundary conditions on the face a:

( , ) ( , ) ( , )i r tz z zE E Eρ ρ ρΦ + Φ = Φ , ( , ) ( , ) ( , )i r tE E Eρ ρ ρρ ρ ρΦ + Φ = Φ ,

( , ) ( , ) ( , )i r tz z zH H Hρ ρ ρΦ + Φ = Φ , ( , ) ( , ) ( , )i r tH H Hρ ρ ρρ ρ ρΦ + Φ = Φ

and get the four unknowns orE , orH , otE and otH . The expressions of them are cumbersome and will be reported in the MATHEMATICA program code (Appendix). By introducing the residues in the m plane− :

We get the following residues in the planeα − :

64

Taking into account the definitions of the minus iX − functions, finally we get:

1 2 ( )o o orR j E Eπ= − +

Φ, 2 2 cot( )( )

2o

o o orR j E Eπ ϕπ= − −

Φ Φ

3 2 ( )sin( )2

oo o orR j E E π ϕπ= +

Φ Φ, 4 2 ( )cos( )

2o

o o orR j E E π ϕπ= −

Φ Φ (A9)

5 2 ( )o o orR j H Hπ= +

Φ, 6 2 cot( )( )

2o

o o orR j H Hπ ϕπ= −

Φ Φ

7 2 ( ) in( )2

oo o orR j H H π ϕπ= − +

Φ Φ, 8 2 ( ) cos( )

2o

o o orR j H H π ϕπ= − −

Φ Φ

As for as the kernel in the FIE is concerned , we observe that it can be shown that the

operator 2 2 2 2

''

'

o o

o o

τ α τ ατ α τ α

α α

+ + − − −

− is compact in the Hilbert space defined on the real

axis α (Daniele 2004a). Appendix B. An application of the Cauchy formula Equations (96) requires to express ( )iQ v in terms of ( )jP v for real values of v and u . This can be accomplished by taking into account that, within a normalization factor, we are facing the same functions defined for different values of the argument. Consequently we can resort to the Cauchy formula:

11 1

1 ( ) 1 ( ( ')) ( ')( ( )) '2 ( ) 2 ( ') ( ) '

F m F m u dm uF m v dm duj m m v j m u m v duγπ π

∞− −

− −∞= =

− −∫ ∫% (B1)

where:

65

( ) cos2

m u juπτπΦ = − + +Φ

, 11 1 1( ) cos

2m v jvπτ

πΦ = − + +Φ

and ( )F m− represent an analytical minus function in the variable m . Equation (B1) holds for conventional minus ( )F m− that present singularities only in the second quadrant (Fig.4). Occasionally the source can produce poles of ( )F m− located in the region between γ and the line that reports 1( )m v for real values of v ; in this case we must slightly modify (B1) according to the residues in these poles. From the equations (B1), taking into account the expressions (17), we get:

1 21

1

sin[ ( ) ] ( ( ))2( ( ))

2 cos[ ( ) ] cos[ ( ) ]2 2r

ju F m uF m v du

ju jv

πππ ππ ε

π π

−∞

− −∞

Φ + +ΦΦ= −

Φ Φ+ +Φ − + +Φ

∫

(B2)

Figure12. The complex values of ( )m u and 1( )m v for real values of u and v . (23πΦ = , 1ok = ,

1 10k = )

Notice that in equation (B1) the roles of the two variables m and 1m cannot be reversed since if we express the function ( )F m− in terms of the values of 1( )F m− , the Cauchy formula would require the evaluation on the branch line contributions located in the second quadrant (Fig. 4). This means that the operator

21

1

sin[ ( ) ] ( ( ))2( , )

2 cos[ ( ) ] cos[ ( ) ]2 2r

ju F m uT T v u

ju jv

πππ ππ ε

π π

−Φ + +ΦΦ

→ = −Φ Φ

+ +Φ − + +Φ (B3)

66

is not invertible. We apply the operator T to the minus functions ( , ), ( , )zV m I mρ+ +− ±Φ − ±Φ that define

the functions ( )iX m+ − , 1( )iX m+ −! through the definitions of the equations (7)-(10). It yields the equations that express 2,4 ( )Q v in terms of 2,4 ( )P v and the following representations that relate 1,3( )Q v to 1,3( )P v :

1 11

11

1

cos( )sin[ ] sin[ ( ) ] ( )1 2 2( )2 cos( )sin[ ] cos[ ( ) ] cos[ ( ) ]

2 2 2r

jvu ju P uQ v dujuv ju jv

ππ π

π ππ π επ π π

∞

−∞

Φ Φ Φ+ + +ΦΦ= −

Φ Φ Φ Φ+ + +Φ − + +Φ

∫

3

1 13

21

1

( )1cos[ ( )]sin[ ] sin[ ( ) ] ( )2 2 2 212 cos[ ( )]sin[ ] cos[ ( ) ] cos[ ( ) ]2 2 2 2 2r

Q vjvju ju P u

dujujv ju jv

π ππ π

π π ππ επ π π

∞

−∞

=Φ Φ Φ− + + + +ΦΦ

= −Φ Φ Φ Φ− + + + +Φ − + +Φ

∫

In practice through a quadrature, the operator T is represented by a matrix and the accuracy of this approximation has been tested numerically by several simulations. By using the representations of 1,2Q in terms of 1,2P one could obtain the integral equations directly through (23) without use of the mapping (B3). This has been proposed in (Daniele, 2005). However the operator T not invertible and in this case the numerical solution involves the inversion of ill-conditioned matrices. Appendix C. Characteristics of the kernels Remark: The material presented in this Appendix refers to the case of normal incidence. The more general cases only require a slight modification of it.

A property of the kernel ( )'

'1( , ')

1

u u

u uu

e e jM u uj ee jπ +

+=

+− is that it satisfies the

condition: ( , ') *( , ')M u u M u u= − − . (C1)

where * means complex conjugation. Consequently by considering the complex conjugate of the equations (22) and (23) we get that 1,2 * ( )P u− − , 1,2 * ( )Q v− − are solutions of the problem too. Since the solution is unique it yields:

1,2 1,2( ) *( )P u P u= − − 1,2 1,2( ) *( )Q u Q u= − − . (C2) The property ( C2) has been ascertained numerically (see also end of section 4). According to the general theory (Daniele,2004a), the kernels of the integral equations occurring in the Fredholm factorizations are compact in a suitable functions space. Now this important behavior will be studied directly in the Hilbert space 2L . We first reduce the infinite interval of integration u−∞ < < ∞ into a segment 1 1x− ≤ ≤ , through the well

67

known transformation ( ) arctanh( )u u x x= = ( do not confuse this new variable x with the geometrical coordinate x of Fig.1). After algebraic manipulations, in the x − domain ( 1 1x− ≤ ≤ ), the integral equations (28-29) becomes:

12 1/ 21 2 2 22 2 1/ 21

1 ( , ')( ) ( (1 ) ) ( ) ( ') ' ( )(1 ) (1 ' )

x xP x j x P x P x dx N xj x xδ δπ −−

∆− − − − =

− −∫& & & (C3)

12

1 1 2 2 22 2 1/ 21

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

rB x xC x P x C x P x P u dxj x xδ δπ −−

+ − =− −∫& & & (C4)

where, being ( , )T v u defined by (B3), we have:

2 1/ 2( ( ))( ) , 1,2

(1 )i

iP u xP x i

x δ+= =−

& , 2 1/ 2 2 1/ 2

2 1/ 2 2 1/ 2( ' ((1 ) (1 ' ) )( , ')

(1 ) ' (1 ' )x x j x xx x

x x x x− + − − −

∆ =− − −

22 2 1/ 2

4( )(1 ) ( (1 ) cos )

o

o

EN xx x j xδ

ππϕ=

Φ − + −Φ

, 1 1

1

( ( ))sin[ ]2( ) ( )sin[ ]

2

jv u x

C x ju xπ

π

Φ Φ+=

Φ Φ+

21

22

1( ( )) sin [ ( ( ))]cosh( ( ( )))2 2 2( ) tan 12 cosh( ( )) sin [ ( ( ( )))]2 2o

jv u x Z ju xv u xC xu x Z jv u x

π π

π

− + − +=

− +

2 2 2( , ') ( ( ( ), ') ( ( '), ') 'rB x x B v u x v R u x v dv∞

−∞= ∫

( )

'

2' 2

cosh( ) ( )( , ') 1( 1)sin [ ( ')]2 2

v v

v v v

v e e jB v ve j e jvπ+

+=

− + − +

212 2

1( ', ') sin [ ( ')] ( ', ')2 2 2o

ZR u v ju T v uZ

ππΦ

= − − +

In the equations (C3) and (C4), the real parameter δ has been introduced for convenience. The range of this parameter has some limitations. For instance we must have 1/ 4δ < , otherwise the vector 2 ( )N x ∉ 2L . Taking into account equation (33), we get:

1/ 22 2

2 1/ 2(1 )( ) [ ], 1,2(1 )

c

ixP x O ix

π

δ

Φ+

+

−= =

−&

Whence 1/ 4δ < yields 2( ) , 1,2iP x L i∈ =& . Taking into account that as u → ±∞ (or 1x = ± ) :

1

1 1 1

log ( )( )2 2

rv u u jπ ε πΦ Φ −Φ≈ − −Φ Φ Φ

(C5)

68

it follows that the matrix 2 1/ 2

1 2

1 ( (1 ) )( ) ( )

j xC x C x

− − − is invertible for every values of

1 1x− ≤ ≤ + . For instance at 1x = ± , it becomes 1

1

1o

jZjZ

−.

Consequently to verify that the system (C3-C4) is Fredholm of second kind, we must study the two functions ( , ')x x∆ and 2 ( , ')rB x x . The function ( , ')x x∆ is not bounded in the square 1 , ' 1x x− ≤ ≤ . However numerical simulations have ascertained that, for non vanishing positive values of the parameter 1δ , it results { }12Max (1 ' ) ( , ')x x xδ− ∆ < ∞ in the square 1 , ' 1x x− ≤ ≤ . Whence we rewrite the kernel of (C3) in the form:

1

1 1

2

1/ 22 2(1 ' ) ( , ')

(1 ) (1 ' )x x x

x x

δ

δ δ δ δ+ − +

− ∆− −

(C6)

Kernel (C6) is compact in 2L if 1

1 1

221 1

1/ 22 21 1

(1 ' ) ( , ') '(1 ) (1 ' )

x x x dx dxx x

δ

δ δ δ δ+ − +− −

− ∆< ∞

− −∫ ∫ . Taking into

account that the max of 122(1 ' ) ( , ')x x xδ− ∆ is finite in the square 1 , ' 1x x− ≤ ≤ , we

easily verify this equation provided that 1104

δ δ< < < . Within the same conditions, also

the kernel 22 2 1/ 2

( , ')(1 ) (1 ' )

rB x xx xδ δ−− −

is compact provided that

2

22 2 2( , ') ( ( ( ), ') ( ( '), ') 'rB x x B v u x v R u x v dv

∞

−∞= < ∞∫ 1 , ' 1x x− ≤ ≤ (C7)

To ascertain (C7) we observe that { }2Max ( , ')B v v < ∞ for every value of , 'v v−∞ ≤ ≤ ∞ . It occurs in the worst case 'v v= − too. Furthermore numerical simulations put in

evidence that 2 ( ( '), ') 'R u x v dv∞

−∞∫ is finite for every 1 ' 1x− ≤ ≤ .

Whence, taking into account (C5), equation (C7) holds. To get the compactness of the kernels involved in system (30)-(31) requires a slight modification of the above proof. In particular the new system is :

12 1/ 23 4 4 42 2 1/ 21

1 ( , ')( ) ( (1 ) ) ( ) ( ') ' ( )(1 ) (1 ' )

x xP x j x P x P x dx N xj x xδ δπ −−

∆− − − − =

− −∫& & & (C8)

1 3 4 4

1/ 4 14

42 1/ 4 2 1/ 21

( ) ( ) ( ) ( )

1 2( 1) ( , ') ( ') ' 0(1 ) (1 ' )1 1

r

C x P x C x P x

B x x P u dxj x xx j x δ δπ + −−

+ +

−− =

− −− + + ∫

& &

& (C9)

where we have:

69

22 1/ 2( ( ))( ) , 3,4

(1 )i

iP u xP x L i

x δ+= ∈ =−

& , 2 1/ 2 2 1/ 2

2 1/ 2 2 1/ 2( ' ((1 ) (1 ' ) )( , ')

(1 ) ' (1 ' )x x j x xx x

x x x x− + − − −

∆ =− − −

42 2 1/ 2

4 sin( )2( )

(1 ) ( (1 ) cos )

oo

o

EN x

x x j xδ

πϕπ

πϕΦ=

Φ − + −Φ

,

1

4

1 1( ( )) cos[ ( ( ( )))] sin[ ( ( ))]2 2 2 2 2( ) tan 1 12 cos[ ( ( ))] sin[ ( ( ( )))]

2 2 2 2o

jv u x jv u x Z ju xC x

ju x Z jv u x

π π π

π π

− + − + − +=

− + − +

4 4 4( , ') ( ( ( ), ') ( ( '), ') 'rB x x B v u x v R u x v dv∞

−∞= ∫

( )

'

4'

1cos[ ( )) ( )2 2( , ') 1( 1)sin[ ( ')]

2 2

v v

v v v

jv e e jB v v

e j e jv

π

π+

− + +=

− + − +

1

4 21( ', ') sin[ ( ')) ( ', ')]

2 2 2o

ZR u v ju T v uZ

ππΦ

= − − +

Taking into account the discussion of the system (C3)-(C4) , the compactness of the kernels of the system (C8)-(C9) reduces to the compactness of the kernel:

1/ 44

2 1/ 4 2 1/ 22( 1) ( , ')

(1 ) (1 ' )1 1rB x x

x xx j x δ δ+ −

−− −− + +

This kernel is compact provided that:

2

24 4 4( , ') ( ( ( ), ') ( ( '), ') 'rB x x B v u x v R u x v dv

∞

−∞= < ∞∫ 1 , ' 1x x− ≤ ≤ (C10)

We observe that { }4Max ( , ')B v v < ∞ for every value of , 'v v−∞ ≤ ≤ ∞ ; furthermore

numerical simulations show that 2 1/ 44 ( ( '), ') ' [(1 ' ) ]R u x v dv O x

∞

−∞= −∫ for every

1 ' 1x− ≤ ≤ . Whence, taking into account (C5), equation (C10) holds. The above discussion shows that the solution of the two systems (28-29) or (30)-(31) exists and is unique. Besides the choice 10A = of the truncation parameter is justified from the fact that 13( ) tanh(10) 1 10x A −= = − . In order to evaluate the truncation error, let’ s rewrite the system (28-29) or (30)-(31) in the abstract form G P N⋅ = with 1G K= + and / cosh( ),iP P u→ ( 1,..,4i = ). Let’s introduce the window functions ( )Aw u and ( )Aw u that are zero-valued in the intervals u A> and u A≤ respectively. It can be shown that the kernels

( , ') ( ')A AKW K u u w u→ and ( , ') ( ')A AKW K u u w u→ are compact and bounded

70

respectively. Since the solution of G P N⋅ = exists, 1(1 )AKW −+ exists and regularizes (1 )K+ :

11 1 (1 ) (1 (1 ) )A A A A AG K KW KW KW KW KW−= + = + + = + + + To get the truncation error let’s compare the exact 1P G N−= and the approximate

1A A AP G N−= ⋅ solution that has been obtained by truncating the integral in u A= ± :

1A A A A AG W W W K W= + , A AP W P= , A AN W N= Using well known equations we get:

211 1

11A A

AA A

G G GG G

G G G

−− −

−

−− ≤

− −

211

1( )

1A A

A A A A AA A

G G GP P N G NW W KW N

G G G

−−

−

−− ≤ + +

− −

where: A A A A A A AG G W GW W GW W GW− = + +

!

Appendix D: The rotating waves representations of the spectra Remark: The material presented in this Appendix refers to the case of normal incidence. The more general cases only require a slight modification of it. Angular transmission lines are useful for studying waves in angular regions (Felsen & Marcuvitz, 1973; Daniele,2003). Of course, they imply the presence of clockwise and counterclockwise angular rotating waves. The Sommerfeld integrals allow the introduction of only one of these two opposite rotating waves. This simplification comes at a price: the presence in the integral representation of a complex and artificial integration path called Sommerfeld contour. Many difficulties which are present in the Sommerfeld-Malyuzhinets method are overcome when we use rotating waves. In particular we safely deal with the rotating waves when several angular regions are present . In terms of the rotating waves 1 2( ) and ( )v w v w the following representations of the spectra ( )dV wϕ and ( )I wϕ+ hold in an angular isotropic region with propagation constant k and impedance Z (Daniele,2003):

( )1 21( ) ( ) ( )dV w v w v wkϕ ϕ ϕ= + + −−

(A1)

( )1 21( ) ( ) ( )I w v w v w

k Zϕ ϕ ϕ+ = + − − (A2)

71

where )(1 wv (clockwise) and )(2 wv (counterclockwise) define the rotating waves in the angular region. We remark that the clockwise )(1 wv is related to the well known Sommerfeld function ( )s w :

1 ( ) ( )v w j s w= − The rotating waves satisfy many properties. For instance the two opposite rotating waves are related by the property

1 2( ) ( )v w v w= − − Taking into account this property the equations (A.1) and (A.2) show that, for every direction ϕ , the spectra are always even functions of w . By inverting the above equations we can get the rotating waves in terms of the spectra :

1

( ) ( )( )

2dV w Z I w

v w k ϕ ϕϕ +−+ = − (A3)

2

( ) ( )( )

2dV w Z I w

v w k ϕ ϕϕ ++− = − (A4)

Putting 0ϕ = , oZ Z= and ok k= in (A3) and (A4) , we get the explicit expression of

1 ( )v w and 2 ( )v w in terms of the axial spectra ˆ ˆ( ), ( )dV w I w+ . Substituting them in (A1) and (A2) we obtain the equations (14) and (15) of the paper . Putting 1w w= , 1Z Z= , ϕ π= and 1k k= in (A3) and (A4), we get the explicit

expression of 1 1( )v w π+ and 2 1( )v w π− in terms of the axial spectra 1 1ˆ ˆ( ), ( )dV w I wπ π + .

Taking into account that 1ϕ π ϕ= − , substituting these expressions in (A1) and (A2) we obtain the equations (16) and (17) of the paper. Observe that in the dielectric wedge, according to the increasing direction of 1ϕ , the role of the two rotating waves is reversed. Appendix E: Discussion on the recursive equations. Remark: The material presented in this Appendix refers to the case of normal incidence. The more general cases only require a slight modification of it. The spectra band that is required for evaluating the spatial field depends on the values of 1,ϕ ϕ . From here on we will assume 1Re[ , ] 2w w π≤ .

In equations (11) we must iterate the functions 1 1ˆ ( 2 )V wπ + − Φ and 1 1 1

ˆ ( ( ) )dV g w +Φ +Φ .

The spectrum 1 1 1ˆ ( ( ) )dV g w +Φ +Φ is correctly evaluated from the starting spectra and

does not requires iterations, provided that :

72

1 1 1Re[ ( )]g w−Φ < +Φ +Φ < Φ If 1Re[ ] 2w π≤ , we have numerically verified the above inequality for every value of

Re[ ] 1rε > and 102π

< Φ < . Whence we must iterate only 1 1ˆ ( 2 )V wπ + − Φ . This iteration

is easy to study For instance the thNπ iteration yields the validity strip

1 1 1Re[ ] 2w Nπ< Φ + Φ . Similar considerations apply to equations (13) for the spectrum

1ˆ ( )I wπ + .

Forcing 1 12 2N Bπ πΦ + Φ > ≤ , it follows that the number Nπ of iterations that provides

spectra of 1ˆ ( )V wπ + and 1

ˆ ( )I wπ + that are accurate in the strip 10 Re[ ]w B< < is given by:

1

1

IntegerPart[ ]2

BNπ−Φ

≥Φ

The same reasoning enlarges the validity strips of spectra of ˆ ( )dV w , ˆ ( )dI w by considering the equations (10) and (12). In particular we observe that in these representations we have accurate spectra of 1

ˆ ( ( ) ))dV g wπ −Φ −Φ and 1ˆ ( ( ) ))I g wπ + −Φ −Φ provided that

1 1 10 Re[ ( ) ] 2g w Nπ< −Φ −Φ < Φ + Φ . If 2Nπ ≥ and Re[ ] 2w π≤ , the above inequality

has been numerically ascertained for every value of of Re[ ] 1rε > and 102π

< Φ < .

Except for the important region near the real point w = −Φ , the recursive equations produce numerical discontinuities in the line Re[ ]w = −Φ . It means that for significant values of 0u ≠ , numerically we have the wrong results : ˆ ˆ( ) ( )d dV ju V ju− +−Φ + ≠ −Φ + ,

( ˆ ˆ( ) ( )d dI ju I ju− +Φ + ≠ Φ + ). In these cases the more accurate spectra are those evaluated in Re[ ]w ju+= −Φ + since they derive from the starting spectra. Similar

consideration apply for 1ˆ ( )dV wπ , 1

ˆ ( )dI wπ . Fortunately the above considerations do not apply to the evaluations of the diffraction coefficients since they require the knowledge of the spectra only on real axis. On this axis the recursive equations do not produces numerical discontinuities. Appendix F. Some exacts solutions of the dielectric wedge a) The case of absence of the wedge: 1rε = ( π ϕ π− ≤ ≤ ):

( )( cos , )

cos coso

z oo o

EV k w jk w

ϕϕ ϕ+ − = −

− − , π ϕ π− ≤ ≤

( )

( )sin

( cos , )cos cos

o oo

o o o

EI k w j

k Z wρ

ϕ ϕϕ

ϕ ϕ+

−− = −

− − , π ϕ π− ≤ ≤

73

b) The case of the PEC wedge: rε = ∞ :

( cos , )

cos cos csc( )sin2 2 2

( ) ( )sin sin sin sin2 2 2 2

z o

oo

o oo

V k wwE w

jw wk

ϕπϕ πϕ ππ

π ϕ πϕ π ϕ πϕ

+ − =

Φ Φ Φ= −− + Φ + − + Φ Φ Φ Φ

, ϕ−Φ ≤ ≤ Φ

( cos , )

( ) ( )cos sin sin 2sin2 2 2 2

( ) ( )2 sin sin sin sin2 2 2 2

o

o oo

o oo o

I k w

w wEj

w wk Z

ρ ϕ

πϕ π ϕ π ϕ πϕπ

π ϕ πϕ π ϕ πϕ

+ − =

− + − + − Φ Φ Φ Φ = −− + Φ + − + Φ Φ Φ Φ

, ϕ−Φ ≤ ≤ Φ

c) The case of the half space dielectric wedge: 1 2π

Φ = Φ = , rε arbitrary:

( ) ( )

( cos , )

cos cos cos cos

z o

o o

o o o o

V k wE Ej j

k w k w

ϕ

ϕ ϕ ϕ ϕ

+ − =Γ

= − −− − + +

, 02πϕ< ≤

( )( )

( )( )

( cos , )

sin sincos cos cos cos

o o o o

o o o o o o

I k w

E Ej j

k Z w k Z w

ρ ϕ

ϕ ϕ ϕ ϕϕ ϕ ϕ ϕ

+ − =

− Γ += − +

− − + +

, 02πϕ< ≤

( )1 11 1

( cos , )cos cos

oz

t

T EV k w jk w

ϕϕ ϕ+ − = −

− − ,

2π ϕ π≤ ≤

( )

( )1 11 1 1

sin( cos , )

cos coso t

t

T EI k w j

k Z wρ

ϕ ϕϕ

ϕ ϕ+

−− = −

− − ,

2π ϕ π≤ ≤

where: 1arcsin sin( )t o

r

ϕ ϕε

=

1

1

cos cos

cos cos

o

t o

o

t o

Z Z

Z Zϕ ϕ

ϕ ϕ

−Γ =

+, 1T = + Γ

Diagrams relevant to other directions ϕ as well as to complex values of w ( 1w ), have also been considered but are not reported here since they present the same characteristics.

74

Appendix G. Some diagrams of diffraction for selected values of rε , Φ and oϕ .

Figure 13a . Diagram of diffraction out the wedge ( ϕ−Φ < < Φ ) (

34πΦ = , / 5oϕ π= and 3rε = ).

- 0.75 - 0.5 - 0.25 0.25 0.5 0.75j 1

- 40- 20

204060

DHj 1,p€€€€€5L

Figure 13b . Diagram of diffraction into the wedge ( 1 1 1ϕ−Φ < < Φ ) (

34πΦ = , / 5oϕ π= and

3rε = ).

75

Figure 14a . Diagram of diffraction out the wedge ( ϕ−Φ < < Φ ) (34πΦ = , 7 /8oϕ π= and

3rε = ).

Figure 14b . Diagram of diffraction into the wedge ( 1 1 1ϕ−Φ < < Φ ) (34πΦ = , 7 /8oϕ π= and

3rε = ).

Figure 15a . Diagram of diffraction out the wedge ( ϕ−Φ < < Φ ) (

78πΦ = , / 8oϕ π= and 3rε = ).

76

Figure 15b. Diagram of diffraction into the wedge ( 1 1 1ϕ−Φ < < Φ ) (78πΦ = , / 8oϕ π= and

3rε = ).

Figure 16a . Diagram of diffraction out the wedge ( ϕ−Φ < < Φ ) (78πΦ = , 4 / 5oϕ π= and

3rε = ).

Figure 16b. Diagram of diffraction into the wedge ( 1 1 1ϕ−Φ < < Φ ) (78πΦ = , 4 / 5oϕ π= and

3rε = ).

77

Figure 17a . Diagram of diffraction out the wedge ( ϕ−Φ < < Φ ) (34πΦ = , / 5oϕ π= and

10rε = ).

Figure 17b . Diagram of diffraction into the wedge ( 1 1 1ϕ−Φ < < Φ ) (34πΦ = , / 5oϕ π= and

10rε = ).

78

Figure 18a . Diagram of diffraction out the wedge ( ϕ−Φ < < Φ ) (78πΦ = , / 8oϕ π= and

10rε = ).

Figure 18b. Diagram of diffraction into the wedge ( 1 1 1ϕ−Φ < < Φ ) (78πΦ = , / 8oϕ π=

and 10rε = ).

79

Figure 19a. Diagram of diffraction out the wedge ( ϕ−Φ < < Φ )

(30, 1,4o oE H π= = Φ = , / 8,

4oπϕ π β= = and 3, 1r rε µ= = ).

Figure 19b. Diagram of diffraction out the wedge ( ϕ−Φ < < Φ )

(34πΦ = ,

30, 1,4o oE H π= = Φ = / 8,

4oπϕ π β= = and 3, 1r rε µ= = ).

80

Figure 19c. Diagram of diffraction into the wedge ( 1 1 1ϕ−Φ < < Φ )

(30, 1,4o oE H π= = Φ = , / 8,

4oπϕ π β= = and 3, 1r rε µ= = ).

Figure 19d. Diagram of diffraction into the wedge ( 1 1 1ϕ−Φ < < Φ ) (30, 1,4o oE H π= = Φ = ,

/ 8,4oπϕ π β= = and 3, 1r rε µ= = ).

APPENDIX H : The MATHEMATICA code for the diffractionof a skew plane wave incident on arbitrary penetrable wedges

<< LinearAlgebra`MatrixManipulation`

Appendice H.nb 1

k1 := λ koµr = 1Eo := 0

Ho := 1k := 1 − Ik1 := λ kko = kβ := π ê4

β1 = ArcCosA ko(((((((k1

Cos@βDE ;

Φ := 3 π((((4

Φ1 := π − Φ

τo = k Sin@βD;ϕo := π ê8;εr := 3 − 0.00001 I

λ =è!!!!!!!!!!!!!

εr µr ;

λe ="###############################################

1 + H−1 + λ2L Csc@βD2 ;

εtr = λe2;

Zo := 377

Z1 := $%%%%%%%%%µr(((((((εr

Zo

Yo := 1êZo

µ =ko Zo((((((((((((((

ω;

µ1 = µ;

ε =ko

(((((((((((ω Zo

;

ε1 =k1

(((((((((((ω Z1

;

A := 10h := 0.5A1 := A

h1 := hA êh

g@w_D := −ArcCosA 1((((λ

Cos@wDEgt@w_D := −ArcCosA 1

(((((((λe

Cos@wDEg1@w1_D := −ArcCos@λ Cos@w1DDgt1@w1_D := −ArcCos@λe Cos@w1DDH∗Sin@g@wDD=−

"#########################1− HCos@wDL2((((((((((((((((((((

εr∗LH∗Sin@g1@w1DD=−

è!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!1−εr HCos@w1DL2 ∗L

Appendice H.nb 2

τo = ko Sin@βD;αo = ko Cos@βD;η = −τo Cos@wDτ1 = λe τo ;α1 = αo;ϕt = Φ + gt@Φ − ϕoD;

ϕr = 2 Φ − ϕo;Eor = −HEo Z1 Zo α12 τo2 Cos@Φ − ϕtD2 + Ho ko Z1 Zo2 α1 τ1 τo Cos@Φ − ϕtD Sin@Φ − ϕoD −

Ho ko Z1 Zo2 α1 τ1 τo Cos@Φ − ϕtD Sin@Φ − ϕrD + Eo ko2 Z1 Zo τ12 Sin@Φ − ϕoD Sin@Φ − ϕrD −

Eo k1 ko Z12 τ1 τo Sin@Φ − ϕoD Sin@Φ − ϕtD − Eo k1 ko Zo2 τ1 τo Sin@Φ − ϕrD Sin@Φ − ϕtD +

Eo k12 Z1 Zo τo2 Sin@Φ − ϕtD2 + Z1 Zo αo τ1 Cos@Φ − ϕoD HEo αo τ1 Cos@Φ − ϕrD −

Eo α1 τo Cos@Φ − ϕtD + Ho ko Zo τ1 Sin@Φ − ϕrD − Ho k1 Z1 τo Sin@Φ − ϕtDL + Z1 Zo αo τ1Cos@Φ − ϕrD H−Eo α1 τo Cos@Φ − ϕtD − Ho ko Zo τ1 Sin@Φ − ϕoD + Ho k1 Z1 τo Sin@Φ − ϕtDLLêHZ1 Zo αo2 τ12 Cos@Φ − ϕrD2 − 2 Z1 Zo α1 αo τ1 τo Cos@Φ − ϕrD Cos@Φ − ϕtD +

Z1 Zo α12 τo2 Cos@Φ − ϕtD2 + ko2 Z1 Zo τ12 Sin@Φ − ϕrD2 −

k1 ko Z12 τ1 τo Sin@Φ − ϕrD Sin@Φ − ϕtD −

k1 ko Zo2 τ1 τo Sin@Φ − ϕrD Sin@Φ − ϕtD + k12 Z1 Zo τo2 Sin@Φ − ϕtD2L;Eot = −HZ1 τ1 H−Eo Zo αo2 τ1 Cos@Φ − ϕrD2 + ko HSin@Φ − ϕoD − Sin@Φ − ϕrDLHHo Zo2 α1 τo Cos@Φ − ϕtD + Eo ko Zo τ1 Sin@Φ − ϕrD − Eo k1 Z1 τo Sin@Φ − ϕtDL +

Zo αo Cos@Φ − ϕoD HEo αo τ1 Cos@Φ − ϕrD − Eo α1 τo Cos@Φ − ϕtD +

Ho ko Zo τ1 Sin@Φ − ϕrD − Ho k1 Z1 τo Sin@Φ − ϕtDL + Zo αo Cos@Φ − ϕrDHEo α1 τo Cos@Φ − ϕtD − Ho ko Zo τ1 Sin@Φ − ϕoD + Ho k1 Z1 τo Sin@Φ − ϕtDLLLêHZ1 Zo αo2 τ12 Cos@Φ − ϕrD2 − 2 Z1 Zo α1 αo τ1 τo Cos@Φ − ϕrD Cos@Φ − ϕtD +

Z1 Zo α12 τo2 Cos@Φ − ϕtD2 + ko2 Z1 Zo τ12 Sin@Φ − ϕrD2 −

k1 ko Z12 τ1 τo Sin@Φ − ϕrD Sin@Φ − ϕtD −

k1 ko Zo2 τ1 τo Sin@Φ − ϕrD Sin@Φ − ϕtD + k12 Z1 Zo τo2 Sin@Φ − ϕtD2L;Hor = H−Ho Z1 Zo α12 τo2 Cos@Φ − ϕtD2 + Eo ko Z1 α1 τ1 τo Cos@Φ − ϕtD Sin@Φ − ϕoD −

Eo ko Z1 α1 τ1 τo Cos@Φ − ϕtD Sin@Φ − ϕrD − Ho ko2 Z1 Zo τ12 Sin@Φ − ϕoD Sin@Φ − ϕrD +

Ho k1 ko Zo2 τ1 τo Sin@Φ − ϕoD Sin@Φ − ϕtD + Ho k1 ko Z12 τ1 τo Sin@Φ − ϕrD Sin@Φ − ϕtD −

Ho k12 Z1 Zo τo2 Sin@Φ − ϕtD2 + αo τ1 Cos@Φ − ϕrDHHo Z1 Zo α1 τo Cos@Φ − ϕtD − Eo ko Z1 τ1 Sin@Φ − ϕoD + Eo k1 Zo τo Sin@Φ − ϕtDL −

αo τ1 Cos@Φ − ϕoD HHo Z1 Zo αo τ1 Cos@Φ − ϕrD − Ho Z1 Zo α1 τo Cos@Φ − ϕtD −

Eo ko Z1 τ1 Sin@Φ − ϕrD + Eo k1 Zo τo Sin@Φ − ϕtDLLêHZ1 Zo αo2 τ12 Cos@Φ − ϕrD2 − 2 Z1 Zo α1 αo τ1 τo Cos@Φ − ϕrD Cos@Φ − ϕtD +

Z1 Zo α12 τo2 Cos@Φ − ϕtD2 + ko2 Z1 Zo τ12 Sin@Φ − ϕrD2 −

k1 ko Z12 τ1 τo Sin@Φ − ϕrD Sin@Φ − ϕtD −

k1 ko Zo2 τ1 τo Sin@Φ − ϕrD Sin@Φ − ϕtD + k12 Z1 Zo τo2 Sin@Φ − ϕtD2L;Hot = Hτ1 HHo Z1 Zo αo2 τ1 Cos@Φ − ϕrD2 − αo Cos@Φ − ϕrDHHo Z1 Zo α1 τo Cos@Φ − ϕtD + Eo ko Z1 τ1 Sin@Φ − ϕoD − Eo k1 Zo τo Sin@Φ − ϕtDL −

αo Cos@Φ − ϕoD HHo Z1 Zo αo τ1 Cos@Φ − ϕrD − Ho Z1 Zo α1 τo Cos@Φ − ϕtD −

Eo ko Z1 τ1 Sin@Φ − ϕrD + Eo k1 Zo τo Sin@Φ − ϕtDL + ko H−Sin@Φ − ϕoD + Sin@Φ − ϕrDLH−Eo Z1 α1 τo Cos@Φ − ϕtD + Ho Zo Hko Z1 τ1 Sin@Φ − ϕrD − k1 Zo τo Sin@Φ − ϕtDLLLLêHZ1 Zo αo2 τ12 Cos@Φ − ϕrD2 − 2 Z1 Zo α1 αo τ1 τo Cos@Φ − ϕrD Cos@Φ − ϕtD +

Z1 Zo α12 τo2 Cos@Φ − ϕtD2 + ko2 Z1 Zo τ12 Sin@Φ − ϕrD2 −

k1 ko Z12 τ1 τo Sin@Φ − ϕrD Sin@Φ − ϕtD −

k1 ko Zo2 τ1 τo Sin@Φ − ϕrD Sin@Φ − ϕtD + k12 Z1 Zo τo2 Sin@Φ − ϕtD2L;

H∗The following expressions providethe axial spectra Hϕ=0L of the exact PEC solution ∗L

Appendice H.nb 3

H∗VPECρp@w_D=

−H2 4 π Csc@βD HEo Cos@ π ϕo((((((((((2 Φ

D Cot@wD Cot@βD Sin@ π w((((((((2 Φ

D+Ho Zo Cos@ π w((((((((2 Φ

D Csc@βDSin@ π ϕo((((((((((

2 ΦDLLêHk Φ HCos@ π w((((((((

ΦD−Cos@ π ϕo((((((((((

ΦDLL;

IPECρp@w_D=4 Ho π Zo Cot@wD Cot@βD Csc@βD Sin@ π w(((((((((Φ D−4 Eo π Csc@βD2 Sin@ π ϕo(((((((((((Φ D(((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((

−k Zo Φ Cos@ π w(((((((((Φ D+k Zo Φ Cos@ π ϕo(((((((((((Φ D ;

VPECzp@w_D=2 4 Eo π Cos@ π ϕo(((((((((((2 Φ D Csc@wD Csc@βD Sin@ π w(((((((((2 Φ D((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((

−k Φ Cos@ π w(((((((((Φ D+k Φ Cos@ π ϕo(((((((((((Φ D ;

IPECzp@w_D=4 Ho π Csc@wD Csc@βD Sin@ π w(((((((((Φ D(((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((

−k Φ Cos@ π w(((((((((Φ D+k Φ Cos@ π ϕo(((((((((((Φ D ;∗LH∗FPECp@w_D=88VPECzp@wD<,8IPECρp@wD<,8IPECzp@wD<,8VPECρp@wD<<∗LH∗The following expressions provide

the axial spectra Hϕ=0L of the exact PMC solution ∗LH∗VPMCρp=−

4 π Csc@βD HEo Cot@wD Cot@βD Sin@ π w(((((((((Φ D+Ho Zo Csc@βD Sin@ π ϕo(((((((((((Φ DL(((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((

k Φ HCos@ π w(((((((((Φ D−Cos@ π ϕo(((((((((((Φ DL ;

IPMCρp=H2 4 Ho π Zo Cos@ π ϕo((((((((((2 Φ

D Cot@wD Cot@βD Csc@βD Sin@ π w((((((((2 Φ

D−

2 4 Eo π Cos@ π w((((((((2 Φ

D Csc@βD2 Sin@ π ϕo((((((((((2 Φ

DLêH−k Zo Φ Cos@ π w((((((((Φ

D+k Zo Φ Cos@ π ϕo((((((((((Φ

DL;

VPMCzp=4 Eo π Csc@wD Csc@βD Sin@ π w(((((((((Φ D(((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((

−k Φ Cos@ π w(((((((((Φ D+k Φ Cos@ π ϕo(((((((((((Φ D ;

IPMCzp=2 4 Ho π Cos@ π ϕo(((((((((((2 Φ D Csc@wD Csc@βD Sin@ π w(((((((((2 Φ D((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((

−k Φ Cos@ π w(((((((((Φ D+k Φ Cos@ π ϕo(((((((((((Φ D ;∗L

v@t_D =4 π JΦ1 − 2 ArcCosA CosA Hπ+2 4 tL Φ((((((((((((((((((((((((2 π E

(((((((((((((((((((((((((((((λe

EN((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((

2 Φ1;

Ma@t_, u_D =Et

(((((((((((((((Et − I

Eu + I

(((((((((((((((((((Et+u + 1

;

Appendice H.nb 4

H∗h1a@u_D=∗Lhae@u_D = 99Cosh@uD CscA Hπ + 2 4 uL Φ

(((((((((((((((((((((((((((((((2 π

E Sech@v@uDD SinA Φ1 Hπ + 2 4 v@uDL((((((((((((((((((((((((((((((((((((((((((

2 πE, 0, 0, 0=,90, CosA 1

((((4

Hπ − 2 4 uLE CscA Hπ + 2 4 uL Φ(((((((((((((((((((((((((((((((

2 πE SecA 1

((((4

Hπ − 2 4 v@uDLE SinA Φ1 Hπ + 2 4 v@uDL((((((((((((((((((((((((((((((((((((((((((

2 πE,

0, 0=, 90,ikjjjjjj4

è!!!!2 Zo $%%%%%%%%%µr

(((((((εr

Cos@βD ikjjjλe CosA Hπ + 2 4 uL Φ(((((((((((((((((((((((((((((((

2 πE − CosA Φ1 Hπ + 2 4 v@uDL

((((((((((((((((((((((((((((((((((((((((((2 π

Ey{zzzCscA Hπ + 2 4 uL Φ

(((((((((((((((((((((((((((((((2 π

E JCoshA u((((2E + 4 SinhA u

((((2ENy{zzzzzz ì

Hλ H4 + Sinh@v@uDDLL,λe "#######µr((((((

εrH4 + Sinh@uDL

(((((((((((((((((((((((((((((((((((((((((((((((((((((((((λ H4 + Sinh@v@uDDL , 0=, 9 1

((((((((((((((((((((((((((((((2 Zo λ "#######µr((((((

εr

ikjjjCos@βD Cosh@uD ikjjj−λe CotA Hπ + 2 4 uL Φ(((((((((((((((((((((((((((((((

2 πE + CosA Φ1 Hπ + 2 4 v@uDL

((((((((((((((((((((((((((((((((((((((((((2 π

E CscA Hπ + 2 4 uL Φ(((((((((((((((((((((((((((((((

2 πEy{zzz

CscA 1((((4

Hπ − 2 4 v@uDLEy{zzz, 0, 0,λe Csc@ 1((((

4Hπ − 2 4 v@uDLD Sin@ 1((((

4Hπ − 2 4 uLD

(((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((λ "#######µr((((((

εr

==;

hbe@u_D =99CosA 1((((4

Hπ − 2 4 uLE CscA Hπ + 2 4 uL Φ(((((((((((((((((((((((((((((((

2 πE SecA 1

((((4

Hπ − 2 4 v@uDLE SinA Φ1 Hπ + 2 4 v@uDL((((((((((((((((((((((((((((((((((((((((((

2 πE, 0,

0, 0=, 90, Cosh@uD CscA Hπ + 2 4 uL Φ(((((((((((((((((((((((((((((((

2 πE Sech@v@uDD SinA Φ1 Hπ + 2 4 v@uDL

((((((((((((((((((((((((((((((((((((((((((2 π

E, 0, 0=,

90,1

((((((((2 λ

ikjjjjjjZo $%%%%%%%%%µr

(((((((εr

Cos@βD Cosh@uD ikjjjλe CotA Hπ + 2 4 uL Φ(((((((((((((((((((((((((((((((

2 πE −

CosA Φ1 Hπ + 2 4 v@uDL((((((((((((((((((((((((((((((((((((((((((

2 πE CscA Hπ + 2 4 uL Φ

(((((((((((((((((((((((((((((((2 π

Ey{zzz CscA 1((((4

Hπ − 2 4 v@uDLEy{zzzzzz,

λe "#######µr((((((εr

Csc@ 1((((4Hπ − 2 4 v@uDLD Sin@ 1((((

4Hπ − 2 4 uLD

((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((λ

, 0=, 9 1((((((((((((((((((((((((((Zo λ "#######µr((((((

εr

ikjjjjCosA 1

((((4

Hπ − 2 4 uLECos@βD ikjjj−λe CotA Hπ + 2 4 uL Φ

(((((((((((((((((((((((((((((((2 π

E + CosA Φ1 Hπ + 2 4 v@uDL((((((((((((((((((((((((((((((((((((((((((

2 πE CscA Hπ + 2 4 uL Φ

(((((((((((((((((((((((((((((((2 π

Ey{zzzCscA 1

((((4

Hπ − 2 4 v@uDLE2y{zzzz, 0, 0,λe H4 + Sinh@uDL

(((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((λ "#######µr((((((

εrH4 + Sinh@v@uDDL ==;

Appendice H.nb 5

H∗T1e@v_,u_D=∗LTae@v_, u_D = 99 Φ Cosh@uD Sech@vD SinA Hπ+2 4 vL Φ1((((((((((((((((((((((2 π

E((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((2 π2 I−CosA Hπ+2 4 uL Φ((((((((((((((((((((

2 πE + λe CosA Hπ+2 4 vL Φ1((((((((((((((((((((((

2 πEM , 0, 0, 0=,

90, −ikjjjΦ CosA 1((((4

Hπ − 2 4 uLE SecA 1((((4

Hπ − 2 4 vLE SinA Hπ + 2 4 vL Φ1(((((((((((((((((((((((((((((((((

2 πEy{zzz ìikjjj2 π2 ikjjjCosA Hπ + 2 4 uL Φ

(((((((((((((((((((((((((((((((2 π

E − λe CosA Hπ + 2 4 vL Φ1(((((((((((((((((((((((((((((((((

2 πEy{zzzy{zzz, 0, 0=, 90,

ikjjjjjj4 Zo $%%%%%%%%%µr(((((((εr

Φ

Cos@βD ikjjj−λe CosA Hπ + 2 4 uL Φ(((((((((((((((((((((((((((((((

2 πE + CosA Hπ + 2 4 vL Φ1

(((((((((((((((((((((((((((((((((2 π

Ey{zzz JCoshA u((((2E + 4 SinhA u

((((2ENy{zzzzzz ìikjjjè!!!!

2 π2 λ ikjjjCosA Hπ + 2 4 uL Φ(((((((((((((((((((((((((((((((

2 πE − λe CosA Hπ + 2 4 vL Φ1

(((((((((((((((((((((((((((((((((2 π

Ey{zzz H4 + Sinh@vDLy{zzz,ikjjjjjjλe $%%%%%%%%%µr(((((((εr

Φ CscA 1((((4

Hπ − 2 4 vLE2

SinA 1((((4

Hπ − 2 4 uLE2

SinA Hπ + 2 4 uL Φ(((((((((((((((((((((((((((((((

2 πEy{zzzzzz ìikjjj2 π2 λ ikjjj−CosA Hπ + 2 4 uL Φ

(((((((((((((((((((((((((((((((2 π

E + λe CosA Hπ + 2 4 vL Φ1(((((((((((((((((((((((((((((((((

2 πEy{zzzy{zzz, 0=,9ikjjjΦ Cos@βD ikjjjλe CosA Hπ + 2 4 uL Φ

(((((((((((((((((((((((((((((((2 π

E − CosA Hπ + 2 4 vL Φ1(((((((((((((((((((((((((((((((((

2 πEy{zzz Cosh@uD CscA 1

((((4

Hπ − 2 4 vLEy{zzz ìikjjjjjj4 π2 Zo λ $%%%%%%%%%µr(((((((εr

ikjjjCosA Hπ + 2 4 uL Φ(((((((((((((((((((((((((((((((

2 πE − λe CosA Hπ + 2 4 vL Φ1

(((((((((((((((((((((((((((((((((2 π

Ey{zzzy{zzzzzz,

0, 0, ikjjjλe Φ CscA 1((((4

Hπ − 2 4 vLE SinA 1((((4

Hπ − 2 4 uLE SinA Hπ + 2 4 uL Φ(((((((((((((((((((((((((((((((

2 πEy{zzz ìikjjjjjj2 π2 λ $%%%%%%%%%µr

(((((((εr

ikjjj−CosA Hπ + 2 4 uL Φ(((((((((((((((((((((((((((((((

2 πE + λe CosA Hπ + 2 4 vL Φ1

(((((((((((((((((((((((((((((((((2 π

Ey{zzzy{zzzzzz==;

ua = DiagonalMatrix@Table@1, 8r, −A êh, A êh<DD êê N;za = DiagonalMatrix@Table@0, 8r, −A êh, A êh<DD êê N;

da = DiagonalMatrixATableA−TanA I h r − π((((2(((((((((((((((((((((((

2E, 8r, −A êh, A êh<EE êê N;

da1 = DiagonalMatrixATableA−TanA I v@ h1 rD − π((((2(((((((((((((((((((((((((((((((((((

2E, 8r, −A1êh1, A1êh1<EE êê N;

Mma = TableA−h

(((((((((π I

Ma@h r, h iD, 8r, −A êh, A êh<, 8i, −A êh, A êh<E êê N;

Mma1 = TableA−h1

(((((((((π I

Ma@v@h1 rD, h1 iD, 8r, −A êh, A êh<, 8i, −A1êh1, A1êh1<E êê N;

Null6

Appendice H.nb 6

Dimensions@MmaD841, 41<hae11 = DiagonalMatrix@Table@hae@h rD@@1, 1DD, 8r, −A êh, A êh<DD êê N;hae32 = DiagonalMatrix@Table@hae@h rD@@3, 2DD, 8r, −A êh, A êh<DD êê N;hae33 = DiagonalMatrix@Table@hae@h rD@@3, 3DD, 8r, −A êh, A êh<DD êê N;hae22 = DiagonalMatrix@Table@hae@h rD@@2, 2DD, 8r, −A êh, A êh<DD êê N;hae41 = DiagonalMatrix@Table@hae@h rD@@4, 1DD, 8r, −A êh, A êh<DD êê N;hae44 = DiagonalMatrix@Table@hae@h rD@@4, 4DD, 8r, −A êh, A êh<DD êê N;

hbe22 = hae11;

hbe11 = hae22;

hbe41 = −εr

(((((((((((((((((Zo2 µr

hae32;

hbe32 = −µr(((((((εr

Zo2 hae41;

hbe33 =µr(((((((εr

hae44;

hbe44 =εr((((((((µr

hae33;

Tae11 = Table@Tae@h r, h iD@@1, 1DD, 8r, −A êh, A êh<, 8i, −A êh, A êh<D êê N;Tae12 = Table@0, 8r, −A êh, A ê h<, 8i, −A êh, A êh<D êê N;Tae13 = Table@0, 8r, −A êh, A ê h<, 8i, −A êh, A êh<D êê N;Tae14 = Table@0, 8r, −A êh, A ê h<, 8i, −A êh, A êh<D êê N;Tae21 = Table@0, 8r, −A êh, A ê h<, 8i, −A êh, A êh<D êê N;Tae22 = Table@Tae@h r, h iD@@2, 2DD, 8r, −A êh, A êh<, 8i, −A êh, A êh<D êê N;Tae23 = Table@0, 8r, −A êh, A ê h<, 8i, −A êh, A êh<D êê N;Tae24 = Table@0, 8r, −A êh, A ê h<, 8i, −A êh, A êh<D êê N;Tae31 = Table@0, 8r, −A êh, A ê h<, 8i, −A êh, A êh<D êê N;Tae32 = Table@Tae@h r, h iD@@3, 2DD, 8r, −A êh, A êh<, 8i, −A êh, A êh<D êê N;Tae33 = Table@Tae@h r, h iD@@3, 3DD, 8r, −A êh, A êh<, 8i, −A êh, A êh<D êê N;Tae34 = Table@0, 8r, −A êh, A ê h<, 8i, −A êh, A êh<D êê N;Tae41 = Table@Tae@h r, h iD@@4, 1DD, 8r, −A êh, A êh<, 8i, −A êh, A êh<D êê N;Tae42 = Table@0, 8r, −A êh, A ê h<, 8i, −A êh, A êh<D êê N;Tae43 = Table@0, 8r, −A êh, A ê h<, 8i, −A êh, A êh<D êê N;Tae44 = Table@Tae@h r, h iD@@4, 4DD, 8r, −A êh, A êh<, 8i, −A êh, A êh<D êê N;

Appendice H.nb 7

Tbe22 = Tae11;

Tbe11 = Tae22;

Tbe41 = −εr

(((((((((((((((((Zo2 µr

Tae32;

Tbe32 = −µr(((((((εr

Zo2 Tae41;

Tbe33 =µr(((((((εr

Tae44;

Tbe44 =εr

((((((((µr

Tae33;

General::spell1 :

Possible spelling error: new symbol name "Tbe22" is similar to existing symbol "Tae22". More…

General::spell1 :

Possible spelling error: new symbol name "Tbe11" is similar to existing symbol "Tae11". More…

General::spell1 :

Possible spelling error: new symbol name "Tbe41" is similar to existing symbol "Tae41". More…

General::stop : Further output of General::spell1 will be suppressed during this calculation. More…

Null6H∗Taee=h1 BlockMatrix@88Tae11,Tae12,Tae13,Tae14<,8Tae21,Tae22,Tae23,Tae24<,8Tae31,Tae32,Tae33,Tae34<,8Tae41,Tae42,Tae43,Tae44<<DêêN;∗LH∗Tbee=h1 BlockMatrix@88Tbe11,Tbe12,Tbe13,Tbe14<,8Tbe21,Tbe22,Tbe23,Tbe24<,8Tbe31,Tbe32,Tbe33,Tbe34<,8Tbe41,Tbe42,Tbe43,Tbe44<<DêêN;∗L

Appendice H.nb 8

Dimensions@Tae32DDimensions@Tbe32D841, 41<841, 41<H∗equazioni partenza diel_skew _new _ok.nb∗LR1o = −

2 4 HEo + EorL π(((((((((((((((((((((((((((((((((((((((

Φ;

R2o = −2 4 HEo − EorL π Cot@ π ϕo(((((((((

2 ΦD

(((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((Φ

;

R3o =2 4 HEo + EorL π Sin@ π ϕo(((((((((

2 ΦD

(((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((Φ

;

R4o =2 4 HEo − EorL π Cos@ π ϕo(((((((((

2 ΦD

(((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((Φ

;

R5o =2 4 HHo + HorL π(((((((((((((((((((((((((((((((((((((((

Φ;

R6o =2 4 HHo − HorL π Cot@ π ϕo(((((((((

2 ΦD

(((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((Φ

;

R7o = −2 4 HHo + HorL π Sin@ π ϕo(((((((((

2 ΦD

(((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((Φ

;

R8o = −2 4 HHo − HorL π Cos@ π ϕo(((((((((

2 ΦD

(((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((Φ

;

H∗ source terms∗Ln1p = − TableA R1o

(((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((H I Sinh@h rD − Cos@ π((((Φ

ϕoDL −

TanA I h r − π((((2(((((((((((((((((((((((

2E

R2o(((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((H I Sinh@h rD − Cos@ π((((

Φ ϕoDL , 8r, −A êh, A êh<E;

n3p = − TableA R3o(((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((H I Sinh@h rD − Cos@ π((((

Φ ϕoDL −

TanA I h r − π((((2(((((((((((((((((((((((

2E

R4o(((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((H I Sinh@h rD − Cos@ π((((

Φ ϕoDL , 8r, −A êh, A êh<E;

Appendice H.nb 9

n5p = − TableA R5o(((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((H I Sinh@h rD − Cos@ π((((

Φ ϕoDL −

TanA I h r − π((((2(((((((((((((((((((((((

2E

R6o((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((k H I Sinh@h rD − Cos@ π((((

Φ ϕoDL , 8r, −A êh, A êh<E;

n7p = − TableA R7o((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((k H I Sinh@h rD − Cos@ π((((

Φ ϕoDL −

TanA I h r − π((((2(((((((((((((((((((((((

2E

R8o((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((k H I Sinh@h rD − Cos@ π((((

Φ ϕoDL , 8r, −A êh, A êh<E;

n1m = TableA−−I 4 π(((((((

Φ Eo

((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((4 Sinh@uD − Cos@ π((((

Φ ϕoD , 8u, −A, A, h<E êê N;

n3m = TableA−I 4 π(((((((

Φ Eo Sin@ π((((((

2 Φ ϕoD

((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((4 Sinh@uD − Cos@ π((((

Φ ϕoD , 8u, −A, A, h<E êê N;

n5m = TableA−I 4 π(((((((

Φ Ho

((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((4 Sinh@uD − Cos@ π((((

Φ ϕoD , 8u, −A, A, h<E êê N;

n7m = TableA−−I 4 π(((((((

Φ Ho Sin@ π((((((

2 Φ ϕoD

(((((((((((((((((((((((((((((((((((((((((((((((((((((((((4 Sinh@uD − Cos@ π((((

Φ ϕoD , 8u, −A, A, h<E êê N;

H∗S1:=−I 4 π((((((((Φ

Eo;H∗=S1 riduzione skew∗LS3:= I 4 π((((((((

Φ Eo Sin@ π((((((

2 Φ ϕoD;H∗=S2 riduzione skew∗L

S5=I 4 π((((((((Φ

Ho;H∗=S3 riduzione skew∗LS7:=− I 4 π((((((((

Φ Ho Sin@ π((((((

2 Φ ϕoD;H∗=S4 riduzione skew∗L∗L

n1 := If@ϕo < Φ ê2, n1m, n1pDn3 := If@ϕo < Φ ê2, n3m, n3pDn5 := If@ϕo < Φ ê2, n5m, n5pDn7 := If@ϕo < Φ ê2, n7m, n7pDnz = Table@0, 8u, −A, A, h<D êê N;sa = Join@n1, nz, n7, nzD;sb = Join@n3, nz, n5, nzD;

Ga32 = −da1.hae32 − h Mma1.Tae32;Ga33 = −da1. hae33 − h Mma1.Tae33;Ga41 = −h Mma1 Tae41 − da1.hae41;Ga44 = −da1.hae44 − h Mma1.Tae44;

Gb32 = −da1.hbe32 − h Mma1.Tbe32;Gb33 = −da1. hbe33 − h Mma1.Tbe33;Gb41 = −h Mma1 Tbe41 − da1.hbe41;Gb44 = −da1.hbe44 − h Mma1.Tbe44;

Appendice H.nb 10

aa = da + Mma;bba = hae11;cca = Ga32;dda = Ga33;eea = Ga41;ffa = hae22;gga = Ga44;

ss1a = −LinearSolve@Hdda − bba.aaL, Hbba.n1 + cca.n7LD;

ss7a = −LinearSolve@Hgga − ffa.aaL, Heea.n1 + ffa.n7LD;

σσa = −LinearSolve@Hdda − bba.aaL, cca.aaD;

ννa = −LinearSolve@Hgga − ffa.aaL, eea.aaD;

p2 = LinearSolve@Hua − σσa.ννaL, Hss1a − σσa.ss7aLD;

P2 = %;

p8 = LinearSolve@Hua − ννa.σσaL, Hss7a − ννa.ss1aLD;

P8 = %;

P1 = n1 − aa.P2;P7 = n7 − aa.P8;

Appendice H.nb 11

Dimensions@P1D841<Q2 = h Tae32.P7 + h Tae33.P2;Q8 = h Tae41.P1 + h Tae44.P8;

aa = da + Mma;

bbb = hbe11;

ccb = Gb32;

ddb = Gb33;

eeb = Gb41;

ffb = hbe22;

ggb = Gb44;

ss3b = −LinearSolve@Hddb − bbb.aaL, Hbbb.n3 + ccb.n5LD;

ss5b = −LinearSolve@Hggb − ffb.aaL, Heeb.n3 + ffb.n5LD;

σσb = −LinearSolve@Hddb − bbb.aaL, ccb.aaD;

ννb = −LinearSolve@Hggb − ffb.aaL, eeb.aaD;

p4 = LinearSolve@Hua − σσb.ννbL, Hss3b − σσb.ss5bLD;

P4 = %;

p6 = LinearSolve@Hua − ννb.σσbL, Hss5b − ννb.ss3bLD;

P6 = %;

P3 = n3 − aa.P4;

Appendice H.nb 12

P5 = n5 − aa.P6;

Q4 = h Tbe32.P5 + h Tbe33.P4;

Q6 = h Tbe41.P3 + h Tbe44.P6;

Appendice H.nb 13

Dimensions@Q8D841<x2f = Function@8n<, P2@@n + HA êhL + 1DDD;

x4f = Function@8n<, P4@@n + HA êhL + 1DDD;

y2f = Function@8n<, Q2@@n + HA1êh1L + 1DDD;y4f = Function@8n<, Q4@@n + HA1êh1L + 1DDD;x6f = Function@8n<, P6@@n + HA êhL + 1DDD;

x8f = Function@8n<, P8@@n + HA êhL + 1DDD;y6f = Function@8n<, Q6@@n + HA1êh1L + 1DDD;y8f = Function@8n<, Q8@@n + HA1êh1L + 1DDD;

H∗da partenza diel_skew _. nb∗Ls1@w_D := IfAϕo <

Φ((((2

,−I 4 π(((((((

Φ Eo

(((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((H Cos@ π((((Φ

wD − Cos@ π((((Φ

ϕoDL ,ikjjjjj R1o(((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((HCos@ π((((

Φ wD − Cos@ π((((

Φ ϕoDL − TanA π w

(((((((((2 Φ

E R2o

((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((H Cos@ π((((Φ

wD − Cos@ π((((Φ

ϕoDL y{zzzzzEs3@w_D := IfAϕo <

Φ((((2

,I 4 π(((((((

Φ Eo Sin@ π((((((

2 Φ ϕoD

((((((((((((((((((((((((((((((((((((((((((((((((((((((((((Cos@ π((((

Φ wD − Cos@ π((((

Φ ϕoD ,ikjjjjj R3o

(((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((HCos@ π((((Φ

wD − Cos@ π((((Φ

ϕoDL − TanA π w(((((((((2 Φ

E R4o

((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((H Cos@ π((((Φ

wD − Cos@ π((((Φ

ϕoDL y{zzzzzEs5@w_D := IfAϕo <

Φ((((2

,I 4 π(((((((

Φ Ho

((((((((((((((((((((((((((((((((((((((((((((((((((((((((((Cos@ π((((

Φ wD − Cos@ π((((

Φ ϕoD ,ikjjjjj R5o

(((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((HCos@ π((((Φ

wD − Cos@ π((((Φ

ϕoDL − TanA π w(((((((((2 Φ

E R6o

((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((H Cos@ π((((Φ

wD − Cos@ π((((Φ

ϕoDL y{zzzzzEs7@w_D := IfAϕo <

Φ((((2

,− I 4 π(((((((

Φ Ho Sin@ π((((((

2 Φ ϕoD

((((((((((((((((((((((((((((((((((((((((((((((((((((((((((Cos@ π((((

Φ wD − Cos@ π((((

Φ ϕoD ,ikjjjjj R7o

(((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((HCos@ π((((Φ

wD − Cos@ π((((Φ

ϕoDL − TanA π w(((((((((2 Φ

E R8o

((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((H Cos@ π((((Φ

wD − Cos@ π((((Φ

ϕoDL y{zzzzzE

Appendice H.nb 14

Y1@w_D =ikjjjjj 1

((((((((τo

ikjjjjj h

((((((((((π I

‚i=−Aêh

Aêh

MaA−I J π((((Φ

w +π((((2N, h i E x2f@iD − s1@wDy{zzzzzy{zzzzz;

Y3@w_D =ikjjjjj 1

((((((((τo

ikjjjjj h

((((((((((π I

‚i=−Aêh

Aêh

MaA−I J π((((Φ

w +π((((2N, h i E x4f@iD − s3@wDy{zzzzzy{zzzzz;

Y5@w_D =ikjjjjj 1

((((((((τo

ikjjjjj h

((((((((((π I

‚i=−Aêh

Aêh

MaA−I J π((((Φ

w +π((((2N, h i E x6f@iD − s5@wDy{zzzzzy{zzzzz;

Y7@w_D =ikjjjjj 1

((((((((τo

ikjjjjj h

((((((((((π I

‚i=−Aêh

Aêh

MaA−I J π((((Φ

w +π((((2N, h i E x8f@iD − s7@wDy{zzzzzy{zzzzz;

Y2@w1_D =ikjjjjj 1

((((((((τo

ikjjjjj−

h1((((((((((π I

‚i=−A1êh1

A1êh1

MaA−I J π(((((((Φ1

w1 +π((((2N, h1 i E y2f@iDy{zzzzzy{zzzzz;

Y4@w1_D =ikjjjjj 1

((((((((τo

ikjjjjj−

h1((((((((((π I

‚i=−A1êh1

A1êh1

MaA−I J π(((((((Φ1

w1 +π((((2N, h1 i E y4f@iDy{zzzzzy{zzzzz;

Y6@w1_D =ikjjjjj 1

((((((((τo

ikjjjjj−

h1((((((((((π I

‚i=−A1êh1

A1êh1

MaA−I J π(((((((Φ1

w1 +π((((2N, h1 i E y6f@iDy{zzzzzy{zzzzz;

Y8@w1_D =ikjjjjj 1

((((((((τo

ikjjjjj−

h1((((((((((π I

‚i=−A1êh1

A1êh1

MaA−I J π(((((((Φ1

w1 +π((((2N, h1 i E y8f@iDy{zzzzzy{zzzzz;

Appendice H.nb 15

Vρp@w_D = H∗− np αo η Y1@wD+np µ ξpè!!!!!!!

τo ω Y7@wD(((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((2 ξp τo2 =∗L−

1((((2

CosA π w(((((((((2 Φ

E JCot@wD Cot@βD SinA π w(((((((((2 Φ

E Y1@wD + Zo Csc@βD Y7@wDN;

Vzp@w_D = H∗ np Y1@wD((((((((((((((((((2 ξp

=∗L−1((((4

Csc@wD SinA π w(((((((((

ΦE Y1@wD;

Iρp@w_D = H∗− np ε ξpè!!!!!!!

τo ω Y3@wD−np αo η Y5@wD(((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((2 ξp τo2 =∗L 1

(((((((((((2 Zo

JCosA π w(((((((((2 Φ

E J−Csc@βD Y3@wD + Zo Cot@wD Cot@βD SinA π w(((((((((2 Φ

E Y5@wDNN;

Izp@w_D = H∗− np Y5@wD((((((((((((((((((2 ξp

=∗L 1((((4

Csc@wD SinA π w(((((((((

ΦE Y5@wD;

Vρπp@w1_D = H∗− n1p αo η Y2@w1D−n1p µ ξ1pè!!!!!!!

τ1 ω Y8@wD(((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((2 ξ1p τ12 =∗L 1

((((((((((((((((((((2è!!!!!!!!!

εtr ikjjCosA π w1

(((((((((((2 Φ1

E ikjj−Cot@w1D Cot@βD SinA π w1(((((((((((2 Φ1

E Y2@w1D + Zo Csc@βD Y8@w1Dy{zzy{zz;

Vzπp@w1_D = H∗ n1p Y2@w1D((((((((((((((((((((((2 ξ1p

=∗L−1((((4

Csc@w1D SinA π w1(((((((((((

Φ1E Y2@w1D;

Iρπp@w1_D = H∗− 1((((((((((((((((((((2 ξ1p τ12 I−n1p ε1 ξ1p

è!!!!!!τ1 ω Y4@w1D−n1p αo η Y6@w1DM=∗L 1

(((((((((((((((((((((((((((2 Zo

è!!!!!!!!!εtr

ikjjCosA π w1(((((((((((2 Φ1

E ikjjεr Csc@βD Y4@w1D + Zo Cot@w1D Cot@βD SinA π w1(((((((((((2 Φ1

E Y6@w1Dy{zzy{zz;

Izπp@w1_D = H∗− n1p Y6@w1D((((((((((((((((((((((2 ξ1p

=∗L 1((((4

Csc@w1D SinA π w1(((((((((((

Φ1E Y6@w1D;

Vρpp@w_D = If@Re@wD < 0, Vρp@wD, Vρp@−wDD;Iρpp@w_D = If@Re@wD < 0, Iρp@wD, Iρp@−wDD;

Vzpp@w_D = If@Re@wD < 0, Vzp@wD, Vzp@−wDD;Izpp@w_D = If@Re@wD < 0, Izp@wD, Izp@−wDD;

Vρπpp@w1_D = If@Re@w1D < 0, Vρπp@w1D, Vρπp@−w1DD;Iρπpp@w1_D = If@Re@w1D < 0, Iρπp@w1D, Iρπp@−w1DD;

Vzπpp@w1_D = If@Re@w1D < 0, Vzπp@w1D, Vzπp@−w1DD;Izπpp@w1_D = If@Re@w1D < 0, Izπp@w1D, Izπp@−w1DD;

H11n@w_, w1_D =88−HCsc@w − ΦD H16 HZ1 − ZoL HZ1 + ZoL Sin@wD Sin@w1D Sin@βD Sin@β1D Sin@w + ΦD +

8 Z1 Zo HCos@w1D Sin@2 βD Sin@2 β1D Sin@ΦD − 2 Cos@β1D2 Sin@βD2 Sin@w + ΦDL +

Z1 Zo Sin@β1D2 H10 Sin@w − ΦD + 3 Sin@w − 2 β − ΦD + 3 Sin@w + 2 β − ΦD −

4 Sin@βD2 H−2 Cos@2 w1D Sin@w + ΦD + Sin@3 w + ΦDLLLLê

Appendice H.nb 16

H4 H−3 Z1 Zo + Z1 Zo Cos@2 βD + 2 Z1 Zo Cos@βD2 Cos@2 β1D + 2 Sin@βD Sin@β1DH−2 HZ12 + Zo2L Sin@wD Sin@w1D + Z1 Zo HCos@2 wD + Cos@2 w1DL Sin@βD Sin@β1DL +

2 Z1 Zo Cos@wD Cos@w1D Sin@2 βD Sin@2 β1DLL, 0, 0, H64 Z1 Zo Csc@w − ΦDSin@wD Sin@βD Sin@β1D HCos@w1D Cos@β1D Sin@βD − Cos@wD Cos@βD Sin@β1DLLêH−2 Z1 Zo Cos@2 β1D H8 Cos@βD2 − 4 HCos@2 wD + Cos@2 w1DL Sin@βD2L −

8 Z1 Zo H−3 + Cos@2 βD + HCos@2 wD + Cos@2 w1DL Sin@βD2L +

32 HZ12 + Zo2L Sin@wD Sin@w1D Sin@βD Sin@β1D −

16 Z1 Zo Cos@wD Cos@w1D Sin@2 βD Sin@2 β1DL<,80, H8 Z1 Zo Cos@β1D2 Sin@βD2 − 8 HZ1 − ZoL HZ1 + ZoL Sin@wD Sin@w1D Sin@βD Sin@β1D +

Z1 Zo H−4 Cos@2 w1D Sin@βD2 Sin@β1D2 + Csc@w − ΦD H4 Cos@w1D Sin@2 βD Sin@2 β1DSin@ΦD + Sin@β1D2 H2 Sin@βD2 Sin@3 w − ΦD − H5 + 3 Cos@2 βDL Sin@w + ΦDLLLLêH6 Z1 Zo − 2 Z1 Zo Cos@2 βD − 4 Z1 Zo Cos@βD2 Cos@2 β1D + 4 Sin@βD Sin@β1DH2 HZ12 + Zo2L Sin@wD Sin@w1D − Z1 Zo HCos@2 wD + Cos@2 w1DL Sin@βD Sin@β1DL −

4 Z1 Zo Cos@wD Cos@w1D Sin@2 βD Sin@2 β1DL, H4 Csc@w − ΦDH2 Z1 Zo Sin@2 wD Sin@2 βD H1 + Cos@2 β1D + H−Cos@2 w1D + Cos@2 ΦDL Sin@β1D2L −

4 Z1 Zo Cos@w1D Sin@wD H2 Cos@βD2 + H−Cos@2 wD + Cos@2 ΦDL Sin@βD2L Sin@2 β1D +

8 HZ1 − ZoL HZ1 + ZoL Cos@βD Sin@wD Sin@w1D Sin@β1D Sin@2 ΦDLLêH2 Z1 Zo Cos@2 β1D H8 Cos@βD2 − 4 HCos@2 wD + Cos@2 w1DL Sin@βD2L +

8 Z1 Zo H−3 + Cos@2 βD + HCos@2 wD + Cos@2 w1DL Sin@βD2L −

32 HZ12 + Zo2L Sin@wD Sin@w1D Sin@βD Sin@β1D +

16 Z1 Zo Cos@wD Cos@w1D Sin@2 βD Sin@2 β1DL, 0<,80, H64 Z1 Zo Csc@w − ΦD Sin@wD Sin@βD Sin@β1DHCos@w1D Cos@β1D Sin@βD − Cos@wD Cos@βD Sin@β1DLLêH−2 Z1 Zo Cos@2 β1D H8 Cos@βD2 − 4 HCos@2 wD + Cos@2 w1DL Sin@βD2L −

8 Z1 Zo H−3 + Cos@2 βD + HCos@2 wD + Cos@2 w1DL Sin@βD2L + 32 HZ12 + Zo2L Sin@wDSin@w1D Sin@βD Sin@β1D − 16 Z1 Zo Cos@wD Cos@w1D Sin@2 βD Sin@2 β1DL,HCsc@w − ΦD H16 HZ1 − ZoL HZ1 + ZoL Sin@wD Sin@w1D Sin@βD Sin@β1D Sin@w + ΦD −

8 Z1 Zo HCos@w1D Sin@2 βD Sin@2 β1D Sin@ΦD − 2 Cos@β1D2 Sin@βD2 Sin@w + ΦDL −

Z1 Zo Sin@β1D2 H10 Sin@w − ΦD + 3 Sin@w − 2 β − ΦD + 3 Sin@w + 2 β − ΦD −

4 Sin@βD2 H−2 Cos@2 w1D Sin@w + ΦD + Sin@3 w + ΦDLLLLêH4 H−3 Z1 Zo + Z1 Zo Cos@2 βD + 2 Z1 Zo Cos@βD2 Cos@2 β1D + 2 Sin@βD Sin@β1DH−2 HZ12 + Zo2L Sin@wD Sin@w1D + Z1 Zo HCos@2 wD + Cos@2 w1DL Sin@βD Sin@β1DL +

2 Z1 Zo Cos@wD Cos@w1D Sin@2 βD Sin@2 β1DLL, 0<, 8H4 Csc@w − ΦDH2 Z1 Zo Sin@2 wD Sin@2 βD H1 + Cos@2 β1D + H−Cos@2 w1D + Cos@2 ΦDL Sin@β1D2L −

4 Z1 Zo Cos@w1D Sin@wD H2 Cos@βD2 + H−Cos@2 wD + Cos@2 ΦDL Sin@βD2L Sin@2 β1D +

8 H−Z12 + Zo2L Cos@βD Sin@wD Sin@w1D Sin@β1D Sin@2 ΦDLLêH2 Z1 Zo Cos@2 β1D H8 Cos@βD2 − 4 HCos@2 wD + Cos@2 w1DL Sin@βD2L +

8 Z1 Zo H−3 + Cos@2 βD + HCos@2 wD + Cos@2 w1DL Sin@βD2L − 32 HZ12 + Zo2L Sin@wDSin@w1D Sin@βD Sin@β1D + 16 Z1 Zo Cos@wD Cos@w1D Sin@2 βD Sin@2 β1DL, 0, 0,H8 Z1 Zo Cos@β1D2 Sin@βD2 + 8 HZ1 − ZoL HZ1 + ZoL Sin@wD Sin@w1D Sin@βD Sin@β1D +

Z1 Zo H−4 Cos@2 w1D Sin@βD2 Sin@β1D2 + Csc@w − ΦD H4 Cos@w1D Sin@2 βD Sin@2 β1DSin@ΦD + Sin@β1D2 H2 Sin@βD2 Sin@3 w − ΦD − H5 + 3 Cos@2 βDL Sin@w + ΦDLLLLêH6 Z1 Zo − 2 Z1 Zo Cos@2 βD − 4 Z1 Zo Cos@βD2 Cos@2 β1D + 4 Sin@βD Sin@β1DH2 HZ12 + Zo2L Sin@wD Sin@w1D − Z1 Zo HCos@2 wD + Cos@2 w1DL Sin@βD Sin@β1DL −

4 Z1 Zo Cos@wD Cos@w1D Sin@2 βD Sin@2 β1DL<<;H12n@w_, w1_D = 88H8 Zo Csc@w − ΦD H4 Z1 Cos@w1D Cos@2 β1D Cos@w1 + Φ1D Sin@wD Sin@βD2 −

2 Z1 H−3 Cos@Φ1D + Cos@2 w1 + Φ1DL Sin@wD Sin@βD2 − Z1 Cos@w1 + Φ1DSin@2 wD Sin@2 βD Sin@2 β1D + 8 Zo Sin@wD2 Sin@βD Sin@β1D Sin@w1 + Φ1DLLêH−2 Z1 Zo Cos@2 β1D H8 Cos@βD2 − 4 HCos@2 wD + Cos@2 w1DL Sin@βD2L −

8 Z1 Zo H−3 + Cos@2 βD + HCos@2 wD + Cos@2 w1DL Sin@βD2L + 32 HZ12 + Zo2L Sin@wDSin@w1D Sin@βD Sin@β1D − 16 Z1 Zo Cos@wD Cos@w1D Sin@2 βD Sin@2 β1DL,

0, 0, −H64 Z1 Zo Csc@w − ΦD Sin@wD Sin@βD Sin@β1DHCos@w1D Cos@β1D Sin@βD − Cos@wD Cos@βD Sin@β1DLLêH−2 Z1 Zo Cos@2 β1D H8 Cos@βD2 − 4 HCos@2 wD + Cos@2 w1DL Sin@βD2L −

8 Z1 Zo H−3 + Cos@2 βD + HCos@2 wD + Cos@2 w1DL Sin@βD2L + 32 HZ12 + Zo2L Sin@wD

Appendice H.nb 17

Sin@w1D Sin@βD Sin@β1D − 16 Z1 Zo Cos@wD Cos@w1D Sin@2 βD Sin@2 β1DL<,80, H16 Z1 HSin@β1D H4 Z1 Sin@wD Sin@w1D Sin@βD + Zo H−HCos@2 w − ΦD − 3 Cos@ΦDLCsc@w − ΦD Sin@wD + Cos@2 βD Cot@w − ΦD Sin@2 wDL Sin@β1DL −

Zo Cos@w1D Cot@w − ΦD Sin@wD Sin@2 βD Sin@2 β1DLLêH−2 Z1 Zo Cos@2 β1D H8 Cos@βD2 − 4 HCos@2 wD + Cos@2 w1DL Sin@βD2L −

8 Z1 Zo H−3 + Cos@2 βD + HCos@2 wD + Cos@2 w1DL Sin@βD2L +

32 HZ12 + Zo2L Sin@wD Sin@w1D Sin@βD Sin@β1D −

16 Z1 Zo Cos@wD Cos@w1D Sin@2 βD Sin@2 β1DL, H8 Z1 Csc@w − ΦD Sin@wD HZo Cos@w − ΦDH3 Cos@Φ1D + 2 Cos@w1D Cos@2 β1D Cos@w1 + Φ1D − Cos@2 w1 + Φ1DL Sin@2 βD +

Zo H−2 Cos@wD Cos@2 βD Cos@w − ΦD + Cos@2 w − ΦD − 3 Cos@ΦDL Cos@w1 + Φ1D Sin@2 β1D +

8 Z1 HCos@β1D Sin@βD Sin@w − ΦD Sin@Φ1D + Cos@βD Sin@β1D Sin@ΦD Sin@w1 + Φ1DLLLêH−2 Z1 Zo Cos@2 β1D H8 Cos@βD2 − 4 HCos@2 wD + Cos@2 w1DL Sin@βD2L −

8 Z1 Zo H−3 + Cos@2 βD + HCos@2 wD + Cos@2 w1DL Sin@βD2L +

32 HZ12 + Zo2L Sin@wD Sin@w1D Sin@βD Sin@β1D −

16 Z1 Zo Cos@wD Cos@w1D Sin@2 βD Sin@2 β1DL, 0<,80, −H64 Z1 Zo Csc@w − ΦD Sin@wD Sin@βD Sin@β1DHCos@w1D Cos@β1D Sin@βD − Cos@wD Cos@βD Sin@β1DLLêH−2 Z1 Zo Cos@2 β1D H8 Cos@βD2 − 4 HCos@2 wD + Cos@2 w1DL Sin@βD2L −

8 Z1 Zo H−3 + Cos@2 βD + HCos@2 wD + Cos@2 w1DL Sin@βD2L + 32 HZ12 + Zo2L Sin@wDSin@w1D Sin@βD Sin@β1D − 16 Z1 Zo Cos@wD Cos@w1D Sin@2 βD Sin@2 β1DL,H8 Z1 Csc@w − ΦD H4 Zo Cos@w1D Cos@2 β1D Cos@w1 + Φ1D Sin@wD Sin@βD2 −

2 Zo H−3 Cos@Φ1D + Cos@2 w1 + Φ1DL Sin@wD Sin@βD2 − Zo Cos@w1 + Φ1DSin@2 wD Sin@2 βD Sin@2 β1D + 8 Z1 Sin@wD2 Sin@βD Sin@β1D Sin@w1 + Φ1DLLêH−2 Z1 Zo Cos@2 β1D H8 Cos@βD2 − 4 HCos@2 wD + Cos@2 w1DL Sin@βD2L −

8 Z1 Zo H−3 + Cos@2 βD + HCos@2 wD + Cos@2 w1DL Sin@βD2L + 32 HZ12 + Zo2L Sin@wDSin@w1D Sin@βD Sin@β1D − 16 Z1 Zo Cos@wD Cos@w1D Sin@2 βD Sin@2 β1DL, 0<,8H8 Zo Csc@w − ΦD Sin@wD HZ1 Cos@w − ΦD H3 Cos@Φ1D + 2 Cos@w1D Cos@2 β1D

Cos@w1 + Φ1D − Cos@2 w1 + Φ1DL Sin@2 βD +

Z1 H−2 Cos@wD Cos@2 βD Cos@w − ΦD + Cos@2 w − ΦD − 3 Cos@ΦDL Cos@w1 + Φ1D Sin@2 β1D +

8 Zo HCos@β1D Sin@βD Sin@w − ΦD Sin@Φ1D + Cos@βD Sin@β1D Sin@ΦD Sin@w1 + Φ1DLLLêH−2 Z1 Zo Cos@2 β1D H8 Cos@βD2 − 4 HCos@2 wD + Cos@2 w1DL Sin@βD2L −

8 Z1 Zo H−3 + Cos@2 βD + HCos@2 wD + Cos@2 w1DL Sin@βD2L + 32 HZ12 + Zo2L Sin@wDSin@w1D Sin@βD Sin@β1D − 16 Z1 Zo Cos@wD Cos@w1D Sin@2 βD Sin@2 β1DL, 0,

0, H16 Zo HSin@β1D H4 Zo Sin@wD Sin@w1D Sin@βD + Z1 H−HCos@2 w − ΦD − 3 Cos@ΦDLCsc@w − ΦD Sin@wD + Cos@2 βD Cot@w − ΦD Sin@2 wDL Sin@β1DL −

Z1 Cos@w1D Cot@w − ΦD Sin@wD Sin@2 βD Sin@2 β1DLLêH−2 Z1 Zo Cos@2 β1D H8 Cos@βD2 − 4 HCos@2 wD + Cos@2 w1DL Sin@βD2L −

8 Z1 Zo H−3 + Cos@2 βD + HCos@2 wD + Cos@2 w1DL Sin@βD2L + 32 HZ12 + Zo2L Sin@wDSin@w1D Sin@βD Sin@β1D − 16 Z1 Zo Cos@wD Cos@w1D Sin@2 βD Sin@2 β1DL<<;

H21n@w_, w1_D = 88H8 Z1 Csc@w1 − Φ1D H4 Zo Cos@wD Cos@2 βD Cos@w + ΦD Sin@w1D Sin@β1D2 −

2 Zo H−3 Cos@ΦD + Cos@2 w + ΦDL Sin@w1D Sin@β1D2 − Zo Cos@w + ΦD Sin@2 w1DSin@2 βD Sin@2 β1D + 8 Z1 Sin@w1D2 Sin@βD Sin@β1D Sin@w + ΦDLLêH−2 Z1 Zo Cos@2 β1D H8 Cos@βD2 − 4 HCos@2 wD + Cos@2 w1DL Sin@βD2L −

8 Z1 Zo H−3 + Cos@2 βD + HCos@2 wD + Cos@2 w1DL Sin@βD2L + 32 HZ12 + Zo2L Sin@wDSin@w1D Sin@βD Sin@β1D − 16 Z1 Zo Cos@wD Cos@w1D Sin@2 βD Sin@2 β1DL,

0, 0, H64 Z1 Zo Csc@w1 − Φ1D Sin@w1D Sin@βD Sin@β1DHCos@w1D Cos@β1D Sin@βD − Cos@wD Cos@βD Sin@β1DLLêH−2 Z1 Zo Cos@2 β1D H8 Cos@βD2 − 4 HCos@2 wD + Cos@2 w1DL Sin@βD2L −

8 Z1 Zo H−3 + Cos@2 βD + HCos@2 wD + Cos@2 w1DL Sin@βD2L + 32 HZ12 + Zo2L Sin@wDSin@w1D Sin@βD Sin@β1D − 16 Z1 Zo Cos@wD Cos@w1D Sin@2 βD Sin@2 β1DL<,80, H8 Zo Sin@w1D HZ1 H4 Cos@w1D Cos@2 β1D Cos@w1 − Φ1D − 2 Cos@2 w1 − Φ1D + 6 Cos@Φ1DL

Csc@w1 − Φ1D Sin@βD2 + 8 Zo Sin@wD Sin@βD Sin@β1D −

2 Z1 Cos@wD Cot@w1 − Φ1D Sin@2 βD Sin@2 β1DLLêH−2 Z1 Zo Cos@2 β1D H8 Cos@βD2 − 4 HCos@2 wD + Cos@2 w1DL Sin@βD2L −

8 Z1 Zo H−3 + Cos@2 βD + HCos@2 wD + Cos@2 w1DL Sin@βD2L + 32 HZ12 + Zo2L Sin@wD

Appendice H.nb 18

Sin@w1D Sin@βD Sin@β1D − 16 Z1 Zo Cos@wD Cos@w1D Sin@2 βD Sin@2 β1DL,H4 Zo H2 Z1 Cos@w + ΦD HHCos@2 w1 − Φ1D − 3 Cos@Φ1DL Csc@w1 − Φ1D Sin@w1D −

Cos@2 β1D Cot@w1 − Φ1D Sin@2 w1DL Sin@2 βD +

2 Sin@w1D H8 Zo Cos@βD Sin@β1D Sin@ΦD + Z1 Cot@w1 − Φ1D Sin@2 β1DHCos@ΦD H3 + Cos@2 βD − 2 Cos@2 wD Sin@βD2L + 2 Sin@2 wD Sin@βD2 Sin@ΦDL +

8 Zo Cos@β1D Csc@w1 − Φ1D Sin@βD Sin@w + ΦD Sin@Φ1DLLLêH−2 Z1 Zo Cos@2 β1D H8 Cos@βD2 − 4 HCos@2 wD + Cos@2 w1DL Sin@βD2L −

8 Z1 Zo H−3 + Cos@2 βD + HCos@2 wD + Cos@2 w1DL Sin@βD2L +

32 HZ12 + Zo2L Sin@wD Sin@w1D Sin@βD Sin@β1D −

16 Z1 Zo Cos@wD Cos@w1D Sin@2 βD Sin@2 β1DL, 0<,80, H64 Z1 Zo Csc@w1 − Φ1D Sin@w1D Sin@βD Sin@β1DHCos@w1D Cos@β1D Sin@βD − Cos@wD Cos@βD Sin@β1DLLêH−2 Z1 Zo Cos@2 β1D H8 Cos@βD2 − 4 HCos@2 wD + Cos@2 w1DL Sin@βD2L −

8 Z1 Zo H−3 + Cos@2 βD + HCos@2 wD + Cos@2 w1DL Sin@βD2L + 32 HZ12 + Zo2L Sin@wDSin@w1D Sin@βD Sin@β1D − 16 Z1 Zo Cos@wD Cos@w1D Sin@2 βD Sin@2 β1DL,H8 Zo Csc@w1 − Φ1D H4 Z1 Cos@wD Cos@2 βD Cos@w + ΦD Sin@w1D Sin@β1D2 −

2 Z1 H−3 Cos@ΦD + Cos@2 w + ΦDL Sin@w1D Sin@β1D2 − Z1 Cos@w + ΦD Sin@2 w1DSin@2 βD Sin@2 β1D + 8 Zo Sin@w1D2 Sin@βD Sin@β1D Sin@w + ΦDLLêH−2 Z1 Zo Cos@2 β1D H8 Cos@βD2 − 4 HCos@2 wD + Cos@2 w1DL Sin@βD2L −

8 Z1 Zo H−3 + Cos@2 βD + HCos@2 wD + Cos@2 w1DL Sin@βD2L + 32 HZ12 + Zo2L Sin@wDSin@w1D Sin@βD Sin@β1D − 16 Z1 Zo Cos@wD Cos@w1D Sin@2 βD Sin@2 β1DL, 0<,8H4 Z1 H2 Zo Cos@w + ΦD HHCos@2 w1 − Φ1D − 3 Cos@Φ1DL Csc@w1 − Φ1D Sin@w1D −

Cos@2 β1D Cot@w1 − Φ1D Sin@2 w1DL Sin@2 βD +

2 Sin@w1D H8 Z1 Cos@βD Sin@β1D Sin@ΦD + Zo Cot@w1 − Φ1D Sin@2 β1DHCos@ΦD H3 + Cos@2 βD − 2 Cos@2 wD Sin@βD2L + 2 Sin@2 wD Sin@βD2 Sin@ΦDL +

8 Z1 Cos@β1D Csc@w1 − Φ1D Sin@βD Sin@w + ΦD Sin@Φ1DLLLêH−2 Z1 Zo Cos@2 β1D H8 Cos@βD2 − 4 HCos@2 wD + Cos@2 w1DL Sin@βD2L −

8 Z1 Zo H−3 + Cos@2 βD + HCos@2 wD + Cos@2 w1DL Sin@βD2L +

32 HZ12 + Zo2L Sin@wD Sin@w1D Sin@βD Sin@β1D −

16 Z1 Zo Cos@wD Cos@w1D Sin@2 βD Sin@2 β1DL, 0, 0,H8 Z1 Sin@w1D HZo H4 Cos@w1D Cos@2 β1D Cos@w1 − Φ1D − 2 Cos@2 w1 − Φ1D + 6 Cos@Φ1DLCsc@w1 − Φ1D Sin@βD2 + 8 Z1 Sin@wD Sin@βD Sin@β1D −

2 Zo Cos@wD Cot@w1 − Φ1D Sin@2 βD Sin@2 β1DLLêH−2 Z1 Zo Cos@2 β1D H8 Cos@βD2 − 4 HCos@2 wD + Cos@2 w1DL Sin@βD2L −

8 Z1 Zo H−3 + Cos@2 βD + HCos@2 wD + Cos@2 w1DL Sin@βD2L + 32 HZ12 + Zo2L Sin@wDSin@w1D Sin@βD Sin@β1D − 16 Z1 Zo Cos@wD Cos@w1D Sin@2 βD Sin@2 β1DL<<;

H22n@w_, w1_D =88HCsc@w1 − Φ1D HZ1 Zo H−HH4 + Cos@2 wD − Cos@2 w1DL Cos@2 βD + H−4 + Cos@2 wD − Cos@2 w1DLCos@2 β1DL Cos@Φ1D Sin@w1D +HCos@w1D H−4 + H−Cos@2 wD + Cos@2 w1DL Cos@2 βD + H−Cos@2 wD + Cos@2 w1D +

4 Cos@2 βDL Cos@2 β1DL + 4 Cos@wD Sin@2 βD Sin@2 β1DL Sin@Φ1DL −H−Z1 Zo HCos@2 wD − Cos@2 w1DL H1 + Cos@2 βD Cos@2 β1DL + 8 HZ1 − ZoLHZ1 + ZoL Sin@wD Sin@w1D Sin@βD Sin@β1DL Sin@w1 + Φ1DLLêH6 Z1 Zo − 2 Z1 Zo Cos@2 βD − 4 Z1 Zo Cos@βD2 Cos@2 β1D + 4 Sin@βD Sin@β1DH2 HZ12 + Zo2L Sin@wD Sin@w1D − Z1 Zo HCos@2 wD + Cos@2 w1DL Sin@βD Sin@β1DL −

4 Z1 Zo Cos@wD Cos@w1D Sin@2 βD Sin@2 β1DL,0, 0, −H64 Z1 Zo Csc@w1 − Φ1D Sin@w1D Sin@βD Sin@β1DHCos@w1D Cos@β1D Sin@βD − Cos@wD Cos@βD Sin@β1DLLêH−2 Z1 Zo Cos@2 β1D H8 Cos@βD2 − 4 HCos@2 wD + Cos@2 w1DL Sin@βD2L −

8 Z1 Zo H−3 + Cos@2 βD + HCos@2 wD + Cos@2 w1DL Sin@βD2L +

32 HZ12 + Zo2L Sin@wD Sin@w1D Sin@βD Sin@β1D −

16 Z1 Zo Cos@wD Cos@w1D Sin@2 βD Sin@2 β1DL<,80, −H16 HZ1 − ZoL HZ1 + ZoL Sin@wD Sin@w1D Sin@βD Sin@β1D +

Z1 Zo H4 + 4 H2 + Cos@2 wDL Cos@2 βD Sin@β1D2 +

Appendice H.nb 19

Csc@w1 − Φ1D H−Sin@2 w + w1 − Φ1D + 8 Cos@wD Sin@2 βD Sin@2 β1D Sin@Φ1D +

Sin@2 w − w1 + Φ1D + 2 Sin@βD2 HSin@3 w1 − Φ1D − 5 Sin@w1 + Φ1DLLL −

Z1 Zo Cos@2 β1D H4 + Csc@w1 − Φ1D H−Sin@2 w + w1 − Φ1D + Sin@2 w − w1 + Φ1D +

2 Sin@βD2 HSin@3 w1 − Φ1D + 3 Sin@w1 + Φ1DLLLLêH4 H−3 Z1 Zo + Z1 Zo Cos@2 βD + 2 Z1 Zo Cos@βD2 Cos@2 β1D + 2 Sin@βD Sin@β1DH−2 HZ12 + Zo2L Sin@wD Sin@w1D + Z1 Zo HCos@2 wD + Cos@2 w1DL Sin@βD Sin@β1DL +

2 Z1 Zo Cos@wD Cos@w1D Sin@2 βD Sin@2 β1DLL, H2 Csc@w1 − Φ1DH−8 Z1 Zo Cos@wD Sin@w1D Sin@2 βD H1 + Cos@2 β1D + H−Cos@2 w1D + Cos@2 Φ1DL Sin@β1D2L +

4 Z1 Zo Sin@2 w1D H2 Cos@βD2 + H−Cos@2 wD + Cos@2 Φ1DL Sin@βD2L Sin@2 β1D −

16 HZ1 − ZoL HZ1 + ZoL Cos@β1D Sin@wD Sin@w1D Sin@βD Sin@2 Φ1DLLêH2 Z1 Zo Cos@2 β1D H8 Cos@βD2 − 4 HCos@2 wD + Cos@2 w1DL Sin@βD2L +

8 Z1 Zo H−3 + Cos@2 βD + HCos@2 wD + Cos@2 w1DL Sin@βD2L −

32 HZ12 + Zo2L Sin@wD Sin@w1D Sin@βD Sin@β1D +

16 Z1 Zo Cos@wD Cos@w1D Sin@2 βD Sin@2 β1DL, 0<,80, −H64 Z1 Zo Csc@w1 − Φ1D Sin@w1D Sin@βD Sin@β1DHCos@w1D Cos@β1D Sin@βD − Cos@wD Cos@βD Sin@β1DLLêH−2 Z1 Zo Cos@2 β1D H8 Cos@βD2 − 4 HCos@2 wD + Cos@2 w1DL Sin@βD2L −

8 Z1 Zo H−3 + Cos@2 βD + HCos@2 wD + Cos@2 w1DL Sin@βD2L + 32 HZ12 + Zo2L Sin@wDSin@w1D Sin@βD Sin@β1D − 16 Z1 Zo Cos@wD Cos@w1D Sin@2 βD Sin@2 β1DL,HCsc@w1 − Φ1D HZ1 Zo H−HH4 + Cos@2 wD − Cos@2 w1DL Cos@2 βD +H−4 + Cos@2 wD − Cos@2 w1DL Cos@2 β1DL Cos@Φ1D Sin@w1D +HCos@w1D H−4 + H−Cos@2 wD + Cos@2 w1DL Cos@2 βD + H−Cos@2 wD + Cos@2 w1D +

4 Cos@2 βDL Cos@2 β1DL + 4 Cos@wD Sin@2 βD Sin@2 β1DL Sin@Φ1DL +HZ1 Zo HCos@2 wD − Cos@2 w1DL H1 + Cos@2 βD Cos@2 β1DL + 8 HZ1 − ZoLHZ1 + ZoL Sin@wD Sin@w1D Sin@βD Sin@β1DL Sin@w1 + Φ1DLLêH6 Z1 Zo − 2 Z1 Zo Cos@2 βD − 4 Z1 Zo Cos@βD2 Cos@2 β1D + 4 Sin@βD Sin@β1DH2 HZ12 + Zo2L Sin@wD Sin@w1D − Z1 Zo HCos@2 wD + Cos@2 w1DL Sin@βD Sin@β1DL −

4 Z1 Zo Cos@wD Cos@w1D Sin@2 βD Sin@2 β1DL, 0<, 8H2 Csc@w1 − Φ1DH−8 Z1 Zo Cos@wD Sin@w1D Sin@2 βD H1 + Cos@2 β1D + H−Cos@2 w1D + Cos@2 Φ1DL Sin@β1D2L +

4 Z1 Zo Sin@2 w1D H2 Cos@βD2 + H−Cos@2 wD + Cos@2 Φ1DL Sin@βD2L Sin@2 β1D +

16 HZ1 − ZoL HZ1 + ZoL Cos@β1D Sin@wD Sin@w1D Sin@βD Sin@2 Φ1DLLêH2 Z1 Zo Cos@2 β1D H8 Cos@βD2 − 4 HCos@2 wD + Cos@2 w1DL Sin@βD2L +

8 Z1 Zo H−3 + Cos@2 βD + HCos@2 wD + Cos@2 w1DL Sin@βD2L −

32 HZ12 + Zo2L Sin@wD Sin@w1D Sin@βD Sin@β1D +

16 Z1 Zo Cos@wD Cos@w1D Sin@2 βD Sin@2 β1DL,0, 0, −H−16 HZ1 − ZoL HZ1 + ZoL Sin@wD Sin@w1D Sin@βD Sin@β1D +

Z1 Zo H4 + 4 H2 + Cos@2 wDL Cos@2 βD Sin@β1D2 +

Csc@w1 − Φ1D H−Sin@2 w + w1 − Φ1D + 8 Cos@wD Sin@2 βD Sin@2 β1D Sin@Φ1D +

Sin@2 w − w1 + Φ1D + 2 Sin@βD2 HSin@3 w1 − Φ1D − 5 Sin@w1 + Φ1DLLL −

Z1 Zo Cos@2 β1D H4 + Csc@w1 − Φ1D H−Sin@2 w + w1 − Φ1D + Sin@2 w − w1 + Φ1D +

2 Sin@βD2 HSin@3 w1 − Φ1D + 3 Sin@w1 + Φ1DLLLLêH4 H−3 Z1 Zo + Z1 Zo Cos@2 βD + 2 Z1 Zo Cos@βD2 Cos@2 β1D + 2 Sin@βD Sin@β1DH−2 HZ12 + Zo2L Sin@wD Sin@w1D + Z1 Zo HCos@2 wD + Cos@2 w1DL Sin@βD Sin@β1DL +

2 Z1 Zo Cos@wD Cos@w1D Sin@2 βD Sin@2 β1DLL<<;

General::spell1 :

Possible spelling error: new symbol name "H21n" is similar to existing symbol "H12n". More…

Appendice H.nb 20

F@w_D = 88Vzpp@wD<, 8Iρpp@wD<, 8Izpp@wD<, 8Vρpp@wD<<;

F1@w_D = F@wD@@1, 1DD;F2@w_D = F@wD@@2, 1DD;F3@w_D = F@wD@@3, 1DD;F4@w_D = F@wD@@4, 1DD;

Fπ@w1_D = 88Vzπpp@w1D<, 8Iρπpp@w1D<, 8Izπpp@w1D<, 8Vρπpp@w1D<<;

F1π@w1_D = Fπ@w1D@@1, 1DD;F2π@w1_D = Fπ@w1D@@2, 1DD;F3π@w1_D = Fπ@w1D@@3, 1DD;F4π@w1_D = Fπ@w1D@@4, 1DD;

F@0.8D884.35636 − 4.35638 #<, 8−2.61236 + 2.61235 #<,8−3.13349 + 3.13349 #<, 8−600.169 + 600.169 #<<

Appendice H.nb 21

Fπ@0.7D886.70473 − 6.70472 #<, 80.0614511 − 0.0614477 #<,80.638805 − 0.638844 #<, 8−11.8909 + 11.891 #<<F1πFIN@w1_D := Which@Abs@Re@w1DD ≤ Φ1, F1π@w1D, Re@w1D > 0, F1πFIN@−w1D,

Abs@Re@w1 + 2 Φ1DD ≤ Φ1, HH21n@gt1@w1 + Φ1D, w1 + Φ1D@@1, 1DDL F1FIN@gt1@w1 + Φ1D + ΦD +HH21n@gt1@w1 + Φ1D, w1 + Φ1D@@1, 2DDL F2FIN@gt1@w1 + Φ1D + ΦD +HH21n@gt1@w1 + Φ1D, w1 + Φ1D@@1, 3DDL F3FIN@gt1@w1 + Φ1D + ΦD +HH21n@gt1@w1 + Φ1D, w1 + Φ1D@@1, 4DDL F4FIN@gt1@w1 + Φ1D + ΦD +HH22n@gt1@w1 + Φ1D, w1 + Φ1D@@1, 1DDL F1π@w1 + 2 Φ1D +HH22n@gt1@w1 + Φ1D, w1 + Φ1D@@1, 2DDL F2π@w1 + 2 Φ1D +HH22n@gt1@w1 + Φ1D, w1 + Φ1D@@1, 3DDL F3π@w1 + 2 Φ1D +HH22n@gt1@w1 + Φ1D, w1 + Φ1D@@1, 4DDL F4π@w1 + 2 Φ1D, Abs@Re@w1 + 2 Φ1DD > Φ1,HH21n@gt1@w1 + Φ1D, w1 + Φ1D@@1, 1DDL F1FIN@gt1@w1 + Φ1D + ΦD +HH21n@gt1@w1 + Φ1D, w1 + Φ1D@@1, 2DDL F2FIN@gt1@w1 + Φ1D + ΦD +HH21n@gt1@w1 + Φ1D, w1 + Φ1D@@1, 3DDL F3FIN@gt1@w1 + Φ1D + ΦD +HH21n@gt1@w1 + Φ1D, w1 + Φ1D@@1, 4DDL F4FIN@gt1@w1 + Φ1D + ΦD +HH22n@gt1@w1 + Φ1D, w1 + Φ1D@@1, 1DDL F1πFIN@w1 + 2 Φ1D +HH22n@gt1@w1 + Φ1D, w1 + Φ1D@@1, 2DDL F2πFIN@w1 + 2 Φ1D +HH22n@gt1@w1 + Φ1D, w1 + Φ1D@@1, 3DDL F3πFIN@w1 + 2 Φ1D +HH22n@gt1@w1 + Φ1D, w1 + Φ1D@@1, 4DDL F4πFIN@w1 + 2 Φ1DDF2πFIN@w1_D := Which@Abs@Re@w1DD ≤ Φ1, F2π@w1D, Re@w1D > 0, F2πFIN@−w1D,

Abs@Re@w1 + 2 Φ1DD ≤ Φ1, HH21n@gt1@w1 + Φ1D, w1 + Φ1D@@2, 1DDL F1FIN@gt1@w1 + Φ1D + ΦD +HH21n@gt1@w1 + Φ1D, w1 + Φ1D@@2, 2DDL F2FIN@gt1@w1 + Φ1D + ΦD +HH21n@gt1@w1 + Φ1D, w1 + Φ1D@@2, 3DDL F3FIN@gt1@w1 + Φ1D + ΦD +HH21n@gt1@w1 + Φ1D, w1 + Φ1D@@2, 4DDL F4FIN@gt1@w1 + Φ1D + ΦD +HH22n@gt1@w1 + Φ1D, w1 + Φ1D@@2, 1DDL F1π@w1 + 2 Φ1D +HH22n@gt1@w1 + Φ1D, w1 + Φ1D@@2, 2DDL F2π@w1 + 2 Φ1D +HH22n@gt1@w1 + Φ1D, w1 + Φ1D@@2, 3DDL F3π@w1 + 2 Φ1D +HH22n@gt1@w1 + Φ1D, w1 + Φ1D@@2, 4DDL F4π@w1 + 2 Φ1D, Abs@Re@w1 + 2 Φ1DD > Φ1,HH21n@gt1@w1 + Φ1D, w1 + Φ1D@@2, 1DDL F1FIN@gt1@w1 + Φ1D + ΦD +HH21n@gt1@w1 + Φ1D, w1 + Φ1D@@2, 2DDL F2FIN@gt1@w1 + Φ1D + ΦD +HH21n@gt1@w1 + Φ1D, w1 + Φ1D@@2, 3DDL F3FIN@gt1@w1 + Φ1D + ΦD +HH21n@gt1@w1 + Φ1D, w1 + Φ1D@@2, 4DDL F4FIN@gt1@w1 + Φ1D + ΦD +HH22n@gt1@w1 + Φ1D, w1 + Φ1D@@2, 1DDL F1πFIN@w1 + 2 Φ1D +HH22n@gt1@w1 + Φ1D, w1 + Φ1D@@2, 2DDL F2πFIN@w1 + 2 Φ1D +HH22n@gt1@w1 + Φ1D, w1 + Φ1D@@2, 3DDL F3πFIN@w1 + 2 Φ1D +HH22n@gt1@w1 + Φ1D, w1 + Φ1D@@2, 4DDL F4πFIN@w1 + 2 Φ1DD

Appendice H.nb 22

F3πFIN@w1_D := Which@Abs@Re@w1DD ≤ Φ1, F3π@w1D, Re@w1D > 0, F3πFIN@−w1D,Abs@Re@w1 + 2 Φ1DD ≤ Φ1, HH21n@gt1@w1 + Φ1D, w1 + Φ1D@@3, 1DDL F1FIN@gt1@w1 + Φ1D + ΦD +HH21n@gt1@w1 + Φ1D, w1 + Φ1D@@3, 2DDL F2FIN@gt1@w1 + Φ1D + ΦD +HH21n@gt1@w1 + Φ1D, w1 + Φ1D@@3, 3DDL F3FIN@gt1@w1 + Φ1D + ΦD +HH21n@gt1@w1 + Φ1D, w1 + Φ1D@@3, 4DDL F4FIN@gt1@w1 + Φ1D + ΦD +HH22n@gt1@w1 + Φ1D, w1 + Φ1D@@3, 1DDL F1π@w1 + 2 Φ1D +HH22n@gt1@w1 + Φ1D, w1 + Φ1D@@3, 2DDL F2π@w1 + 2 Φ1D +HH22n@gt1@w1 + Φ1D, w1 + Φ1D@@3, 3DDL F3π@w1 + 2 Φ1D +HH22n@gt1@w1 + Φ1D, w1 + Φ1D@@3, 4DDL F4π@w1 + 2 Φ1D, Abs@Re@w1 + 2 Φ1DD > Φ1,HH21n@gt1@w1 + Φ1D, w1 + Φ1D@@3, 1DDL F1FIN@gt1@w1 + Φ1D + ΦD +HH21n@gt1@w1 + Φ1D, w1 + Φ1D@@3, 2DDL F2FIN@gt1@w1 + Φ1D + ΦD +HH21n@gt1@w1 + Φ1D, w1 + Φ1D@@3, 3DDL F3FIN@gt1@w1 + Φ1D + ΦD +HH21n@gt1@w1 + Φ1D, w1 + Φ1D@@3, 4DDL F4FIN@gt1@w1 + Φ1D + ΦD +HH22n@gt1@w1 + Φ1D, w1 + Φ1D@@3, 1DDL F1πFIN@w1 + 2 Φ1D +HH22n@gt1@w1 + Φ1D, w1 + Φ1D@@3, 2DDL F2πFIN@w1 + 2 Φ1D +HH22n@gt1@w1 + Φ1D, w1 + Φ1D@@3, 3DDL F3πFIN@w1 + 2 Φ1D +HH22n@gt1@w1 + Φ1D, w1 + Φ1D@@3, 4DDL F4πFIN@w1 + 2 Φ1DD

F4πFIN@w1_D := Which@Abs@Re@w1DD ≤ Φ1, F4π@w1D, Re@w1D > 0, F4πFIN@−w1D,Abs@Re@w1 + 2 Φ1DD ≤ Φ1, HH21n@gt1@w1 + Φ1D, w1 + Φ1D@@4, 1DDL F1FIN@gt1@w1 + Φ1D + ΦD +HH21n@gt1@w1 + Φ1D, w1 + Φ1D@@4, 2DDL F2FIN@gt1@w1 + Φ1D + ΦD +HH21n@gt1@w1 + Φ1D, w1 + Φ1D@@4, 3DDL F3FIN@gt1@w1 + Φ1D + ΦD +HH21n@gt1@w1 + Φ1D, w1 + Φ1D@@4, 4DDL F4FIN@gt1@w1 + Φ1D + ΦD +HH22n@gt1@w1 + Φ1D, w1 + Φ1D@@4, 1DDL F1π@w1 + 2 Φ1D +HH22n@gt1@w1 + Φ1D, w1 + Φ1D@@4, 2DDL F2π@w1 + 2 Φ1D +HH22n@gt1@w1 + Φ1D, w1 + Φ1D@@4, 3DDL F3π@w1 + 2 Φ1D +HH22n@gt1@w1 + Φ1D, w1 + Φ1D@@4, 4DDL F4π@w1 + 2 Φ1D, Abs@Re@w1 + 2 Φ1DD > Φ1,HH21n@gt1@w1 + Φ1D, w1 + Φ1D@@4, 1DDL F1FIN@gt1@w1 + Φ1D + ΦD +HH21n@gt1@w1 + Φ1D, w1 + Φ1D@@4, 2DDL F2FIN@gt1@w1 + Φ1D + ΦD +HH21n@gt1@w1 + Φ1D, w1 + Φ1D@@4, 3DDL F3FIN@gt1@w1 + Φ1D + ΦD +HH21n@gt1@w1 + Φ1D, w1 + Φ1D@@4, 4DDL F4FIN@gt1@w1 + Φ1D + ΦD +HH22n@gt1@w1 + Φ1D, w1 + Φ1D@@4, 1DDL F1πFIN@w1 + 2 Φ1D +HH22n@gt1@w1 + Φ1D, w1 + Φ1D@@4, 2DDL F2πFIN@w1 + 2 Φ1D +HH22n@gt1@w1 + Φ1D, w1 + Φ1D@@4, 3DDL F3πFIN@w1 + 2 Φ1D +HH22n@gt1@w1 + Φ1D, w1 + Φ1D@@4, 4DDL F4πFIN@w1 + 2 Φ1DD

Appendice H.nb 23

F1FIN@w_D := Which@Abs@Re@wDD ≤ Φ, F1@wD, Re@wD > 0, F1FIN@−wD, Abs@Re@w + 2 ΦDD ≤ Φ,HH11n@w + Φ, gt@w + ΦDD@@1, 1DDL F1@w + 2 ΦD + HH11n@w + Φ, gt@w + ΦDD@@1, 2DDL F2@w + 2 ΦD +HH11n@w + Φ, gt@w + ΦDD@@1, 3DDL F3@w + 2 ΦD + HH11n@w + Φ, gt@w + ΦDD@@1, 4DDL F4@w + 2 ΦD +HH12n@w + Φ, gt@w + ΦDD@@1, 1DDL F1πFIN@gt@w + ΦD + Φ1D +HH12n@w + Φ, gt@w + ΦDD@@1, 2DDL F2πFIN@gt@w + ΦD + Φ1D +HH12n@w + Φ, gt@w + ΦDD@@1, 3DDL F3πFIN@gt@w + ΦD + Φ1D +HH12n@w + Φ, gt@w + ΦDD@@1, 4DDL F4πFIN@gt@w + ΦD + Φ1D, Abs@Re@w + 2 ΦDD > Φ,HH11n@w + Φ, gt@w + ΦDD@@1, 1DDL F1FIN@w + 2 ΦD + HH11n@w + Φ, gt@w + ΦDD@@1, 2DDL

F2FIN@w + 2 ΦD + HH11n@w + Φ, gt@w + ΦDD@@1, 3DDL F3FIN@w + 2 ΦD +HH11n@w + Φ, gt@w + ΦDD@@1, 4DDL F4FIN@w + 2 ΦD + HH12n@w + Φ, gt@w + ΦDD@@1, 1DDL

F1πFIN@gt@w + ΦD + Φ1D + HH12n@w + Φ, gt@w + ΦDD@@1, 2DDL F2πFIN@gt@w + ΦD + Φ1D +HH12n@w + Φ, gt@w + ΦDD@@1, 3DDL F3πFIN@gt@w + ΦD + Φ1D +HH12n@w + Φ, gt@w + ΦDD@@1, 4DDL F4πFIN@gt@w + ΦD + Φ1DD

F2FIN@w_D := Which@Abs@Re@wDD ≤ Φ, F2@wD, Re@wD > 0, F2FIN@−wD, Abs@Re@w + 2 ΦDD ≤ Φ,HH11n@w + Φ, gt@w + ΦDD@@2, 1DDL F1@w + 2 ΦD + HH11n@w + Φ, gt@w + ΦDD@@2, 2DDL F2@w + 2 ΦD +HH11n@w + Φ, gt@w + ΦDD@@2, 3DDL F3@w + 2 ΦD + HH11n@w + Φ, gt@w + ΦDD@@2, 4DDL F4@w + 2 ΦD +HH12n@w + Φ, gt@w + ΦDD@@2, 1DDL F1πFIN@gt@w + ΦD + Φ1D +HH12n@w + Φ, gt@w + ΦDD@@2, 2DDL F2πFIN@gt@w + ΦD + Φ1D +HH12n@w + Φ, gt@w + ΦDD@@2, 3DDL F3πFIN@gt@w + ΦD + Φ1D +HH12n@w + Φ, gt@w + ΦDD@@2, 4DDL F4πFIN@gt@w + ΦD + Φ1D, Abs@Re@w + 2 ΦDD > Φ,HH11n@w + Φ, gt@w + ΦDD@@2, 1DDL F1FIN@w + 2 ΦD + HH11n@w + Φ, gt@w + ΦDD@@2, 2DDL

F2FIN@w + 2 ΦD + HH11n@w + Φ, gt@w + ΦDD@@2, 3DDL F3FIN@w + 2 ΦD +HH11n@w + Φ, gt@w + ΦDD@@2, 4DDL F4FIN@w + 2 ΦD + HH12n@w + Φ, gt@w + ΦDD@@2, 1DDL

F1πFIN@gt@w + ΦD + Φ1D + HH12n@w + Φ, gt@w + ΦDD@@2, 2DDL F2πFIN@gt@w + ΦD + Φ1D +HH12n@w + Φ, gt@w + ΦDD@@2, 3DDL F3πFIN@gt@w + ΦD + Φ1D +HH12n@w + Φ, gt@w + ΦDD@@2, 4DDL F4πFIN@gt@w + ΦD + Φ1DD

Appendice H.nb 24

F3FIN@w_D := Which@Abs@Re@wDD ≤ Φ, F3@wD, Re@wD > 0, F3FIN@−wD, Abs@Re@w + 2 ΦDD ≤ Φ,HH11n@w + Φ, gt@w + ΦDD@@3, 1DDL F1@w + 2 ΦD + HH11n@w + Φ, gt@w + ΦDD@@3, 2DDL F2@w + 2 ΦD +HH11n@w + Φ, gt@w + ΦDD@@3, 3DDL F3@w + 2 ΦD + HH11n@w + Φ, gt@w + ΦDD@@3, 4DDL F4@w + 2 ΦD +HH12n@w + Φ, gt@w + ΦDD@@3, 1DDL F1πFIN@gt@w + ΦD + Φ1D +HH12n@w + Φ, gt@w + ΦDD@@3, 2DDL F2πFIN@gt@w + ΦD + Φ1D +HH12n@w + Φ, gt@w + ΦDD@@3, 3DDL F3πFIN@gt@w + ΦD + Φ1D +HH12n@w + Φ, gt@w + ΦDD@@3, 4DDL F4πFIN@gt@w + ΦD + Φ1D, Abs@Re@w + 2 ΦDD > Φ,HH11n@w + Φ, gt@w + ΦDD@@3, 1DDL F1FIN@w + 2 ΦD + HH11n@w + Φ, gt@w + ΦDD@@3, 2DDL

F2FIN@w + 2 ΦD + HH11n@w + Φ, gt@w + ΦDD@@3, 3DDL F3FIN@w + 2 ΦD +HH11n@w + Φ, gt@w + ΦDD@@3, 4DDL F4FIN@w + 2 ΦD + HH12n@w + Φ, gt@w + ΦDD@@3, 1DDL

F1πFIN@gt@w + ΦD + Φ1D + HH12n@w + Φ, gt@w + ΦDD@@3, 2DDL F2πFIN@gt@w + ΦD + Φ1D +HH12n@w + Φ, gt@w + ΦDD@@3, 3DDL F3πFIN@gt@w + ΦD + Φ1D +HH12n@w + Φ, gt@w + ΦDD@@3, 4DDL F4πFIN@gt@w + ΦD + Φ1DDF4FIN@w_D := Which@Abs@Re@wDD ≤ Φ, F4@wD, Re@wD > 0, F4FIN@−wD, Abs@Re@w + 2 ΦDD ≤ Φ,HH11n@w + Φ, gt@w + ΦDD@@4, 1DDL F1@w + 2 ΦD + HH11n@w + Φ, gt@w + ΦDD@@4, 2DDL F2@w + 2 ΦD +HH11n@w + Φ, gt@w + ΦDD@@4, 3DDL F3@w + 2 ΦD + HH11n@w + Φ, gt@w + ΦDD@@4, 4DDL F4@w + 2 ΦD +HH12n@w + Φ, gt@w + ΦDD@@4, 1DDL F1πFIN@gt@w + ΦD + Φ1D +HH12n@w + Φ, gt@w + ΦDD@@4, 2DDL F2πFIN@gt@w + ΦD + Φ1D +HH12n@w + Φ, gt@w + ΦDD@@4, 3DDL F3πFIN@gt@w + ΦD + Φ1D +HH12n@w + Φ, gt@w + ΦDD@@4, 4DDL F4πFIN@gt@w + ΦD + Φ1D, Abs@Re@w + 2 ΦDD > Φ,HH11n@w + Φ, gt@w + ΦDD@@4, 1DDL F1FIN@w + 2 ΦD + HH11n@w + Φ, gt@w + ΦDD@@4, 2DDL

F2FIN@w + 2 ΦD + HH11n@w + Φ, gt@w + ΦDD@@4, 3DDL F3FIN@w + 2 ΦD +HH11n@w + Φ, gt@w + ΦDD@@4, 4DDL F4FIN@w + 2 ΦD + HH12n@w + Φ, gt@w + ΦDD@@4, 1DDL

F1πFIN@gt@w + ΦD + Φ1D + HH12n@w + Φ, gt@w + ΦDD@@4, 2DDL F2πFIN@gt@w + ΦD + Φ1D +HH12n@w + Φ, gt@w + ΦDD@@4, 3DDL F3πFIN@gt@w + ΦD + Φ1D +HH12n@w + Φ, gt@w + ΦDD@@4, 4DDL F4πFIN@gt@w + ΦD + Φ1DD

H∗Plot@Im@k Sin@wD F1FIN@wDD,8w,−Φ,0<D∗LH∗ Plot@Im@k F2FIN@wDD,8w,−Φ,0<D ∗LH∗ ∗L H∗Plot@Im@k F3FIN@wDD,8w,−Φ,0<D ∗LH∗Plot@Im@k F4FIN@wDD,8w,−Φ,0<D ∗LH∗Plot@Im@k F3FIN@wDD,8w,−2 π,0<D

∗L

Appendice H.nb 25

H∗Plot@Im@k F4FIN@wDD,8w,−2 π,0<D ∗LH∗Plot@Im@k Sin@wD F1πFIN@wDD,8w,−Φ1,0<D ∗LH∗Plot@Im@k F2πFIN@wDD,8w,−Φ1,0<D ∗LH∗Plot@Im@k F1FIN@wDD,8w,−2 π,0<D ∗LH∗ Plot@Im@k F2πFIN@wDD,8w,−Φ1,0<D∗LH∗ Plot@Im@k Sin@wD F1FIN@wDD,8w,−2 π,0<DH∗Vzd∗L∗LH∗Vzd∗LH∗si accorda con il programma caso normale∗LH∗Plot@Im@k F2FIN@wDD,8w,−2 π,0<D ∗LH∗ Plot@Im@k F1πFIN@wDD,8w,−2 π,0<D∗LH∗ Plot@Im@k F2πFIN@wDD,8w,−2 π,0<D∗LH∗ Plot@Im@k F3πFIN@wDD,8w,−2 π,0<D∗LH∗ Plot@Im@k F4πFIN@wDD,8w,−2 π,0<D∗LVFIN@w_D := Sin@wD F1FIN@wDIFIN@w_D := F2FIN@wDIzop@w_D := F3FIN@wDVzop@w_D := F1FIN@wDIρop@w_D := F2FIN@wDVρop@w_D := F4FIN@wDIzoπp@w_D := F3πFIN@wDVzoπp@w_D := F1πFIN@wDIρoπp@w_D := F2πFIN@wDVρoπp@w_D := F4πFIN@wD

Appendice H.nb 26

Vzϕd@w_D := H∗ 1((((2

Sin@wD H Csc@wD H−Zo Cos@βD Cos@w−ϕD Izop@w−ϕD+Zo Cos@βD Cos@w+ϕD Izop@w+ϕD+ZoHIρop@w−ϕD−Iρop@w+ϕDL Sin@βD+Sin@w−ϕD Vzop@w−ϕD+Sin@w+ϕD Vzop@w+ϕDLL∗L1((((2

H−Zo Cos@βD Cos@w − ϕD Izop@w − ϕD + Zo Cos@βD Cos@w + ϕD Izop@w + ϕD +

Zo HIρop@w − ϕD − Iρop@w + ϕDL Sin@βD + Sin@w − ϕD Vzop@w − ϕD + Sin@w + ϕD Vzop@w + ϕDLIzϕd@w_D :=H∗ Sin@wD((((((((((((((

2 Zo HCsc@wD HZo Izop@w−ϕD Sin@w−ϕD+Zo Izop@w+ϕD Sin@w+ϕD+Cos@βD HCos@w−ϕD

Vzop@w−ϕD−Cos@w+ϕD Vzop@w+ϕDL+Sin@βD H−Vρop@w−ϕD+Vρop@w+ϕDLLL∗L1

(((((((((((2 Zo

HZo Izop@w − ϕD Sin@w − ϕD + Zo Izop@w + ϕD Sin@w + ϕD + Cos@βDHCos@w − ϕD Vzop@w − ϕD − Cos@w + ϕD Vzop@w + ϕDL + Sin@βD H−Vρop@w − ϕD + Vρop@w + ϕDLLVzϕdd@w1_D :=

1((((2

HZ1 Cos@β1D Cos@w1 − ϕ1D Izoπp@w1 − ϕ1D − Z1 Cos@β1D Cos@w1 + ϕ1D Izoπp@w1 + ϕ1D +

Z1 H−Iρoπp@w1 − ϕ1D + Iρoπp@w1 + ϕ1DL Sin@β1D +

Sin@w1 − ϕ1D Vzoπp@w1 − ϕ1D + Sin@w1 + ϕ1D Vzoπp@w1 + ϕ1DLIzϕdd@w1_D :=

1(((((((((((2 Z1

HZ1 Izoπp@w1 − ϕ1D Sin@w1 − ϕ1D + Z1 Izoπp@w1 + ϕ1D Sin@w1 + ϕ1D +

Cos@β1D H−Cos@w1 − ϕ1D Vzoπp@w1 − ϕ1D + Cos@w1 + ϕ1D Vzoπp@w1 + ϕ1DL +

Sin@β1D HVρoπp@w1 − ϕ1D − Vρoπp@w1 + ϕ1DLLH∗Since we assumed Eo=0, Ho=1,we have Deh=I τo Vzϕd@−πD

Dhh=I τo Izϕd@−πD∗LDeh = Plot@20 Log@10, Abs@I τo Vzϕd@−πDDD, 8ϕ, −Φ, Φ<D

-2 -1 1 2

-50

50

100

% Graphics %

Appendice H.nb 27

Dehd = Plot@20 Log@10, Abs@I τ1 Vzϕdd@−πDDD, 8ϕ1, −Φ1, Φ1<D

-0.75 -0.5 -0.25 0.25 0.5 0.75

20

40

60

80

100

120

% Graphics %

Dhh = Plot@20 Log@10, Abs@I τo Izϕd@−πDDD, 8ϕ, −Φ, Φ<D-2 -1 1 2

-125

-100

-75

-50

-25

25

50

% Graphics %

Dhhd = Plot@20 Log@10, Abs@I τ1 Izϕdd@−πDDD, 8ϕ1, −Φ1, Φ1<D

-0.75 -0.5 -0.25 0.25 0.5 0.75

-40

-20

20

40

60

80

% Graphics %

Appendice H.nb 28