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DAYS on DIFFRACTION' 2011

1

Derivation of mo dified Smyshlyaev's formulae using integral transform of Kontorovich-Leb edev type
Valery Yu. Valyaev, Andrey V. Shanin
Physics Department, Moscow State University, Russia; e-mail: valery-valyaev@yandex.ru
The aim of this work is to fill the gap b etween the emb edding formulae for cones and the mo dified Smyshlyaev's formulae. Emb edding formulae for cones represent the directivity of the scattered field as multiple integrals over spatial variables. Mo dified Smyshlyaev's formulae represent the same directivity as a single contour integral over parameter . This situation resembles the convolution theorem for Fourier transform: multiple convolutions can b e represented as a single integral over frequency. Originally, mo dified Smyshlyaev's formulae have b een "guessed" and then proved by study of the p oles of the integrands instead of b eing regularly derived. Extension of the analogy with Fourier transform allows to obtain a regular metho d for deriving the mo dified Smyshlyaev's formulae. To p erform this extension we intro duce integral transform of Kontorovich - Leb edev typ e and prove for it the analogues of Plancherel's and convolution formulae. Using the develop ed technique we demonstrate the p ossibility to derive the mo dified Smyshlyaev's formulae.

1

Introduction

1.1 Motivation

As an example we are considering the scalar problem of plane wave diffraction by a quarter plane. As usual, our main goal is to find the diffraction coefficient of the scattered field. General approach to this kind of diffraction problems is separation of radial variable and studying the Laplace-Beltrami problem on the unit sphere. This approach has been significantly developed by Smyshlyaev and co- 1.2 Basic ideas workers [1]. He has obtained the following formula There are embedding formulae of two sorts: for the diffraction coefficient: f ( , 0 ) = i e-
i

as a limiting case of Green's function g as the source location approaches edge of the scatterer. Integrals over separation parameter in these modified Smyshlyaev's formulae have better convergence properties than one in (1). Derivation of these formulae consists of three steps. At first an embedding operator is applied to the total field and the result is expressed in terms of edge Green's functions in 3D space. This expression is called embedding formula. Then the directivities of edge Green's functions in 3D space are represented as the series over the eigenvalues of Laplace-Beltrami operator. By using the embedding formula one obtains the series for the diffraction coefficient of the scattered field. Final step is transformation of the series to the contour integral. In fact the resulting formula had to be guessed and then checked by the study of the poles of the integrands instead of being regularly derived. This approach can also be applied to more complicated problems, for example to the problem of diffraction by the wedge of the cube [3]. Guessing of the formulae here becomes more difficult and potentially lead to errors. This paper fills the mentioned gap between the embedding formulae for cones and the modified Smyshlyaev formulae, giving a regular way of deriving the later from the former.

g ( , 0 , ) d,

(1)



f ( , 0 ) =
0

f1 ( , r)f2 (0 , r)

dr r

(2)

where 0 and are directions of incidence and scattering, is the separation parameter and g is the and Green's function of the of the spherical problem. Further extension of this approach has been f ( , 0 ) = achieved in work [2] in which the formulae of the same type as (1) were obtained. In these formudr = lae integrand is constructed from so called spherr 0 ical edge Green's functions, which can be treated



f1 ( , r)g (r, r )f2 (0 , r )
0

dr r

(3)


2 with f1,2 of the form (15) (possibly multiplied by r-n for some integer n), and g (r, r ) of the form (17). Note that f1,2 and g are expressed through contour integrals over parameter . Formally, a direct implementation of (2) or (3) leads to calculation of three (in case of (2)) or five (in case of (3)) nested integrals over and r. However, fortunately, these integrals can be converted into modified Smyshlyaev's formulae expressing f ( , 0 ) as a single integral over the parameter . The possibility of reducing several integrals to a single one reminds the well-known properties of Fourier transform. Let us explain this analogy. Let ^ F1,2 ( ) be transforms of the functions F1,2 (x). Let ^ also G( ) be transform of function G(x), and we introduce the function G(x, y ) = G(x - y ). Then


DAYS on DIFFRACTION' 2011 integration. As the result, the functions participating in the representation are no longer orthogonal. However, for our needs the orthogonality (and even the uniqueness and invertibility of the representation) is not relevant, we need only the analogs of Plancherel formula and convolution formula. That is why, we prove only these important formulae without using orthogonality and demonstrate the possibility of deriving the modified Smyshlyaev's formulae. 2 Basic relations

2.1

Problem statement

We are considering the scalar Dirichlet problem of diffraction by quarter plane Q = {(x, y , z )|x 0, y 0, z = 0} (see Fig. 1).

F1 (x
-

)F2

(x)dx =
-

^ ^ F1 ( )F2 ( )d

(4)
u
sc

which is the Plancherel's theorem, and
F1 (x)G(x, y )F2 (y )dxdy = - -

=
-

^ ^ ^ F1 ( )G( )F2 ( )d ,

(5) Figure 1: Geometry of the problem. Let the Helmholtz equation
2 u + k0 u = 0

which is a combination of the Plancherel's theorem and the convolution theorem. Here superscript star stands for complex conjugation. If F1,2 (x) and ^ G(x, y ) are expressed as integrals containing F1,2 ( ) ^ ( ) then the left-hand side of (4) contains and G three integrals, and the left-hand side of (5) contains five integrals. In both cases the right-hand side contains only one integral. The most straightforward way to extend this analogy to the conical case is to use the Kontorovich-Lebedev transform. However, we cannot use it directly because of convergence problems. In particular, for the classical Kontorovich-Lebedev procedure it is necessary for the parameter k0 of the Helmholtz equation to be purely imaginary, which is hardly interesting from the practical point of view. That is why, we develop a slightly different approach. Instead of the Kontorovich-Lebedev transform we use another representation that differs by the choice of the cylindrical function (Bessel instead of Hankel), and, more important, by the contour of

(6)

be valid in the 3D space (x, dence of all variables is of omitted henceforth. The Dirichlet boundary both surfaces of the quarter

y , z ). The time depenthe form e-it and is conditions fulfilled on plane is of the form: (7)

u|Q = 0.

Let the incident field have the form of a plane wave coming from direction 0 : uinc = e
-ik0 (
0x

x+

0y

y +

0z

z)

.

(8)

Beside the governing ditions, the radiation, should be imposed to mulation. We do not and refer reader to [2].

equation and boundary conedge and vertex conditions make a proper problem fordiscuss these matters here


DAYS on DIFFRACTION' 2011 Our main interest is to find the diffraction coeffi- Here we present one of them, namely cient f ( , 0 ) of the scattered field usc = u - uinc , which we define as the amplitude of the spherical 4 2 i fx ( ; X )fx (0 ; X )dX f= 2 wave diffracted by the tip of the quarter plane: k0 (y + 0y ) eik0 r u (r, ) = 2 f ( , 0 ) + O ( r k0 r
sc 0 -2

3

(12)

), r . (9) which is of the form (2). Embedding formula of type (3) is also presented in [2], but we do not conIt depends not only on the scattering direction sider it here for the sake of brevity. but also on the direction 0 from which the incident wave comes.

2.2 Edge Green's functions in 3D space

2.3 Edge Green's functions on the unit sphere

Solving the problem (6)-(8), after the separation Let us consider Greens function G(x, y , z ; x0 , y0 , z0 ) of the radial variable one comes to the Laplaceof our problem, i.e. the function which obeys the Beltrami problem on the unit sphere S with a cut equation Sq produced by the quarter plane Sq = S Q (see Fig. 3). 2 G + k0 G = (x - x0 ) (y - y0 ) (z - z0 ) (10) and the same boundary, edge, vertex and radiation conditions as field u does. We define the pair of edge Green's functions in 3D space Gx (x, y , z ; X ) and Gy (x, y , z ; Y ) as following limits: Gx (x, y , z ; X ) = lim and Gy (x, y , z ; Y ) = lim
0

0

G(x, y , z ; X, -, 0) G(x, y , z ; -, Y , 0),

q

(11)

Figure 3: Geometry of the problem on the sphere. We introduce the Green's function g ( , 0 , ) of this sphere as the solution of the following problem 1 ~ + 2 - 4 g = ( - 0 ), (13)

i.e. as fields produced by sources lying on the edges of the scatterer (see Fig. 2).

which obeys Dirichlet conditions on the cut: g |Sq = 0 and Meixner conditions at the ends of the cut (see [2]). It is the function participating in (1). Let us define the edge Green's functions on the sphere vx ( , ) and vy ( , ) as the following limits: Figure 2: To the definition of the edge Green's function Gy . vx ( , ) = lim g ( , x , ) and g ( , y , ),

(14) Hereafter we consider the diffraction coefficients vy ( , ) = lim fx ( , X ) and fy ( , Y ) of the edge Green's functions 0 Gx and Gy correspondingly, defined in the same way as f in (9). where x is the point with conventional spherical In [2] the embedding formulae which connects di- coordinates = /2 and = - (see Fig. 4) and rectivity f with directivities fx and fy are derived. similarly for y .

0


4

DAYS on DIFFRACTION' 2011
I m[ ]



Re[ ]

q

Figure 5: Contour . Figure 4: To the definition of the edge Green's function on the sphere. Our aim now is to transform (12) into the modified Smyshlyaev's formula, i.e. the contour integral over parameter from the function involving vx ( , ). In order to do it in the regular way we introduce the integral transform of KontorovichLebedev type. 3 The representation of KontorovichLebedev type Note that there is a considerable difference between representations (15) and (17). In (15) a function of one variable is represented through another function of one variable, while in (17) a function of two variables is represented through a function of one variable. Thus, representation (17) exists for a very restricted class of functions. We do not need the transformation converting h(r) into ( ). We also do not need uniqueness of the transformants in (15) and (17). The functions that have these representations emerge naturally from solving the Helmholtz equation in conical coordinates [2]. Let us now prove some important for our goals properties of the representation. Properties of the representation

3.1 Definition of the representation

We introduce the representation for two types of functions. For a function of a single variable h(r), 4 r > 0 the representation is as follows: h(r) = 1 2 e
-i /2

Here we study two types of integrals emerging in (15) embedding formulae. The first one is an analog of a convolution property. Here ( ) is the transformant of h(r). Contour Theorem 1 Let h(r) and g (r, r ) be functions havis shown in Fig. 5. ing representations (15) and (17) with transforWe assume that mants and , respectively. Then 1. function ( ) is even J (k0 r)( ) d. (- ) = ( ); 2. on the 3. |Im[ ]| Let admit g (r, r where obeys called (16)


g (r, r )h(r )

0 singularities of ( ) are only isolated poles 1 real axis, and ( ) is regular at = 0; = e-i /2 J (k0 r)( ) ( ) d, (18) function ( ) decays exponentially as 2 . function g (r, r ), r, r > 0 of two variables i.e. the integral in the l.-h.s. of (18) has the representation of the form (15) with the transformant the following representation: ( ) ( ). 1 (1) J (k0 r< )H (k0 r> ) ( ) d, (17) )= The proof is as follows. Transform the contours 2 of integration in representations of and to Å r< = min(r, r ) and r> = max(r, r ) If ( ) and correspondingly (see Fig. 6). Then conconditions 1-3 listed above, this function is vert the product g (r, r )h(r ) to double integral over the transformant of g . cartesian product ç Å .

dr = r


DAYS on DIFFRACTION' 2011
I m[ ]

5 Substitute (23) into (19). Split the double integral into sum of two terms and convert them to iterated integrals.
Re[ ]

Å

K (r) = =-

e-i 2 Å JÅ Å (Å) ( ) d dÅ + 2 - Å2


i 2

Figure 6: Contours Å and . Denote the integral in the l.-h.s. of (18) by K (r). 1 4




Å





+



Å

e J Å (Å) ( ) d dÅ . (24) Å2 - 2

-i 2

In the first term transform the contour of integration over into shown at Fig. 7 (the part with çÅ 0 (1) ç JÅ (k0 r )J (k0 r< )H (k0 r> ) dÅ d (19) Im[ ] < 0 is symmetrically reflected with respect to zero). Due to relation (16) this change does not Change the integration order and calculate the in- affect the integral. For each non-zero Å the integral over can be then taken by residue method after tegral over r first. Namely, find closing the integration path at +i.


K (r) =

dr r

e

-i Å 2

Å (Å) ( )ç

I=
0

JÅ (k0 r )J (k0 r< )H

(1)

(k0 r> )

dr . r

(20)

I m[ ]

Å Re[ ]

Represent I as a sum of two integrals
r

I=H

(1)

(k0 r)
0

dr JÅ (k0 r )J (k0 r ) + r


+ J (k0 r)
r

JÅ (k0 r )H

(1)

dr (k0 r ) . r

I m[ ]

(21)

Å Re[ ]

Use a well-known formula
r

Z

(1) Å

(k0 r )Z =

(2)

(k0 r )

dr = r Figure 7: Contours Å and .

In the second term the contour Å is deformed into Å . It is also closed in the upper half-plane. -Z (k0 r)Z (k0 r) , (22) The first term gives non-trivial poles at = ÁÅ, and the second term compensates the singularity of where Z (1) and Z (2) stand for general cylindrical functions (i.e. they can be replaced by J or H (1) in the first term at Å = = 0. The result is (18). The second result reminds of the Plancherel's our formulae). Performing all computations, get equation. 2i JÅ (k0 r) - ei(Å- )/2 J (k0 r) I= . (23) Å2 - 2 Theorem 2 Let h1 (r) and h2 (r) be functions that Note that I is regular at Å = . have the representation (15) with transformants 1
(1) Å (2) +1

1 (2) Z (1) (k0 r)Z (k0 r)- Å+ Å k0 r (1) (2) -2 Z (k0 r)Z (k0 r)- Å - 2 Å+1


6 and 2 respectively. Then


DAYS on DIFFRACTION' 2011 Thus, h(r) ik 0 = r 4 e-
i

h1 (r)h2 (r)
0

dr = r = 1 4 1 ( )2 ( ) d. (25)

e
+1

-i /2

J (k0 r)( - 1)d -

-
-1

e

-i /2

J (k0 r)( + 1)d . (29)

To prove it let us follow the procedure used for proving the previous theorem, i.e. deform the contours into Å and , change the order of integration, then use well-known formula


JÅ (k0 r)J (k0 r)
0

dr 2 sin[ (Å - )/2] = . r Å2 - 2

(26)

Now note that the only pole of ( - 1) on (0, 1] is 1 - while ( + 1) is regular at it, and vice versa for the point - 1. This allows us to deform contours Á 1 to taking care of singularities and ~ write (27). Note that ( ) obeys all the conditions imposed on proper transformant. Now let us show how the described above methods help in derivation of modified Smyshlyaev's formulae.

After that we split the integral into two, using the 5 Example of derivation of a symmetry deform the contour of inner integration modified Smyshlyaev's formula into Å or and calculate the inner integral by residues. As the result, we get (25). We will be working with embedding formula (12): Concluding this section let us prove the theo rem describing how multiplication by 1/r affects 4 2 i the representation of the function. f= 2 fx ( ; X )fx (0 ; X )dX. (30) k0 (y + 0y ) 0 Theorem 3 Let h(r) have representation (15) with the transformant . Let be the only pole of ( ) In [2] the following formulae are proven for fx on the segment (0, 1]. and vx : Then h(r) 1 = r 2 e
+ -i /2

~ J (k0 r)( ) d

(27)

fx ( , X ) = = k0 e-i 2 X
3 4



k0 ~ where ( ) = i2 [( - 1) - ( + 1)] and additional contour shown at Fig. 8 consists of two loops encircling points - 1 and 1 - .

Cj j ( )Jj (k0 X )e-
j =1

i 2

j

, (31)

vx ( , ) = 2
j =1

Cj j ( ) 2, 2 - j

(32)

I m[ ] - - 1 -1 1 -




1

Re[ ]

where j and j ( ) are eigenvalues and eigenfunctions of Laplace-Beltrami operator on sphere with a Dirichlet cut and Cj are some constants describing the behavior of j ( ) near the ends of the cut. From (31) and (32) it obviously follows, that fx ( ; X ) =

Figure 8: Contour .

=

k0 e-i 4 2 2 X



e


-i 2

J (k0 X )vx ( , ) d

(33)

The proof is as follows. Let us take into account i.e. the well-known formula J J (z ) = z
+1

(z ) + J 2

-1

(z )

(28)

fx ( ; X ) =

A h( ; X ) , X

A=

k0 e-i 2

/4

(34)


DAYS on DIFFRACTION' 2011 where h( ; X ) for each fixed has representation (15) with the transformant vx ( , ). Thus, 2/ k0 f ( , 0 ) = + 0


7
I m[ ] - - 1 -1 -
1 2 1 2

Å 1 Re[ ]

0

h(0 ; X ) dr h( ; X ) X r

(35)

1-



i.e. the integral has the form of (25). To proceed transform h(0 ; X )/X according to (27). Here we use the empirical fact that the only pole of vx on (0, 1] is 1 . Then deform the contour + in its representation into contour Å (see Fig. 9). In order to assure convergence of forthcoming integrals over X let us shift the contour in representation of h( ; X ) to = + 1/2. It is possible, since 1/2 < < 1 and thus vx ( , ) has no poles on (0, 1/2].
I m[ ] - - 1 -1 -
1 2 1 2

Figure 10: Deformation of the contour 6 Conclusion

Å 1 Re[ ]

1-



Figure 9: Deformation of the contour + Now let us follow the "Plancherel's" theorem. uct h( ; X )h(0 ; X )/X to Å ç , substitute it into tegration over X first. As following:

Let us briefly summarize the work. A new integral transform of Kontorovich-Lebedev type was introduced. Analogues of convolution and Plancherel's theorems were proven for it without demands for orthogonality, uniqueness and invertibility. Developed technique gives a neat method of transformation of spatial integrals emerging in embedding formulae for conical problems into contour integrals of Smyshlyaev's type. As an example of usage of this technique a modified Smyshlyaev's formula for the problem of diffraction of plane wave by a Dirichlet quarter plane was derived in a way different from the original work [2].

References procedure of proof of We convert the prod- [1] Smyshlyaev, V.P., 1990, Diffraction by conithe double integral over cal surfaces at high frequencies, Wave motion, the (35) and do the inVol. 12, pp. 329-339. a result one obtains the [2] Shanin, A.V., 2005, Modified Smyshlyaev's formulae for the problem of diffraction of a plane 1 wave by an ideal quarter-plane, Wave motion, f ( , 0 ) = ç 2 2 (x + 0x ) Vol. 41, pp. 79-93. e-i - e-iÅ ç vx ( , )(Å)Å d dÅ, (36) [3] Skelton E.A. et al., 2010, Embedding formulae Å2 - 2 for scattering by three-dimensional structures, Å ç Wave motion, Vol. 47, pp. 299-317. where (Å) = vx (0 , Å - 1) - vx (0 , Å + 1) . 2Å

Now let us transform + 1/2 to (Fig. 10) and follow the rest of the procedure. As a result we obtain the sought modified Smyshlyaev's formula: f ( , 0 ) = i/4 x + 0 e
x + -i Å

vx ( , Å)ç (37)

ç [vx (0 , Å - 1) - vx (0 , Å + 1)]d.