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Embedding formulae for Laplace-Beltrami problems on the sphere with a cut.
Valery Valyaev, Andrey Shanin
Department of Physics, Moscow State University, Leninskie Gory, Moscow 119992, Russia

Abstract We consider the problem of diffraction of a plane by a quarter-plane with Dirichlet boundary conditions. For this problem exist various expressions for diffraction co efficient. These expressions have the form of contour integrals over separation parameter. Integrands are constructed from the solutions of Laplace-Beltrami problems on the unit sphere with a cut pro duced by the quarter- plane. In this paper we derive embedding formulae which connect these solutions and show the possibility to derive expressions for diffraction co efficient from one another. Keywords: Diffraction by cones, Embedding formulae 1. Introduction We are considering the scalar problem of plane wave diffraction by a quarter-plane. Our main goal is to find the diffraction co efficient of the scattered field. Since a plane sector is a degenerated case of an elliptic cone, this problem has an explicit solution in sphero-conal co ordinates [1, 2]. This solution is a series of Lam´ functions. Computations with this series are quite ineffective e and it is difficult to extract from it the structure of the diffracted field. Another approach to problems of diffraction by cones is separation of radial variable and studying the Laplace-Beltrami problem on the unit sphere


Corresponding author Email address: valery-valyaev@yandex.ru (Valery Valyaev)

Preprint submitted to Wave Motion

January 23, 2011


for each value of separation parameter. This approach has been significantly developed by Smyshlyaev and co-workers [3]. Applying the BesselSommerfeld technique he has obtained the following formula for the diffraction co efficient: ^ i f ( , 0 ) = e-i g ( , 0, ) d, (1 )


where 0 and are directions of incidence and scattering, is the separation parameter and g is the Green's function of the of the spherical problem. To compute with this formula one has to solve an integral equation for Green's function for each [4]. Integral over is rapidly convergent only in the domain of directions in which propagates only the spherical wave diffracted by the tip of the cone. It diverges in domain of directions where geometrically reflected wave or the waves diffracted by the edges of the quarterplane exist. It is still possible to use Smyshlyaev's formula in the domain of divergence [5] but in this case the integral should be understo o d in sense of Abel-Poisson limit, which is difficult for numerical computations. In works [6, 7] the formulae of the same type as (1) were obtained. In these formulae integrand is constructed from so called spherical edge Green's functions, which are the fields of singular sources lying at the edges of the scatterer. These formulae are based on application of embedding operators to the field in 3D space. These operators cancel the geometrically reflected wave and one of the waves diffracted by the edges (or both of them). Thus the domain of convergence of contour integral in these formulae is wider than than one in (1). When both waves diffracted by the edges are canceled the domain of divergence consists of directions in which propagate waves consequently scattered by both edges. For some directions of incidence this domain do es not even exist. Computation of spherical edge Green's functions can be performed by using integral equations [8], but there exist much more effective way to compute them based on the equations with multidimensional time [9]. So these formulae are more effective than Smyshlyaev's formula. Basic steps of derivation of mo dified Smyshlyaev's formulae in [7] are as follows. Application of an embedding operator to the field in 3D space allows to express the diffraction co efficient in terms of directivities of edge Green's functions in 3D space. This expression is called embedding formula. The directivities of edge Green's functions in 3D space can be represented as contour integrals over the separation parameter. Final step is substitution of these representations into the embedding formula and transformation of 2


resulting multiple integral to a single contour integral over the separation parameter. In this paper we use a different technique of derivation of these formulae. We apply embedding operators directly to solutions of Laplace-Beltrami problems on the unit sphere. This approach allows to obtain non-trivial relations between these solutions and consequently derive all the formulae from Smyshlyaev's formula (1). The paper is organized as follows. Section 2 contains the problem formulation and intro duces the notions used throughout the paper. In section 3 we intro duce embedding operators on the unit sphere and, their properties and derive the embedding formulae. In section 4 we use these formulae for derivation of MSF from Smyshlyaev's formula (1). 2. Basic relations 2.1. Problem formulation We seek the scalar field u which satisfies the Helmholtz equation
2 u + k0 u = 0

(2 )

in the 3D space (x, y , z ). The time dependence of all variables is of the form e-it and is omitted henceforth. The scatterer is the quarter-plane Q = {(x, y , z )|x 0, y 0, z = 0} (see Fig. 1). The field u satisfies the

Figure 1: Geometry of the problem.

Dirichlet boundary conditions on the quarter-plane: u | Q = 0. 3 (3 )


The incident field uin is the plane wave coming from direction defined by the unit vector 0 : uin (r ) = e-ik0 (0 r) . (4 ) This problem can be symmetrized in a standard way. The field u is represented as a sum of even and o dd functions of z . Solution for the o dd part contains only the incident wave and geometrically reflected wave. In what follows we denote as u only the even part which obeys the homogeneous ~ Neumann conditions on the complement Q of the quarter-plane Q to the whole xy plane: u = 0. (5 ) n Q ~ Beside the governing equation and boundary conditions, the radiation, edge and vertex conditions should be imposed to make a proper problem formulation. We do not discuss these matters here for the sake of brevity and refer the reader to [6, 7]. The most important feature of the field u is the diffraction co efficient f ( , 0) of its scattered part usc = u - uin . It can be defined as the amplitude of the spherical wave diffracted by the tip of the quarter plane: usc (r, ) = 2 This definition is ical wave. It can which the geomet the quarter-plane eik0 r f ( , 0 ) + O (r - 2 ), k0 r as r . (6 )

valid for directions in which propagate only the spherbe analytically continued in the domain of directions in rically reflected wave or waves diffracted by the edges of exist.

2.2. Spectrum of the spherical problem A natural way to solve our problem is to separate the radial variable and to study the spherical problem for each value of separation constant. This leads to the following eigenvalue problem on the unit sphere S with the cut

4


Sq = S Q "pro duced" by the quarter plane Q (see Fig. 2): ~ ( ) = 0, |Sq = 0, = 0, n Sq ~
1/2 1 1/2 2

(7 ) (8 ) (9 ) (1 0 ) (1 1 )

1 + O ( 2 2 + O ( sin 2 sin

3/2 1 3/2 2

), ),

as 1 0, as 2 0.

~ Here Sq is the complement of Sq to are the spherical co ordinates shown is the Laplace-Beltrami operator on spherical co ordinates ( , ) has the f ~ = 1 sin

the whole equator; (1 , 1 ) and (2 , 2 ) ~ ~ ~ on Fig. 2; = + 2 - 1/4, and the unit sphere, which in conventional o rm + 1 2 . sin2 2 (1 2 )

sin

q

Figure 2: Geometry of the problem on the sphere.

This problem ±n , n = 1, 2... adjoint operator 1/2 1 2

has a solution only for a discrete set of real values of = which form the spectrum of the problem. Since the self~ - has a positive discrete spectrum [3] we can write that . . . n .

2.3. Edge Green's functions on the unit sphere Besides the eigenfunctions an important role in solution of our problem play the Green's function and spherical edge Green's functions. Let us intro duce them. The definitions below exactly repeat the ones in [3] and [6]. 5


which obeys Dirichlet conditions on the cut: g |Sq = 0, Neumann conditio g on the rest of the equator n |Sq = 0, and Meixner conditions at the ends the cut (see [6]). It is the function participating in (1). Note that g ( , 0, is even function of and points = ±n are its poles [3]. We define the spherical edge Green's functions v1 ( , ) and v2 ( , ) the following limits: v 1 ( , ) = lim v 2 ( , ) = lim
0

The Green's function g ( , 0, ) is the solution of the following LaplaceBeltrami problem ~ g ( , 0, ) = ( - 0 ), (1 3 )

ns of ) as

0

g ( , x , ) g ( , y , ),

and

(1 4 ) (1 5 )

where x is the point with spherical co ordinates 1 = and 1 = (see Fig. 3) and similarly for y . One can prove [9, 6] that these func-

x
q

Figure 3: To the definition of the edge Green's function on the sphere.

tions have the following asymptotics near 1 -1/2 1 v 1 ( 1 , 1 , ) = - 1 sin 2 1 2C2 ( ) 1/2 1 v 2 (1 , 1 , ) = 1 sin 2 2 1 -1/2 sin v 2 ( 2 , 2 , ) = - 2 2 1 2C2 ( ) 1/2 2 v 1 (2 , 2 , ) = 2 sin 2 6

the edges of the cut: + O (
1/2 1

), ),

as 1 0, as 1 0, as 2 0, as 2 0.

(1 6 ) (1 7 ) (1 8 ) (1 9 )

+ O ( + O (

3/2 1

1/2 2

), ),

+ O (

3/2 2


1 1 Here C2 ( ) is an unknown co efficient. Note that v 1,2 ( , ) and C2 ( ) are even functions of and points = ±n are their poles. From definition of edge Green's functions and recipro city principle follow the asymptotics of Green's function at the edges of the cut:

v 1 ( 0 , ) v 2 ( 0 , ) g ( 2 , 2 ; 0 , ) = g ( 1 , 1 ; 0 , ) =

1/2 1 1/2 2

1 + O ( 2 2 + O ( sin 2 sin

3/2 1 3/2 2

), ),

as 1 0, as 2 0.

(2 0 ) (2 1 )

Edge Green's functions participate in the following formulae for diffraction co efficient, which we call mo dified Smyshlyaev's formulae [6, 7]. ^ i/4 f ( , 0 ) = e-iµ v 2 ( , µ)2(0 , µ)µdµ; (2 2 ) x + 0x + ^ i/4 f ( , 0 ) = e-iµ v 1 ( , µ)1(0 , µ)µdµ; (2 3 ) y + 0y
+

i/8 f ( , 0 ) = (x + 0x )(y + 0y )
1 1

^
+

e-

i

[V 1 ( , 0, ) + V 2 ( , 0, )+

+ 2 0x v ( , ) (0 , ) + 2 0y v 2 ( , )2 (0 , )]d, (24) i/8 f ( , 0 ) = (x + 0x )(y + 0y ) ^
+

e-

i µ

1 C2 (µ)[1 ( , µ)2(0 , µ)-

- 1 (0 , µ)2( , µ)]µdµ. (25) Here we use the following notation: v k ( , µ - 1 ) - v k ( , µ + 1 ) , k = 1, 2 . µ V k ( , 0 , ) = v k ( , + 1)v k (0 , - 1)- k ( , µ ) = - v k ( , - 1)v k (0 , + 1), (2 6 ) (2 7 )

k = 1, 2 .

Contour of integration + is shown on Fig. 4. Contour consists of two lo ops encircling points 1 - 1 and 1 - 1. 7


Im 1 - 1 -1 0 1 - 1 1 Re

Figure 4: Contour + .

3. Embedding formulae on the unit sphere 3.1. Embedding operators We intro duce operators on the unit sphere X and Y by the following identities. -1/2 r ( ) = r -3/2 X [( )], (2 8 ) x -1/2 r ( ) = r -3/2 Y [( )], (2 9 ) y We call X and Y embedding operators. These operators have the following representations in spherical co ordinates (1 , 1 ) and (2 , 2 ) shown on Fig. 2. X = Y = X = Y = 1 2 1 - 2 1 - 2 1 - 2 - cos 1 - sin
1

, 1
1

(3 0 ) , 1 , 2 (3 1 ) (3 2 ) (3 3 )

sin 1 - 1 sin 1 sin 2 sin 2 cos 2 + cos 2 cos 2 - 2 sin 2 cos 2 - sin 2 . 2 sin 1 cos 1 + cos 1 cos

Let us formulate two important properties of these operators. Lemma 1. If some function ( ) satisfies the boundary conditions ( )|Sq = 0, ~ n then X [( )] and Y [( )] also satisfy these conditions. ( )|
S
q

= 0,

and

(3 4 )

8


Proof. From conditions of lemma it follows that combination r -1/2 ( ) satisfies homogeneous Dirichlet conditions on Q and homogeneous Neumann ~ conditions on Q. Since differentiation with respect to x and y preserve these conditions, from definition of X and Y we obtain the statement of the lemma. Lemma 2. If some function ( ) satisfies the equation ~ ( ) = h( ), then X [( )] and Y [( )] satisfy the fol lowing equations ~ -1 X [( )] = X -2 [h( )], ~ -1 Y [( )] = Y -2 [h( )], (3 6 ) (3 7 ) (3 5 )

Proof. We will prove the property only for operator X . Pro of for Y is literally the same. From conditions of lemma it follows that combination r -1/2 ( ) satisfies the equation [r
-1/2

( )] = r

-5/2

h( ).

(3 8 )

Since Laplacian commutes with differentiation with respect to x, we can write that -5/2 r h( ) = [ r -1/2 ( )], (3 9 ) x x Applying the definition of X we transform this equation into the following r
-7/2

X

-2

[h( )] = [r

-3/2

X [( )]].

(4 0 )

Statement of the lemma directly follows from the last equation. 3.2. Embedding formulae Properties of the operators X and Y formulated above allow us to prove two theorems which are main results of this paper. Theorem 1. If and ± 1 do not belong to the spectrum, then the fol lowing formulae are valid. 2 x v 1 ( , ) = [v 1 ( , - 1) + v 1 ( , + 1)]+ 1 +C2 ( )[v 2 ( , - 1) - v 2 ( , + 1)], 2 y v 2 ( , ) = [v 2 ( , - 1) + v 2 ( , + 1)]+ 1 +C2 ( )[v 1 ( , - 1) - v 1 ( , + 1)]. 9 (4 1 ) (4 2 )


Proof. We will prove only the first formula. Pro of of the second is the same. Let us consider the function X [v 1 ( , )]. Applying the representations (30) and (32) to asymptotics (16) and (19) we obtain the following asymptotics of X [v 1 ( , )] at the edges of the cut: -1/2 1 1/2 X [v 1 (1 , 1 , )] = - 1 sin + O ( 1 ), 2 1 2 C2 ( ) -1/2 1/2 1 2 sin + O ( 2 ), X [v (2 , 2 , )] = - 2 Thus combination X [v 1 ( , )] - Meixner conditions at at the edg definition of edge Green's functio neous Dirichlet conditions on Sq , and equation as 1 0, as 2 0. (4 3 ) (4 4 )

1 v 1 ( , - 1) - C2 ( )v 2 ( , - 1) satisfies es of the cut. From lemmas 1 and 2 and ns it follows that it also satisfies homoge~ homogeneous Neumann conditions on Sq

1 ~ -1 [X [v 1 ( , )] - v 1 ( , - 1) - C2 ( )v 2 ( , - 1)] = 0.

(4 5 )

Appealing to the uniqueness theorem we conclude that this combination is identically zero. Thus, taking into account representation (30), we can write - 1 2 cos 1 v 1 ( , ) - sin
1

1 v ( , ) = 1 1 = v 1 ( , - 1) + C2 ( )v 2 ( , - 1). (46)

Substituting in this equation - instead of and taking into account even1 ness of edge Green's functions and C2 ( ) we obtain - - 1 2 cos 1 v 1 ( , ) - sin 1 v ( , ) = 1 1 = - v 1 ( , + 1) + C2 ( )v 2 ( , + 1). (47)
1

Subtracting these equations and taking into account that cos 1 = x we obtain (41). In what follows we will omit arguments and 0 of Green's function g where it do esn't lead to a confusion. We denote spherical co ordinates 0 (1,2 , 1,2) of 0 as (1,2, 0,2 ). 1 10


Theorem 2. If and ± 1 do not belong to the spectrum, then the fol lowing formulae are valid. 2 x g ( ) = 0x [g ( - 1) + g ( + 1)] + v 2 2 30x 0 - + sin 1 0 [g ( 2 1 2 y g ( ) = 0y [g ( - 1) + g ( + 1)] + v 1 2 30y 0 - + sin 2 0 [g ( 2 2 Proof. We will prove only the first formula. Let us consider the function X [g ( , 0, ) (30) and (32) to asymptotics (20) and (21) totics of X [g ( , 0, )] at the edges of the X [g (1 , 1 ; 0 , )] = O (1 ), v 2 ( 0 , ) X [g (2 , 2 ; 0 , )] = - 2 Thus combination g ( , 0, ) := X [g ( , 0, )] - v 2 (0 , )v 2 ( , - 1)/2 satisfies Meixner conditions at at the edges of the cut. From lemmas and definition of Green's function it follows that it also satisfies homo Dirichlet conditions on Sq , homogeneous Neumann conditions on equation ~ -1 g ( , 0, ) = X -2 [ ( - 0 )]. Taking into account representation (30) we can write X
-2 1/2

( 0 , ) 2 ( , )- - 1) - g ( + 1)]. ( 0 , ) 1 ( , )- - 1) - g ( + 1)].

(4 8 )

(4 9 )

Pro of of the second is the same. ]. Applying the representations we obtain the following asympcut: a s 1 0 , (5 0 )
1/2 2

-1/2 2

sin

2 + O ( 2

),

a s 2 0 . (5 1 )

(5 2 ) 1 and 2 geneous ~ Sq and (5 3 )

[ ( - 0 )] =

-

5 cos 1 ( - 0 ) - sin 1 ( - 0 ). ( 5 4 ) 2 1

Using the properties of delta function this expression can be transformed as follows X
-2

[ ( - 0 )] =

-

3 2

0 cos 1 ( - 0 ) - sin

0 1

( - 0 ). ( 5 5 ) 0 1

11


Thus for g ( , 0, ) we obtain g ( , 0 , ) = - 3 2
0 cos 1 - sin 0 1



0 1

g ( , 0, - 1).

(5 6 )

From representation (30) it follows, that - 1 cos 1 - sin 1 2 + g ( , 0, ) = v 2 (0 , )v 2 ( , - 1)/2+

1

3 0 0 cos 1 - sin 1 0 g ( , 0, - 1). (57) 2 1 Substituting in this equation - instead of and taking into account evenness of Green's functions we obtain - - - 1 2 cos 1 - sin +
1



g ( , 0, ) = v 2 (0 , )v 2 ( , + 1)/2+
1

3 0 0 cos 1 - sin 1 0 g ( , 0, + 1). (58) 2 1 Subtracting these equations and taking into account that cos 1 = x and 0 cos 1 = 0x we obtain (41). - - 4. Derivation of modified Smyshlyaev's formulae Now let us apply embedding formulae obtained above to derivation of mo dified Smyshlyaev's formulae (22) ­ (25). 4.1. From formula (1) to (22) Let us multiply formula (48) by e-i and integrate the result over the contour + . Since g ( ) is regular at points 1 - 1 and 1 - 1 we can write ^ ^ -i e g ( ) d = e-i g ( ) d. (5 9 )
+

Let us consider integral of the first term on the right-hand side of (48). Changing the variable of integration to µ = ± 1 we obtain ^ e-i [g ( - 1) + g ( + 1)] d =
+

=-

^

e

-i µ

g (µ)(µ - 1)dµ -

^

e-

i µ

g (µ)(µ + 1)dµ. (60)

++1

+-1

12


Contour of integration in the first term, + + 1, is shown on Fig. 5. Since Im +1 1 0 1 2 - 1 Re

Figure 5: Contour + + 1.

integrand is regular at point 2 - 1 and 1 is its only pole on [0, 1] we can deform this contour into . Performing the same pro cedure with contour + - 1 we get ^ ^ -i e [g ( - 1) + g ( + 1)] d = -2 e-iµ g (µ)µdµ. (6 1 )
+

Consideration of the third term on the right-hand side of (48) is essentially the same. As a result we get ^ e-i [g ( - 1) - g ( + 1)]d = 0. (6 2 )
+

Combining all these results we obtain 2
x

^


e

-i

g ( ) d = -2

0x

^


e-

i

g ( ) d + ^
+

+

e-

i

2 v ( 0 , ) 2 ( , )d . ( 6 3 ) 2

Using (1) we get (22). Formula (23) can be obtained in the same way.

13


4.2. From formula (22) to formula ( Using the recipro city principle let forms: ^ i/4 f ( , 0 ) = x + 0x + ^ i/4 f ( , 0 ) = x + 0x
+

25) us write formula (22) in two equivalent
i µ 2

e- e-

v ( , µ ) 2 ( 0 , µ )µ d µ , v ( 0 , µ ) 2 ( , µ )µ d µ .

(6 4 ) (6 5 )

i µ 2

Multiplying the first formula by 2x and the second by 20x and using embedding formula (42) we obtain ^ i/4 2 x f ( , 0 ) = e-iµ [v 2 ( , µ - 1) + v 2 ( , µ + 1)+ x + 0x (6 6 ) +
1 +C2 (µ)1 ( , µ)]2(0 , µ)µdµ,

i/4 20x f ( , 0) = x +

0x

^
+

e-

i µ

[v 2 (0 , µ - 1) + v 2 (0 , µ + 1)+
1 +C2 (µ)1 (0 , µ)]2 ( , µ)µdµ.

(6 7 )

Adding up these formulae and using the expression for 2 we get i/4 â 2(x + 0x )f ( , 0) = x + 0x ^ 1 â e-iµ C2 (µ) 1 ( , µ)2(0 , µ) + 1 (0 , µ)2 ( , µ) µdµ+
+

+

^

e-

i µ

v 2 ( , µ - 1)v 2(0 , µ - 1)- - v 2 ( , µ + 1)v 2 (0 , µ + 1) dµ . (68)

+

The second integral is equal to zero. This can be obtained in the same way as (61). Thus, we get (25). Formula (25) can be obtained from (24) by using (41) and (42) in the same way. 14


References [1] J. Bo ersma, J. Jansen, Electromagnetic field singularities at the tip of an elliptic cone, Technical Report EUT 90-01, Department of Mathematics and Computing Science, Eindhoven University of Technology, Eindhoven, 1990. [2] L. Kraus, L. Levine, Diffraction by an elliptic cone, Research report EM156, Institute of mathematical sciences, Department of electromagnetic research, New York University, New York, 1960. [3] V. Smyshlyaev, Wave motion 12 (1990) 329­339. [4] V. Babich, V. Smyshlyaev, D. Dement'ev, B. Samokish, IEEE transactions on antennas and propagation 44 (1996) 740­746. [5] V. Babich, D. Dement'ev, B. Samokish, V. Smyshlyaev, SIAM J. Appl. Math. 60 (2000) 536­573. [6] A. Shanin, Wave motion 41 (2005) 79­93. [7] V. Valyaev, A. Shanin, Wave motion ?? (2011) ???­??? [8] E. Skelton, R. Craster, A. Shanin, V. Valyaev, Wave motion 47 (2010) 299­317. [9] A. Shanin, Q. J. Mechanics Appl. Math. 58 (2005) 289­308.

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