Документ взят из кэша поисковой машины. Адрес оригинального документа : http://acoustics.phys.msu.su/teachers/shanin_files/~shanin/papers/cone_asymptotics.pdf Дата изменения: Fri Nov 25 16:37:00 2011 Дата индексирования: Mon Oct 1 20:34:23 2012 Кодировка: Поисковые слова: south pole

DAYS on DIFFRACTION' 2011

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Diffraction series on a sphere and conical asymptotics
Andrey V. Shanin
Department of physics, Moscow State University, Russia; e-mail: a.v.shanin@gmail.com
The problem of finding field comp onents for a conical diffraction problem is studied. All comp onents except the spherical wave diffracted by cone tip are under consideration. As a starting p oint, the integral formula (7) derived by Babich et.al. is used. A geometrical optics approximation of the spherical Green's function is constructed in the form of diffraction series. There is a finite set of terms of the diffraction series on sphere, to each of which the conical field comp onents corresp ond. Formula (7) is simplified, giving a convenient field representation (26). In many cases further simplification can b e p erformed, giving formula (30) directly converting the terms of the diffraction series on sphere into the field comp onent in the 3D space.

1

Introduction

The problem of scattering of a plane harmonic wave by a conical obstacle with ideal boundary conditions (Dirichlet or Neumann) is studied. As an example of such a cone here we take a flat cone with the angle at the vertex. However, most of the consideration performed here can be easily applied to any cone of polygonal cross-section. Some ideas can also be useful for the cones of arbitrary crosssections. Usually, the main sub ject of investigation for the conical obstacles is the diffraction coefficient, which is the amplitude of scattering into the spherical wave. The diffraction coefficient depends on two directions: the direction of incidence and the direction of scattering. Both directions can be treated as points on a unit sphere. The problem of computing the diffraction coefficient has been studied by Smyshlyaev and co-authors [4]. The result is the so-called Smyshlyaev's formula expressing the diffraction coefficient as a contour integral of the Green's function on a sphere with scatterers. The spherical wave is a very interesting part of the diffracted wave, but there are many other types of waves emerging as as a result of diffraction of a plane wave by a conical obstacle. Namely, if a polygonal cone is considered, one can mention the geometrically reflected plane waves, cylindrical waves scattered by the edges and multiply diffracted waves (i.e. the waves scattered by one edge and then re-scattered by another edge). Each

of these components can be computed by the ray technique, however there is a lot of penumbral zones corresponding to the geometrical boundaries of visibility of each wave component, and finding of the asymptotics in the penumbral zones is a sophisticated problem. Here we are interested in all wave components except the spherical wave. The case studied in details in the literature is the "conical" penumbral zone where the wave front of the cylindrical scattered wave touches the wave reflected from the surface of a smooth cone. This field is known to have structure described by the function of parabolic cylinder or generalized fresnel integral [1, 2, 3]. Our aim here is to find a convenient formula to describe the wave asymptotics in the numerous penumbral zones. For this we follow Smyshlyaev's approach, and express the field in terms of the Green's function on the sphere. For the spherical problem we construct an asymptotics in the form of diffraction series, i.e. we consider the sequence of successive diffractions acts, and each diffraction term on the sphere corresponds to some wave contribution for the initial conical problem. For each term we restrict the consideration to only the leading asymptotics. 2 Problem formulation and representation of the field Let the (1)

The problem is formulated as follows. Helmholtz equation u + k 2 u = 0

be valid in the 3D space (x, y , z ), from which a conical obstacle is removed. The tip of the cone coincides with the origin of the coordinate system The time dependence of all values has form e-2f t , and it is omitted henceforth. The direction of incidence (the direction to the source) is denoted by 0 , the direction of scattering is denoted by . Both and 0 can be treated as points on a unit sphere. The incident wave has form uin = exp{-ik r cos(( , 0 ))}, (2)


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DAYS on DIFFRACTION' 2011

where ( , 0 ) is the angular distance between the Green's function on the sphere for the Laplace­ points and 0 . Here r = x2 + y 2 + z 2 is the ra- Beltrami equation dial variable. The total field is a sum of the incident 1 field and the scattered field: ~ g = ( - 0 ). (8) + 2 - 4 u = uin + usc . (3) ~ where is the Laplace­Beltrami differential opThe total field obeys Dirichlet or Neumann erator on sphere acting on variable . Function boundary conditions on the boundary . Also, g obeys Dirichlet or Neumann boundary condition (depending on the initial problem) on the contour the field obeys vertex conditions ~ , which is the boundary of the cross-section of - 3 /2 u = o(r ). (4) the conical body by the unit sphere. Equation ~ (8) is valid outside the cross-section . Also the scattered field should obey the radiation condition, and the total field should obey edge conditions on the edges of the cone . In our simplest case is the flat cone (see Fig. 1). = {(x, y , z ) | z = 0, x > 0, 0 < y < x tan }. (5) We assume here that < /2. is the "opening angle" of the flat cone. This cone has two edges, for which the edge condition has form u = O (
1 /2

Re[n] g Re[n]

Figure 2: Contour of integration in (7)

Formula (7) can be understood as the result of (6) variables separation. Contour encircles the singularities of g on the positive half-axis corresponding ~ where is the distance between the observation to the spectrum of the operator - 1/4 in the com~ with corresponding boundary condiplement of point and the edge. tions. ), 3
w
0

Ray asymptotics Green's function

of

the

spherical

w

w

w
F

0

~ ¶W

W

To simplify formula (7) we should make some statements related to the asymptotics of the Green's function g as a function of . For this, consider the process, developing in time, i.e. study the wave equation on the sphere 2 ~ - 2 t w = 0. (9)

Figure 1: Diffraction by a flat cone

The total field obeying the conditions of the problem can be found by using the formula taken from Let the field be generated by a delta pulse emitted by a point source located at the point 0 . It is [4]: quite clear that the pulse travels with a unit veloc2 ity along the sphere. The pulse first travels from в u(r, ; 0 ) = 2e3i/4 kr the source directly, then it hits the scatterer (the ~ contour ) and is scattered for the first time. As J (k r)e-i /2 g ( , 0 , ) d (7) a result, the reflected wave and the edge wave scat tered by one of the edges is produced. Then this The observation point is described by the coordi- wave hits another edge producing the secondary nates (r, ), J is the Bessel's function. Contour diffracted wave etc. Let the whole signal w(t) be is shown in Fig. 2. Function g ( , 0 , ) is the "recorded" at the spherical observation point .


DAYS on DIFFRACTION' 2011

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The function g ( , 0 , ) is the Fourier transform and the ray Ansatz on the sphere. The result is as of w(t) taken at the point 2 - 1/4. follows: According to this concept, the function g has the iei( )(21 +21 ) TD,N (1 , 2 ) following asymptotics (14) g2 ( , 0 , ) = 4 ( ) sin(21 ) sin(22 ) g ( , 0 , ) gm ( , 0 , ) = ei( )m Gm ( ), where 11 and 12 are distances between the source m m (10) and the edge, and between the receiver and the where the summation is held over all diffraction tra- edge. The value TD,N (1 , 2 ) is the trigonometrical jectories going from 0 to , m is the length of the factor. The angles 1 , 2 describe locally the scattering process. These angles are shown in Fig. 3. m-th tra jectory, The values TD,N are taken from the planar theory , Im[ ] > 0, of diffraction by a half-line, and they are equal to (11) ( ) = -, Im[ ] < 0, cos(1 /2) TD (1 , 2 ) = - sin((2 + )/2) - sin(1 /2) Gm ( ) are slowly varying functions of (comparatively to the exponentials). The terms are sorted cos(1 /2) according to the rule m+1 m . The first term , (15) sin((2 - )/2) - sin(1 /2) (we assign index m = 0 to it) is the direct wave coming along the shortest tra jectory. cos(2 /2) TN (1 , 2 ) = - We will see below that the terms obeying the sin((2 + )/2) - sin(1 /2) relation m < correspond to the scattered field cos(2 /2) components of the initial conical problem. , (16) sin((2 - )/2) - sin(1 /2) Let us compute several first terms of the series (10) in some cases. The zeroth term and the reflected wave. The direct wave has form ei/4 ei g0 ( , 0 , ) = - , 2 2 sin (12)

w

w
0

where is the angular distance between 0 and . This expression was obtained by standard ray technique. Namely, the ray Ansatz ei was matched with the outer (far field) asymptotics of the Green's function of an entire plane. The asymptotics of the reflected ray is as follows g1 ( , 0 , ) = -R e
i /4

q21 f1 f2

q22

Figure 3: Diffraction by an edge Secondary diffraction. Consider the wave going along the tra jectory shown in Fig. 4. The wave goes from 0 to the first edge of the scatterer, then goes along the scatterer, approaches the second edge and goes to . In fact, the ray travels along two different tra jectories. The first tra jectory goes along the upper face of the scatterer, and the second one goes along the lower face. For the Neumann case the representation is as follows: e-i/4 в g3 ( , 0 , ) = 4 [TN ( , 1 )TN (2 , ) + TN (- , 1 )TN (2 , - )]в (2 )3
/2

2



ei1 . 2 sin 1

(13)

Here R is the reflection coefficient for the surface (R = 1 for the Neumann boundary and R = -1 for the Dirichlet boundary), 1 is the length of the reflected ray. This asymptotics is not valid when 1 . Diffracted edge wave. This is the ray diffracted a single time by an edge of the flat cone. To compute this term we use the locality principle: we assume that the wave approaching the edge has approximately the structure of a plane wave on a plane. Then we study the process of diffraction of a plane wave by a half-line. Finally, we perform matching between the diffracted cylindrical wave

ei(31 ++32 ) , (sin(31 ) sin() sin(32 ))1/2

(17)


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DAYS on DIFFRACTION' 2011

w w
0

q31 f1 F

q32 f2

f2 w f q w'0 f1

Figure 4: Secondary edge diffraction

w0
where = ( ). In the Dirichlet case the formula is a bit more complicated. Penumbral zone. The expansion (14) is valid only when the geodesic arc connecting 0 and pass far enough from both edges of the scatterer. We are also interested in the case when this ray passes quite close to this point. In this case a penumbral field can be observed on the sphere. Using the technique of phase integrals we obtain the following representation of the penumbral field: gpen ( , 0 , ) = - Here = ( ), 2e-i I (z ) =
/4

Figure 5: Observation point in the proximity of the focal point The field in this case is found using the formula gfo ei ( , 0 , ) = - 4

2

c

exp{-i cos( - )}d .
1

(20) 4 Analysis of the integral (7)

To simplify (7) substitute Bessel's function by I (± ). (18) Sonin's integral. As the result, get 4 2 sin() e
i +i /4

u(r, ) = e e d , (19)
ikr cos 1 +2

3 i/4

2 в kr

e
z

i

2

e

i -i

g ( , 0 , ) d d . (21)

where contour is shown in Fig. 6.

is the difference between the lengths of the ray going from 0 to through the edge, and the direct ray from 0 to . is the length of the direct ray. Sign + is chosen if the direct ray is obstructed by the scatterer, and sign - is chosen when the direct ray is not obstructed. The asymptotics (19) is valid if the point is far from the point opposite to 0 , i.e. is not close to . Fo cal p oint proximity. Finally, consider the case when is close to , and thus is close to the point 0 opposite to 0 . Introduce local polar coordinates near the point 0 . Let the point have coordinates (, ) in this system. A set of geometrical rays goes from 0 to 0 . Some of the rays are obstructed by the scatterer. Let the rays which are not obstructed occupy the angles in the set (1 , 2 ) (see Fig. 5).

Im[t]

G 0 G' p

Re[t]

Figure 6: Contour of integration for the Bessel function Here we split contour into two parts: 1 is a part of contour going from infinity to 0 below the real axis, and 2 goes from 0 to infinity along above the real axis. Split the inner integral into two integrals I1 and I2 , corresponding to integration along 1 and 2 , respectively. According to the asymptotic estimation


DAYS on DIFFRACTION' 2011 performed above, we can expect that the function I1 has singularities at the points m = + m , 2 cos m 2

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exp ik r - cos m - sin m +

,

(25) (22) = - m . Substituting this approximation into the integral and changing the order of integration, where the m are the ray lengths from (10). Respec- obtain a representation for the terms of (24): tively, I2 should have singularities at the points 2 -ikr cos m um (r, ) = 2e3 i/4 e в m = - m . (23) kr First consider the integral of the form (21) containing only I1 , i.e. with integration over 1 . Since there are no singular points on the segment (0, ), one can deform contour into a contour lying in the areas of the exponential decay of the function exp{ik r cos t} (see Fig. 6). Thus, the integral describes a linear combination of the functions behaving as eikr and e-ikr for large r. It is well known that the function behaving as e-ikr disappears when both components I1 and I2 are taken into account. Thus, it can be ignored. The component behaving as eikr (taken from I1 + I2 ) is the spherical wave diffracted by the tip of the cone. This wave is well studied by Smyshlyaev and coauthors, so we do not study it here (we remind that here we study all waves except the spherical wave). This means that the part with I1 contains no components we are interested in. Now consider the integral with I2 . Since some of the values m can be located between 0 and , we cannot deform into . Instead, we put the straight part of the contour onto the segment (0, ) of the real axis of bypassing above the singularities. We base our consideration on the assumption that for large r the integral (21) can be estimated as a sum of contributions of the singular point belonging to the segment (0, ), and each contribution is provided by a small proximity of the singular points, i.e.


K (k r, , m )e
0

-i

m

+ gm ( , 0 , ) d

(26)

where K (z , , ) = 1 2


-

i exp i( - z sin ) + z 2 cos 2

m

d ,

(27) + and gm is the asymptotics of gm taken in the upper half-plane (i.e. with ( ) = ). Performing the integration, obtain that K (z , , ) = e-i/4 ( - z sin ) exp -i 2z cos -2 z cos
2

.

(28) The integral formula (26) with the kernel (28) is the main result of the paper. We claim that together + with the asymptotics gm obtained above this formula describes uniformly the most interesting field components. 5 Some properties of formulas (26), (28)

Ray asymptotics. The function K defined by (28) and treated as a function of is a complex (oscillating) Gaussian centered at z sin and hav ing width z cos . We remind that according to (26) z = k r, thus the width of the Gaussian grows as k , however, the center position grows um (r, ). (24) faster. If sin is not close to zero and if g + behaves u(r, ) = m m as exp{i m } m as , (29) The terms um correspond to the terms of the spherical Green's function gm , for which 0 < m < , i.e. then for k r large enough one can neglect variation + there is finite number of terms in the sum (24). of gm ( ) in the area covered by the width of the Consider a vicinity of the singularity m = - m Gaussian and substitute the function in the integral belonging to the segment (0, ). Near this point (26) by its value at = k r sin . In this case K approximate the cosine function by its Taylor series acts as a delta-function ( - z sin ), giving the up to the quadratic term: following approximation exp{ik r cos } um (r, ) 2e3i/4 2 k re-ikr cos m в


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+ gm ( , 0 , k r sin m ) sin m . (30) This formula is very important. It transfers the ray (or sometimes penumbral) asymptotics on the unit sphere into asymptotics in the 3D space. A necessary condition of validity of this formula is as follows: k r sin m | cos m | (31)

DAYS on DIFFRACTION' 2011 Proximity of the shadow of cone tip. Consider the particular case of 0 located at the South pole, i.e. having coordinate = . Look for the field in the proximity of the North pole, i.e. for the point = (, ) with 0. Use formula (20) for + gm and substitute the result into (26). The result is as follows: u(, ) = -

2

This relation for large k r denotes that the point is far from a singular ray corresponding to the conical penumbra, i.e. to the set of points where the spherical diffracted wave touches some other wave front (namely, the front of wave corresponding to the m-th diffracted wave). Beside the relation (31), one should study the + behavior of the function gm ( ) in each particular case to find whether the function is smooth com paratively to the scale k r sin m . Connection with the parab olic cylinder + functions. If gm is a power function then the integral (26) is closely connected with the function of parabolic cylinder. Namely, if
+ gm = A( , 0 )e i
m

e

i /2 ikr

e в 2 k r



exp i

1

0

2 + (k r)2 - i cos( - ) d d . 2k r

(35) This expression matches the Fresnel integral written for the screen with a quarter-plane cut from it. 7 Conclusions



s

then um = -A 2i
i s/4

(k r)s/2 (- cos m )(s+1) (s + 2)(-s - 2) sin( s) sin2 m 2 cos m kr - cos m . в

/2

Approximate formula (26) is the main formula connecting the spherical Green's function and the asymptotics of the components of the field (32) diffracted by a cone. In many cases (the ma jor exception is the conical penumbra) this formula can be reduced to a simpler expression (30). Author is grateful to Prof. V.M. Babich and Prof. в M.A. Lyalinov for fruitful discussions and motivation of the work. References (33) [1] V.A. Borovikov. Diffraction by polygons and by polyhedra. Moscow, "Nauka", 1966, 456 p. [2] V.M. Babich, On PC Ansatz, Jounal of Mathematical Sciences, V. 132, N. 1, pp. 2­10, 2006. [3] A. Popov, A. Ladyzhensky, S. Khoziosky, Uniform asymptotics of the wave diffracted by a cone of arbitrary cross-section, Russian Journal of Mathematical Physics, V. 16, N. 2, pp. 296­299, 2009. [4] V.M. Babich, D.B. Dement'ev, B.A. Samokish, and V.P. Smyshlyaev, On evaluation of the diffraction coefficient for arbitrary "nonsingular" directions of a smooth convex cone, SIAM J. Appl. Math. V. 60, N. 2, pp. 536­573, 2000.

exp -ik r cos m + D e
3 i/4

-s-2

sin

m

Thus, the asymptotics built here is closely connected with the conical penumbral asymptotics constructed and studied in [1, 2, 3] 6 Some examples of asymptotics obtained with the new formula

Wave diffracted by edge. Conical p enumbra. Consider the conical penumbra. Use formula (14) + for the gm component and (26) without any additional simplifications. As the result, get the expression in terms of the Fresnel integral: TD,N (1 , 2 )eikr+i/4 I u=- 2 2 k r sin 21 sin 22 where I is defined by (19). ± kr sin . 2 (34)