Diffraction by a nonplanar screen

WAVE MOTION 3 (1981) 71-80 @ NORTH-HOLLAND PUBLISHING

DIFFRACTION

COMPANY

BY A NONPLANAR

SCREEN

V.B. PHILIPPOV Academy

of Sciences, Leningrad Branch of the V.A. Steklov Mathematical Institute, Leningrad,

U.S.S.R.

Received 25 July 1980

This paper is concerned with diffraction of short waves by a nonplanar screen (two-dimensional case, Dirichlet boundary condition). The high-frequency asymptotic approximation to the solution is obtained. First the wave field of the primary wave is found in a neighbourhood of the screen edge and then this field is continued along the boundary. Secondary waves arise here as the consequence of interaction between the edge and the primary wave. The secondary wave is diffracted by another edge of the screen, and a third order wave arises, and so on. This process gives the formulas for the wave field in a neighbourhood of the screen. Green’s formula is used to continue the solution outside of this neighbourhood.

1. Introduction

We consider a nonplanar screen S illuminated by a high-frequency field zinc. The picture of the scattered wave field is complicated. The wave field consists of: the incident field zinc; the field z/, reflected by the screen; the edge waves creeping along the convex side of the screen and rereflecting from the concave side of the screen (see Fig. 1). Some of these will generate secondary diffracted waves, and so on. There are known asymptotic methods which describe some of these processes. Kirchhoff’s method (see, for example, [l]) can be used to find the main part of the scattered field (the reflected field). Keller’s geometrical theory of diffraction [2] gives the primary edge waves. Uniform asymptotic theory of diffraction [3] makes it possible to calculate the field in a penumbra. It is found, however, that these methods are not applicable in some regions outside the screen S. In those regions it is necessary to take into account the whispering gallery waves and creeping waves. In this paper a method for the calculation of the high-frequency asymptotic approximation for all kinds of waves is suggested. The asymptotic approximation is found successively: first the asymptotic approximation to the primary diffracted waves is obtained, then the asymptotic approximation to the secondary diffracted waves, and so on. It turns out, that for a large screen the values of the field and of the current noticeably decrease as the number of diffractions increases. Thus in order to find the asymptotic approximation with desirable accuracy it is necessary to complete a finite number of steps. Since the field outside the screen has a complicated behaviour (there are caustics, which may be broken, penumbras, and so on) the following method [5] for the calculation of the asymptotic approximation is used: first the asymptotic approximation to the field in a small neighbourhood of the screen S is sought, and then the asymptotic approximation to the current on the screen S is found. After that we obtain the asymptotic approximation to the field outside the screen S by using Green’s formula. The two-dimensional Dirichlet problem is considered. The wave field u satisfies the following equations:

(d+k2)u =o

outside S,

(l.la)

u=o

on S,

(l.lb) 71

72

V.B. Philippov / Diffraction by a nonplanar screen

Fig. 1. Geometry of the diffracted rays.

&(a/&-ik)(u u = O(l),

-uinc)+O r &4/& = o(l)

when r = IxI-,oo,

(l.lc)

when r = Ix - x(P&] -* 0.

(i.ld)

It is supposed that dimensions of both the screen and the aperture are large in comparison with the wavelength (kp >>1, ka >>1, where k is a wave number, p is a curvature radius, a is the distance between edges of the screen S). We assume also that the incident field possesses the ray expansion Uinc(X,k) = eik9(x) X0 ajix)k-‘,

x 6 R2

(1.2)

and that rays of incident wave are not tangent to the screen S.

2. Solution in a nei~~~hood

of a screen edge

We shall construct the asymptotic (k + 03) solution of the problem (1.1) in a neighbourh~od of a point PI on the edge of the screen. Let us continue smoothly the curve S, and denote this continuation as St. We shall use coordinates 11,s, where /FZ 1is the distance from the curve S + SipII > 0 at a convex side, II < 0 at a concave side, s is the length alongS+S~,~=Ointhepoint~~,~>OonS,~cOonS~. The ray method allows us to find the field reflected by the boundary S + Sj as u~(x, k) = eikrZfx)C bj(x)k?. j&I

(2.11

Let us define the geometrical optics solution u’(x, k) =

@‘(x, k) + u”(x, k)

in the illuminated region,

0

in the shadow region.

I

(2.2)

The solution ug obviously satisfies the equation (l.la), the boundary condition (l.lb) and Meixner’s condition (l.ld). But the function ug is not smooth outside of the curve S. The normal derivative of up possesses a jump on S + S, so that a fictitious current arises on Si, We seek the function fi = u - up (n, s are small) which neutralizes this fictitious current. Let us suppose that the function tZ possesses the following representation: 6 zjFo

Oj(c,v)k-',

where CT= ks, v = kn.

(2.3)

V.B. Philippov / Diffraction by a nonplanar screen

73

The function ug may be written in an analogous form ug = & v: (a, V)k? The equations for the function fij are obtained from (1.1). So we find for j = 0, (#/&r* + a*/&* + 1) &, = 0

when YZ?0,

(2.4a)

&=o

when Y =O, a>O,

(2.4b)

when Y=O,U
(2.4~)

[a&+]

= -alY;/av

where [fl is the jump of the function J The condition (2.4~) arises from the requirement function

of continuity of the

auJav=air,/av+aufi/avon& In order to obtain the unique solution &, we shall add to conditions (2.4) Meixner’s condition (1.2d) and the condition of limited absorption: C?,,= O(1) when Y+ 00, Im v = E Re Y, E > 0. It is easy to see that the function U0 = fiO + UE is the solution of the problem of plane wave diffraction by the half-plane which is tangent to the curve S in the point PI. The solution of this problem is well known u~(~, r,) = i eik71(Pt) a&W+ 2m A

a0

O”exp{i[ma * J~?zJ]} I -co

(CX - cro)Ji+cu

da (2.5)

when Yg 0, a0 = ari/& for s = 0. The regular branch of &??is defined (the plane (Yis cut along Im (Yg 0, Re (Y= f 1) by the condition m > 0 for -1~ (Y< 1. The contour of integration rounds the point CY= cyo from above. The next approximation fil is found in an analogous manner (see [6]). The geometrical optics solution ug is defined by the formula (2.2) in a neighbourhood of the curve S for any s (0 c s G 1, 1 is the length of the curve S). But the series (2.3) loses its asymptotic behavior when the value v becomes large. So the problem arises to construct the continuation of the solution for large u.

3. Expression for the diffraction field in the domain ks >>1, In I<< 1 The diffracted field li is described by different expressions at different sides of the curve S. There are creeping waves at the convex side of the screen, and at the concave side there are rereflected waves and whispering gallery modes. The determination of the diffracted field at a convex and at a concave side of the screen may be carried out independently. The construction of asymptotics in our case is analogous to the case of an infinite curve S. The latter problem was investigated in a previous set of works (see [4,7-g]). Following these works, let us seek the diffracted fields in the form: ; = eiksUl ,

(3.1)

where the sign ‘+’ (‘-‘) corresponds to a convex (concave) side of S. For the case of the Dirichlet problem it is possible to construct the function U* for any s, but for the case of the Neumann problem it is necessary to consider two cases separately: for small values of s and for values of s of order unity (see [9]).

74

V.B. Philippoo / Difiaction

by a nonplanar screen

Let us suppose that the value of s is small (s CC1). We shall seek the function u* in the following form: (3.2) where (T’= (k/2)“3P-2’3(0)S,

V’= /k2’3(2/p(o))“3n

and p(O) is the radius of curvature of S at the point Pi. The equations for the functions C$ are found from (1 .la). So one may find that the function Vi satisfies the equation (i a/&r’ + a*/a(v’)* + v’) V,’ = 0

when V’B 0

and the boundary condition Vc =0

when y’=O.

We shall have the unique solution U,“, assuming that V$ satisfies the condition of limited absorption tY,‘=O(l)

whenv’+co,Imv’=cRev’,E>O.

It is possible to verify (see, for example, [4]) that the functions V,’ are cc

u,’

=

c+

eiu’L w(t

I

-

Y’)

w(5)

-02

(3.3a)

d&

(3.3b) where u(l) = X& Ai is the Airy integral, wl,* are first and second Airy functions. The expressions (3.1), (3.3) and (2.5) are to coincide when C#CC1, CY>>1. This condition makes it possible to obtain the constants c * (see [lo]). It appears that the values c* are i

CfZZ c(ao)=-

4&

*‘“JiG 2 0 ( kp(0) > 1-Q

e~hw~o(pl).

(3.4)

If the value (T’increases (the value Y’is bounded) the functions uf decrease exponentially and they can be neglected. But the function UT does not decrease and the essential domain of integration of the integral tY7 enlarges. The expansion (3.2) may lose asymptotic character when a’ becomes too large. There is, however, a transformation of the integrals 17; (see [4]) which gives the possibility to use the formula (3.2) if ~9 is large. This transformation follows from the identity 1 -= u(5)

2i Wl(5) -

=-$Y[$$+&[$$

(3.5)

w*(L)

Using the identity (3.5) for the integral (3.3b), we obtain: M-l 6

=

C m=O

2 C j=l

gjnz +

GM,

(3.6a)

V.B. Philippov / Diffraction by a nonplanar screen

75

where gj,

=

d5>

C(cUlJ)(-1)'

(3.6b)

(3.6~)

and the contour 9 is defined by equation arg 4 = $7~. Let us suppose that in the essential domain of integration for the integral gj, we have 151~ 1, when CT’>>1, V’= O(1). This supposition allows us to use the asymptotic expansions of Airy functions: WI([) - t-“4 exp{i[$t3’2 + $rr]}

(3.7a)

w2(5) - t-“4 exp{-i[$t”“+$r]}

(3.7b)

where t=-5, largtl<$r-c,E>O. From (3.6b) we find gj, - c((yO)(-l)j eim(m+l)/2 e’%(‘) dt, Ia where ~jm(t)=-V’t+$(WZ+l)t3’2+(-l)i-1V’Jt.

(3.8b)

The integral (3.8a) may be calculated by the method of stationary phase. The stationary point is defined by

(3.9) The method of stationary phase will be valid if the stationary point will be far enough from the origin. From this we obtain the upper bound for the value M: M + 1 < a’/2A,

(3.10)

where A >>1. The lower bound for the value M can be obtained from the assumption that the series (3.2) should be asymptotical. The stationary contour 9jm in the integral gj, is shown in Fig. 2. The region where Im djm > 0 is shaded. Using the method of stationary phase, we obtain gjm

ns(-l)j-’ “‘C(CfCK

exp( i[ -A

&+(-l)i2(~‘:l)+~(m

-$)I].

(3.11)

It may be shown (see [8]) that eiksgjm is the expression for the geometrical optics wave, emitted by the screen’s edge and reflected m +i - 1 time by the screen’s surface. If the value C’ is not very large, it is necessary to compute the function GM numerically. Such calculations for the current (the value ~GM/~v’ for V’= 0) are presented in [4]. If the value (T’is large, it is possible to make a certain transformation of the integral GM in order to simplify the expression for GM (see the annotation in [ 111).

V.B. Philippoo / Diffraction by a nonplanar screen

77

The functions Im &(t) (when Im t > 0) and Im &-i(t) (when Im t < 0) are positive in the shaded domain (see Fig. 3) and increase when [Im t] increases. When the contour 9 is deformed into the contour _!&, it intersects N poles of the integrand in (3.6~) (roots of the function v(l)). So the residues of these poles appear. Thus we obtain the following formula for the integral: GM= g

(3.15a)

~,+RMN,

n=l

where V, = 2ni$$

n

ei”‘tv([n -v’), M

e &o) RMN = -id’ 2i I rsNw2(04f)

H([, d)[s

d[.

(3.15c)

I

It is possible to replace the Airy functions in integral R MN by their asymptotic expansions (3.7). As a result we have iv’J;_

ei&&)

,-iv’\r

Ce 1 1 - exp{i[$t3’2 + &r]} dt’

R MN_ -c(ao) eiv(M+l)/2

(3.16)

The function C&M(~) is defined by (3.14~). It is easy to see that the essential domain of integration of the integral RMN is the small segment of the contour zN, where Im f = 0(1/&N), since the integrand decreases rapidly, when ]Im t] increases. Let us simplify the expression for the integral (3.16). For this purpose we use the following expansions in the essential domain of integration: t 3’2 = az2 +i&$

+0(x2/&&,

(3.17a)

Jt = JON+ 0(X/J&),

(3.17b)

where x = i(aN - t). Substituting these expansions into (3.16), we obtain: eiv’J;;;;

RMN -

c ((~0)

_ e-i&&

eisrM12

eibMcaN)[F(aN9

&)+F@-aN,

PN)]

I

2&

-e i~~-l’“-‘[F(l-a,~~)+~(l+a~,P~)]],

(3.18a)

where G>

P) = jo’ 1 _,,I;;:,

+x2

d-5

The error in this formula is of order (M + l)aG3” or a;;‘.

(3.18b)

78

V.B. Philippov / Diffraction by a nonplanar screen

We note that another formula for the remainder R MN is presented in [ 121, but it seems that the latter is not sufficiently justified. So for cr’>>1 we have the following representation for the diffracted fields at the concave side of the screen M-l fi -e

Iks

[

2

C C m=O j=l

gjm + Fl

Vn + RMN]

(3.19)

*

The functions in equation (3.19) are defined by (3.11), (3.15b), and (3.18a). These formulae are obtained if the point of observation is not far from the edge of the screen (s <<1). Let us consider the problem of continuing this expression to arbitrary values of s. The geometrical optics field eiksgjmmay be continued by the ray formula in both forms: usual elksgjm

=

e ikTi-(x) f

aljm(x)k-‘,

XER’

I=0

and exponential e’ksgjm

(see [8]) =

e

ikTi,,,

,

where

The expressions for the normal modes u, = eiksV,, may be continued in several ways. (1) The asymptotics of the whispering gallery modes in the boundary layer may be found by Cherry’s method (see [4]). According to this method, we seek the solution in the following form: u, = exp{ikA(v, s, &, k)M&,

(3.20)

s, 5”, k)),

where A = f

j=O

aj(v, S, l”)k-“3,

B = f

j=O

bj(Y, S, l”)k-i’3

and aj and bj are polynomials in v u =

k2’3[2/p(s)]“3n

with the coefficients depending on s and 5” (5” are the roots of the Airy integral v(l)). If we substitute the expression (3.20) into (l.la) and equate the coefficients with the same powers k, we obtain the equations for the coefficients aj and b+ (2) In accordance with the boundary-layer method (see [7]) the expressions for the normal modes are sought in the following form: u, =exp{iks +ik”3&r1(s)}{u(&

-v) ,fo Gj(v, S, &)k-“3

+ ~‘(5” - U) f

j-0

&(V,S, t)k-“3}p

(3.21)

where 6j and 6j are polynomials in v with the coefficients depending on s and 1;1.These functions are found successively from the recurrence system of equations.

V.B. Philippov / Diffraction by a nonplanar screen

79

It is possible to define the remainder term R m for arbitrary values of s by the formula (3.15c), by substituting for the expressions exp{iks + i~‘~}wr,&- Y’) expressions similar to (3.20) or (3.21).

4. Multiple diffraction and the wave field outside of a neighborhood of the screen The diffracted waves, emitted by an edge of the screen, may be rediffracted by another edge. Mathematically it means that the asymptotic solution, corresponding to the diffracted field, is not a smooth function since there is a jump of the normal derivative when n = 0. This jump defines the current on the screen S. On the continuation of the screen S the fictitious current arises. In order to neutralize this current the additional solution (the secondary diffracted field) is found. Each term in (3.19) will generate a corresponding term in the secondary diffracted field. These solutions are found in a neighbourhood of the edge at first and then they are continued along the boundary S (see [lo]). The determination of these solutions is analogous to the one for the primary diffracted field. It may be found that the ratio of the value of the secondary diffracted field to the primary one is of order (k2ap)-1’2 . Thus if the aperture a is not small, the secondary diffracted field can be neglected in the first approximation. Above we have obtained the asymptotics of the field in the neighbourhood of the screen S. If we calculate the normal derivative of this solution, we shall obtain the asymptotics of the current p(s, k). Let us substitute this expression for the current into Green’s formula i&x, k) =

uinc(x,k)-f j- ?P(s,k)Hb”@R(s)) ds s

(4.1)

where R(s) = Jx -x(s)l, x(s) E S. The formula (4.1) gives the asymptotics of the field at an arbitrary point. Using the asymptotic methods for the calculation of the integral (4.1) we may find expressions for the different types of waves. We obtain the primary reflected waves by the method of stationary phase. The contribution from the ends of the contour of integration gives the edge waves. In a penumbra the stationary point approaches the end of the contour of integration. In this case the asymptotics of the integral (4.1) are expressed through the Fresnel function. The behaviour of the diffracted field is most complicated in a neighbourhood of a ray which is tangent to the screen in the end point P 1,2. In this case the asymptotics are expressed through the incomplete Airy function.

Acknowledgment The author gratefully acknowledges Prof. V.M. BabiE, who has read this paper and has made important comments. References [l] H. Honl, A. Maue, and K. Wbstpfahl, Theory OfDiffraction, Mir, Moskow (1964). [2] J.B. Keller, “Geometrical theory of diffraction”, .I. Opt. Sot. Amer. 52, 116-130 (1962). [3] D.S. Ahluwalia, “Uniform asymptotic theory of diffraction by the edge of a three-dimensional body”, SZAMJ. Appl. Math. 18 (2) 287-301 (1970). [4] V.B. BabiE, V.S. Buldyrev, Asymptotic Methods in Shortwave Diffraction Problems, Nauka, Moskow (1972). [5] V.S. Buslaev, “On the asymptotic behavior of spectral characteristics of exterior problems for the Schriidinger operator”, Izv. A&ad. Nauk SSR (ser. Mat.) 39 149-235 (1975), .l. Sov. Math. USSR Zzv. 9 139-223 (1975).

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V.B. Philippov / Diffraction by a nonplanar screen

[6] V.B. Philippov, “Diffraction from a curved half-plane” Zap. Nauchn. Sem. Leningrad. Otd. Mat. Inst. ANSSSR 42,244-249 (1974); J. Sov. Math. 9, 626-632 (1978). [7] V.M. BabiT, N.Y. Kirpicnicova, The Boundary-Layer Method in Diffraction Problems, Springer, Berlin (1979). [8] V.M. BabiE, “Oscillations of a high-frequency point source near a concave mirror”, Zap. Nauchn. Sem. Leningrad. Otd. Mat. Inst. ANSSSR 51, 5-19 (1975); J. Sov. Math. 11, 361-371 (1979). [9] V.M. BabiE, “Oscillating point source near a concave mirror”, Zap. Nauchn. Sem. Leningrad. Otd. Mat. Inst. ANSSSR 89,3-13 (1978). [lo] V.B. Philippov, “Shortwave asymptotics of the current in the problem of diffraction from nonplanar screens”, Zap. Nauchn. Sem. Leningrad. Otd. Mat. Inst. ANSSSR 51, 172-182 (1975); J Sov. Math. 11,479-487 (1979).

[ll]

V.B. Philippov, “Shortwave asymptotics of the current and the angle diagram for the problem of diffraction by nonplanar screens”, VZZZSoviet Symposium on Problems of Diffraction and Wave Propagation, Rostow-na-Dony 1 (1977) 47-50. [12] T. Ishihara, L.B. Felsen, A. Green, “High-frequency fields excited by a line source located on a perfectly conducting concave cylindrical surface”, IEEE Trans. Ant. Prop. 26 (6), 757-767 (1978).