Diffraction by a polyhedral angle of a spherical wave gliding along an edge

DlFFRAC~ON

BY A POLYHEDRAL ANGLE OF A SPHERICAL WAVE GLIDING ALONG AN EDGE* A. F. FILIPPOV Moscow (Received 28 December 1972)

THE problem of the diffraction by a polyhedral angle of a wave with a spherical front, whose centre is situated on an edge of the polyhedral angle, is considered. The diffraction wave is the sum of a radial series of which the first and second terms are found. The radial expansion of a spherical wave arising in the diffraction by a polyhedral angle of an arbitrary non-stations wave, possessing, close to the vertex of the polyhedral angle, a convergent radial expansion, was found in [I] . The case where one of the rays of the incident wave slides along an edge of the polyhedral angle has not yet been investigated. This case occurs, for example, in diffraction by a polyhedron. When a wave is diffracted at any of the vertices of the polyhedron a wave with a spherical front with centre at this vertex is produced. When this wave reaches another vertex of the polyhedron, joined by an edge to the first vertex, a new diffraction wave is produced who whose radial expansion cannot be found by the method described in [ I] in the general case. It is proved below that this wave is the sum of radial series, for which the first and second terms are calculated. The degrees of the first terms of these series differ from the degree of the fust term of the incident wave, in general by a fractional number depending on the magnitude of the dihedral angle for this edge. In [2] a standard wave method is proposed for solving the problem considered. But the reasoning presented there is erroneous. Thus, the solution of the wave equation constructed on p. 415, possessing for t
1. Structure of a harmonic function close to the edge of a dihedral angle Lemma 1 For any /3ZO the solution of the problem U,, + ?=-ru, - pW% + U,, = 0, u(!?, s)=*(s), u=O(1),

@0.9 *h) =x* (I^)t r-+0,

jZf
Zh. vj?chisI.Mat. mat. Fiz., 14, 1, 157-165, 1974.

156

o
/,zl
Diffraction by a polyhedral angle of a spherical wave

157

exists and close to the point r=O, z=O it can be expanded in the absolutely convergent series m 7.L(r, 2) =

c,pr@+2nzP,

c

(1.2)

n, p-0

whose coefficients satisfy the conditions

4n(n + I$cnp= - (P + 1)(P + 3 (It is assumed that x+EL,

with weight r, *EL,.

)

Proo_f We seek the solution in the form u=u+w, bounded as r + 0,

v(q, z)=O,

where v, w satisfy Eqs. (1 .l), and are

w(r, *h)=O.

w(q, 2)=$(z),

u(r, ih)=x*(r),

(1.3)

cn-I, pt2.

We seek the function v by Fourier’s method in the form u = ZRk (r) Zk (2). We obtain Z,(z>=[~sh pk(h-z)+a_ksh pLR(h+z)]lsh2&?, where ti=x&q, is the k-th zero of the Bessel function J,, and Nk is a normalizing factor. As is well-known, Rk(r)=NJB(pkr),

k~+b-1
-=

p>O,

k=1,2

(1.4)

xgh

,...,

(1.5)

[Jp (pkr)12r dr =

It follows from the boundary conditions that a *tk are the coefficients of the expansion of the functions x+ (r) in the system & (r) , orthonorms (with weight r), therefore %_& = 11 x+- [I2< OO. It is easy to show that 2”Vz!l?(~+l+n)~(2n)!~(/3+l), E>O is arbitrary

X*=IIX+ll+llX-II,

Z,(z) ac2x* exp (-pkh) exp (Clkz),

where *: is the sign of majorization of power series. Therefore

(1.6)

r-b

(r, 2) <

cQf

c

exp

[pk

(8 -

h)l

exp

[vk

tr +

z)l.

(1.7)

k=l

By (1.5) the series converges for rfzth-e.

Therefore, for r-l- 1z 1
the function

r-Q (r, z) is analytic.

We also seek the function w by the Fourier method. We obtain w=ZRk*Zk*,

Zk*=sin v,(z+h),

Rk*=bAZB(vkr),

vk=kn/2h.

A. F. Filippov

158

The majorant for the function r-@Js(v&r) = (ir) -“is ( Z&J-)is similar to (I .6), and sin va (z-l-h)
w (r, 2) =

c

ak

k-_l

IP(vkr) -sin IP @k!?)

vk

(z + hf.

Therefore, for r-l- I.2 1cmin (h; q] -8 the function u =~-i-r~ is expanded in a series of the form (1.2). Substituting this series in Eq. (1 .l) and equating to zero the coefficient of ~s+2”-2zp, we obtain Eq. (1.3). By the same method as for the IXrichfet problem in a rectangle, it can be proved that the function u ~tis~es Eq. f 1, 1) and the boast control.

The harmonic function Y,bounded in the cylindrical sector O
O+Ga,

zi
is expanded close to any point Z=Z,E: (zi, z,) of the edge r = 0 in a series of the form

0.9)

where, respectively, a) s, =sin prnq, pm==mn / ct; b) s,,,==cos p,,rp, firn==(m-l) c) 8, =sin &$I+ Brn= (m-ilz> z / CL &OQ~ Consider case a). Let z,,=O. Fourier series:

rt / ct;

(The proof is similar in the other cases.) We expand Yin a

co

u=

c

urn(r, 4

sinPmCP.

(1.10)

rn.=l

It foBows from 1Y 1C=cthat 1zz, 1Gicln. isubstituting Eq. fl JO) in Laplace’s equation, we obtain that every function u, sin pm9 is harmonic and that u, satisfies Eq. (I +I) for p=&. By Lemma 1, u, cm be expanded in a series of the form (1.2). The series (1.10) also converges absolutely, since the harmonic function Yremains analytic for odd continuation across the boundaries rp=Oand (~=a, on which v=O.

Diffraction by a polyhedral angle of a spherical wave

159

2. Structure of a sphericalwave close to the edge of a diiedral angle Let 0 be the origin of coordinates, x, y, z rectangular, r, cp, z cylindrical, and p, 0, rp spherical coordinates. The Laplace operator in p, 8, cp coordinates is of the form A,J.z~z~~+u~ ctg 0+u,sin-2

Au~u,,+2p-‘~,+p-~A,u,

0.

(2.1)

In 121, p. 276, it was proved that if the function w (q, 0, rp) in the spherical coordinates q, 8, q satisfies Laplace’s equation, the function

is a solution of the wave equation u,*=Au in spherical coordinates p, 8, rp.This solution is a function of degree -% homogeneous in t, p . If w (1, 0, cp) =0, the function u, defined as zero for pbt, will be a generalized solution of the wave equation. For any Q=-0 the operator m

s

I~LU(t,f-v)~

u(t-s,

SF1

N)

0

-as

r (CL)

(2.3)

(where N is the point (p, 8, VP); by the above the integrand equals zero for s>t-p) transforms the solution into a solution of a higher degree of homogeneity in ~1.Differentiation of the solution with respect to t lowers the degree of homogeneity. By [2] , when a plane wave is diffracted by a polyhedral angle with vertex 0 waves arise which are reflected from the faces, and diffracted by the edges, and a spherical wave scattered by the vertex of the polyhedral angle, of the form (2.2) or (2.3), where u is of the form (2.2). Lemma 3 Let the and rp=a of zero for t
solution (2.2) of the wave equation in the domain OCcpC a satisfy on the faces cp=O the dihedral angle one of the boundary conditions of the form a), b) or c), and equal Then close to the point p=t, 8=O of intersection of the front I)= t of the wave with of the dihedral angle, the solution u for t>p is expanded in the series: 1c

u=

c

c 72

u my

U m=

VI:1

k. n=o

r

ct-

p)“+k%nk p,i2n s, (cp), (h + k + 1) pkfl (sin ‘)

(2.4)

where A=‘/,, Pn and s, (cp) are the same as in (1.9). For small t-p and 0 the double series converges absolutely. Remark. Each of the waves u and u, is a simple wave, that is, its radiative expansion is completely determined by its first term (for example, the term(t-p)Xa(8,cp)/r(h+l)pfor u, see [l], Eq. (20), or [2], p. 29). Thereby

a@,(P)= The coefficients nmllk,kal,

Z m

am (sin

0) h

(Cp),

a, (sin f3)=

amnO(sin 6) @m+2n. z

n

are expressed in terms of a,,~, P=O, 1, . . ,

(2.5)

160

proof: In the space with spherical coordinates q, 6, ip we perform an inversion with centre at the point r)=l, 8=n and ra~us~2, The point with cylind~~al coor~~ates r,===qsin 8, rpo, z,=q cos 6 becomes the point r, cp, z,where I‘=

30

roa + ($04

q2 7

.7-+-i=

2 (20+ 1)

r*a+

(20 + ly

’

cp =rp,*

(2.7)

The sphere q=% becomes the plane +=&and the h~o~c function w (q, 0, cp>passes into the harmonic function u(r, rp, Z>=pl-*w(t~, 8, cp), where pt- [P’+ (z+l)“]““, zT0, The bounda~ monitions forcp==@and 9-a are retained. But if w (1, 8, cp)=a(@, tp) *it”o,then by Eq. (2.2), u] ta;p==a(8, cp)p-‘“. Since u=O for t
Then u = Zu,,where t=+p *

every u, is obtained from w, as in Eq. (2.2). From Eq. (2.2) it follows that for

031#

0,

%= t - p,

~erefore, ZZ,is expanded in a series which may differ from Eqs. (2.4) only by the presence in addition to the fractional powers (t-p) k*‘h of integral powers also. The series for (sin 0) -@mu, converges absolutely close to the point ~v==O,8=0 by the theorem of the su~rposit~on of analytic functions. From the radial relations (see, for example, Eq. (1 ,181 in [2] ) it follows that the c~f~cients of integral powers of t-p are equal to zero and the series is of the form (2.4). Remark, A wave, homogeneous with respect to t, e, of degree h-l, is obtained from the wave (2.2) by the application of the operator (2.3) with ~J=X-*/~, ifO+*lzof the operators (2.3) and a,+%, if hc%. Thetafore, Eqs. (2.4) is valid for homogeneous sphe~c~ waves of degree h- 1> -2, equal to zero ahead of the front,

Diffraction by a polyhedral angle of a sphen’calwave

161

3. Diffraction by the polyhedral angle Let p, 0, cpbe spherical coordinates with centre 0 (p=O) at the vertex of the polyhedral angle, let the straight line 8=0 be directed along the edge OA, and let the faces of the dihedral angle with the edge OA lie in the planes q=O and ~=cc, the domain O<(r
It is required to find a solution of the wave equation 7z!,=An outside the polyhedral angle, satisfying on each face a given boundary condition IZ=0 or &J / o’rz=O (on different faces the boundary conditions may be different) and for O
(3.1)

where A,, is similar to A,, in (2.1) with the replacement of 0, cpby Cl,, Q,. In particular, if a spherical wave with centre A has been generated by the diffraction at the polyhedral angle with vertex A of an arbitrary wave possessing a radial expansion, this spherical wave is, by [l] , p. 583-584, represented by the sum Z(D). of simple waves of the form (3.1). Therefore, we consider only the case where the given wave ue with centre A is of the form (3.1). In [l] , p. 581-584, the diffraction of such a wave by a polyhedral angle with vertex 0 was considered in the case where the function u (oo)in (3.1) is analytic close to the ray AO, ero=ero*. In the case where the ray A0 is directed along an edge of the polyhedral angle, this condition is not in general satisfied, and Eq. (74) of [l] for the diffracted wave is inapplicable. However, in this case also the value of the solution u for t=t,+t, is expressed in terms of the value of u and ut for t=to by means of Green’s function G of Eq. (65) of [ 1] :

The integral is taken over the domain where / M,+-Idt,, G#O, n#O. We assume that t,td, since for t,
162

A. F. Filippov

where R and w are unit vectors, OM==RQ, #N-PO, and each function cDris expressed by the series obtained from (3.1) by replacing the arguments h, t, pu,wo, a ( oo) by 1, t,-P, R, Q, b,; the function s is the same as in (66) of [ 1 ] , and in the general case may be found by numerical methods (this requires Laplace’s equation to be solved in some spherical sector). Therefore, the required diffraction wave is of the form

(3.4) It is here impossible to evaluate these integrals by the method described in [l] . We evaluate their principal parts close to the wave front, expanding u. and CDlin series of the form (2.4). By the orthogonality of the functions s, (cp),

HereR, 8, cpand po, 80, cpare the spherical coordinates of the point M relative to the points 0 and A, andD is the domain poO. In the case of the boundary condition du / dn=O on both faces cp=O and cp=awe have si (cp) =1 and the coefficient of II, in the first of the equations (3.5) has to be doubled. We change in Eqs. (3.5) to the variable of integration ~0, R. If 1OA 1=d, we have R = 1Oil4 1 = l/(r2 + zz), p() =

1AM 1 = V(7.2 + (d -

6’ (PO, R)

z)“),

rdrdz

poR = d dr, dR,

sin Cl0= C,

d --2gv

3 (r2, 4 sine=

PO

LH’

By the formula for the height in the triangle OiK4 r’= [ (R+pJ

“-d”] [d”-

(K-p*)

“1 / 4d2.

to+ti-p-d=T, we obtain that the domain of integration D is Writing to-po=ro, tl-~-R==~I, defined by the inequalities T~>O, zi>O, T~-+T~(z and that

1

1 -=I’d

p _ -

6, -

To

’

-

I f-1

5 Pd + 4 (2d 4d3

.-z

d -

1 to +- z -

2i, + x) (iit, -

x),

~1 ’ 0 = 7-

zu -

Zl.

Diffrction by a polyhedral angle of a spherical wave

163

The functions 1/ pO, 1 / R and r2 for sufficiently small r in the domain D are expanded in power series in z, to, zi with coefficients dependent on the parameters d, t,, t,=const>O. Then the integrals (3.5) are represented by absolutely and u~formly convergent series of integrals of the form

u 1)

ZOOTlb (z -

D

t,, -

Z,)” dT,,dT, =

r (a +

1) r (b+ 1)r cc + 1)

I‘(a+Gc

-t3)

Tat*++9

“.

(3.6)

This method enables us to find any number of terms of the expansion of II, in a series of powers of ‘c. The first two terms are of the form

where P=Pm, P==H-E+~,+~, and ao=nmoor bO=bmOO, a,=~,~,, bi=b,,,o are coefficients of the series (3.5) (the coefficients amoi and bmei are expressed in terms of ee, ho, as, bl by means of (2.6); of course, b,, nodepends on 1, p and w), Summing the results obtained from Eqs. (3.4) and (3.5) and noting that the integral II” also . equals the right side of Eq. (3.7) ( smce II* is obtained from II by replacing h-l and 1 by X and Z-l), we obtain

(3 -8)

where T==-p-d; p, o are the spherical coordinates of the point N, as in Eqs. (3.3), p, is the same as in Eqs. (1.9), clJLis independent of r and the summation extends over all integers k&1, n>O, n+?G2. In the case of the boundary conditions du/dn=O on both faces rg=O and ?=a the first term on the right side of Eq. (3.8) has to be doubled because of what was said above after Eq. (3.5). The numbers u=uloa, p -‘b (0) = bioo are defined by relations similar to Eqs. (2.5) in terms of the functions~(~~)=a(0~, rp), b,(Q; o, p) = b. (13, q; o, p) in expansions of the form (3.1) of the given function ue and the function a0 of Eqs. (3.3). More precisely, let oo*, Q* and w be unit vectors directed from A to 0, from 0 to A and from 0 to N respectively. Then

be, p and s (52, o) are the same as in Eqs. (3.3), B0 and 8 are the angles formed by the vectors w. and 52 with the straight line OA. Using Eq. (3.7), it is easy to write down in Eq. (3.8) all the terms containing 2 in powers less than k+p,+X

164

A. F. Fihppov

Therefore, if the number x/a is an integer, the diffraction wave is expressed by one radial series, and if it is not an integer, by the sum of radial series whose principal terms contain T to powers differing from one another by x/a. If n/a is rational, there are a finite number of these series, if it is irrational the set of them is enumerable. For example, for diffraction by a cube or rectangular parallelopiped the dihedral angles are right angles, a=3n/2, n/a=2/3. The diffraction wave of the type discussed is expressed, by (34, as the sum of three radial series with principal terms containing T’+“~, T”“~, rki3 respectively (if the principal term of the incident wave contained zOL,as in Eq. (3.1), and the boundary conditions were of the form u=O) . In the similar stationary problem, that is, for u=e- ‘%I (5, y, z), where w is the frequency, the asymptotic expansion of the diffraction wave will contain terms with the factors w-“-~“‘~,~ m, n=l, 2.3,. . , . Translated by J. Berry REFERENCES 1.

FILIPPOV, A. F., Diffraction by dihedral and polyhedral angles. M&tern.Sb., 70,4,562-590,

1966.

2.

BOROVIKOV, V. A., Diffraction by Polygons and Polyhedra (Difraktsiya na mnogougol’nikakh mnogogrannikakh), “Nauka”, Moscow, 1966.

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