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A direct method for problems of diffraction by a locally inhomogeneous body
A DIRECT METHOD FOR PROBLEMS OF DIFFRACTION BY A LOCALLY INHOMOGENEOUS BODY* A. G. SVESI-INIKOV and A. S. IL’INSKII
(Received 3 January 1970)
IN this paper we propose a direct numerical diffraction
problems.
problems
of diffraction
This method
method of investigating
enables universal
algorithms
a certain class of
for the solution
by bodies of fairly general form and arbitrarily
of
variable compo
sition to be constructed. Problems of diffraction in a theoretical
investigation.
by a transparent
inhomogeneous
Even in problems
of diffraction
refractive index, results have been obtained
body cause great difficulties by bodies with constant
only for bodies of the simplest shape.
We will consider the following problem of the diffraction of electromagnetic waves. Within the domain D botnded by the surface S let there be a medium c$aracterized by the tensors E(M) and p,(M), which can be represented in the form E - d + e^‘, i= fl+$, where ^el and 6’ are Hermitian tensors, and e(IM) and /.A@!)are scalar functions with positive imaginary part. Outside the domain D the medium is described by the constant parameters ao, uo. The electromagnetic field f is excited by a system of local currentsf. We will assume that the characteristics of the medium and the specified currents determining the method of excitation are functions smooth enough for a classical solution of the problem formulated to exist [l] . In this formulation the mathematical problem reduces to the determination of a bounded solution of an inhomogeneous system of Maxwell’s equations 0)
rot E =
i&Y,
rot H = --i&E
+ j
in the whole of space, satisfying the radiation conditions, which consists of the require ment of the absence of waves arriving from infinity. For what follows it is convenient for us to formulate these conditions as partial radiation conditions in a form similar to that given in [2] . To formulate these conditions we require the concept of normal spherical waves. It is well known [3], that the system of homogeneous Maxwell’s equations in an *Zh. vjkhisl. Mat. mat. Fiz., 11,4,
960-968,
1971.
180
A direct method for problems of diffraction
unbounded solutions
medium with constant {E,, H,},
characteristics
satisfying the radiation
181
e. and cc0 has a denumerable
conditions.
system of
This system of waves is uniquely
defined by a system of scalar functions equation
in a spherical coordinate
q,(W), which is the solution of the scalar wave system with centre at an arbitrary point:
Here the Ykm (0, r+~) are spherical functions. Here and below we understand by the subscript n an arbitrarily ordered sequence of pairs of numbers {k, m}, and by bi(kor) we denote the spherical Bessel functions
g,,‘(kd-) =
[-$-]“*H:+s(k,r),
where ko2 =
02&ouo.
The fields of normal waves are connected iup0
E ,,=-rotH,,
Ef)=
-
(2b)
E”ln=-
E,
imp.0
$,,(M)
for waves of “magnetic
%I!‘= _ &rot
rot(R$,),
and for waves of “electric
1 -rot
H,=
ho2 and are expressed in terms of the function follows:
(24
by the relations
(roW%J
type” as
1,
type”:
rrot
rot(R$,),
H'f
=
-$mt(R&).
0
Here R = ril,
where ir is the unit vector of the spherical coordinate
system.
Any solution of the homogeneous system of equations (1) satisfying the radiation conditions is representable as a superposition of fields of normal waves of electric and magnetic types:
and since the function
~@,,(r,‘0, cp) satisfies the radiation
represents
form of the radiation
the analytic
conditions,
formula (3)
conditions.
The fields of normal waves {Ed, H,j} are orthogonal to any sphere with centre at the origin of coordinates. Indeed, orthogonality to one another of fields of electric and magnetic types follows from formulas (2). Orthogonality of fields of one type is a consequence of the definition of the function $,,(_M) :
182
A. G. Sveshnikov and A. S. Il’inskii
In formulas (4) for the spherical functions the normalization
has been used, where Ve& =
1
dU
au.
-iie + ---_ sin8 aq " de
Using the orthogonal&y conditions (4), we can put the radiation conditions (3) in the equivalent integral form (5)
$
[EH,*] i, da = pan,
n =
1,2,.
...
8R
From the relations (3) and (5) there follows an expression for the energy flow of the Complete Wave through the surface SR :
(6)
Cj[EH’] i, da = C Ian1‘Bna BR
n
Applying the lemma of Lorentz to the functions E, R* and E*, H in a domain bounded by a sphere of radius R and enveloping the domain D and the domain of definition of the currents f, we obtain the relation
Considering that
183
A direct method for problems of diffraction
the relation
Equation
(7) can be rewritten
in the form
(7) has the sense of the law of conservation
of energy.
In order to construct an approximate solution of problem (1) we will use a method developed for application to problems of the theory of irregular waveguides [2]. We construct an approximate solution satisfying the energy relation (8). This solution can be obtained by solving the boundary value problem for a system of ordinary differential equations. For what follows it is convenient for us to introduce a system of vectors defined on the unit sphere and complete on it in the sense that any vector tangential to the sphere can be represented as a linear combination of fields of electric and magnetic types. This system can be chosen in the form [3] e, for magnetic
i = [VwYn,Ll,
h,’
=
Ve*Y,
fields and
v tlJn,
en2=
hn’ = [V,,Yn, irl
for electric fields. The system of functions e,, h, is proportional normal spherical waves on the unit sphere. We define an approximate the form
solution
of the boundary
N
E,“'
=
11c,” (4
Here the radial components equations
6,
H,
(0, cp),
of the approximate
(rot E” - iopHN) T = 0, (9) To determine the coefficients cnN(r) and following integral relations: (rot EN 9 r=ron>t
iohH”)
to the tangential
value problem
are calculated
(rot HN + i&E
,h,,* dQ = 0,
of
(1) and (5) in
=A bflN(r)h,(O,(p).
solution
b,“‘(r)
components
-
from Maxwell’s
j) r =
0.
we require the satisfaction
of the
A. G. Sveshnikov and A. S. Il’inskii
184
O
(10)
n=i,...,N;
(rot H”+
tt
i&EN)
len* dQ =
9
(je,*)t
dQ,
r=cu,,st
r=const
n=
1,... N, and for r = R we require the satisfaction of conditions (5), after writing them in terms of the functions (en, hn). We notice that in a homogeneous medium for r > R the fields EN, iP constructed in this way are the solutions of a homogeneous system of Maxwell’s equations satisfying the radiation conditions. Since the functions (e,, h,) are proportional to the tangential components of the normal waves (2), the following relations are satisfied:
O
EtN(r) =
(11)
~u.~[E.~,J.
GN(r)
= 2
c,N(r) e,,
r > R
n=i
7l=l
are the coefficients of the expansion of the field EN for r > R terms of En. If account is taken of the orthogonality of the fields En and ek, the relations (11) imply
where the
anN
a, N =
u,,c,,~
in
(R),
where a,
=
s
[e,,H,,*]
i,
do.
EI3 The relations can be written
(10) form a system of first-order
ordinary
differential
equations
in the form
where the coefficient matrices Ank, Bnk, A,+ Bd are expressed in terms characteristics of the medium and basis functions by relations of the form
of the
which
A direct
The right sides of the equations
method
for problems
18fi
of’ diffraction
are expressed in terms of the currents by means of the
similar relations: fn = Thereby
-
1
the coefficients
of the equations
and the right sides are given functions
independent variable r. Relation (5) gives the boundary convenient to write it in the following form:
condition
[EN-~,*] i, do = finanN = pnancnN(R),
(12)
of the
for r=R. It is
n=
i,...,N.
8 R Using the condition of the boundedness of the solutions at zero, it is easy to construct a numerical algorithm for the solution of the boundary value problem (10) and (12) posed. We show that the functions
(E”,
HN),
defined as the solution
of problem
(10)
and (12) always exist, the boundary value problem (10) and (12) has only a unique solution, and as N -+ 00 the approximate solution tends to the solution of the original electromagnetic boundary value problem. Following the method of [2], we show that EN, HN satisfy the energy relation (8), which enables the existence and uniqueness of the functions
defined by
EN, HN to be proved.
Multiplying the first of the relations (10) by bnN’, and the second by c,,~*, summing with respect to n and integrating with respect to r, we obtain (rot E” -
5
io;H”) lHt”’ do = 0,
RR
J
(rot H” + io;E” - j) t E,N’ do = 0.
IR
From relations
(9) it follows that
J
(rot E” -
h&P),
H,“’ do = 0,
RR
s
(rot P’ + ico;E-’ -
j) ,ETN’ do = 0.
RR
Applying
(13)
the formula of Lorentz to EN, N”*and
E”*, HN in the domain KR , we obtain
186
A. G. Sveshnikov and A. S. Il’inskii
From the relation approximate
(13), taking into account
solution
the fact that on the sphere SR the
satisfies the conditions
(12)
there follows the relation
On the basis of the general properties of linear boundary value problems for ordinary differential equations the relations (14) imply that a homogeneous boundary value problem has only a unique solution, and consequentIy, an inhomogeneous boundary value problem is always solvable. We consider the functions
JYN and
aN
&YN=
E - EN and sN
=
H - HN. The functions
satisfy the relations
(rot $ *I-
zfN- Iowan) h; (
do =
0,
I
(rot
EN-
nEgIv
iocHN)t h*,do, n>N,
r
?z
0, $ (rot ZN + i~&$‘~)~ e,* do =
ST The functions
5)
BN and ZN are expanded
(rotH
N-
imEN -j)tezdo,
n>N.
in the series
We also write m
PN=
c
c,e,,
KfS
where c,,, b, are the coefficients For
r >
R the functions
QN =
2 b&n, iV+t
of the expansion gPN and ZN
of the exact solution E, H. satisfy Maxwell’s equations
and the
187
A direct method for problems of diffraction
radiation conditions (5). Again using the lemma of Lorentz, we obtain for the functions .BN and G%SNthe relation Re$
(15)
[8N%“*]i,d~+
[ (k,Im~~~N~2+k~Imp(~N)z)du=
LI
%I
8R
= Re_I[it&*] ~TJ+ Im m 5 [ (I;,H”)t’QN- (;-EN) ,P,*]
du.
=R
=R
Here we have used the fact that Sp” and %‘” satisfy Maxwell’s radial equations. radiation conditions (S), we finally rewrite relation (15) in the form
=
j ( jtPN*) da + Im w s [ (I;*HN) ifQN -
,P,‘]da.
*a
=l3
For the convergence of the approximate solution sufficient to show that the right side of equation We estimate the right side as N +
ofN,
The first factor is independent
(47)
(kN)
ENH” to the exact solution it is (16) tends to zero as N+ 00,
00 :
and to estimate the second we consider
P,%u-Jd’$pt-&+o=J~ JI s Ji 1 ll=t
R
R
We notice that the eigenvectors
~c,pzr. N+l
e, and h,, satisfy the following relations: for electric waves,
-
divh,
(rot&), h n= We partition
= (rot en)? =
= dive,,
=
h “y n
h
for magnetic waves;
-&La
for electtic waves,
O
for magnetic waves.
k(k + 1).
the sum (17) into electric
Cjcn(r)tz N+i
Using the
=
2 i=K(N)
and magnetic Icy’(r)
I”+
waves: 2
!+=aqq
IcF’
12.
188
A. G. Sveshnikov and A. S. Il’inskii
We notice that 9
(48)
;(diveqEJ Y,da = -c,‘(r),
” 1 d diver =--gine+----_-; sin0 de (rot&)
,Y,
1
8
sin 0 arp
da = c,2(r),
sr
(19)
(rot,,.=&+4& From the relations
(18), (19) we obtain
the estimates
The energy relations (7) imply the integrability in KR of the fields E and H, which by Maxwell’s equations and the assumptions of the integrability in KR of the given current and div j and for sufficient smoothness of the functions E and l.~ enables us to conclude that the first factors in formulas (20) are bounded. Because of the assumed normalization
of the functions
from which it also follows that the right side of equation The following terms are estimated similarly since J I (;tHN)t]2dU, =l, are uniformly
bounded
Y,, we obtain
(17) tends to zero as Iv -+ 00.
5 I @Wt]2du =R
with respect to N.
In conclusion we mengon tha_t in this paper an algorithm for solving the diffraction problem for the case where e and II. are continuous functions of the coordinates has been considered in detail. It can easily be generalized to the case of a discontinuous composition. It is then necessary to introduce into relation (10) terms allowing for the discontinuities in the medium. The justification is similar in this case (see [5]). Translated by J. Berry.
A direct method for problems of diffraction
189
REFERENCES 1.
MIRANDA, C. Partial Differential Equations of Elliptic Type (Uravncniya s chastnymi proizvodnymi ellipticheskogo tipa), Izd-vo in. ht., Moscow, 1957.
2.
SVESHNIKOV, A. G. A substantiation of methods for computing the propagations of clcctromagnetic oscillations in irregular waveguides, Z/r. vjkhisl. Mat. mat. Fiz. 3, 2, 314G326, 1963.
3.
STRATTON, 3. A. Electromagnetic
4.
MORSE, P. M. and FESHBACH, H. Methods of Theoretical Physics. (Metody teoreticheskoi Izd-vo in. ht., Moscow, 1960.
5.
SV~SHNIKOV, A. G. A justification of methods used to study the propagation of electromagnetic oscillations in waveguides with anisotropic filling,Zh. v&hi& Mat. mat. Fiz. 3,5, 953-955, 1963.
Theory (Teoriya elektromagnetizma),
OGIZ, Moscow, 1948. fiziki),
(Received 3 January 1970)
IN this paper we propose a direct numerical diffraction
problems.
problems
of diffraction
This method
method of investigating
enables universal
algorithms
a certain class of
for the solution
by bodies of fairly general form and arbitrarily
of
variable compo
sition to be constructed. Problems of diffraction in a theoretical
investigation.
by a transparent
inhomogeneous
Even in problems
of diffraction
refractive index, results have been obtained
body cause great difficulties by bodies with constant
only for bodies of the simplest shape.
We will consider the following problem of the diffraction of electromagnetic waves. Within the domain D botnded by the surface S let there be a medium c$aracterized by the tensors E(M) and p,(M), which can be represented in the form E - d + e^‘, i= fl+$, where ^el and 6’ are Hermitian tensors, and e(IM) and /.A@!)are scalar functions with positive imaginary part. Outside the domain D the medium is described by the constant parameters ao, uo. The electromagnetic field f is excited by a system of local currentsf. We will assume that the characteristics of the medium and the specified currents determining the method of excitation are functions smooth enough for a classical solution of the problem formulated to exist [l] . In this formulation the mathematical problem reduces to the determination of a bounded solution of an inhomogeneous system of Maxwell’s equations 0)
rot E =
i&Y,
rot H = --i&E
+ j
in the whole of space, satisfying the radiation conditions, which consists of the require ment of the absence of waves arriving from infinity. For what follows it is convenient for us to formulate these conditions as partial radiation conditions in a form similar to that given in [2] . To formulate these conditions we require the concept of normal spherical waves. It is well known [3], that the system of homogeneous Maxwell’s equations in an *Zh. vjkhisl. Mat. mat. Fiz., 11,4,
960-968,
1971.
180
A direct method for problems of diffraction
unbounded solutions
medium with constant {E,, H,},
characteristics
satisfying the radiation
181
e. and cc0 has a denumerable
conditions.
system of
This system of waves is uniquely
defined by a system of scalar functions equation
in a spherical coordinate
q,(W), which is the solution of the scalar wave system with centre at an arbitrary point:
Here the Ykm (0, r+~) are spherical functions. Here and below we understand by the subscript n an arbitrarily ordered sequence of pairs of numbers {k, m}, and by bi(kor) we denote the spherical Bessel functions
g,,‘(kd-) =
[-$-]“*H:+s(k,r),
where ko2 =
02&ouo.
The fields of normal waves are connected iup0
E ,,=-rotH,,
Ef)=
-
(2b)
E”ln=-
E,
imp.0
$,,(M)
for waves of “magnetic
%I!‘= _ &rot
rot(R$,),
and for waves of “electric
1 -rot
H,=
ho2 and are expressed in terms of the function follows:
(24
by the relations
(roW%J
type” as
1,
type”:
rrot
rot(R$,),
H'f
=
-$mt(R&).
0
Here R = ril,
where ir is the unit vector of the spherical coordinate
system.
Any solution of the homogeneous system of equations (1) satisfying the radiation conditions is representable as a superposition of fields of normal waves of electric and magnetic types:
and since the function
~@,,(r,‘0, cp) satisfies the radiation
represents
form of the radiation
the analytic
conditions,
formula (3)
conditions.
The fields of normal waves {Ed, H,j} are orthogonal to any sphere with centre at the origin of coordinates. Indeed, orthogonality to one another of fields of electric and magnetic types follows from formulas (2). Orthogonality of fields of one type is a consequence of the definition of the function $,,(_M) :
182
A. G. Sveshnikov and A. S. Il’inskii
In formulas (4) for the spherical functions the normalization
has been used, where Ve& =
1
dU
au.
-iie + ---_ sin8 aq " de
Using the orthogonal&y conditions (4), we can put the radiation conditions (3) in the equivalent integral form (5)
$
[EH,*] i, da = pan,
n =
1,2,.
...
8R
From the relations (3) and (5) there follows an expression for the energy flow of the Complete Wave through the surface SR :
(6)
Cj[EH’] i, da = C Ian1‘Bna BR
n
Applying the lemma of Lorentz to the functions E, R* and E*, H in a domain bounded by a sphere of radius R and enveloping the domain D and the domain of definition of the currents f, we obtain the relation
Considering that
183
A direct method for problems of diffraction
the relation
Equation
(7) can be rewritten
in the form
(7) has the sense of the law of conservation
of energy.
In order to construct an approximate solution of problem (1) we will use a method developed for application to problems of the theory of irregular waveguides [2]. We construct an approximate solution satisfying the energy relation (8). This solution can be obtained by solving the boundary value problem for a system of ordinary differential equations. For what follows it is convenient for us to introduce a system of vectors defined on the unit sphere and complete on it in the sense that any vector tangential to the sphere can be represented as a linear combination of fields of electric and magnetic types. This system can be chosen in the form [3] e, for magnetic
i = [VwYn,Ll,
h,’
=
Ve*Y,
fields and
v tlJn,
en2=
hn’ = [V,,Yn, irl
for electric fields. The system of functions e,, h, is proportional normal spherical waves on the unit sphere. We define an approximate the form
solution
of the boundary
N
E,“'
=
11c,” (4
Here the radial components equations
6,
H,
(0, cp),
of the approximate
(rot E” - iopHN) T = 0, (9) To determine the coefficients cnN(r) and following integral relations: (rot EN 9 r=ron>t
iohH”)
to the tangential
value problem
are calculated
(rot HN + i&E
,h,,* dQ = 0,
of
(1) and (5) in
=A bflN(r)h,(O,(p).
solution
b,“‘(r)
components
-
from Maxwell’s
j) r =
0.
we require the satisfaction
of the
A. G. Sveshnikov and A. S. Il’inskii
184
O
(10)
n=i,...,N;
(rot H”+
tt
i&EN)
len* dQ =
9
(je,*)t
dQ,
r=cu,,st
r=const
n=
1,... N, and for r = R we require the satisfaction of conditions (5), after writing them in terms of the functions (en, hn). We notice that in a homogeneous medium for r > R the fields EN, iP constructed in this way are the solutions of a homogeneous system of Maxwell’s equations satisfying the radiation conditions. Since the functions (e,, h,) are proportional to the tangential components of the normal waves (2), the following relations are satisfied:
O
EtN(r) =
(11)
~u.~[E.~,J.
GN(r)
= 2
c,N(r) e,,
r > R
n=i
7l=l
are the coefficients of the expansion of the field EN for r > R terms of En. If account is taken of the orthogonality of the fields En and ek, the relations (11) imply
where the
anN
a, N =
u,,c,,~
in
(R),
where a,
=
s
[e,,H,,*]
i,
do.
EI3 The relations can be written
(10) form a system of first-order
ordinary
differential
equations
in the form
where the coefficient matrices Ank, Bnk, A,+ Bd are expressed in terms characteristics of the medium and basis functions by relations of the form
of the
which
A direct
The right sides of the equations
method
for problems
18fi
of’ diffraction
are expressed in terms of the currents by means of the
similar relations: fn = Thereby
-
1
the coefficients
of the equations
and the right sides are given functions
independent variable r. Relation (5) gives the boundary convenient to write it in the following form:
condition
[EN-~,*] i, do = finanN = pnancnN(R),
(12)
of the
for r=R. It is
n=
i,...,N.
8 R Using the condition of the boundedness of the solutions at zero, it is easy to construct a numerical algorithm for the solution of the boundary value problem (10) and (12) posed. We show that the functions
(E”,
HN),
defined as the solution
of problem
(10)
and (12) always exist, the boundary value problem (10) and (12) has only a unique solution, and as N -+ 00 the approximate solution tends to the solution of the original electromagnetic boundary value problem. Following the method of [2], we show that EN, HN satisfy the energy relation (8), which enables the existence and uniqueness of the functions
defined by
EN, HN to be proved.
Multiplying the first of the relations (10) by bnN’, and the second by c,,~*, summing with respect to n and integrating with respect to r, we obtain (rot E” -
5
io;H”) lHt”’ do = 0,
RR
J
(rot H” + io;E” - j) t E,N’ do = 0.
IR
From relations
(9) it follows that
J
(rot E” -
h&P),
H,“’ do = 0,
RR
s
(rot P’ + ico;E-’ -
j) ,ETN’ do = 0.
RR
Applying
(13)
the formula of Lorentz to EN, N”*and
E”*, HN in the domain KR , we obtain
186
A. G. Sveshnikov and A. S. Il’inskii
From the relation approximate
(13), taking into account
solution
the fact that on the sphere SR the
satisfies the conditions
(12)
there follows the relation
On the basis of the general properties of linear boundary value problems for ordinary differential equations the relations (14) imply that a homogeneous boundary value problem has only a unique solution, and consequentIy, an inhomogeneous boundary value problem is always solvable. We consider the functions
JYN and
aN
&YN=
E - EN and sN
=
H - HN. The functions
satisfy the relations
(rot $ *I-
zfN- Iowan) h; (
do =
0,
I
(rot
EN-
nEgIv
iocHN)t h*,do, n>N,
r
?z
0, $ (rot ZN + i~&$‘~)~ e,* do =
ST The functions
5)
BN and ZN are expanded
(rotH
N-
imEN -j)tezdo,
n>N.
in the series
We also write m
PN=
c
c,e,,
KfS
where c,,, b, are the coefficients For
r >
R the functions
QN =
2 b&n, iV+t
of the expansion gPN and ZN
of the exact solution E, H. satisfy Maxwell’s equations
and the
187
A direct method for problems of diffraction
radiation conditions (5). Again using the lemma of Lorentz, we obtain for the functions .BN and G%SNthe relation Re$
(15)
[8N%“*]i,d~+
[ (k,Im~~~N~2+k~Imp(~N)z)du=
LI
%I
8R
= Re_I[it&*] ~TJ+ Im m 5 [ (I;,H”)t’QN- (;-EN) ,P,*]
du.
=R
=R
Here we have used the fact that Sp” and %‘” satisfy Maxwell’s radial equations. radiation conditions (S), we finally rewrite relation (15) in the form
=
j ( jtPN*) da + Im w s [ (I;*HN) ifQN -
,P,‘]da.
*a
=l3
For the convergence of the approximate solution sufficient to show that the right side of equation We estimate the right side as N +
ofN,
The first factor is independent
(47)
(kN)
ENH” to the exact solution it is (16) tends to zero as N+ 00,
00 :
and to estimate the second we consider
P,%u-Jd’$pt-&+o=J~ JI s Ji 1 ll=t
R
R
We notice that the eigenvectors
~c,pzr. N+l
e, and h,, satisfy the following relations: for electric waves,
-
divh,
(rot&), h n= We partition
= (rot en)? =
= dive,,
=
h “y n
h
for magnetic waves;
-&La
for electtic waves,
O
for magnetic waves.
k(k + 1).
the sum (17) into electric
Cjcn(r)tz N+i
Using the
=
2 i=K(N)
and magnetic Icy’(r)
I”+
waves: 2
!+=aqq
IcF’
12.
188
A. G. Sveshnikov and A. S. Il’inskii
We notice that 9
(48)
;(diveqEJ Y,da = -c,‘(r),
” 1 d diver =--gine+----_-; sin0 de (rot&)
,Y,
1
8
sin 0 arp
da = c,2(r),
sr
(19)
(rot,,.=&+4& From the relations
(18), (19) we obtain
the estimates
The energy relations (7) imply the integrability in KR of the fields E and H, which by Maxwell’s equations and the assumptions of the integrability in KR of the given current and div j and for sufficient smoothness of the functions E and l.~ enables us to conclude that the first factors in formulas (20) are bounded. Because of the assumed normalization
of the functions
from which it also follows that the right side of equation The following terms are estimated similarly since J I (;tHN)t]2dU, =l, are uniformly
bounded
Y,, we obtain
(17) tends to zero as Iv -+ 00.
5 I @Wt]2du =R
with respect to N.
In conclusion we mengon tha_t in this paper an algorithm for solving the diffraction problem for the case where e and II. are continuous functions of the coordinates has been considered in detail. It can easily be generalized to the case of a discontinuous composition. It is then necessary to introduce into relation (10) terms allowing for the discontinuities in the medium. The justification is similar in this case (see [5]). Translated by J. Berry.
A direct method for problems of diffraction
189
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MORSE, P. M. and FESHBACH, H. Methods of Theoretical Physics. (Metody teoreticheskoi Izd-vo in. ht., Moscow, 1960.
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