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A comparative investigation of the surface radiation condition in electromagnetics
Wave Motion 16 (1992) ! 21 Elsevier
A comparative investigation of the surface radiation condition in electromagnetics M. Teymur* Department of Mathematics and Computer Science. The Unioersity, Dundee. DD! 4HN. Scotland. UK Received ! 1 October 1989, Revised 22 October 1991
This paper assesses a vector surface radiation condition (SRC) in electromagnetics by reference to a canonical problem, namely the scattering of waves by a perfectly conducting sphere• The method furnishes two coupled differential equations for the tangential components of the surface field and so the calculation of the scattered field is reduced to a quadrature. The differential equations are first solved exactly and the distant scattered field is determined analytically. Later, to examine the effect of introducing a high frequency asymptotic expansion the solutions are constructed approximately• Comparisons are made between the exact answer of the problem, the SRC solutions, and the physic~d optics (PO) appreximafion. It is obs~ar',,ed that, although the SRC method is essentially a high frequency method, it produces surprisingly good results for lower frequencies. In the high frequency range the method is superior to PO approximation and also the combination of the SRC and high frequency perturbation expansion yields better results than PO.
I. |ntroduction
The SRC method in electromagnetic scattering was first introduced and applied t\'~r two dimensional . the field aad problems [ I 4] • The me~hod is based on the application of a radiation condition, ~.~. "~nnc,.,~ne "" its normal derivative, directly on the surface of the scattering object to determine approximately the surface field or its derivative in terms of the given field. The calculation of the scattering field is then reduced to quadratures. The derivation of the radiation condition which is used in these investigations is based on the expansion of the scattering field satisfying the two dimensional Helmholtz equation and the Sommerfeld radiation condition [5]. The generalization of the method for the application to an arbitrary convex cylinder has been made by applying the circular cylindrical form of the SRC operator locally at each point on the scattering surface by considering that an osculating circle can approximate the surface locally. This idea later has been applied to obtain a three dimensional SRC in acoustics and the results of the method for a soft sphere are compared with the exact answer of the problem [6]. It has been observed that fi'om moderate to high freauencies the method is effective in obtaining the magnitude of t.be normal derivative, but it is less effective for its phase in the shadow part of the surface. Nevertheless, the magnitude of the normal derivative is relatively small in this region, therefore the defect in the phase may be quite tolerable in practical applications. Later, in [7], by using the methodology presented in [1 ], a slightly different operator than the one given in [6] has been introduced to investigate the scattering of an acoustic plane wave by a * Present address" Marmara Scientific and Industrial Research Center, Department of Mathematics, P.O. Box 21, 41470 Gebze, Kocaeli, Turkey. 0923-5965/92/$05.00 ~" 1992 Elsevier Science Publishers B.V. All rights reserved
2
M. Teymur / Surface radiation conditions
sp~':: :'.-al target which possesses a surface impedance. The surface pressure is constructed by the WKBJ m.~'~.d and then utilizing this result an approximatqon for the backscattering cross section is deduced. F,-,m3 cemp~ria~ns with the eaa~,t backscattering closs sections for various spemal cases it is concluded that the method is effective from the middle frequency range. Later, a general method of obtaining surface radiation conditions which is based on an asymptotic expansion similar to the Luneburg-Kline expansion and the assumption that the surface of the scattering object is a phase front, has been introduced in [8]. In this work a vector SRC for electromagnetic waves which connects the tangential components of the field on a phase front is also derived. The relative merits of the tv, o forms of the SRC introduced by different approaches in [7] and [S] have been examined by reference to acoustic scattering by a hard sphere in [9]. The SRC equations have been solved exactly and the far field b"~ been obtained analytica:lly. From a comparison between the exact answer of the problem and solutions constructed by the SRC techniques, it has been observed that both approaches almost equivalently provide the surface and the scattered field very accurately in the lower frequency range, i.e. for ka<~5, where k is the wave number and a is the radius of the scattering sphere. Note that a high frequency analysis misses this point [7]. When frequency increases beyond ka = 5 the results become less accurate on the shadow side of the sphere. This affects the far field results and although the phase remains remarkably accurate, the relative error in the magnitude of the far field grows in the forward scattering region; nevertheless, the results are still qualitatively satisfactory. In [9], the effect of introducing a high frequency perturbation expansion and an iterative technique have also been investigated. It has been observed that, as frequency increases, the first order solutions obtained by these methods approach the exact series solutions of the SRC equations thereby indicating that both technique might be efficient supplementary techniques in applying the SRC concept for arbitrary smooth convex obstacles. It has been already pointed out in [8] that the vector SRC introduced there reduces to the formula given in [l] when specialized to two dimension. The aim of this work is further examination of this general form of the SRC via a three dimensional canonical problem in electromagnetics, namely the scatteri,~g of a plane wave by a Ferfectly conducting sphere. The method offers two coupled differential equatmns for the iangential components of the surface field. These equations are first solved exactly by series expansion and ~he results are compared with the exact answer to the problem. As in the acoustical case, [9]. there is good agocement at tow flcquencies. When the frequency increases the results become less accurate on the shadow side of the sphere; that is the surface field predicted by the SRC fails to account for the creeping wave physics adequately. This type of deficiency has been also observed in the previous applications of the SRC concept in two dimensional electromagnetics and three dimensional acoustic scattering problems [I, 6, 7, 9]. This should be expected, since the derivations of the conditions is either based on the assumption of outgoing waves which are subdominant in the shadow, near the surface, from around moderate frequencies [1, 7]- or is obtained by an asymptotic expansion similar to the geometrical optics (Luneburg-Kline) expansion and the assumption that the surface of the scattering object is a phase front [8] ; but the scattering surt:ace is a caustic of the creeping wave rays and the asymptotic expansion of the geometrical optics is not umik~rmly valid around a caustic. It should be also noted here that in the soft sphere and circular cylinder ,'~..e. ..... :"- of the "--" ..... wave) problems, the ampiitudes of the surface fields provided by . . . . . .::~. . . TM ,.,.-,1.,.-: ~,,-,,,,,,.~.-~,~,, ~c~ucnt ~hc SRC become remarkably close to the exact answers in the shadow region with increasing frequency tr¢ , ~. I 1. This is due to the fact tha~ creeping waves arc less pervasive |br soft objects. On the other hand, the agreement at low frequencies is poor by comparison to the hard sphere and circular cylinder (i.e. for TE polarization of the incident wave) and perfectly conducting sphere problems. These results indicate the necessity of further research for improvements of the SRC methods for better predictions of the surface ticMs.
M. Teymur / Surface radiation conditions
3
To solve the surface differential equations is an important step of the method and exact solutions of these equations can be constructed by series expansion for only a limited number of geometries where the Maxwell's equations are separable. Therefore, to apply the method for an arbitrary convex object a supplementary technique ;s needed to determine the surface field. Here, for this purpose, the effect of introducing a high frequency asymptotic perturbation expansion is investigated. The first order solutions are presented and it is observed that as frequency increases these solutions approach to the exact series solutions of the SRC equations. Thus, it ap?ears that the perturbation analysis is a reliable supplementary method in order to apply the SRC concept for arbitrary smooth convex obstacles. Later, the solutions for the tangential components of the surface field are employed along with an integral representation of the scattering field to deduce the far field. For the series solution the integrals are performed analytically whereas for the perturbation solution they are calculated numerically. Comparisons are made between the exact far field omp'ffudes, the series and perturbation results constructed via the SRC method and the PO approximation. It is observed that the SRC method produces surprisingly good results for lower frequencies as in the acoustical sphere having a hard boundary [9]. When frequency increases, in the forward scattering region, the relative error in the far field amplitude grows, but the results are still qualitatively quite satisfactory and they are better than obtained by the PO approximation. It is also observed that as frequency increases the far field amplitude obtained by the perturbation technique approaches to the series solution result. Moreover, the combination of the SRC method and perturbation expansion yields better results than the PO approximation. The paper is organised as follows. The formulation of the problem is described in Section 2. In Section 3, the exact and perturbation solutions of the SRC equations are presented and the results are compared with the exact answer. The scattering far field in calculated in Section 4 and comparisons are made between the SRC results, exact answer and PO approximation. Section 5 contains some concluding remarks. The PO approximation for the problem is presented in an appendix.
2. Formulation Consider the scattering of time harmonic electromagnetic waves by a perfectly conducting obstacle bounded by a surface Z of finite area. The exterior of Z, which will be denoted by Z ~, is free space with permittivity e0 and permeability po. In the absence of impressed currents and charges, the scattered electric and magnetic intensities E ~ and/./5 are governed by the equations, [10], curl
E ~+ ikh S =
O,
curl
h ~- ik E " = 0
(2.1)
where k = c o ( p o S o ) ~''2 is the wave number, co is the angular frequency, h'=(yo/so) ~ 2H' and the time dependence is taken as m, ;~t). Since the scattering o',;~;.mle is considered to be a perfectly conducting object, on its surface S. E" is subject to the condition n x E" = - - ( n
x E i)
(2.2)
and at infinity it must have the form of a radiating wave, where E ~is the incident electric field falls on the body from 27e and n is the unit outward normal to Z. Then the scattered electromagnetic field E", h" at a
4
M. Teymur / Surface radiation conditions
point x in 27* can be represented as, [10], E"(x) = - ( i / k ) ( g r a d div + k 2) f , n x h ( y ) V ( x , y) dZ>,,
(2.3)
Z
h~(x) = curl f,, n x h ( y ) g ( x , y ) d2;>,
(2.4)
6,.
where h is the total magnetic field and V , ( x , y ) = e x p ( - i k l x - y l ) / 4 n l x - y l . From (2.3) and (2.4) one can calculate the scattering field satisfying the boundary and radiation conditions provided that n x h is known on X. This requires the solution of an integral equation for n x h which can be derived from (2.4). On the other hand, there are also certain ways to avoid solving an integral equation. One way is to employ the physical optics approximation. This is based on the assumption that at high frequencies, each portion of the geometrically illuminated side of Z, say 2; ~, reflects as if it were locally plane, whereas over the shadowed portion of r the surface field is zero. Thus for a perfectly conducting body nxh=2nxh
i
on Z i,
nxh=0
on ~ _ ~ i
(2.5)
and then the determination of the scattered field is reduced to quadratures [11]. The approximation of physical optics is known to be quite effective at high frequencies, although a refined transition between the illuminated and shadow side of the scattering obstacle is needed for a complete coverage of the radiation pattern. On the other hand, with decreasing frequency the distinction between the illuminated and shadowed sides begins to fade away and the question of where to apply the equations (2.5) arises. The surface radiation concepts attempts to introduce a modification in this direction. For this purpose, in [8], the following approximate formula is derived to link the tangential components fo the electromagnetic field on a wavefront
< i2) n > ' ( h - n x E ~
(1-
'
. . .D. ' L [ n x C u r l ( n x h ) + { C u r l ( n x E ) } , ]
1
2k 2 [Grad Div(n x h) + {Grad Div(n x E) } x #]
(2.6)
where Curl, Div, and Grad are surface curl, divergence and gradient operators, respectively; H is the mean curvature of the wavefront and the suffix t indicates tangential components. When deriving (2.6) for any kind of wave i.e. reflected, scattered ect. the wavefronts are locally treated as parallel surfaces and therefore they are expressed as X=Xo(O-~, cr2) +s= where cr mand cr2 denote the coordinates in these surfaces and s is the variable measuring the distance along a normal = to the wavefront. The surface gradient is defined as G r a d u = g J a 0u ~x Oo.j O.r~
(2.7)
~vherc g'a denmes contravariant metric tensor. The surface divergence of the tangential vector 7"= T j C~x/ ~ ' and the surface cur~ of the x.... " '~t c~"' + ,~ aaa are also defined a:~ follows e~:~c~r .4 = A' cx. Div T -
g
,
S~r /
g 2TJ)
Cuff A := Grad A 3 x n -.vPgp;_Aj + n Div(,4 × n)
(2.8) (2.9)
M. Teymur / Surface radiation conditions
5
where
-
b,
Since the relation (2.6) is based on behaviour local to a wave front, it should be valid anywhere where there is reasonable wave front. The main assumption leading to the SRC is that a wave scattered by an obstacle will obey this rule on the surface of the scatterer. This is the extension of the assumption suggested by Kriegsmann et al. in two dimensions [ 1]. Hence for a perfectly conducting object since the condition (2.2) is valid, this assumption yields the following vector differential equation to evaluate n x h ~ on the scattering object 1 - i~k) n x ( h ~ + n x E i ) = 9 ~ l 1k [ n x C u r l ( n x h ~ ) - { C u r l ( n x E i ) } , ]
1 [Grad Div(n × h " ) - {Grad Div(n x E~)} x hi. 2k 2
(2.1o)
Thus, since (2.10) is applied over the whole Z and not just some part of it, the dilemma of physical optics is also circumvented. On the other hand, since n x h ~ is known on 2~, (2.10) can be reduced to a differential equation for the total field n x h and then the determination of the scattered field is eventually reduced to quadratures as in the case of physical optics approximation. It is evident that, to construct the solution of (2.10) for a given scattering object and incident field, is an important step in the method. Exact solutions may be obtained by the series expansion for only limited number of geometries where the Maxwell's equations are separable. For an object of more general shape the surface field can be found by an approximate solution method such as a high frequency perturbation expansion. Here, to assess the vector SRC (2.10) we will consider a canonical problem, namely the scattering of a plane wave incident in the direction of the negative z-axis, i.e. E i = i eik',
hi= _] eik-',
(2.11 )
by a perfectly conducting spherical object aad present exact and perturbation solutions of the SRC equation (2.10). Then by employing these solutions in the integral representation, the scattered field will be calculated. For parallel spherical surfaces, the surface operators defined in (2.7), (2.8) and (2.9) take the following respective forms 1 0 1 0t~ Div T - - (to sin 0) + - , r sin 0 00 r sin 0 0~b 1
1
F
/r
1 0u 1 0u Grad u = - - - eo + - e~ r 00 r sin 0 ~4~
(2.12)
Cur! ~ =w-.~A ~ ~ ~ " + - - A o e ~ - - ~ " 0 + " r ~ ; " ( ~ y "~ ~
lk.,~.~au.~..s
.~l,,..j
--
~e
.
dA
~
,
wD
.~-"~
v
,a
-
"
,,,my
where T and A are defined as
T=toeo+t~e~,
A=Aoeo+A~,ee~+Arn
(2.13)
and 0, ~b and r the usual spherical coordinates with 0 ~<0 ~
6
M. Teymur / Surface radiation conditions
Using (2.12) in the general relation (2.10), on a perfectly conducting scatterer of radius a, for the incident plane wave (2.11), the following coupled differential equations to calculate the tangent;al components of the total field h are obtained;
d-O ~
(sin 0 h~ )-/~o
(2.14)
+ a(e)h~=f(O)
1 {d~(sinOh~,)-ho}+a(e)ho=f(O)
(2.15)
sin" 0 where
ho = - s i n ~b/~o,
ho= -cos q~/~,
e=ka,
a(e) = 2 e ( e - i),
1 dE d2E f(O)~- + a(e)(1 +cos 0)E, sin 0 dO d 0 2
(2.16) E=exp(i~ cos 0).
and it is required that/~o and/~ must be bounded functions of 0 for 0 ~<0 ~
3. The surface field In this section, first the exact series solutions of the SRC equations d:rived for perfectly conducting sphere problem is presented. Then as a supplementary technique the effect of introducing a high frequency perturb Jon expansion is examined.
3.1. Series solutions ,)f the SR(" equations Let us introduce a function D" by the relation W= 1-~sin 0 td,dO(sin0/~ ) -/~o} •
(3.i)
It then follows from (2.14) and (2.15) that W satisfies the equation sin0 d-d- sin0 -W+a(,:)sin20 W=sin0 d0, -d-b-
{sinOf(O)}-f(O)
(3.2)
and h,, h~, are given as
~. .
.
.
1 If(O )
.
.
.
u~)t
.
.
.
.
.
I IV! .
.
.
sin0
.
,
~,
1 t f( q)-- dW~
. . . . . .
~xts~'
(3.3)
d0j"
Note that the solutions of the equation
(
:t
sin0 dO d sin0d-d-O-J . A. = -n(n + !) sin 2 0 A,, .
.
.
.
.
.
.
.
.
•,
n = 1 2, 3, ,~
°
°
.
(3.4)
M. Teymur / Surface radiation conditions
7
which are fimte and have also finite derivatives at 0 = 0, n are proportional to tLc associated Legendre polynomials P~, and any such two distinct solutions of (3.4) satisfy the orthogonality relation
i
n
i
I
P,,,P,, sin 0 d0 = 0, for m ~n,
(3.5)
t, 0
and they constitute a complete set in 0 ~<0 ~
W = E W,,P~,(cos 0).
(3.6)
n=l
It ~hould be noted that, since tt(e) is complex for every real e then for every n
a(e)~n(n+ 1)
(3.7)
and therefore apon substitution (3.6) into (3.2) and than considering (3.4), for I~',, we get
i W"'--a(e)-n(n+ 1) F,
(3.8)
where F,, are defined by the relation
n = I
, F,,P,,=F(O)
(sin O f ) - "
1 sin0
¢,.9)
We now determine the coefficients F, of the above expansion. To do this we first perform an ex~an~io.-, for f(O). and then we use this expansion to fix the F,'s. By employing the I%llowing addition theorem. [10]. e '':~°~'°= ~ (2n+ 1)(i)')',,(¢)P,(cos 0)
(3. i0)
n - 0
the first two t e r m s o f /gt' tv,¢'~\ j can be expressed as d2E~ -
1
dE
d0 2 s i n 0 d 0
-
I ~
(
e,=0
\d0
~ (2n+ 1)(i)"~.(e) dP~,+
P~/
(3.11)
sin0/
where ~,,(e)= ej,,(e) and j, is the spherical Bessels function of the first kind. To construct a similiar expansion for the other two terms o f f ( 0 ) we will make use of the following representation of the plane wave (2.11). [10]; w~. --
~,.,,+ I) [
tl
~
|
,li,
,u',l•
. •I
=,
, ~n(n+ l) Eo--sin~)E=sin~
dp~ = n(n+l) (i)" ~,, dO
sin 0
sin O)J
d0 )J
(3.13)
(3.14)
M. Teymur / Surface radiation conditions
where q~;,.=d~,,,/ds. Hence, it is seen that, from (3.13) and (3.14) we may write
cos
0 E + E= --
I '- ( 2 n + 1) ,Y,-- - - ~ g ._ n ( n + l )
(i)"(~,,,- i~,,,)(dP~'+ P!' \ ' sm,,,-T£-~]" \d0
(3.15)
..
It then follows that
f(0)=
Y.
n= I
f,\d o
(3.16)
+ s i n O/
where
1 ( 2 n + l)(i)" f,,= ~:
{
o
V,,
}
n ( n + 1) ( V , , - I V [ , )
(3.17)
•
U p o n inserting this result in (3.9) a n d considering that d"P I, c o s 0 d P I ..........':+ " dO: sin 0 dO
p1
--n(n+
(3.18)
1)P~,,
sin" 0
we obtain F,,= - n ( n + l)f,.
(3.19)
n ( n + 1) H;.= .............................................f,. ' a - n(n + l )
(3.20)
Hence
Then using (3.16) a n d (3.6) in (3.3). together whh (3.17) and (3.20), for h,, and h,~ we get
£,= (a
A. e . + 8.
,,FI
Z A. ---dO]"
sin O
,,-I
8. dO
(3.21) sin O/
where
| A,,-- a " - n i ; , + l ) f''
B,, = a! f ' '
(3.22)
Note that the solutio~s secureG here via the S R C method are similar to the series solutions of the scattering problem as far as their lbrms are c~.ncerned. In the latter case the coefficients A, and B,, are given as, [11]. .4.~ =
] .,,.~ ,~:
1
2n+l
B~;=IF'+"
2n+l
]
( 3 ...~) '~"
n~ n ~- 1 ~ { sh~.7 ~' ~'"
where the supcrs.r:p, • - -c " ' c r~,:l:~, l( the exact answer .:rod hI,,-~ is the spherical H a n k e l function defined by
1/'~' , ~,,:) = i"'i ,,. ..~ e . .,.,. . . c" .;
.. ( n + - ~ ) ! ,p m!{n-- n,~}~
(2i,)-".
(3.24)
M. Teymto" /Surface radiation conditions
9
By using the Wronskian relation ('r
•
h~2).,
j,,h,;- -
i
. j ,,
62 ,
(.~. ;.~ )
A,~ and B,~, can be rewritten as
{
}
2n + 1 /; n(n + 1) - 6j,, + (eht,2)) ' (6J") ,
i"
' '
2,,+,{ .,-,, } ""
B, =
i"
n(n + 1)
- i t,~-', j'' ~" ~j" " ..,;"
(3.26)
A,, and B,, can also be rewritten in the forms A,,
-
i" - 2 n- + l n(n+
{-6j.+
1)
(6j.)'}.
ia
B . = i " 2n+______~l{ e n ( n + l ) a -( 6 n ( n + 1) ea
1)
a-n(n+
-
i ) j . + 9 .., ,,}. (3.27)
Then comparisons between A,, and A~,, and B. and B~ reveal that in the perfectly conducting sphere problem the SRC method is equivalent to introducing the following approximations ~h},')
ia
h},2)'
. 6 n ( n + 1) - a ( e ... ~ n,"(2) 6a
(eh~,..)) , .~. a - n(n + 1) ,
i) (3.28)
"
Consequen! j, the accuracy of the m e t h o d will be bound up with the accuracies of these approximations. Let us denote the left hand sides of the expressions given in '~ ~o~ by D .~ and E . , respectively. Then by ~.,.<~,, employing the definition of the spherical Hankei function n,, t, ~z~, we find that -6
" ( 2 n - m ) ! _.,..,, ~ 12~,:.) ..... o ( n - m ) ! m !
~t
X-" ( i e + n - m } ,,, ~:o
(2n-m)!
(3.29}
{i(i + n - m ) - e } E t)
.~. ,,,
(n - m ) ! m !
( 2 n - m ) ! (2is)" (n - m)!m!
- - 2 " =o
"~ -(2-n---!-n-)-! (2ig)"'+' ,,,-o(n- m)!m! i.e., D,~ and E~, are the ratios of the two polynomials of degree n + 1. On tile other hand the right-hand sides of the expressions given in (3.29), say D'), and E ) , . are respectively the ratios of two polynomials of degree two and three, in general. Thus the method in this special problem approximates the ratio of two polynomials of degree n + 1 by the ratio of two polynomials of degree two and three, respectively. Therefore, as it should be expected, the range ef these approximations would be limited• This point will be investigated that for n = 0 and n = 1; D~ = L'){',= i,
E,)~-- E,)"
~ -
s- i ..........
as
•
D~ = D ~ -
is: + 6
I ......
T
g .......
............. ,
s'-le- !
g2- 2 i s - 2 E{ = E'~ = --:--a...........
( 3..,0)"
~6-+c
Hence, it is seen that the first two terms of the exact answers and the approximate solutions obtained by the SRC method for h~, and h~ are identically equal. Therefore. since for a sufficiently accurate numerical
M. Teymur / Surface radiation conditions
10
value, approximately, the first ka + 4 terms of the infinite series (3.21) are needed, for lower frequencies the SRC method gives accurate results everywhere on the sphere. Consequently, the scattered far field also behaves similarly, they are accurate in both backward and forward scattering regions. To make a quantitative comparison for higher frequencies and tbr n > 1 the real and imaginary parts of (ej.)'D~., (sj.)'D~. and j.E~., j.E~. are computed for various ka values and have been shown in Figs. l(a)(c) and Figs. 2(a)-(c) as functions of n, where the thick and thin solid curves are represents, respectively, the real and imaginary parts of the exact values, whereas the thick and thin dotted curves are the SRC results. Note that, for a given ka, (ej.)'D~., (ej.)'E~., j.D,~, and j.E~. are discrete functions taking the values only on integer n's, although they are presented as continuing curves between these values. From these figures, it can be observed that, for ka = 1, the approximations are quite accurate for all n, and therefore as can be seen from the Fig. 3(a) and Fig. 4(a), the SRC method produces the surface and consequently the far fields very accurately for ka = 1. When frequency increases, the deviations of approximate values from the exact ones become important for n > 0.7ka approximately, and hence the error in D~ and E,] will introduce inaccuracies in the approximate surface field. Note that, for a given ka the coefficients A. and B. of the infinite series (3.21) will be fixed for every 0, i.e. for every observation direction. In spite of this fact - as can be observed from Figs. 3(b)-(d) where the modulus of the total surface field obtained by the series solution of the SRC equations have been presented by dotted curves for ka = 5, 10 and 20, respectively - in the shadow, i.e. for 0
o.°°'~.. .4 ka:
-
1
~5 .sl
~)
0
=
4
N
I!
= ~:
.,~
-!
-.6
la)
(b) !
~
.5
l, }J tcl Vig. I. Variatkms of rca~ and imaginaD, parts of (ej,,)'D~ ( ........... ) and (;;j,t'D~ ( ) vs n for various values of k a (real parts heavy. imaginary paris light curves)' (a) k a = 1' (b) ka=:5: and (c) ka=20.
M. T~ ymur / Surface radiation conditions .1 0
.2
i
..-" ........................
~.........
/ / = ,-..,,
-
11
I
.
=
.1
ka =: ,5
.3
E
"
$
ka= 1
ol
"%1
A-_...-,~
1 ......
.
..... I
8
=
IO Ill
~-.7 .
-.9
2
{a)
1
-
(b)
,...,
A"
.03 =
101 __ ~15
ka
=
20 20
25
3O
t
~,
- .03
.05L
(c) Fig. 2. V a r i a t i o n s o f real a n d i m a g i n a r y p a r t s ofj,,E~. ( - - ) a n d j'.E~ ( ) vs n for v a r i o u s v a l u e s o f ka (real p a r t s h e a v y . i m a g i n a r y p a r t s light c u r v e s ) : (a) ka = 1" ( b ) ka = 5; a n d (c) ka = 20.
i.e. for ~,/2 < 0 < re. This means that, the variations of P~,/sin 0 and dP~,/dO with 0 are also effecting the total error of the method and the alternating behaviours of these angular functions cause the accumulation of the errors introduced by the approximations (3.28), i.e. by the SRC method, towards and over the dark region. This observation indicate~ the nonuniformity of the SRC approximation with respect to the angular variable and therefore it may then be concluded that to include the effect of creeping waves more accurately in the high frequency range it is necessary to modify the tangential operator of the SRC.
3.2. As),mptotic solutions of the sRc equations The SRC equations (2.14) and (2.15) given for the tangential components of the total surface field h can be reduced to the following equations for the tangential components of the scattered surface field h ~ (3.31) sin 0 d0
shl 2 0
(sh! 0 h,~) -:~. . + ahT~= c~i +
dO 2
where A
h~%= -sin 4~ h~%,
h~ = - c o s 4) ~ .
( 3.33)
12
M.
Teymur
/ Surface radiation conditions
e s . s,,.
2 , r.,
--.y
.=_~ ka ~
1
'
..~1
I
I
SO
I
I
I
120
(a)
I
I
J
,
0
180
o m
"7
_
"\ ~
Ifiol
l
t
o
i
1~
180
120
{c)
'.
\
._'i,
120
180
O(degrees)
~
lfiol
"
I
60
, 60
"';"
{b}
21 l,g=~°m
,
Nv
.(degrees)
m
""" : ~
,,o, i N
lfi.l 0
~
0
60
0
ldegre sl
{d)
120
180
OIdegr e l
Fig. 3. Magnitudes of/~,, an, ~. % vs 0 for various values of ,~a. Exact answers ( ..... 1, the series ( ..... ) and perturbation ( ......... ) solutions of the SRC equations, and the physical optics results ( ..................... ).
We will now seek solutions for the S R C equations (3.31) and (3.32) for large e = k a by applying the W K B J technique which was used by K r i e g s m a n n and M o o r e [7], in c o n j u n c t k m with a scalar surface radiation condition, to stud) ~ the scattering of acoustic waves by an impedance coated sphere. F o r this purpose let us first define the functions H ° and H ~ as
H°=hbexp(-iecosO),
HC'=h;exp(-iecosO)
(3.34)
and expand these functions in the following asymptotic series in s -~
H°~- ~ ,. ,~,,, n =0
~ Y'. s-'H°,.
(3.35)
n =0
If we employ the expansions (3.35) in (3.31) and (3.32), then by collecting the terms of like powers in s -~ we get the follov~ing hierarchy of e q u a t i o n s from which it is possible to determine H ° and H~ successively; • ~ i2-san-0)H,~,-i
te s i n 0 dH;,~.-~ d(i +2c°sOIt°'"+2H~u
_ Uo -"~t
+ d-H,,~. ........... 2 + c o s 0 dH~_ 2 d0 ~ sin0 dO
...................................... 0 it2-, l dH~[ :, ........................ 1 I ! ~ .... + - cos : - ; ......... sinO dO sin:O " sin'0 ' " .. cos 0)&,, - i( l + ~. ~ cos O)SB. . . . . .~ . . =. . 0, 1 "~
(3.36)
M. Teymur / Surface radiation condilions
13
°l
/ o = ~/2
j
. . . . . . .
1
.-..~.
~° : ka = 5 t=
\ / o=0 ,~,../
.
1
J
/ ~..
y
Z ~ ) ~'
-.-. :....
I~ ~.. . . . -~ I I
L¢/ •
II
.1
.2
t
l
l
I
l
60
0
....I
.
I
...
l
(a)
.02
l
180
120
-
,[
I
0
!
I
I
,,, i
I
(b)
O(degrees)
I
!
120
60
180
#{degrees)
1000
100
100 10
,, = : / 2 10
,, = x / 2 oj I
%
%
l=
i=
"-
100
ka = 10
10
•
_ .
ka = 20
'I
.01
I
60
0
120
1$0
_
i
_a
_ _ l ...........
60
0
O(degrees)
ic) FiF. 4. N o r m a l i z e d b i s t a t i c r a d a r cross s e c t i o n s
-t~H.-l+
}+
I
I
........
l
L .....
120
I
180 0(degrees)
or(O, x/2)/rta 2 a n d cr(0, O)/rca "~vs 8 f o r v a r i o u s v a l u e s o f k a . E x a c t a n s w e r s ( ........... ~, vi:~
]
~ ~,~
sin 0
=(2-sin 2 0)80.-i(2+cos
|
cos 0
. . . . . ~m: + _ _
dO
0)81.,
sin" O
I H~_ 2_ _ _
sh~2 0
H~_ :
n = 0 , l, 2, . . .
where H~., = H~_1 = H°__: = H°_~ -=0, and 6o., 81. are usual Kronecker symbols.
(3.37)
M. Teymur / Surface radiation conditiort~
14
From (3.36) and (3.37), for n = 0, the zeroth order solutions are found to be Ho~ =--,2c A
H0 =_A 2
(3.38)
c = cos 0,
A = 1 + c".
(3.39)
where
Then, by using the zeroth order solutions (3.38) in (3.36) and (3.37), for n= 1, after some algebra we get H~=i
1 1 2 A
2c 4 4c 8ca 8c" 8c4] =i/t~, " A A"~-A-5+ + a 3 a"/ (3.40)
H°=i~
+c"-c-I
=i/4 °.
Proceeding to the nigher orders requires similar effort. But here an attempt will not be made towards obtain!ng them, since the higher order solutions become less effective as e increases. Thus, up to first order in e-~, the tangential components of the total surface field h are found to be h o ~ - c o s ~/~o=-cos ~(1 + Ho~+ i~:H~) exp(i~: cos 0), (3.41) h 0 ~ - s i n ~b/~o= - s i n
q~(c+H°+ i~/-t°) exp(i~: cos 0).
3.3. Comparisons For comparisons, the modulus of the total surface field components/~0 and/~o are computed from the exact answer of the problem, the PO approximation presented in the Appendix, and the exact and perturbation solutions of the SRC equations for various ka values. They have been shown in Figs. 3(a)-(d) as functions of 0 by full, dashed-dotted, dotted and dashed curves, respectively. Let us consider the modulus of the surface field obtained by the exact solutions of the SRC equations. From the Fig. 3(a), it can be observed that, for ka= 1, for both I/~01and Ih01, over the entire surface, there is an excellent agreement between the approximate and the exact answers. As frequency increases, as it is seen from Figs. 3(b)-(d), the results become less accurate in the shadow where creeping waves dominate the surface field. That is, as is mentioned in the introduction section, the SRC begins to fail to account for the creeping wave physics adequately. On the illuminated side, as frequency increases, although the approximate Ihol-curves follow the exact ones with increasing accuracy, the same thing is not valid for the Ih,lcurves. They first oscillate around the exact curves in the middle frequency range and later, for higher frequencies, the agreement between the exact and approximate results is improving. From Figs. 3( .O ., I . .~ Q. } it C a l l be observed that in the :l l 'l U"l 't U-t:l a.t r.~.u . .a region, k~.,.;..,.;,.,~ ts,.,E, l t t t t a t t ~ ¢,.~,,~ t A,,.,l,a ka= 5, ,h,~ .... perturbation results are very satisfactory, but for ka = 5, in the shadow beyond 0 = 100° there is a considerable discrepancy between the perturbation and exact SRC curves. However, as frequency increases, this difference diminishes and the perturbation curves converge to the exact curves fo the SRC. It is also seen that on the illuminated side of the surface the combination of the SRC method and high frequency perturbation expansion gives better results than the PO approximatior', from moderate to higher frequencies. This approach also yields a non-vanishing surface field in the shadow, except at 0 = 180°, thus providing
M. Teymur / Surface radiation comfitions
15
a smooth transition between the illuminated and shadow sides. As it will be observed later, this significantly affects the accuracy of the far field results by comparison to the PO approximation.
4. The far field
In this section we calculate the scattering field by employing the tangential components of the surface field obtained in the previous section by different approaches, along with the integral representation of the scattered electrical field intensity E s (2.3). Considering that ~,(x, y) ~
1
4nr
exp(-ikr + ik.~. y)
(4.1)
as r = Ixl ~ oo, where .~ is the unit vector in the direction of observation point x and, x and y are x = r(sin 0 cos O, sin 0 sin O, cos 0),
y = a ( s i n 0' cos O', sin 0' sin O', cos 0')
(4.2)
from (2.3) the distant scattering field E s is tbund to be
E S~(Soeo + S,t,e¢ ) e-i*'/kr
(4.3)
where eo, e# are unit vectors in the directions 0- and 0-increasing, respectively, and
So and
S~ are given
as
So = - ~ ie21~~fo~ 19 exp(ie/4re,
) sin O' dO' dO',
(4.4)
S¢,= - - -ie2;; " f" ~ 4rt
o
exp(ieF) sin O' dO' dO'
with F = ~..P = sin 0 sin 0' cos(0 - @') + cos 0 cos 0', O = {cos 0 cos 0' cos(0 - 0') + sin 0 sin 0' } cos O'/~#"+ {cos 0 sin(0' - 0) } sin @' ho,
(4.5)
= {cos 0' sin(0' - 0) } cos 0 ' / ~ ' - {cos(0 - 0') } sin @'/~o,.
4.1. Series solution We first present the scattering far field obtained by using the series ~ '-"" - this case ~he integrals in (4.4) can be performed in closed forms. To this end the tangential components/~, and/~# of the surface field from (3.21) are substituted in (4.4) and then resulting integrals are evaluated by making use of the recurrence and orthogonality relations of the Legendre and associated Legendre polynomials. Calculations are tedious but straightforward and details can be found in [ ! 2]. The results are So=cos 0 S0,
S0=sin 0 S~
(4.6)
M. Teymur/ Surface radiation conditions
16
where
I
d P'l'+iB,,v,, p,! l ,
g o = i t ~ (i) "÷' A,,V~
t
,,:,
g+=ie E
,,=t
dO
~,.~J
(4.7)
( i)"+' A,,~,, P'' +iB,,~. . " sin0 d0J
Note that the far field secured here via the SRC methed is also similar to the one produced by the series solution of the scattering problem as far as their form are concerned. It can be checked that the substitution of A,, and B, in (4.7) from (3.23) yields the exact expressions for the .~0 and S",.
4.2. Perturbation solution We now present the scattering far field by employing the first-order asymptotic solutions of the tangential components of the surface field obtained in Section 3.2 by the combination of a high frequency perturbation expansion and the SRC. Substituting/~o and h+ from (3.41) in (4.4) and then by employing the expansion, [10],
=
{(o)
ei~t'= Y' (2/+ l)(i)~l(K) Pt cos
Pl(cos 0')
1=0
., m...(o) cos ..
+2 ,,,~ I (!+ m)!
P~"(cosO')cosm(dp-tk')
}
(4.8)
where t: = 2e cos(O/2),
/~= sin(O/2)sin O' cos(~b- 4¢) + cos(O/2) cos O'
and later performing the integrations with respect to ~b' we get (details of these calculations are given in
[i31) So
ic: , = - - 4 - ( ( J l + J 2 + J 4 - J s ) cosO+2J3sillO},
is 2 Se~=~(J,-Jz+J3+J4)
(4.9)
where
( 0/( i
.;,= ~" (21+ l)(i)~/,(~:)P, cos~/ A I + - A
J2-- ~] (i)~b(~c)P~ cos +=. (1- 1)(1+ 1)(1+ 2)
( 7 )B~+ i- B~ ,
_
(4.1o) J4 ~ _
(2l+1)(i t--o
s(x)P~ cos
D ~r + - D t i , E
Js= X" (21+1) (i)~t(l,')P~ cos +~'='.,( i - l ) / ( / + 1)(/+ 2)
E,+ i E , e
Surface radiation
M. Teymur /
condit.ms*
17
with A~= a~l°) + 2a~°°) - 2a~°1),
B~ = b~l°) + 2b~°°) - 2b~Ol),
D [ = >:'a~°°) + a~!o) + 1_2,1_~2o),
E~ = 2011 ~.(o0~+ b~,o) + ~'b~2O),
A~= - '--(10)..m . aP ') . 2a~°°)+. 6a~°l) .+12aP 2) 4a~°2)
C" = a~OO)+ 2a~! t,,
16aP s),
B~ = --2or I_zAIO) rill) -.-ut ,~t,¢oo)+6b}Ol) ,-jt,(12~ ~.~o2)- 1 6 b P 3), --01 + ~.~01 -'*or
C~= -2u1 ,_(oo~- a~O,)- 2aP,)+ D~=
E~=
(4.!1)
12a~°2~+4ap 2) 16a~°s),
I_..(oo) ,..(Io) 4~..(2o;+ a~,l) 2M I ~ ~.t~I -- .u I
I_i.(oo)-- 201 I_j.(Io)+ ~u ~l.(2O)+b~tt) I
--201
in which a~"") and b~""), m = O, 1, 2, n = O, 1, 2, 3; are defined as a~°°)= 26ot,
a~'°)= ~6,t,
b[~)= u21+! ~.¢to) =4, b~2°)-0.8,
a~20)= ~621+ 36ot,
~.(0o) _ ~.(~o)~ - 0~j2o) =0 021+1--u21 21+1
;'(2°)=4 021
for 1=2, 3,
•
•
for 1= 1, 2, 3,
•
-gin
Z~ - n p 2 1 d x ,
u21"(0n)+I ---- 0- (, 0I n2)1- -=
u2,"(In)+,
=2jo
xA-"P21+ ! dx,
v
v
bl °") = 2
(4.12)
r~
t"
a~ on) =
•
A-"P~dx,
~.(o.) ~.(t.)=0, o21+~=02~
~.(I.) o21+~=2
Io
xA
P.;t+, dx.
The integrals appearing in (4.12) can be performed analytically or numerically, but for large / the analytical evaluation becomes quite tedious. This is also not desirable m the computational sense, since these integrals can easily be evaluated numerically with sufficient accuracy. It is, therefore, preferable to employ a quadrature and the most appropriate one seems to be the Gauss-Legendre quadrature. For this purpose, the N A G library routine D 0 1 A R F is used [14]. When performing the summations of the series (4.9) to evaluate the scattered far field, the spherical Bessel functions j , are calculated by backward iterations by using their recurrence relation as described in [ 15]. The Legendre and associated Legendre polynomials are also calculated from their recurrence relations by forward iterations. 4.3. Comparisons The bistatic radar cross section (RCS) is related to So(0) and .~(O) by the expression or(0, q ~ = ) 4~~ [l~0l z cos" 4'+ ISoI 2 sin" q~].
(4.13)
To make comparisons, the magnitudes of the bistatic RCS for ~b= 0 and ~ = n/2 are calculated for various ka values. In Figs. 4(a)-(d) cr(0, ~)/~a 2 has been shown as a function of 0, where the full, dashed dotted, dotted and dashed curves correspond respec,:vely to the exact answer, PO approximation, exact and perturbation solutions obtained by the SRC. We first consider the results constructed by the exact solutions of the SRC equations. From the Fig. 4(a), it is seen that, for k a = 1, for both q~=0 and ~b=n/2 directions, the agreements between these results and exact answers are excellent. On the other hand, as ka increases,
18
M. Teymur / Surface radiation conditions
in the forward direction the difference between the magnitude grows; but it should be noted that there are only a few directions where the relative error exceeds the average error percentage. Also, around these directions, the modulus of the bistatic RCS is comparatively small. Nevertheless, the far field results predicted by the SRC are qualitatively quite satisfactory and ~hcy are superior to the PO results. From Figs. 4(b)-(d) it is also seen that as frequency increases, perturbation curves begin to follow the exact curves of the SRC, very closely. For example, at ka = 20 they almost match everywhere, and therefore the curves could not be depicted separately Perturbation expansion also produces better results than the PO approximation from moderate to high frequencies. Thus, it appears that combination of the SRC and a high frequency perturbation expansion may be a reliable technique m order to apply the SRC method ¢. . . . . . . . . . . . . . . . ' ----~o~h object.
5. Conclusions
To assess the vector SRC introduced in [8], its predictions for a perfectly conducting sphere have been compared with those of the exact theory. :, has been obse~ed that, although the SRC is essentially based on a high frequency expansion it yields the surface and far fields with considerable accuracy ill the low frequency range. When ka increases, the surface field becomes less accurate on the shadow side of the sphere, although on the illuminated side it remains remarkably close to the exact answer. Accordingly, similar behaviour occurs in the far field and the modulus of the scattering field becomes less accurate in the forward scattering region. Nevertheless, the predictions of the method are still qualitatively quite satisfactory, and they are better than the PO approximation at all frequencies. The loss of accuracy on the shadow side indicates that the SRC method begins to fail to account adequately for the creeping wave physics with increasing frequency. As has been pointed out before this stems from the nonuniformity of the SRC approximation with respect to the angular variable and to include the creeping wave contribul+~on more accurately in the high frequency range, it seems that it is necessary to medify the tangential operator in the SRC. Introduction of a higher order SRC might i,,prove the results and this will be examined in a future work. Here, the effect of introducing a high frequency perturbation expansion has been also investigated. The conclusion then is that the combination of the SRC and high frequency perturbation expansion yields better results than PO. Although this analysis is restricted to a special problem, it is likely that similar behaviour occurs in scattering problems for less symmetrical smooth convex objects with perfectly conducting boundary conditio~l. As an application of the method for such a case an investigation on spheroidal objects is under consideration.
Acknowledgments I would like to thank Professor D.S. Jones for his many helpful comments and suggestions provided during the preparation of this work. This work has been carried out with the support of the Procurement Executive Ministry of Defence. I would also like to thank the referees for their constructive comments and detailed suggestions for improvements to this paper.
M. Teymur / Surface radiation conditions
19
References [11 G.A. Kriegsmann, A. Taflove and K.R. Umashankar, "A new formulation of electromagnetic wave scatterir, g using on-surtacc radiation boundary condition approach", IEEE Trans. on A.&P. AP-35, 153 161 (1987).
[21 T.G. Moore, G.A. Kriegsmann and A. Taflove, "'An application of the WKBJ technique to the on-surface radiation condition", IEEE Trans. on A.&.P. AP-36, 1329-1331 (1988).
[31 I.D. King, "Application of on-surface radiation condition to electromagnetic scattering by conducting strip", lEE Electr. Le'ters 25, 56-57 (1989).
[41 J.-M. Jing, J.J. Volakis and V.V. Liepa, "A comparative study of the OSRC approach in electromagnetic scattering", IEEE Trans. on A.&P. AP-37, i 18-124 (1989). [51 T.G. Moore, J.G. Blaschak, A. Taflove and G.A. Kriegsmann, "Theory and application of radiation boundary operators", IEEE Trans. on A.&P. AP-36, 1797-1812 (1988). [6! D.S. Jones, "An approximate boundary conditic,n in acoustics", J. Sound Vibration 121, 3745 (1988). ,, [71 G.A. Kriegsmann and T.G. Moore, "An application of the on-surface radiation t,ondition to the scattering of acoustic waves by a reactively loaded sphere", Wave Motion !0, 277-284 (1988). [81 D.S. Jones, "Surface radiation conditions", IMA J. Appl. Maths. 41, 21 30 (1988). [91 M. Teymur, "Assessment of the surface radiation condition by reference to acoustic scattering by a hard sphere", IM.4 J. Appi. Math. (to appear). [101 D.S. Jones, Acoustic and Electromagnetic Waves, Clarendor. Press, Oxford (1986). [111 J.J. Bowman, T.E.A. Senior and P.L.E. UslenghL Electromagnetic and Acoustic Scattering by Simple Shapes, North-Holland, Amsterdam (1969). [121 M. Teymur, An investigation of the scattering of electromagnetic waves by a perfectly conducting sphere via SRC method, Univ. of Dundee, UK, Technical Report, 1989. [|31 M. Teymur, Application of a high frequency perturbation technique to the surface radiation condition in electromagnetic scattering, Univ. of Dundee, UK, Technical Report, 1989. [141 NAG, Fortran Libra[y Manual, ! Chapter D01, Mark 13 (1988). [15l M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions, Dover, New York (1965). [161 T.M. MacRobert, Spherical Harmonics, Pergamon, Oxford (1967).
Appendix The physical optics approximation for the total magnetic field h on a perfectly conducting sphere subjected to the incident plane wave (2.8) yields h=2hi=-2jexp(iscosO),
for0~<0~<~/2;
h=0,
forn/2~<0~
(A.I)
T h e tangential c o m p o n e n t s of h on the lit side of the surface are written explicitly as ho = - s i n ~b ho,
he = 2 cos 0 exp(ie cos 0),
h~ = - c o s ~b t ~ ,
h~ = 2 exp(ie cos 0). (A.2)
T h e n employing these results in (3.1) we get ie 2 So
(cos 0 cos q~ KI + sin 0 1~.2 ' ' ), 2rt
S, =
is: ~ ( - s i n t~ ~ ),
;,' z"~' - ~ . . ,"~ p
where KI =
cos
sin
e x p ( i x F ) d0'dq~',
/%= j
t'2" ¸¸'2sin 2 0' cos 4,, exp(iK'P) dO' dq~' o
(A.4)
20
M. Teymur / Surface radiation conditions
and ~" and/~ are given after (4.8). Now, by using the expansion (4.8) and then considering that if m = 0 ifm~0
f~~ cos m(0 - 0') d O '= (0, / 2n'
(A.5)
fo '~co5 0' cos m(0 -
0') dO' = In (0, cos 0,
ifm=l ifm~l
Kt and K2 can be reduced to the following forms gl=2n
(2•+ 1)(i)~h(s¢
cos
g~ ,
I
(A.6)
K2=2nc°s0
t
1(1+1)
where
/~'] =
f
nl2
cos
0'
Pl(cos 0') sin
0'
dO',
g~
f
n/2
sin 0' P~ (cos 0') sin 0' dO'.
(A.7)
~e0
~'0
For the first integral, for g~= I/2,
1=0, 1 we obtain (A.8)
gl=l/3
whereas for 1> l, if I is odd (A.9)
g~ =0, but if I is even, [16],
R] = {(l- 2),}/ {(-1)q+2'/22t(~+ l ), (~- l),}. Since, P] = - ( 1 - x 2 ) t/2 /<~= -
(A.10)
dPt/dx the second integral can now be written as
fo'
( l - x 2) __dPtdx dx"
(A.1 l)
and then the integration by parts yields
~'~ = e , ( o ) - 2R~.
(A,!2)
Since, [ 16], for I even P,(0)=(-l)'/2{l • 3 .....
(1- 1 ) } / ( 2 . 4 . . . .
1)
(A.13)
and for I odd
P,(0) =0
(A.14)
21
M. Teymur / Surface radiation conditions then we get
R~= ( - 1)t (2/- 1)!! (20!!
(21-2)! (-1)l+!22t(l+ l)!(l- 1)!'
R~ = -2/3,
K;j+ t --0,
1=1,2,...
(A.15) w h e r e ( 2 1 - l)!! = I . 3 . . . . .
( 2 1 - l), ( 2 0 ! ! = 2 . 4 . . . . .
(2/).
A comparative investigation of the surface radiation condition in electromagnetics M. Teymur* Department of Mathematics and Computer Science. The Unioersity, Dundee. DD! 4HN. Scotland. UK Received ! 1 October 1989, Revised 22 October 1991
This paper assesses a vector surface radiation condition (SRC) in electromagnetics by reference to a canonical problem, namely the scattering of waves by a perfectly conducting sphere• The method furnishes two coupled differential equations for the tangential components of the surface field and so the calculation of the scattered field is reduced to a quadrature. The differential equations are first solved exactly and the distant scattered field is determined analytically. Later, to examine the effect of introducing a high frequency asymptotic expansion the solutions are constructed approximately• Comparisons are made between the exact answer of the problem, the SRC solutions, and the physic~d optics (PO) appreximafion. It is obs~ar',,ed that, although the SRC method is essentially a high frequency method, it produces surprisingly good results for lower frequencies. In the high frequency range the method is superior to PO approximation and also the combination of the SRC and high frequency perturbation expansion yields better results than PO.
I. |ntroduction
The SRC method in electromagnetic scattering was first introduced and applied t\'~r two dimensional . the field aad problems [ I 4] • The me~hod is based on the application of a radiation condition, ~.~. "~nnc,.,~ne "" its normal derivative, directly on the surface of the scattering object to determine approximately the surface field or its derivative in terms of the given field. The calculation of the scattering field is then reduced to quadratures. The derivation of the radiation condition which is used in these investigations is based on the expansion of the scattering field satisfying the two dimensional Helmholtz equation and the Sommerfeld radiation condition [5]. The generalization of the method for the application to an arbitrary convex cylinder has been made by applying the circular cylindrical form of the SRC operator locally at each point on the scattering surface by considering that an osculating circle can approximate the surface locally. This idea later has been applied to obtain a three dimensional SRC in acoustics and the results of the method for a soft sphere are compared with the exact answer of the problem [6]. It has been observed that fi'om moderate to high freauencies the method is effective in obtaining the magnitude of t.be normal derivative, but it is less effective for its phase in the shadow part of the surface. Nevertheless, the magnitude of the normal derivative is relatively small in this region, therefore the defect in the phase may be quite tolerable in practical applications. Later, in [7], by using the methodology presented in [1 ], a slightly different operator than the one given in [6] has been introduced to investigate the scattering of an acoustic plane wave by a * Present address" Marmara Scientific and Industrial Research Center, Department of Mathematics, P.O. Box 21, 41470 Gebze, Kocaeli, Turkey. 0923-5965/92/$05.00 ~" 1992 Elsevier Science Publishers B.V. All rights reserved
2
M. Teymur / Surface radiation conditions
sp~':: :'.-al target which possesses a surface impedance. The surface pressure is constructed by the WKBJ m.~'~.d and then utilizing this result an approximatqon for the backscattering cross section is deduced. F,-,m3 cemp~ria~ns with the eaa~,t backscattering closs sections for various spemal cases it is concluded that the method is effective from the middle frequency range. Later, a general method of obtaining surface radiation conditions which is based on an asymptotic expansion similar to the Luneburg-Kline expansion and the assumption that the surface of the scattering object is a phase front, has been introduced in [8]. In this work a vector SRC for electromagnetic waves which connects the tangential components of the field on a phase front is also derived. The relative merits of the tv, o forms of the SRC introduced by different approaches in [7] and [S] have been examined by reference to acoustic scattering by a hard sphere in [9]. The SRC equations have been solved exactly and the far field b"~ been obtained analytica:lly. From a comparison between the exact answer of the problem and solutions constructed by the SRC techniques, it has been observed that both approaches almost equivalently provide the surface and the scattered field very accurately in the lower frequency range, i.e. for ka<~5, where k is the wave number and a is the radius of the scattering sphere. Note that a high frequency analysis misses this point [7]. When frequency increases beyond ka = 5 the results become less accurate on the shadow side of the sphere. This affects the far field results and although the phase remains remarkably accurate, the relative error in the magnitude of the far field grows in the forward scattering region; nevertheless, the results are still qualitatively satisfactory. In [9], the effect of introducing a high frequency perturbation expansion and an iterative technique have also been investigated. It has been observed that, as frequency increases, the first order solutions obtained by these methods approach the exact series solutions of the SRC equations thereby indicating that both technique might be efficient supplementary techniques in applying the SRC concept for arbitrary smooth convex obstacles. It has been already pointed out in [8] that the vector SRC introduced there reduces to the formula given in [l] when specialized to two dimension. The aim of this work is further examination of this general form of the SRC via a three dimensional canonical problem in electromagnetics, namely the scatteri,~g of a plane wave by a Ferfectly conducting sphere. The method offers two coupled differential equatmns for the iangential components of the surface field. These equations are first solved exactly by series expansion and ~he results are compared with the exact answer to the problem. As in the acoustical case, [9]. there is good agocement at tow flcquencies. When the frequency increases the results become less accurate on the shadow side of the sphere; that is the surface field predicted by the SRC fails to account for the creeping wave physics adequately. This type of deficiency has been also observed in the previous applications of the SRC concept in two dimensional electromagnetics and three dimensional acoustic scattering problems [I, 6, 7, 9]. This should be expected, since the derivations of the conditions is either based on the assumption of outgoing waves which are subdominant in the shadow, near the surface, from around moderate frequencies [1, 7]- or is obtained by an asymptotic expansion similar to the geometrical optics (Luneburg-Kline) expansion and the assumption that the surface of the scattering object is a phase front [8] ; but the scattering surt:ace is a caustic of the creeping wave rays and the asymptotic expansion of the geometrical optics is not umik~rmly valid around a caustic. It should be also noted here that in the soft sphere and circular cylinder ,'~..e. ..... :"- of the "--" ..... wave) problems, the ampiitudes of the surface fields provided by . . . . . .::~. . . TM ,.,.-,1.,.-: ~,,-,,,,,,.~.-~,~,, ~c~ucnt ~hc SRC become remarkably close to the exact answers in the shadow region with increasing frequency tr¢ , ~. I 1. This is due to the fact tha~ creeping waves arc less pervasive |br soft objects. On the other hand, the agreement at low frequencies is poor by comparison to the hard sphere and circular cylinder (i.e. for TE polarization of the incident wave) and perfectly conducting sphere problems. These results indicate the necessity of further research for improvements of the SRC methods for better predictions of the surface ticMs.
M. Teymur / Surface radiation conditions
3
To solve the surface differential equations is an important step of the method and exact solutions of these equations can be constructed by series expansion for only a limited number of geometries where the Maxwell's equations are separable. Therefore, to apply the method for an arbitrary convex object a supplementary technique ;s needed to determine the surface field. Here, for this purpose, the effect of introducing a high frequency asymptotic perturbation expansion is investigated. The first order solutions are presented and it is observed that as frequency increases these solutions approach to the exact series solutions of the SRC equations. Thus, it ap?ears that the perturbation analysis is a reliable supplementary method in order to apply the SRC concept for arbitrary smooth convex obstacles. Later, the solutions for the tangential components of the surface field are employed along with an integral representation of the scattering field to deduce the far field. For the series solution the integrals are performed analytically whereas for the perturbation solution they are calculated numerically. Comparisons are made between the exact far field omp'ffudes, the series and perturbation results constructed via the SRC method and the PO approximation. It is observed that the SRC method produces surprisingly good results for lower frequencies as in the acoustical sphere having a hard boundary [9]. When frequency increases, in the forward scattering region, the relative error in the far field amplitude grows, but the results are still qualitatively quite satisfactory and they are better than obtained by the PO approximation. It is also observed that as frequency increases the far field amplitude obtained by the perturbation technique approaches to the series solution result. Moreover, the combination of the SRC method and perturbation expansion yields better results than the PO approximation. The paper is organised as follows. The formulation of the problem is described in Section 2. In Section 3, the exact and perturbation solutions of the SRC equations are presented and the results are compared with the exact answer. The scattering far field in calculated in Section 4 and comparisons are made between the SRC results, exact answer and PO approximation. Section 5 contains some concluding remarks. The PO approximation for the problem is presented in an appendix.
2. Formulation Consider the scattering of time harmonic electromagnetic waves by a perfectly conducting obstacle bounded by a surface Z of finite area. The exterior of Z, which will be denoted by Z ~, is free space with permittivity e0 and permeability po. In the absence of impressed currents and charges, the scattered electric and magnetic intensities E ~ and/./5 are governed by the equations, [10], curl
E ~+ ikh S =
O,
curl
h ~- ik E " = 0
(2.1)
where k = c o ( p o S o ) ~''2 is the wave number, co is the angular frequency, h'=(yo/so) ~ 2H' and the time dependence is taken as m, ;~t). Since the scattering o',;~;.mle is considered to be a perfectly conducting object, on its surface S. E" is subject to the condition n x E" = - - ( n
x E i)
(2.2)
and at infinity it must have the form of a radiating wave, where E ~is the incident electric field falls on the body from 27e and n is the unit outward normal to Z. Then the scattered electromagnetic field E", h" at a
4
M. Teymur / Surface radiation conditions
point x in 27* can be represented as, [10], E"(x) = - ( i / k ) ( g r a d div + k 2) f , n x h ( y ) V ( x , y) dZ>,,
(2.3)
Z
h~(x) = curl f,, n x h ( y ) g ( x , y ) d2;>,
(2.4)
6,.
where h is the total magnetic field and V , ( x , y ) = e x p ( - i k l x - y l ) / 4 n l x - y l . From (2.3) and (2.4) one can calculate the scattering field satisfying the boundary and radiation conditions provided that n x h is known on X. This requires the solution of an integral equation for n x h which can be derived from (2.4). On the other hand, there are also certain ways to avoid solving an integral equation. One way is to employ the physical optics approximation. This is based on the assumption that at high frequencies, each portion of the geometrically illuminated side of Z, say 2; ~, reflects as if it were locally plane, whereas over the shadowed portion of r the surface field is zero. Thus for a perfectly conducting body nxh=2nxh
i
on Z i,
nxh=0
on ~ _ ~ i
(2.5)
and then the determination of the scattered field is reduced to quadratures [11]. The approximation of physical optics is known to be quite effective at high frequencies, although a refined transition between the illuminated and shadow side of the scattering obstacle is needed for a complete coverage of the radiation pattern. On the other hand, with decreasing frequency the distinction between the illuminated and shadowed sides begins to fade away and the question of where to apply the equations (2.5) arises. The surface radiation concepts attempts to introduce a modification in this direction. For this purpose, in [8], the following approximate formula is derived to link the tangential components fo the electromagnetic field on a wavefront
< i2) n > ' ( h - n x E ~
(1-
'
. . .D. ' L [ n x C u r l ( n x h ) + { C u r l ( n x E ) } , ]
1
2k 2 [Grad Div(n x h) + {Grad Div(n x E) } x #]
(2.6)
where Curl, Div, and Grad are surface curl, divergence and gradient operators, respectively; H is the mean curvature of the wavefront and the suffix t indicates tangential components. When deriving (2.6) for any kind of wave i.e. reflected, scattered ect. the wavefronts are locally treated as parallel surfaces and therefore they are expressed as X=Xo(O-~, cr2) +s= where cr mand cr2 denote the coordinates in these surfaces and s is the variable measuring the distance along a normal = to the wavefront. The surface gradient is defined as G r a d u = g J a 0u ~x Oo.j O.r~
(2.7)
~vherc g'a denmes contravariant metric tensor. The surface divergence of the tangential vector 7"= T j C~x/ ~ ' and the surface cur~ of the x.... " '~t c~"' + ,~ aaa are also defined a:~ follows e~:~c~r .4 = A' cx. Div T -
g
,
S~r /
g 2TJ)
Cuff A := Grad A 3 x n -.vPgp;_Aj + n Div(,4 × n)
(2.8) (2.9)
M. Teymur / Surface radiation conditions
5
where
-
b,
Since the relation (2.6) is based on behaviour local to a wave front, it should be valid anywhere where there is reasonable wave front. The main assumption leading to the SRC is that a wave scattered by an obstacle will obey this rule on the surface of the scatterer. This is the extension of the assumption suggested by Kriegsmann et al. in two dimensions [ 1]. Hence for a perfectly conducting object since the condition (2.2) is valid, this assumption yields the following vector differential equation to evaluate n x h ~ on the scattering object 1 - i~k) n x ( h ~ + n x E i ) = 9 ~ l 1k [ n x C u r l ( n x h ~ ) - { C u r l ( n x E i ) } , ]
1 [Grad Div(n × h " ) - {Grad Div(n x E~)} x hi. 2k 2
(2.1o)
Thus, since (2.10) is applied over the whole Z and not just some part of it, the dilemma of physical optics is also circumvented. On the other hand, since n x h ~ is known on 2~, (2.10) can be reduced to a differential equation for the total field n x h and then the determination of the scattered field is eventually reduced to quadratures as in the case of physical optics approximation. It is evident that, to construct the solution of (2.10) for a given scattering object and incident field, is an important step in the method. Exact solutions may be obtained by the series expansion for only limited number of geometries where the Maxwell's equations are separable. For an object of more general shape the surface field can be found by an approximate solution method such as a high frequency perturbation expansion. Here, to assess the vector SRC (2.10) we will consider a canonical problem, namely the scattering of a plane wave incident in the direction of the negative z-axis, i.e. E i = i eik',
hi= _] eik-',
(2.11 )
by a perfectly conducting spherical object aad present exact and perturbation solutions of the SRC equation (2.10). Then by employing these solutions in the integral representation, the scattered field will be calculated. For parallel spherical surfaces, the surface operators defined in (2.7), (2.8) and (2.9) take the following respective forms 1 0 1 0t~ Div T - - (to sin 0) + - , r sin 0 00 r sin 0 0~b 1
1
F
/r
1 0u 1 0u Grad u = - - - eo + - e~ r 00 r sin 0 ~4~
(2.12)
Cur! ~ =w-.~A ~ ~ ~ " + - - A o e ~ - - ~ " 0 + " r ~ ; " ( ~ y "~ ~
lk.,~.~au.~..s
.~l,,..j
--
~e
.
dA
~
,
wD
.~-"~
v
,a
-
"
,,,my
where T and A are defined as
T=toeo+t~e~,
A=Aoeo+A~,ee~+Arn
(2.13)
and 0, ~b and r the usual spherical coordinates with 0 ~<0 ~
6
M. Teymur / Surface radiation conditions
Using (2.12) in the general relation (2.10), on a perfectly conducting scatterer of radius a, for the incident plane wave (2.11), the following coupled differential equations to calculate the tangent;al components of the total field h are obtained;
d-O ~
(sin 0 h~ )-/~o
(2.14)
+ a(e)h~=f(O)
1 {d~(sinOh~,)-ho}+a(e)ho=f(O)
(2.15)
sin" 0 where
ho = - s i n ~b/~o,
ho= -cos q~/~,
e=ka,
a(e) = 2 e ( e - i),
1 dE d2E f(O)~- + a(e)(1 +cos 0)E, sin 0 dO d 0 2
(2.16) E=exp(i~ cos 0).
and it is required that/~o and/~ must be bounded functions of 0 for 0 ~<0 ~
3. The surface field In this section, first the exact series solutions of the SRC equations d:rived for perfectly conducting sphere problem is presented. Then as a supplementary technique the effect of introducing a high frequency perturb Jon expansion is examined.
3.1. Series solutions ,)f the SR(" equations Let us introduce a function D" by the relation W= 1-~sin 0 td,dO(sin0/~ ) -/~o} •
(3.i)
It then follows from (2.14) and (2.15) that W satisfies the equation sin0 d-d- sin0 -W+a(,:)sin20 W=sin0 d0, -d-b-
{sinOf(O)}-f(O)
(3.2)
and h,, h~, are given as
~. .
.
.
1 If(O )
.
.
.
u~)t
.
.
.
.
.
I IV! .
.
.
sin0
.
,
~,
1 t f( q)-- dW~
. . . . . .
~xts~'
(3.3)
d0j"
Note that the solutions of the equation
(
:t
sin0 dO d sin0d-d-O-J . A. = -n(n + !) sin 2 0 A,, .
.
.
.
.
.
.
.
.
•,
n = 1 2, 3, ,~
°
°
.
(3.4)
M. Teymur / Surface radiation conditions
7
which are fimte and have also finite derivatives at 0 = 0, n are proportional to tLc associated Legendre polynomials P~, and any such two distinct solutions of (3.4) satisfy the orthogonality relation
i
n
i
I
P,,,P,, sin 0 d0 = 0, for m ~n,
(3.5)
t, 0
and they constitute a complete set in 0 ~<0 ~
W = E W,,P~,(cos 0).
(3.6)
n=l
It ~hould be noted that, since tt(e) is complex for every real e then for every n
a(e)~n(n+ 1)
(3.7)
and therefore apon substitution (3.6) into (3.2) and than considering (3.4), for I~',, we get
i W"'--a(e)-n(n+ 1) F,
(3.8)
where F,, are defined by the relation
n = I
, F,,P,,=F(O)
(sin O f ) - "
1 sin0
¢,.9)
We now determine the coefficients F, of the above expansion. To do this we first perform an ex~an~io.-, for f(O). and then we use this expansion to fix the F,'s. By employing the I%llowing addition theorem. [10]. e '':~°~'°= ~ (2n+ 1)(i)')',,(¢)P,(cos 0)
(3. i0)
n - 0
the first two t e r m s o f /gt' tv,¢'~\ j can be expressed as d2E~ -
1
dE
d0 2 s i n 0 d 0
-
I ~
(
e,=0
\d0
~ (2n+ 1)(i)"~.(e) dP~,+
P~/
(3.11)
sin0/
where ~,,(e)= ej,,(e) and j, is the spherical Bessels function of the first kind. To construct a similiar expansion for the other two terms o f f ( 0 ) we will make use of the following representation of the plane wave (2.11). [10]; w~. --
~,.,,+ I) [
tl
~
|
,li,
,u',l•
. •I
=,
, ~n(n+ l) Eo--sin~)E=sin~
dp~ = n(n+l) (i)" ~,, dO
sin 0
sin O)J
d0 )J
(3.13)
(3.14)
M. Teymur / Surface radiation conditions
where q~;,.=d~,,,/ds. Hence, it is seen that, from (3.13) and (3.14) we may write
cos
0 E + E= --
I '- ( 2 n + 1) ,Y,-- - - ~ g ._ n ( n + l )
(i)"(~,,,- i~,,,)(dP~'+ P!' \ ' sm,,,-T£-~]" \d0
(3.15)
..
It then follows that
f(0)=
Y.
n= I
f,\d o
(3.16)
+ s i n O/
where
1 ( 2 n + l)(i)" f,,= ~:
{
o
V,,
}
n ( n + 1) ( V , , - I V [ , )
(3.17)
•
U p o n inserting this result in (3.9) a n d considering that d"P I, c o s 0 d P I ..........':+ " dO: sin 0 dO
p1
--n(n+
(3.18)
1)P~,,
sin" 0
we obtain F,,= - n ( n + l)f,.
(3.19)
n ( n + 1) H;.= .............................................f,. ' a - n(n + l )
(3.20)
Hence
Then using (3.16) a n d (3.6) in (3.3). together whh (3.17) and (3.20), for h,, and h,~ we get
£,= (a
A. e . + 8.
,,FI
Z A. ---dO]"
sin O
,,-I
8. dO
(3.21) sin O/
where
| A,,-- a " - n i ; , + l ) f''
B,, = a! f ' '
(3.22)
Note that the solutio~s secureG here via the S R C method are similar to the series solutions of the scattering problem as far as their lbrms are c~.ncerned. In the latter case the coefficients A, and B,, are given as, [11]. .4.~ =
] .,,.~ ,~:
1
2n+l
B~;=IF'+"
2n+l
]
( 3 ...~) '~"
n~ n ~- 1 ~ { sh~.7 ~' ~'"
where the supcrs.r:p, • - -c " ' c r~,:l:~, l( the exact answer .:rod hI,,-~ is the spherical H a n k e l function defined by
1/'~' , ~,,:) = i"'i ,,. ..~ e . .,.,. . . c" .;
.. ( n + - ~ ) ! ,p m!{n-- n,~}~
(2i,)-".
(3.24)
M. Teymto" /Surface radiation conditions
9
By using the Wronskian relation ('r
•
h~2).,
j,,h,;- -
i
. j ,,
62 ,
(.~. ;.~ )
A,~ and B,~, can be rewritten as
{
}
2n + 1 /; n(n + 1) - 6j,, + (eht,2)) ' (6J") ,
i"
' '
2,,+,{ .,-,, } ""
B, =
i"
n(n + 1)
- i t,~-', j'' ~" ~j" " ..,;"
(3.26)
A,, and B,, can also be rewritten in the forms A,,
-
i" - 2 n- + l n(n+
{-6j.+
1)
(6j.)'}.
ia
B . = i " 2n+______~l{ e n ( n + l ) a -( 6 n ( n + 1) ea
1)
a-n(n+
-
i ) j . + 9 .., ,,}. (3.27)
Then comparisons between A,, and A~,, and B. and B~ reveal that in the perfectly conducting sphere problem the SRC method is equivalent to introducing the following approximations ~h},')
ia
h},2)'
. 6 n ( n + 1) - a ( e ... ~ n,"(2) 6a
(eh~,..)) , .~. a - n(n + 1) ,
i) (3.28)
"
Consequen! j, the accuracy of the m e t h o d will be bound up with the accuracies of these approximations. Let us denote the left hand sides of the expressions given in '~ ~o~ by D .~ and E . , respectively. Then by ~.,.<~,, employing the definition of the spherical Hankei function n,, t, ~z~, we find that -6
" ( 2 n - m ) ! _.,..,, ~ 12~,:.) ..... o ( n - m ) ! m !
~t
X-" ( i e + n - m } ,,, ~:o
(2n-m)!
(3.29}
{i(i + n - m ) - e } E t)
.~. ,,,
(n - m ) ! m !
( 2 n - m ) ! (2is)" (n - m)!m!
- - 2 " =o
"~ -(2-n---!-n-)-! (2ig)"'+' ,,,-o(n- m)!m! i.e., D,~ and E~, are the ratios of the two polynomials of degree n + 1. On tile other hand the right-hand sides of the expressions given in (3.29), say D'), and E ) , . are respectively the ratios of two polynomials of degree two and three, in general. Thus the method in this special problem approximates the ratio of two polynomials of degree n + 1 by the ratio of two polynomials of degree two and three, respectively. Therefore, as it should be expected, the range ef these approximations would be limited• This point will be investigated that for n = 0 and n = 1; D~ = L'){',= i,
E,)~-- E,)"
~ -
s- i ..........
as
•
D~ = D ~ -
is: + 6
I ......
T
g .......
............. ,
s'-le- !
g2- 2 i s - 2 E{ = E'~ = --:--a...........
( 3..,0)"
~6-+c
Hence, it is seen that the first two terms of the exact answers and the approximate solutions obtained by the SRC method for h~, and h~ are identically equal. Therefore. since for a sufficiently accurate numerical
M. Teymur / Surface radiation conditions
10
value, approximately, the first ka + 4 terms of the infinite series (3.21) are needed, for lower frequencies the SRC method gives accurate results everywhere on the sphere. Consequently, the scattered far field also behaves similarly, they are accurate in both backward and forward scattering regions. To make a quantitative comparison for higher frequencies and tbr n > 1 the real and imaginary parts of (ej.)'D~., (sj.)'D~. and j.E~., j.E~. are computed for various ka values and have been shown in Figs. l(a)(c) and Figs. 2(a)-(c) as functions of n, where the thick and thin solid curves are represents, respectively, the real and imaginary parts of the exact values, whereas the thick and thin dotted curves are the SRC results. Note that, for a given ka, (ej.)'D~., (ej.)'E~., j.D,~, and j.E~. are discrete functions taking the values only on integer n's, although they are presented as continuing curves between these values. From these figures, it can be observed that, for ka = 1, the approximations are quite accurate for all n, and therefore as can be seen from the Fig. 3(a) and Fig. 4(a), the SRC method produces the surface and consequently the far fields very accurately for ka = 1. When frequency increases, the deviations of approximate values from the exact ones become important for n > 0.7ka approximately, and hence the error in D~ and E,] will introduce inaccuracies in the approximate surface field. Note that, for a given ka the coefficients A. and B. of the infinite series (3.21) will be fixed for every 0, i.e. for every observation direction. In spite of this fact - as can be observed from Figs. 3(b)-(d) where the modulus of the total surface field obtained by the series solution of the SRC equations have been presented by dotted curves for ka = 5, 10 and 20, respectively - in the shadow, i.e. for 0
o.°°'~.. .4 ka:
-
1
~5 .sl
~)
0
=
4
N
I!
= ~:
.,~
-!
-.6
la)
(b) !
~
.5
l, }J tcl Vig. I. Variatkms of rca~ and imaginaD, parts of (ej,,)'D~ ( ........... ) and (;;j,t'D~ ( ) vs n for various values of k a (real parts heavy. imaginary paris light curves)' (a) k a = 1' (b) ka=:5: and (c) ka=20.
M. T~ ymur / Surface radiation conditions .1 0
.2
i
..-" ........................
~.........
/ / = ,-..,,
-
11
I
.
=
.1
ka =: ,5
.3
E
"
$
ka= 1
ol
"%1
A-_...-,~
1 ......
.
..... I
8
=
IO Ill
~-.7 .
-.9
2
{a)
1
-
(b)
,...,
A"
.03 =
101 __ ~15
ka
=
20 20
25
3O
t
~,
- .03
.05L
(c) Fig. 2. V a r i a t i o n s o f real a n d i m a g i n a r y p a r t s ofj,,E~. ( - - ) a n d j'.E~ ( ) vs n for v a r i o u s v a l u e s o f ka (real p a r t s h e a v y . i m a g i n a r y p a r t s light c u r v e s ) : (a) ka = 1" ( b ) ka = 5; a n d (c) ka = 20.
i.e. for ~,/2 < 0 < re. This means that, the variations of P~,/sin 0 and dP~,/dO with 0 are also effecting the total error of the method and the alternating behaviours of these angular functions cause the accumulation of the errors introduced by the approximations (3.28), i.e. by the SRC method, towards and over the dark region. This observation indicate~ the nonuniformity of the SRC approximation with respect to the angular variable and therefore it may then be concluded that to include the effect of creeping waves more accurately in the high frequency range it is necessary to modify the tangential operator of the SRC.
3.2. As),mptotic solutions of the sRc equations The SRC equations (2.14) and (2.15) given for the tangential components of the total surface field h can be reduced to the following equations for the tangential components of the scattered surface field h ~ (3.31) sin 0 d0
shl 2 0
(sh! 0 h,~) -:~. . + ahT~= c~i +
dO 2
where A
h~%= -sin 4~ h~%,
h~ = - c o s 4) ~ .
( 3.33)
12
M.
Teymur
/ Surface radiation conditions
e s . s,,.
2 , r.,
--.y
.=_~ ka ~
1
'
..~1
I
I
SO
I
I
I
120
(a)
I
I
J
,
0
180
o m
"7
_
"\ ~
Ifiol
l
t
o
i
1~
180
120
{c)
'.
\
._'i,
120
180
O(degrees)
~
lfiol
"
I
60
, 60
"';"
{b}
21 l,g=~°m
,
Nv
.(degrees)
m
""" : ~
,,o, i N
lfi.l 0
~
0
60
0
ldegre sl
{d)
120
180
OIdegr e l
Fig. 3. Magnitudes of/~,, an, ~. % vs 0 for various values of ,~a. Exact answers ( ..... 1, the series ( ..... ) and perturbation ( ......... ) solutions of the SRC equations, and the physical optics results ( ..................... ).
We will now seek solutions for the S R C equations (3.31) and (3.32) for large e = k a by applying the W K B J technique which was used by K r i e g s m a n n and M o o r e [7], in c o n j u n c t k m with a scalar surface radiation condition, to stud) ~ the scattering of acoustic waves by an impedance coated sphere. F o r this purpose let us first define the functions H ° and H ~ as
H°=hbexp(-iecosO),
HC'=h;exp(-iecosO)
(3.34)
and expand these functions in the following asymptotic series in s -~
H°~- ~ ,. ,~,,, n =0
~ Y'. s-'H°,.
(3.35)
n =0
If we employ the expansions (3.35) in (3.31) and (3.32), then by collecting the terms of like powers in s -~ we get the follov~ing hierarchy of e q u a t i o n s from which it is possible to determine H ° and H~ successively; • ~ i2-san-0)H,~,-i
te s i n 0 dH;,~.-~ d(i +2c°sOIt°'"+2H~u
_ Uo -"~t
+ d-H,,~. ........... 2 + c o s 0 dH~_ 2 d0 ~ sin0 dO
...................................... 0 it2-, l dH~[ :, ........................ 1 I ! ~ .... + - cos : - ; ......... sinO dO sin:O " sin'0 ' " .. cos 0)&,, - i( l + ~. ~ cos O)SB. . . . . .~ . . =. . 0, 1 "~
(3.36)
M. Teymur / Surface radiation condilions
13
°l
/ o = ~/2
j
. . . . . . .
1
.-..~.
~° : ka = 5 t=
\ / o=0 ,~,../
.
1
J
/ ~..
y
Z ~ ) ~'
-.-. :....
I~ ~.. . . . -~ I I
L¢/ •
II
.1
.2
t
l
l
I
l
60
0
....I
.
I
...
l
(a)
.02
l
180
120
-
,[
I
0
!
I
I
,,, i
I
(b)
O(degrees)
I
!
120
60
180
#{degrees)
1000
100
100 10
,, = : / 2 10
,, = x / 2 oj I
%
%
l=
i=
"-
100
ka = 10
10
•
_ .
ka = 20
'I
.01
I
60
0
120
1$0
_
i
_a
_ _ l ...........
60
0
O(degrees)
ic) FiF. 4. N o r m a l i z e d b i s t a t i c r a d a r cross s e c t i o n s
-t~H.-l+
}+
I
I
........
l
L .....
120
I
180 0(degrees)
or(O, x/2)/rta 2 a n d cr(0, O)/rca "~vs 8 f o r v a r i o u s v a l u e s o f k a . E x a c t a n s w e r s ( ........... ~, vi:~
]
~ ~,~
sin 0
=(2-sin 2 0)80.-i(2+cos
|
cos 0
. . . . . ~m: + _ _
dO
0)81.,
sin" O
I H~_ 2_ _ _
sh~2 0
H~_ :
n = 0 , l, 2, . . .
where H~., = H~_1 = H°__: = H°_~ -=0, and 6o., 81. are usual Kronecker symbols.
(3.37)
M. Teymur / Surface radiation conditiort~
14
From (3.36) and (3.37), for n = 0, the zeroth order solutions are found to be Ho~ =--,2c A
H0 =_A 2
(3.38)
c = cos 0,
A = 1 + c".
(3.39)
where
Then, by using the zeroth order solutions (3.38) in (3.36) and (3.37), for n= 1, after some algebra we get H~=i
1 1 2 A
2c 4 4c 8ca 8c" 8c4] =i/t~, " A A"~-A-5+ + a 3 a"/ (3.40)
H°=i~
+c"-c-I
=i/4 °.
Proceeding to the nigher orders requires similar effort. But here an attempt will not be made towards obtain!ng them, since the higher order solutions become less effective as e increases. Thus, up to first order in e-~, the tangential components of the total surface field h are found to be h o ~ - c o s ~/~o=-cos ~(1 + Ho~+ i~:H~) exp(i~: cos 0), (3.41) h 0 ~ - s i n ~b/~o= - s i n
q~(c+H°+ i~/-t°) exp(i~: cos 0).
3.3. Comparisons For comparisons, the modulus of the total surface field components/~0 and/~o are computed from the exact answer of the problem, the PO approximation presented in the Appendix, and the exact and perturbation solutions of the SRC equations for various ka values. They have been shown in Figs. 3(a)-(d) as functions of 0 by full, dashed-dotted, dotted and dashed curves, respectively. Let us consider the modulus of the surface field obtained by the exact solutions of the SRC equations. From the Fig. 3(a), it can be observed that, for ka= 1, for both I/~01and Ih01, over the entire surface, there is an excellent agreement between the approximate and the exact answers. As frequency increases, as it is seen from Figs. 3(b)-(d), the results become less accurate in the shadow where creeping waves dominate the surface field. That is, as is mentioned in the introduction section, the SRC begins to fail to account for the creeping wave physics adequately. On the illuminated side, as frequency increases, although the approximate Ihol-curves follow the exact ones with increasing accuracy, the same thing is not valid for the Ih,lcurves. They first oscillate around the exact curves in the middle frequency range and later, for higher frequencies, the agreement between the exact and approximate results is improving. From Figs. 3( .O ., I . .~ Q. } it C a l l be observed that in the :l l 'l U"l 't U-t:l a.t r.~.u . .a region, k~.,.;..,.;,.,~ ts,.,E, l t t t t a t t ~ ¢,.~,,~ t A,,.,l,a ka= 5, ,h,~ .... perturbation results are very satisfactory, but for ka = 5, in the shadow beyond 0 = 100° there is a considerable discrepancy between the perturbation and exact SRC curves. However, as frequency increases, this difference diminishes and the perturbation curves converge to the exact curves fo the SRC. It is also seen that on the illuminated side of the surface the combination of the SRC method and high frequency perturbation expansion gives better results than the PO approximatior', from moderate to higher frequencies. This approach also yields a non-vanishing surface field in the shadow, except at 0 = 180°, thus providing
M. Teymur / Surface radiation comfitions
15
a smooth transition between the illuminated and shadow sides. As it will be observed later, this significantly affects the accuracy of the far field results by comparison to the PO approximation.
4. The far field
In this section we calculate the scattering field by employing the tangential components of the surface field obtained in the previous section by different approaches, along with the integral representation of the scattered electrical field intensity E s (2.3). Considering that ~,(x, y) ~
1
4nr
exp(-ikr + ik.~. y)
(4.1)
as r = Ixl ~ oo, where .~ is the unit vector in the direction of observation point x and, x and y are x = r(sin 0 cos O, sin 0 sin O, cos 0),
y = a ( s i n 0' cos O', sin 0' sin O', cos 0')
(4.2)
from (2.3) the distant scattering field E s is tbund to be
E S~(Soeo + S,t,e¢ ) e-i*'/kr
(4.3)
where eo, e# are unit vectors in the directions 0- and 0-increasing, respectively, and
So and
S~ are given
as
So = - ~ ie21~~fo~ 19 exp(ie/4re,
) sin O' dO' dO',
(4.4)
S¢,= - - -ie2;; " f" ~ 4rt
o
exp(ieF) sin O' dO' dO'
with F = ~..P = sin 0 sin 0' cos(0 - @') + cos 0 cos 0', O = {cos 0 cos 0' cos(0 - 0') + sin 0 sin 0' } cos O'/~#"+ {cos 0 sin(0' - 0) } sin @' ho,
(4.5)
= {cos 0' sin(0' - 0) } cos 0 ' / ~ ' - {cos(0 - 0') } sin @'/~o,.
4.1. Series solution We first present the scattering far field obtained by using the series ~ '-"" - this case ~he integrals in (4.4) can be performed in closed forms. To this end the tangential components/~, and/~# of the surface field from (3.21) are substituted in (4.4) and then resulting integrals are evaluated by making use of the recurrence and orthogonality relations of the Legendre and associated Legendre polynomials. Calculations are tedious but straightforward and details can be found in [ ! 2]. The results are So=cos 0 S0,
S0=sin 0 S~
(4.6)
M. Teymur/ Surface radiation conditions
16
where
I
d P'l'+iB,,v,, p,! l ,
g o = i t ~ (i) "÷' A,,V~
t
,,:,
g+=ie E
,,=t
dO
~,.~J
(4.7)
( i)"+' A,,~,, P'' +iB,,~. . " sin0 d0J
Note that the far field secured here via the SRC methed is also similar to the one produced by the series solution of the scattering problem as far as their form are concerned. It can be checked that the substitution of A,, and B, in (4.7) from (3.23) yields the exact expressions for the .~0 and S",.
4.2. Perturbation solution We now present the scattering far field by employing the first-order asymptotic solutions of the tangential components of the surface field obtained in Section 3.2 by the combination of a high frequency perturbation expansion and the SRC. Substituting/~o and h+ from (3.41) in (4.4) and then by employing the expansion, [10],
=
{(o)
ei~t'= Y' (2/+ l)(i)~l(K) Pt cos
Pl(cos 0')
1=0
., m...(o) cos ..
+2 ,,,~ I (!+ m)!
P~"(cosO')cosm(dp-tk')
}
(4.8)
where t: = 2e cos(O/2),
/~= sin(O/2)sin O' cos(~b- 4¢) + cos(O/2) cos O'
and later performing the integrations with respect to ~b' we get (details of these calculations are given in
[i31) So
ic: , = - - 4 - ( ( J l + J 2 + J 4 - J s ) cosO+2J3sillO},
is 2 Se~=~(J,-Jz+J3+J4)
(4.9)
where
( 0/( i
.;,= ~" (21+ l)(i)~/,(~:)P, cos~/ A I + - A
J2-- ~] (i)~b(~c)P~ cos +=. (1- 1)(1+ 1)(1+ 2)
( 7 )B~+ i- B~ ,
_
(4.1o) J4 ~ _
(2l+1)(i t--o
s(x)P~ cos
D ~r + - D t i , E
Js= X" (21+1) (i)~t(l,')P~ cos +~'='.,( i - l ) / ( / + 1)(/+ 2)
E,+ i E , e
Surface radiation
M. Teymur /
condit.ms*
17
with A~= a~l°) + 2a~°°) - 2a~°1),
B~ = b~l°) + 2b~°°) - 2b~Ol),
D [ = >:'a~°°) + a~!o) + 1_2,1_~2o),
E~ = 2011 ~.(o0~+ b~,o) + ~'b~2O),
A~= - '--(10)..m . aP ') . 2a~°°)+. 6a~°l) .+12aP 2) 4a~°2)
C" = a~OO)+ 2a~! t,,
16aP s),
B~ = --2or I_zAIO) rill) -.-ut ,~t,¢oo)+6b}Ol) ,-jt,(12~ ~.~o2)- 1 6 b P 3), --01 + ~.~01 -'*or
C~= -2u1 ,_(oo~- a~O,)- 2aP,)+ D~=
E~=
(4.!1)
12a~°2~+4ap 2) 16a~°s),
I_..(oo) ,..(Io) 4~..(2o;+ a~,l) 2M I ~ ~.t~I -- .u I
I_i.(oo)-- 201 I_j.(Io)+ ~u ~l.(2O)+b~tt) I
--201
in which a~"") and b~""), m = O, 1, 2, n = O, 1, 2, 3; are defined as a~°°)= 26ot,
a~'°)= ~6,t,
b[~)= u21+! ~.¢to) =4, b~2°)-0.8,
a~20)= ~621+ 36ot,
~.(0o) _ ~.(~o)~ - 0~j2o) =0 021+1--u21 21+1
;'(2°)=4 021
for 1=2, 3,
•
•
for 1= 1, 2, 3,
•
-gin
Z~ - n p 2 1 d x ,
u21"(0n)+I ---- 0- (, 0I n2)1- -=
u2,"(In)+,
=2jo
xA-"P21+ ! dx,
v
v
bl °") = 2
(4.12)
r~
t"
a~ on) =
•
A-"P~dx,
~.(o.) ~.(t.)=0, o21+~=02~
~.(I.) o21+~=2
Io
xA
P.;t+, dx.
The integrals appearing in (4.12) can be performed analytically or numerically, but for large / the analytical evaluation becomes quite tedious. This is also not desirable m the computational sense, since these integrals can easily be evaluated numerically with sufficient accuracy. It is, therefore, preferable to employ a quadrature and the most appropriate one seems to be the Gauss-Legendre quadrature. For this purpose, the N A G library routine D 0 1 A R F is used [14]. When performing the summations of the series (4.9) to evaluate the scattered far field, the spherical Bessel functions j , are calculated by backward iterations by using their recurrence relation as described in [ 15]. The Legendre and associated Legendre polynomials are also calculated from their recurrence relations by forward iterations. 4.3. Comparisons The bistatic radar cross section (RCS) is related to So(0) and .~(O) by the expression or(0, q ~ = ) 4~~ [l~0l z cos" 4'+ ISoI 2 sin" q~].
(4.13)
To make comparisons, the magnitudes of the bistatic RCS for ~b= 0 and ~ = n/2 are calculated for various ka values. In Figs. 4(a)-(d) cr(0, ~)/~a 2 has been shown as a function of 0, where the full, dashed dotted, dotted and dashed curves correspond respec,:vely to the exact answer, PO approximation, exact and perturbation solutions obtained by the SRC. We first consider the results constructed by the exact solutions of the SRC equations. From the Fig. 4(a), it is seen that, for k a = 1, for both q~=0 and ~b=n/2 directions, the agreements between these results and exact answers are excellent. On the other hand, as ka increases,
18
M. Teymur / Surface radiation conditions
in the forward direction the difference between the magnitude grows; but it should be noted that there are only a few directions where the relative error exceeds the average error percentage. Also, around these directions, the modulus of the bistatic RCS is comparatively small. Nevertheless, the far field results predicted by the SRC are qualitatively quite satisfactory and ~hcy are superior to the PO results. From Figs. 4(b)-(d) it is also seen that as frequency increases, perturbation curves begin to follow the exact curves of the SRC, very closely. For example, at ka = 20 they almost match everywhere, and therefore the curves could not be depicted separately Perturbation expansion also produces better results than the PO approximation from moderate to high frequencies. Thus, it appears that combination of the SRC and a high frequency perturbation expansion may be a reliable technique m order to apply the SRC method ¢. . . . . . . . . . . . . . . . ' ----~o~h object.
5. Conclusions
To assess the vector SRC introduced in [8], its predictions for a perfectly conducting sphere have been compared with those of the exact theory. :, has been obse~ed that, although the SRC is essentially based on a high frequency expansion it yields the surface and far fields with considerable accuracy ill the low frequency range. When ka increases, the surface field becomes less accurate on the shadow side of the sphere, although on the illuminated side it remains remarkably close to the exact answer. Accordingly, similar behaviour occurs in the far field and the modulus of the scattering field becomes less accurate in the forward scattering region. Nevertheless, the predictions of the method are still qualitatively quite satisfactory, and they are better than the PO approximation at all frequencies. The loss of accuracy on the shadow side indicates that the SRC method begins to fail to account adequately for the creeping wave physics with increasing frequency. As has been pointed out before this stems from the nonuniformity of the SRC approximation with respect to the angular variable and to include the creeping wave contribul+~on more accurately in the high frequency range, it seems that it is necessary to medify the tangential operator in the SRC. Introduction of a higher order SRC might i,,prove the results and this will be examined in a future work. Here, the effect of introducing a high frequency perturbation expansion has been also investigated. The conclusion then is that the combination of the SRC and high frequency perturbation expansion yields better results than PO. Although this analysis is restricted to a special problem, it is likely that similar behaviour occurs in scattering problems for less symmetrical smooth convex objects with perfectly conducting boundary conditio~l. As an application of the method for such a case an investigation on spheroidal objects is under consideration.
Acknowledgments I would like to thank Professor D.S. Jones for his many helpful comments and suggestions provided during the preparation of this work. This work has been carried out with the support of the Procurement Executive Ministry of Defence. I would also like to thank the referees for their constructive comments and detailed suggestions for improvements to this paper.
M. Teymur / Surface radiation conditions
19
References [11 G.A. Kriegsmann, A. Taflove and K.R. Umashankar, "A new formulation of electromagnetic wave scatterir, g using on-surtacc radiation boundary condition approach", IEEE Trans. on A.&P. AP-35, 153 161 (1987).
[21 T.G. Moore, G.A. Kriegsmann and A. Taflove, "'An application of the WKBJ technique to the on-surface radiation condition", IEEE Trans. on A.&.P. AP-36, 1329-1331 (1988).
[31 I.D. King, "Application of on-surface radiation condition to electromagnetic scattering by conducting strip", lEE Electr. Le'ters 25, 56-57 (1989).
[41 J.-M. Jing, J.J. Volakis and V.V. Liepa, "A comparative study of the OSRC approach in electromagnetic scattering", IEEE Trans. on A.&P. AP-37, i 18-124 (1989). [51 T.G. Moore, J.G. Blaschak, A. Taflove and G.A. Kriegsmann, "Theory and application of radiation boundary operators", IEEE Trans. on A.&P. AP-36, 1797-1812 (1988). [6! D.S. Jones, "An approximate boundary conditic,n in acoustics", J. Sound Vibration 121, 3745 (1988). ,, [71 G.A. Kriegsmann and T.G. Moore, "An application of the on-surface radiation t,ondition to the scattering of acoustic waves by a reactively loaded sphere", Wave Motion !0, 277-284 (1988). [81 D.S. Jones, "Surface radiation conditions", IMA J. Appl. Maths. 41, 21 30 (1988). [91 M. Teymur, "Assessment of the surface radiation condition by reference to acoustic scattering by a hard sphere", IM.4 J. Appi. Math. (to appear). [101 D.S. Jones, Acoustic and Electromagnetic Waves, Clarendor. Press, Oxford (1986). [111 J.J. Bowman, T.E.A. Senior and P.L.E. UslenghL Electromagnetic and Acoustic Scattering by Simple Shapes, North-Holland, Amsterdam (1969). [121 M. Teymur, An investigation of the scattering of electromagnetic waves by a perfectly conducting sphere via SRC method, Univ. of Dundee, UK, Technical Report, 1989. [|31 M. Teymur, Application of a high frequency perturbation technique to the surface radiation condition in electromagnetic scattering, Univ. of Dundee, UK, Technical Report, 1989. [141 NAG, Fortran Libra[y Manual, ! Chapter D01, Mark 13 (1988). [15l M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions, Dover, New York (1965). [161 T.M. MacRobert, Spherical Harmonics, Pergamon, Oxford (1967).
Appendix The physical optics approximation for the total magnetic field h on a perfectly conducting sphere subjected to the incident plane wave (2.8) yields h=2hi=-2jexp(iscosO),
for0~<0~<~/2;
h=0,
forn/2~<0~
(A.I)
T h e tangential c o m p o n e n t s of h on the lit side of the surface are written explicitly as ho = - s i n ~b ho,
he = 2 cos 0 exp(ie cos 0),
h~ = - c o s ~b t ~ ,
h~ = 2 exp(ie cos 0). (A.2)
T h e n employing these results in (3.1) we get ie 2 So
(cos 0 cos q~ KI + sin 0 1~.2 ' ' ), 2rt
S, =
is: ~ ( - s i n t~ ~ ),
;,' z"~' - ~ . . ,"~ p
where KI =
cos
sin
e x p ( i x F ) d0'dq~',
/%= j
t'2" ¸¸'2sin 2 0' cos 4,, exp(iK'P) dO' dq~' o
(A.4)
20
M. Teymur / Surface radiation conditions
and ~" and/~ are given after (4.8). Now, by using the expansion (4.8) and then considering that if m = 0 ifm~0
f~~ cos m(0 - 0') d O '= (0, / 2n'
(A.5)
fo '~co5 0' cos m(0 -
0') dO' = In (0, cos 0,
ifm=l ifm~l
Kt and K2 can be reduced to the following forms gl=2n
(2•+ 1)(i)~h(s¢
cos
g~ ,
I
(A.6)
K2=2nc°s0
t
1(1+1)
where
/~'] =
f
nl2
cos
0'
Pl(cos 0') sin
0'
dO',
g~
f
n/2
sin 0' P~ (cos 0') sin 0' dO'.
(A.7)
~e0
~'0
For the first integral, for g~= I/2,
1=0, 1 we obtain (A.8)
gl=l/3
whereas for 1> l, if I is odd (A.9)
g~ =0, but if I is even, [16],
R] = {(l- 2),}/ {(-1)q+2'/22t(~+ l ), (~- l),}. Since, P] = - ( 1 - x 2 ) t/2 /<~= -
(A.10)
dPt/dx the second integral can now be written as
fo'
( l - x 2) __dPtdx dx"
(A.1 l)
and then the integration by parts yields
~'~ = e , ( o ) - 2R~.
(A,!2)
Since, [ 16], for I even P,(0)=(-l)'/2{l • 3 .....
(1- 1 ) } / ( 2 . 4 . . . .
1)
(A.13)
and for I odd
P,(0) =0
(A.14)
21
M. Teymur / Surface radiation conditions then we get
R~= ( - 1)t (2/- 1)!! (20!!
(21-2)! (-1)l+!22t(l+ l)!(l- 1)!'
R~ = -2/3,
K;j+ t --0,
1=1,2,...
(A.15) w h e r e ( 2 1 - l)!! = I . 3 . . . . .
( 2 1 - l), ( 2 0 ! ! = 2 . 4 . . . . .
(2/).