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PROCEEDINGS O F T H E I E E E
Digital Com puter Solut ions of the Rigorous Equationsfor
Scattering Prob lems
Abstract-A survey of recently develop ed techniques for
solvingthe rigorous equations that rise in scattering problems is
presented.These metho ds generate a system of linear equations for
the un-known current density by enforcing the boundary conditions
at dis-crete points in the scatter ing body or on its surface. This
approachsho ws promise of leadin g to a systematic solution for a
dielectr ic orconducting body of arbitrary siz e and s hape.The
rela tive merits of the linear-eq uation solution and the
varia-tional solutions are discussed andnumerical results, obtained
bythese twome thod s, are pre sen ted for straight wire s of finite
length.The computation effort required with the linear-equation
solutioncan be reduced by expanding the current distribution in a
seri es ofmo de s of the proper type, by making a change of va
riables for in-tegration, and by employing interpolation
formulas.
Solutions are readily obtained for a scattering body placed in
aninciden t plane-wave field or in the near-zone of a sourc e.
Examp lesare included for both case s, using a straight wire of
finite length a sthe scattering body.The application of the se tec
hniqu es to scatter ing by a dielectricbody is illustrated with
dielectr ic rods of finite length.
RI. NTRODUCTION
IGOROUSSOLUTIONS existor lane-wavescattering by the perfectly
conducting plane, cir-cularcylinder [I], ellipticcylinder [2] ,
sphere[3], and he prolate spheroid [4]. These solutions areobtained
by the method of separation of variables. Thewave equation, given
byV2# + k2# = 0is separable only in the following eleven coordinate
sys-tems [SI ectangular, circular cylinder, elliptic
cylinder,parabolic cylinder, spherical, conical, parabolic,
prolatespheroidal,oblatespheroidal, ellipsoidal, andparabo-loidal.
Thus , the numberf scattering problems that canbe solved by this
classical method is severely limited,A comparable solution s not
possible for he hemisphere,thecircularcylinder of finite
ength,andothersuchbodies whose surfaces do no t coincide with a
comple teconstant-coordinate surface in oneof the systems
listedpreviously. In fact,difficulty is experienced in
obtainingnumericaldatawith he classical olution even forspheroids
[4] and arge spheres.Variationalandquasi-static olutionshave
hownconsiderable success for scatterersf various shapes, butthese
techniques have been limited to bodies which aresmall in comparison
with the wavelength or are on the
Manuscript received March 4, 1965; revised April 30, 965.The
author s with the Antenna Lab., Dept. of ElectricalEn-gineering, Th
e Ohio Sta te University, C olumbus, Ohio.
order of one wavelength in maximum diameter. Largescatte rers
are handled with the aid of physical optics,geometric optics, and
the geometrical theory of diffrac-tion. These optical solutions
provide reliable data onlywhen the scatter er has a diamete r or
width which islarge in comparison with the wavelength.
Complicationsarise when a portion of the surface is concave as,
forexample, with thehollow hemisphere. Furthermore, thesolution for
each new scattering shape requires a greatdeal of thought and
ingenuity.In the pastew years, with the widespread availabiliof
high-speed digitalcomputers,attentionhasbeengiven t o a distinct
approach to the scatteringproblem.First, a system of linear
equations is obtained by en-forcing the boundary conditions at many
points withinthe scatterer or on its surface. Next, with the aid of
adigita l computer, this system of equations is solved todetermine
the current distribution on the surface or coefficients in the mode
expansion for he sca ttered ield.Finally, one computes the distant
scattering pattern.This linear-equation technique is valid for
scatterersof a ny convex or concave shape, and the exact
solutioncan be approached simply by enforcing the
boundaryconditions at a sufficiently large number of points. T
hecomputa tion ime is least for small scatterers (in theRayleigh
region) bu t i t is reasonable even for bodies ofresonan t size or
la rger, depending on the capacityf thecomputer.Solutionscan
eadilybeobtained orper-fectly conducting, imperfectly conducting,
and dielectbodies. If the bod y is placed in the near-zone field of
asource,he olutionproceedsnhe ame traight-forward manner as in the
plane-wavecase.Thus, the linear-equation solution shows promise
foraccurate, systematic calculations forodies of arbitrarymaterial,
size, and shape. This paper eviews briefly therecent progress in
thisechnique, discusseseveralmethods for reducing the computation
effort, and illus-trates the techniques by considering a wave to be
in-cidenton a straight wireor a dielectric od of finitelength.
11. SURVEYF RECENT ROGRESSSome of the recent applicationsof the
linear-equationtechnique follow:Meind Van Blade1 [ 6 ] calculated
the surface-current density and the scattering patternsof
perfectlyconducting rectangular cylinders of infini te length.
Re-
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Richmond: Computer Solutions of Scattering Problems
797sultswereobtained for incidentplanewaveshavingparallel and
perpendicular polarization with respect tothe cylinder
axis.Andreasen [ 7 ] solved for the surface currents and thescat
tered fields of perfectly conducting cylindersf ellip-tic and
parabolic cross sections, and arrays of parallelcylinders. A plane
wave was assumed t o be normallyincident on these cylinders f
infinite length, and resultswere published for parallel and
perpendicular polariza-tion.Cylinderswithacircumferenceup oabout
40wavelengths could be handled with the aid of a C D C1604
computer.Richmond [SI obtainedsolutionsforarrays of thinparallel
wires of infini teength nd or onductingcylinders of semicircular
and I-beam cross section. Th eincident plane wave was assumed t o
propagate at an yangle 8 with respect to the cylinder axis, with
the inci-dentmagnetic field intensity erpendicularohecylinder
axis.Richmond [9 ] calculated he field distr ibution n-duced in a
dielectric cylinder of infinite length, and thedistant scattering
patterns. The incident plane wave wasassumed t o be polarized
parallel with the cylinder axis.Results were published for
dielectric cylindrical shellsof circular and semicircular cross
section and for planedielectric slabsof infinite height and finite
thickness andwidth. Both lossless and low-loss dielectric
cylindricalshells were considered, and results are given for
homo-geneous and inhomogeneous cases. In one case, the in-cident
field is that of a parallel line source near the di-electric
cylinder.Mullin, Sandburg, and Velline [ l o ] calculated the
bi-static echo widthof conducting elliptic cylinders for
in-cidentplanewaveshavingparallelandperpendicularpolarization. The
scattered ield external to the cylinderwas expanded in a series of
outward-travel ing modes. Inits present form, this technique gives
accurate resultsonly for cylinders which do not depart greatlyrom
thecircular cross-section shape.Kennaugh [ l l ] madeone of
theearlieststudies ofdigital-computer olutions of scatter
ingproblems nwhich the boundary conditions are enforced a t
discretepoints. Successful calculations were carried ou t for
pro-late and oblate spheroids illuminated by a plane waveincident
along the symmetry axis. system of 21 linearequations was obtained
by setting the tangential electricfield intensity equal to zero a t
evenly spaced points onthe surface. Eight coefficients were
determined for thescattered field expansion in spherical modes by
obtain-ing a least-squares solution for the 21 equations.Andreasen
[12] has set up a computer program forcalculating he current
distribution and he scatteredfield of a perfectly conducting body
of revolution. Theincidentplanewavemayhavearbitrarypolarizationand
angle of incidence. The current dist ribution is ex-
perimeter of the conducting body is approximately
20wavelengths.Waterman [I31 hasalsodevelopedaprogram orsolving for
the current distribution and he scatteredfield of a perfectly
conducting figure of revolution. Thescattered field is expressed as
an integral of the surfacecurrentdensity.Theelectric field
intensity s forcedto vanish hroughout a portion of the nterior of
theconductingbody oobtainasystem of linearequa-tions. Greens
identity is employed to decouple the pairof integral equations, to
reduce the number f unknowncurrents by a factor f 2 , and, after
calculating them forone polarization, to obtain by inspection the
currentx-pansion coefficients for the orthogonal
polarization.Schultz,Ruckgaber,Richter,nd chlindler [I41have
employed the linear-equation technique to obtainthe scattering
patternsof conducting cones with spheri-cal caps. The ield was
expanded in spherical mode func-tions of the proper type for each
of the two regions ofspace. Relations were obtained for the field
expansioncoefficients by applying the appropriate boundary
con-ditions.The nfiniteseries for the field wasapproxi-mated with
the finite series of spherical modes, and thesystem of linear
equations for theoefficients was solvedwith the aid of an IB M 7090
computer.BaghdasarianndAngelakos [15] have btainedsolutions for the
current induced on a circular conduct-ing loop by an incident plane
wave, andor the scatteredfield of the loop. Excellent
esultswereobtained ornormal and oblique incidence.Two distinct
approaches have been employed to ob-tain a system of linear
equations for hese scatter ingproblems. One may expand the
scatteredield in a seriesof mode functions (cylindrical modes,
spherical modes,etc.) and obtain a systemof linear equations for
the co-efficients in this series. Alternat ively, the surface
currentdensity can be expanded in a series of mode functionsand a
systemof linear equations s then obtained for thecoefficients in
this series. In the latter case, the scatteredfield is expressed as
an integral over the surface currentdensity and one is led to an
integral equation. The inte-gral expression for the scat tered
field is valid a t everypoint in space, whereas a mode expansion
for the scat-tered field is usually valid only in a particular
region.Mode-series expansions for he scat tered field have
beenemployed by Sommerfeld [16] , JIullin [ lo] , Kennaugh[ l l ]
,Schultz [14] , and others. Integral expressions forthe
scatteredfield were used by Mei and Van Blade1 [6],Andreasen [ 7 ]
, [ l 2 ] , Richmond [8] ,9 ] , Waterman[ I s ] , and Baghdasarian
and Angelakos [15] .
111. AN EXAMPLE: THE SLENDER STRAIGHTWIREOF FINITE ENGTHConsider
a harmonicelectromagneticwave n reespace incident on a slender,
finite, perfectly conductingpressed- asaseries of
modecurrents.Themaximumwireas llustrated nFig. 1 . The
imedependence eiw*
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798 P R O C E E D I N G S O F T H E I E E E A ugustis
understood. The wire radius a is assumed to be muchsmaller than the
length L and the wavelength X. Thewire is considered to be a hollow
metal tube, open a tboth ends. The scattered field may be generated
by thesurface current Jon the circular cylinder=a , radiatingin
free space. With good accuracy, the surface currentdensity on the
thin wire can be considered to have onlyan axial component.
Furthermore, the current density(representing the sum of the
currents on the inner andouter surfaces of the thin-wall metal
tube) can be as-sumed to be distributed uniformly around the
circum-ference of the wire. Th a t is.
J = U(z) (1)where i represents a unit vector parallel to theire
axis.Solutions can also be obtained for solid or hollow wiresof
large radii, but the purpose here is to illustr ate thetechniques
by considering a relatively simple example.
1PA L2.Fig. 1. A plane wave has oblique ncidence on a
conductingwireof length L and radius a.
The z component of t he field scattered by the wire isgive n by
the following expression:
where p and z are the cylindrical coordinates of the
ob-servation point, (a ,+, ) are the cylindrical coordinatesof the
source point, r is the distance between these twopoints:I = d p z +
a2 - 2ap cos 4 + (z - )? (3 )
F ( r , z , 2) = [2r2(1 + k r )- 3 + 3jkr - 2r2) ( r2 2 - ) ~ )
] -+- (4)e- jk r
and k = 2a/X. The angular coordinate + of the observa-tion point
has been taken equal to zero in 3 ) , since thescattered field is
independent of + under the assumedconditions. Equation ( 2 ) can be
derived with the aid ofthe vector potential or by starting with the
expressionfor the field of an infinitesimal electric dipole and
em-ploying the superposition theorem.The current density is related
to the current I(z) by
If the observation point s on the axis of the wire, p = O ,and
the preceding expression for the scattered field re-duces to
wherer = d a z + (2 - 2). (7 )
The interior region of the hollow wire (p
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1965 Richmond: Computer Solutions of Scattering Problems 799and
in nalyticntegration.Althoughheoptimumchange of variab le may be
difficult to determine, oneth at ha s been found very helpful in
the problem of t heslender finite wire is the following:
z - = a tan@ (10)dz' = a sec2e' de'. (11)
With this change of variables, the integral equation (6)for the
slender wire becomes:
[ jk a + cos e ' ) (2 - 3 cos2 ') + k2a2cos e ' ] d8' (12)= -
E,"O, z).The limitsof integration are
0.5L + zae = - an-'
0.5L -8; = t a r 1 aIf the ntegra l is evaluatednumericallywith
heNewton-Cotes formulas [17], the integrand is sampled
at equally spaced points of the variable of in tegration.The
ntegrandvariesmost apidlywhen he ourcepoint is in he vicinity of
the observation point. Thenumber of termsrequired nnumerical
ntegration isfixed by herequirement hat he ntegrandbeade-quately
ampled n he regionwhere itvariesmostrapidly. If the ntegra tion is
on z' as in 6),a argenumber of term s re required to bta in
dequatesampling. If onemakes hechange of variablesde-scribed
reviously,qualntervalsreakenn 0'(instead of in z'),
providingadensesampling n th evicinity of z' = z where
thentegrandvariesmostrapidly, and tapering to lighter sampling as
one movesaway from this region. Thus, fewer terms are requiredin
the numerical integration.V. MODEEXPANSIONF THE
CKRRENTISTRIBUTION
The current on a hollow wire with thin walls mustvanish at the
ends .A suitable expansion for the currentis the Fourierseries
given by
I, cos (2n - 1)-+ I,,' sin-rz 2nrz] . (15)n=l L LInserting this
expression in (12) we obta in:Adz- . G(0') cos (2n -8h2a2 l)I
e'rd
whereG(#) = e-jka/cos 8'
.[(cosO' +jka)(2 - 3 cos2e') + k2a2cose']. (17)The limits of
integration are again given by (13) and
If (16) is enforced at N points z =z1, z2, - , z . ~longthe wire
axis, a system f N linear equations s obtainedwhich have the
following form:
(14).
.v( C m J n + CmntInt ) = - z ' ( 0 , (18)
n=l
C,, represents the scatteredfield a t a point (0, z,) gen-erated
by the current distribution = cos (2%- )az/L,and C,, is the
scattered field a t zm generated by thecurr ent I=si n 2nazlL . The
complex mode amplitudesIn nd I,' are obtained by solving this
system of linearequations. The methodof Crout [18] has been found
tobeconvenientand efficient forsolving the system oflinear
equations on a digital computer.TheFourie r series or thecurrent
I(z) convergesrapidly at first,and henmore slowly, as shown nTable
I. In calculating the data shown in Table I, theintegrals in (16)
were evaluatedwith he ifth-orderNewton-Cotes formula (which is
exact for integrals offifth-degree polynomials) using 1000 terms
for the in te-gration along thewire.
TABLE IFOCRIEROEFFICIEKTSOR THE C U R R E N T( z ) FO R A N
INCIDENT PLANE LvA4VE: L=O.Sx; U=0.005X.I . ei = 30 0
123456781091112131514
0.0800 135.8 0.02300.0235 140. 0 0.00640.0132 141.0 0.00350.0092
141.3 0.00240.0072 141.5.00190.0065 -38.4 0.00170.0060 141.6
0.00160.0056 -38.3 0.00150.0053 141.7 0.00140.0050 -38.3 0.0007
1.4272 -35.3.14080.0373 -41.1.01010.016939.3
0.00450.010838.8.00290.008038.6.0021
179.1179.1179.1179.1179.1179.1179.1179.1
~~
-0 .9-0 .9-0 .9-0 .9-0.9-0.9-0.9
ei = 90"-1--I-
In order to reduce thenumber of lin
3 ,4762 -36. 10.084934.20.1700 146.90 .OS47
145.40.039934.80.0256 -35 .O0.0312 145.10.0218 145.00.0169
144.90.0152 -35.10.0140 144.90.0130 -35.10.0121 144.90.0115
-35.1
0.019035.1
ear equationswhich must be solved, it would be advantageous
toex-pand the current I ( z) n a series of functions which
con-vergesmorerapidly han heFourier seriesgiven nTable I. Table I1
lists the coefficients for the currentexpansion in several types of
eries. The Chebyshev andLegendre series appear most promising,but
this matter= - E,'(O, z) (16) requiresurthernvestigation.
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800 PROCEEDINGS O F T H E I E E E A ugustT A B L E I 1
COEFFICIENTSOR THE C U R R E N T( z ) FOR AN INCIDENT LANEWAVE:
L =O S X ; = O . @ X X ; e i=900"",' No' FourierMaclaurin hebyshev
ermite Legendre
1 3.476.374.7589.2929.27623 0.085 3.128.0319.4135.065545 0.055
4.101 0 .0112 0 .3453 0.04210 . 0 4 0 1.871 0 .0146 0 .0073 0
.03722 0 . 1 7 0 4.037.5581 14.3644 Z.ioOs
Table I 1 gives the mag nitude of In n mA for the following
series:
wherex = 2z/L.VI. THEDISTANT CATTEREDIELD
The distant scattered field of a slender wire of finitelength
has only a 8 component which is given byEa(@,)=- -jkro sin e
I(z')eikz' 01 &'w4aYo SLI P'2 (19)
where ro is the distance rom the origin (at the center fthe
wire) and 8, is the scat tering angle measured fromthe axis of t he
wire.If thecurrent I ( z ) is epresentedby piecewiseuniform
function,' the dis tant scat tered ield is given by
where I , is the currenton segmentn, , is the coordinateof the
centerof segment n, nd N is the total number fsegments.If the
current distribution is expressed a s a Fourierseries as in (15),
the distant scatte red field is given by
+ 2jnI,' sin (aL' cos e4n2 - 2L' cos eJ2where L' = L/X.tude
EO,he echo areaof the wire is given byI f the incident field is th
at of a plane wave of ampli-
a piecewise uniform function for the current.See Baghd asarian
and Angelakos 1151 for an example employing
Figure 2 shows several calculated scattering patternsbased on
the Fourier series expansion for the current .The resul ts shown in
this figure correspond to t he c ur-rent distr ibution listed in Ta
ble I. Fifteen cosine modesand 15 sine modes were used, and 1000
terms were em-ployed in each integra tion along the wire.A
comparison of the measured and calculated back-scatter echo areaf s
tra ight wires is illustrated in Fig. 3.The. measured data are
those published by KOUYOUMjian [19] . In the calculations,15 cosine
terms were usedin the Fourierseries for the current and 000 terms
wereemployed in the numerical integrations. The variat
ionalsolution of Tai [20] is also shown. It may be noted thathe
inear-equation olution howsbetter greementwith the experimental
measurements.In principle, a high degree of accuracy may be
ob-tained with the variational solution by including a suf-ficient
number of te rms in the trial function or the
cur-rentdistribution.Inpractice,however, hecomputa-tional effort
increases rapidly as the numberof terms isincreased. In he catt
ering problem or the lenderwire, only two terms are usually
included in the trialfunction for normal incidence and four terms
or obliqueincidence. On the other hand, t is practical to include
amuch greater numberof terms in the cur rent expansionwith the
linear-equation technique.An interesting property of the zero-order
variationalsolution [20] is th at it yields for normal ncidence
anecho area which is independent of the wire radius whenthe wire
length s 0.5 X , 1.5 X, etc. Experimental measurements [21],
however, show th at th e echo area is defi-nitely a functionf the
radius even when the lengths anoddnumber of half
wavelengths.Higher-ordervaria-tional calculations by Hu 22] show
quite accurately thdependence on he radius forwire engths up to
twowavelengths. The linear-equation solution also shows
adependenceon he adius whichagrees with experi-mental data and he
calculati ons of Hu. This depen-dence is slight for the half-wave
wire,but not or the 3/2wavelength case.Van Vleck, Bloch, and
Hamermesh [23], King [24],and Dike and King [25] have obtained an
approximatesolution for the thin wire by solving the integral
equa-tion to obtain the leading erms n he seriesfor
thecurrentdistribution.AlthoughLindroth [26] hasob-tained useful
results by this method or wire lengths upto 2.5 wavelengths, it
does not appear to show muchpromise for longerwires. The rigorous
solution of HallCn[27] has apparently not yet been exploited to
obtain acurate solutions or scattering by ong wires.Solutions were
readily btainedwithheinear-equationechnique or wires of length L =
2 . 8 6 5 X(kL 18 ) and radii = 0.00415 X and 0.0105 X (ka= 0
.026and 0.066) . For normal incidence, the backscatter echoareas
were found o be .6 5 and 2.63 square
wavelengths,respectively,howingxcellentgreementwith
themeasurementsby As andSchmitt [21]. For a length
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1965 Richmond: Computer Solutions of Scatteringroblems 801
ScatteringAngle os (Degrees)Fig. 2. Calculated bistatic echo
area of s trai ht wire vs. scatteringangle for incident plane wave
with angle of incidence of 30,60a nd 90".
o+ --- xperimental Data From8 8 8 Linear-Equationolution
Variational Solution By T a io p - I I I I I0.30 0.35 0.40.45.50
0.55 0.60 0.65 0.70
WireLength L/AFig. 3. Measured and calculated backscatter echo
area of s traigh twire for normal incidence.
L = 3.82 X (kL 24) and radius a = 0.0035 X the calcu-lated
result is 2.8 square wavelengths and the measuredresult by Sevick
[2 1 ] is 2.01 square wavelengths. Thedifference is believed to
arise from phase errors along thewire in the experimental
measurements.VII. THERECIPROCITYHEOREM
When a plane wave is incident on a straight wire, thedis tan t
scat tered field is a function of t he angle of inci-dence Qi nd
the angle of sca ttering 8,. The reciprocitytheorem assures us th
at the phase and strength of thescattered field is unchanged when t
he angle of incidenceand the angle of scattering are interchanged.
Thus,
This provides a useful check on the accuracyof numeri-cal
scatteringcalculations,except in caseswhere thereciprocity theorem
has been incorporated into the
solu-tion.Furthermore,heeciprocityheorem an eutilized to speed up
th ecalculations. Of course, the reci-procity theorem is
automatically satisfied b y any rigor-ous solution.
In the linear-equation technique, the currents on thewire
satisfy a systemof equations of the following type :N
CmnIn = - E,'(&, Z m ) (24)n= l
where Ezi (0 i , ,) represents the tangential electric
fieldintens ity of the incident pl ane wave . The , may repre-sent
the mode-current amplitudes or the piecewise uni-form currents
induced on the wire. From (24) it is obvi-ous tha t th e cu rre nts
will be a function of the angle ofincidence Bi. The dist ant scatt
ered field in any angulardirection 0, is given by (19)-(21), which
can be wri ttenin the following general form:
N- w e i , 0,) = C In(eJsn(e8>. (25)n=l
The scattering functions S,(e,) aregiven by (20) and(21) when
the piecewise uniform or the Fourier seriesrepresentation is
employed for the c urrent.From (23) and (25), it is required thatN
N
In(SJSn(eJ= In(e,)Sn(e;>. (26)n= l n-1
I t does not necessarily follow th at I n ( e J =A.S,,(Bi),
al-though this is the most obvious solution of (26). I t ispossible
to determine by means of (26) the currents I ,for all angles of
incidence once they have been calcu-lated for one angle f
incidence. However, this approachwill not be pursued here since the
technique descr ibedin Section VI11 accomplishes the same end
result withequal efficiency.I t is not difficult to show tha t th e
eciprocity theoremand (26) are satisfied precisely by the solution
using thepiecewise uniform representation, even when the solu-tion
is inaccurate as a result of dividing the wire into asmall number
of segments. Thus the reciprocity heo-rem does not in this case
provide a useful check on theaccuracy. On the other hand, the
reciprocity theoremsnot precisely atisfied by hesolution nSection
V,which employs a Fouriereries expansion or the current.For the
case illustrated in Table I and Fig. 2, the reci-procity theoremis
satisfied to a high degree of accuracy,bu t it s not satisfied so
accurately when the same prob-lem is solved with an inadequate
numberf terms in th eFourier series and in the numerical
integration. This isillustrated in Table111.
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8 02 P R O C E E D I N G S O F THE I E E E A ugustT A B L E
I11
ECHO REA OF STRAIGHTIRE WITH P L A N E - W A V E IXCIDEXTL =
0.5x a = 0.005xNumbe r of Te rmsumbe r of Termsalculatedch oin
Integration in Fourier Series " Area/X*
300300 4 os and 4 in 60 30 0.074284 cos a n d 4 in 30 "0"
0.07390loo0 15 cos and 15 sin 30 60 0.067631000 15 c os a nd 15 sin
60 30 0.06762
VIII. REPEATED OLUTIONSOR DIFFERENTANGLES F INCIDENCEMost of the
computation effort in the l inear-equationtechnique is
concernedwith the ntegrations over hesurface of t he conducting
body and the solution of thesystem of
linearequations.Onceasolutionhasbeenobtained for the currents on a
given wire with a givenincident field, very little additional
effort is required toobtain he solut ion for a new incident field
or a newangle of incidence. The ntegrat ions need not be e-
peated, since they depend only on the wire length andradiusand
he requency of the ncidentwave. Thematrix nversion for he inear
equations need not berepeated,since he coefficients dependonlyon
hesesame factors. Thus, itwould be inefficient to repeat theentire
calculat ion for each new angle of incidence.This can be
illustrated with a simple example. Con-sider he olut ion of the
following ystem of linearequations:C d l + C d z = El ( 2)C2111 +
C2212 = E2. ( 2 8 )
In solving such a system by the method of Cro ut [18],one first
calculates an "auxiliary" matrix C' whose ele-ments aredefined as
follows:Clll = c11 ( 2 9 )C12' = C12/Cll (3 0 )Czl' = C2l (
31)c22f= c22 - C2llC12'. ( 3 2 )
The preceding calculations for the auxiliary matrixeedbe carried
out only once, since the elements of the ma-trices C and C' depend
only on the scattering structure.The elements E1 a n d E2 in ( 2 7
) and ( 2 8 ) represent theincident field intensity a t certain
points along the axisof th e wire. For each new incident field (or
each newangle of incidence n theplane-wavecase),one
irstcalculates
El' = EI/CI{ (33)Ez' = (E2 - Cz{E?)/C22'. (34)
The solution for the system of linear equations is thengiven by1
2 = E ; (35)11 = El' - Ez'C14. (36)
When a large systemof linear equations is involved,
theefficiency is greatly increased by repeating only the
cal-culations ike hose in ( 3 3 ) through ( 3 6 ) for each
newincident field or angle of incidence, instead of t rea tingeach
one like an entirely ew problem.This methodof Crout is easily
programmed for a digital computer. Even for large systemsf linear
equations,i t has been found to befficient with respect o
computa-tion time and memory storage requirements. (As eachelement
in the auxiliary matrix s calculated, it is storedin the location
previously occupied by the correspondelement of the original
matrix.)The ncident field E,i(O, z) enters nto he problemonly in
the right-hand sidef the linear equations. Thusit is
straightforward to solve for the currents induced a wire by any
type of incident wave. Section I X de-scribes the results obtained
when he ncident field isgenerated by a short dipole parallel with a
slender wireas a function of the distance between the wire and
thedipole.
Ix. S C A T T E R IN G BY WIRE IN NE.4R-ZONE FIELDFigure 4 shows
the current distribution induced on awire when the source of the
inc ident field is an electricdipole parallel with the wire. If the
dipole is a t a dis-tance d from the wire, as shown in Fig.4, the
tangentialcomponent of the incident field is given byK e - j k
T
r 5Egl(O,) =- ( l+ j k r ) ( 2 z 2 - d 2 )+ k2r2d2]
(37)where
r = d z z + d 2 . ( 38)For the solutions shown in Fig. , the
dipole strengthwas adjusted to maintain the incident field E,' at
unitstrength at the center of the wire as the distance d wasvaried.
It may be noted in Fig. that the current distri -bution approaches
that for an incident plane wave asthe distance to theipole
increases. I t is interesting thatthe current distribution in the
plane-wave case differsnoticeably from the cosine current generally
assumedfor the half-wave wire in the variational solution.I t is
believed that considerableeffort would bere-quired to set up a
variational solution for the straightwire near a dipole source. In
particular, a trial functionwould hav e o be ound or
hecurrentdistributionwhich would permit a reasonably accurate
representa-tion of the true current function and at the sa me
timewould contain not more than about four unknown con-stants. On
the other hand, a minimum of effort is re-quired with the
linear-equation formulation for eachewtype of incident field
considered.
X. SCATTERINGY A FINITE IELECTRIC RODThe linear-equation
technique applies equally well incalculating the scattered fields
of a dielectric body. Ifthe permeability of the body is the same as
tha tof freespace, the scattered ield may be generated by an
equiv
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1965 Richmond: Computer Solutions of Scatteringroblems
WireLength: t = 0 .5XWireRadius : a = 0 .005X
00 0.05 0.10 0.15 0.20 0.25
2-XFig. 4. Currentdis t r ibution n duced on s t ra ightwire by
parallelelectric dipole.
alent electric current density J radiating in free space,whereJ
= ~ W ( E o)E. (39)
E represents the total electric field intensity in the
di-electricbody,given by hesum of the ncidentandscattered field
intensities as follows:E = E i + E a . (40)
The techniques may be illustrated by considering theproblem of
scattering bya dielectric rod of finite length.From a study of the
rigorous solution for a plane-waveincident on a dielectric rod of
infinite length, it ma y bededuced th at th e fields in a slender
rod are nearly inde-pendent of the angle 4. On he axis of the rod
thetangential scatte red ield is given, with the aidf (39)
byEZa(O.) = 0.5(Er- 1) l u E z ( p , ' )F(r , P ) dpdz'
-LIZ
= E,(O, 2) - E,"O, 2 ) (41)where a denotes the radius of the
rod, L is the length ,eI is the relative permittivity,
e- jk rF ( r , p ) = p[(l + j k Y ) ( 2 Y 2 - p 2 ) + k2p*r2]-
42)Y 5
andr = z / p 2 + (2' - 2 (43)
0.016 I I IFrequency: 9 .5 Gc
- - - hysical Optics /inear-Equation SolutionDielectr
icConstant: 2 .54/adius Of DielectricRod! 0.0625" /
0.014 /- Rod Material: Polystyrene //Approximation /o'o'2-
Experimentalesults /
803
0Rod Lenqth lnches)
Fig. 5. Calcula tedan dmeasuredechoarea of dielectric ro d as
afunction of length.
Let it be ssumed th at a plane wave is normally inci-den t on
the dielectric rod with the incident electric ieldvector parallel
with the axis f the rod. In this case, thetotal field E E ( p l )
must be an even function of z . Withthe aid of Maxwell's equations,
t can be shown thatthe total field in the rod can be expanded in a
modeseries as follows:
where Jo(x) represents the Bessel function of zero orderand L' =
L/X. In a rigorous solution, an infinite numberof modes would be
employed in (44), ut here we shallinclude only a finite number N .
The summation in (44)is substituted for E,(O, z ) and E, (p, z ' )
in (41).The ntegral equation (41) s enforced at N pointsz= zm on
the axis of the rod (between z = 0 and z = L/2)to obtain N linear
equations for the mode amplitudesEn. he integrals in (41)may be
evaluated numericallywith the Newton-Cotes formulas, but it must be
ecog-nized that a singularity occurs at z' = and p = 0 wherer goes
through zero. The difficulties involved in handlingthis integrable
singularity may be avoided by changingthe limits of integrat ion o
6
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804 PROCEEDINGSner or polys tyrenerods of +-inchdiameter a t
afre-quency of 9.5 Gc. It maybe noted that the calculationsshow
excellent agreement with he experimental mea-surements which are
also presented in Fig. 5 . The doubleintegration (on p and e ) in
(41) was performed with thefifth-order Sewton-Cotes
formula.Comparatively poor esultsareobtained rom hephysical-optics
solution for this problem, shown by th edashed curve in Fig. 5 . In
th e physical-optics solution,the field in the rod is taken to be
the same as that in arod of infinite ength. The scat tered field s
then con-sidered to be the field generated by the equivalent
cur-rent (39) radiating in free space. This current s assumedto
exist only in the region occupied by the fin ite dielec-tric rod.
Evidently, the field in the finite rod differs sig-nificantly from
tha t in the infinite rod.
X I. CONCLUSIONSIn recent years, we have begun to exploit the
high-speed digital computer to obtain accurate solu tions for
the fields scat tered by bodies of complexshapes.Ap-proaches are
now feasible which overcome the limita-tions inherent in the method
of separation of
variables,thevariationalsolutions,physicaloptics,geometricaloptics,
and the geometrical theory of diffraction.In
hemethodconsideredhere,asys tem of linearequations is obtained by
enforcing the boundary condi-tions a t a finite number of points on
the surface or inthe interior of the scattering body. The solution
of thisset of equations yields the current dis tribution inducedon
the conducting surface.This paper surveys the recent progress in
this areaand discusses several techniques for increasing the
effi-ciency of the calculations and extending the maximumdimensions
of the scatteringbodies that can be handled.Among these techniques
are the change of variables forintegration, xpansion of thecurrent
in a eries ofmodes, and interpolation.These echniques re
llustratedby onsidering acomparatively s imple example: the slender
ire of finitelength. Numerical results are shown for the curr ent
dis-tribution induced on such wires and for the backscatterand
bistatic echo area. Calculations are illustrated fora half-wave
wire near an electric dipole which ac ts asthe source of the
field.This inear-equation echnique is alsoapplicable
todielectricbodies. This is illustratedbyconsidering aplane-wave
incident on a dielectric rod of finite length.The resu lts how
excellent agreement with experimentalmeasurements.
ACKNOWLEDGMENTThe autho r wishes t o express his sincere
appreciationfor the extensive digital computations pravided by Th
eOhio State University Computer Center in support ofthis
nvestigation. The experimental measurements of
backscatter from finite dielectric rods were carried out
O F THE IEEEby S. A. Redick of T he Ohio State University
AntennaLaboratory, whose ssistance is gratefully cknowl-edged.
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