42
GEOPHYSICAL. RESEARCH PAPERS No. 15 OF ELECTROMAGNETIC WAVES FROM SPHERES AND SPHERICAL SHELLS A. L. ADEN July 1952 Geophysia Research Division
AD~26688
GEOPHYSICAL. RESEARCH PAPERS
No. 15
~BACK-SCATTERING OF ELECTROMAGNETIC WAVES
FROM SPHERES AND SPHERICAL SHELLS
A. L. ADEN
July 1952
Geophysia Research Division
GEOPHYSICAL RESEARCH PAPERS
No.15
BACK·SCA TTERING OF ELECTROMAGNETIC WAVES
FROM SPHERES AND SPHERICAL SHELLS
A. L. ADEN
July 1952
Geophysics R...ardt Division
Air Force Cambridge Research Center
Cambridge, Massachusetts
THIS DOCUMENT IS BEST QUALITY AVAILABLE. THE COPY
FURNISHED TO DTIC CONTAINED
A SIGNIFICANT NUMBER OF
PAGES WHICH DO NOT
REPRODUCE LEGIBLYo
ABSTRACT
Th dulical the.y of the IC&tkrinJ of a plane e&ectromapetic wave b~· a tphere it re
'riewed, aad abe ditic:ultiel in &ettin« numerical antwen from thi11 formal tolution are dia
a.ed.. It it lhown bow the oomputatioat may be amplified b' uaiq tuitably defined
lopritluaie derivativt' fuuctioat and the recurrence formula• due to lnfeld. B~ thit method,
nwaerical aa.wera for the IC&tterinJ amplitude coefticientt of an~ order can be eomputed
euetly, enn if the indelt of refraction it complex.
Tbe tedmiqae mentioned above haa been uted to determine the IC<eriDC amplitude
eoelieienta and abe bac:k-t~catterinc eroet eection for one tpecial caae involviq water tpherea
with .._ eomparable to the wavelength. The theoretical retulta are eompared with thole
obtaiaed e~tperimentally.
A ripoul tolutioa it alao pven for the ~e&tterinc of a plane electromapetic wave &om
two coacentrie apbern of dilferent dieleetric ooaatant. Thia problem ia formulated in a
IIIAilDel' aimilar to that for a uncle aphere, and the IC&tteriac amplitude eoef&eienta are ell·
~ in ten111 of 8pherieal Bettel functiona and tbe ~arithmie derivative function&.
The applieatioa to a particular phylical problem ia indialted.
ACKNOWLEDGMENTS
Part of the preeent paper review• work which wa11 performed while the author wa• at Harvard Univenit~ •• an RCA 1-'ello" in Eleetronin undn the 'ational Reteareh Council. 11ae author i• happ~ to aclmo" ~~~ the helpfuladvft and oon1tant enroura~ment of Prof_,.. R. W. P. King o( llananl l,;nh·enit~. Thank• are ai~WJ due Mi11 J. Klima• and Mn. E. llaurwit& for allli•tin~ "·ith 11~ c·ompulalion&.
Acs:NOWLBDe.ll:l'f'n •
CLOUAII'I' ••
CONTENTS
ol
I~
I. ll'l'l'llODUCI'ION II
2. C.OIIII SECTIONS .,. ScA1TBiliNG AND AB!!Oilri'ION 1:!
3.
2.1. Total Croee Section• ..... 0
2.2. The Bac:k.SC.tteriat CroN Section
2.3. Restrietiona Gil the em. Section•
3.1. lntroduetioa
12
13
II
H
II
3.2. Formulation of the Problem J.;
3.3. The Far-Zone Field aDd Total Powf!r Relatiou :!0
3.4. The Back.SC.ttea'in! Cro.. Section . . . . . . :!2
3.5. The Sean~ Amplitude Coeflieienta 0 • • 0 • 23
3.6. Evaluatioa of the Scatterin1 Amplitude Coeflicientt Ullin1 Loprithmic: Derivative Functiona • . . • . . 0 :!(•
3.7 • .!'iwnerical Computatioa• 24J
4. EXPE•r•ENTAL DBTD .. NAnoN or THE BA.ck.-Sca1TUII'fC Faow W \TU SPIIKIIKS 0\ND Co•· POtiON WITH Tllr.oay :t I
4.1. lntroduetioa 3.J
4.2. The Esperimeatal Problem
4.3. The Esperimental Tecbnio{UN
4.4. Caaapui1011 of the Theoretical and Ellperimental Re~~~lt•
5. TIIB TBII:Oa'l' OP THII: ScAft'BiliNG OF A PLO, .. E W.\VE BY .\ SPHERE WITH ' eo .. cK~IC
35
35
SPBBIUCAL Sarw. :ti
5.1. lntrodaetioa ............ .
5.2. Formulation aod Solution of the Problem
Buuoca .................. . II
8
l1pw
1. s.••rical Coanliute s,..._ 2. Tile •• F--. far B..J Arp.eat
s. ne •• Fa.etioa • .... Arpmeat
4. R•l P.n altiMI ,_ F1111etiaa
5. 1••....,. Pan ol the ,. Fa~~etioa .
ILLUSTRATIONS
6. n. •• F-.:tiaa far C..piR Arp.eat
7. The _.. F....-. far C.oaaplex .\rpmeat
8. The •• Flllldioa for Complex Arpmeat
9. The •• Faaetioa for C.omplex Arpmeat
10. The •4 Fuetioa for Complex Arpmeat
11. The •• Fuetioa for Complex Arpmeat
12. The •• Faaetimt for Complex Arpmeat
13. The .. , •• Fmldiaa for Complelt A~"KUIIIeDI .
14. The Seatteriac Aapltude Caeliciata. at" ........
J 5. The Seatteriat Amplitude Coelicieatr.. ••" aDd ••' .
16. The Sr.atterii!J Ampltude Ceelicieata. •• •, ... aDd .,.
17. The Seatteriat Aaplitllde Coeltcieata. 6t• aDd... . .
18. The Seat..._ Amplitude CoeAcieata. 6,• aDd 6.· . .
19. The Sr.atteriaf Aaplitude Coelic:ieata. ............ ., •.
20. :&.k Sr.atteriac From Water Spbera . . . . . . . .
21. Plane Wue laeideat Upon a Spheft With a CoDc:eatric sp.riW SheD ...
Pate 16
30
30
30
30
31
31
31
31
32
32
32
32
33
33
33
33
34
34
36
38
• .. · ..... • B D.(x)
E H. II> (s)
•""t .... i J.(x)
l:
"· '"· I.m,a -.n N N. N,
P. ·coo. •> P. P. p, Q. Q, Q, , R.(s), S.(x)
s ... .,. "' JJ', JJ',
S., y, I •• ~(s) Z.(s)
"' ..
GLOSSARY
R.di ... of the ephere
Scatteria« amplitude coefticiente
Radiue of the epherieallheU
Fundamental mapetie vector
Geaeral Be-l dealit,- Cunetion
Fundameatal electric vector
llaakel funetion of the ~d k.ind
Roman !etten indicate u..U vec:tore alon• the x, _,. ancl1 nn
Square root of min ... one
Beeeel funetion of the fint lind
..,-ne number of llpAft
Complex wave number of the ~
Complex wave number of the ipllerical .hell
Orthopaal epherieal vector wave Cuoctione
lntepr indicet. Complex index of refraetion of the ephere
Complex iade:x of refraction of the ephere (Set·tion 5)
Complex inde:~ of refraction of the epherk·al .hell
Auociated Lepndre pol~nomial
Total abeorbed power
Total ecattered power
Total power removed from the inrident ,.ave
Total abeorption croM ~ec:tion
Total ecatterin1 creMe aection
Total attenuation creMe eet·tion
Spherical coordinatf'
Be.el deality functione
Colllple11 Po)ntiq vector
Characterietic velocit~ of •pat~
Complex dlaraeterWtic velocit~ of the .,a.err Complex dlarac:terietic velocit~ of the epheric·alllllf'll
Power dealit~ incident upon the ecatterer
Power deneit~ of the l.ar liiOIIC M'&Uered field in •lirf't'lion of •~ttl'n'
Carteeian eoordiaatee
Geaenl epberical Beeeel function
Geaeral eyliodrical Be.el function
EWtrieal liae of the iphere Elet-trieal li~~e of the llflherM'-al .hell
General Josarithmic derivative funrticMI
l>ielet-ti'M· con11ant ofllpM"e
(I nil•
radilln 1 meter
radilla i meter
raclia•/meter
•limelllionlf'M
dimenaionle.
tlimetllionii'N
watt
-•tt -all ..quare aaeter
..quare aaeter
*Juare ..eter
mrter
-·au/equate meter
meter/eeruad
meter/.,...ca
rMter I leCOIId watt/~ meter
radian
radian
9
10
•• r. '·. A
•• E E1 Et E, p.(~) .... (z)
fl
0
"' Subecript i Subecriptl Subeeript I .-
GLOSSARY (Continual)
Real eft'eetin abeolute dielectric coaatant
Chanc:wriatic: impedaace of apace Spherieal ooordiaatee W nelracth Pfli'Bieability of apace
Mqaetic: eonatant of apace Comples dielectric factor
Comple:s dielectric factor of lhe 1phere (Section 5)
Comples dielectric factor of the apherical aheU
Relatin complex dielectric factor
Lopridunie derivative functiona
Real deetive conductivity Back..:atterin~ croee eeetioa
Solid aqle Aaplu frequency Relen to the incident field
Relen to the aeoondary field outaide the aeanerer Helen to the 1e0011dary field iaaide the aeatterer
Alaumed time dependence
U.nt.
farad/ meter
ohm
radiaa meter
henry /meter
meter/henry
farad/meter farad/meter farad/meter dimenaionlea
mho aquare meter
1teradian
radian/ 1ee011d
BACK-SCATJ'ERI'G OF ELECTRO~IAG,ETIC W\H·:s FROM SPHERES A~D SPHERICAL SHELLS
1. 1:'\TRODUCfiO~
II
When an:'· object ia placed in the field ol an ewtroma,.wtic wave it remona erwr~~ frmn the field.
In pneral, 110111e of the e~y ia diaair-tfif intcrnall~· •• heat, and .ome of the er~er1~ ia reradiatf'fl to pro· duce a lftXNidary or ICAttered field. One of the important probleiiUI of elect rod~ namit·• ia to determine
the amount of ellei'IY abeorbed .ad the amount and diatribution of the ec.ttered ener~~ for a JH!rtirular ~atterin~ objeet illWDinat~ under aperified condi1ion1. One of the limpleat problema of ahia 1~ Jlf' to viauali&e ia that of a plane wave incident upon a apherical panicle. AlthouKh the ~tn~etr~ of lhia IM'ftLwm
ia limple, the problem itaelf ia very important and ha1 ft<'eived conaiderable aUentitm from innati~aton
throupout the ~·ean.
~ in tbia field, T~n•ldl (1869a) ahowed e~periiiM'ntall~ the relatittn of tlw Mizr of au,.pendecl
apberical particle. to the intenlit~· and polarisation of ecattered li,J1t. .~lthou1h he "all mainl~ intere•tecl in atudyiq the chemical action of lipt. he di~euaaed 1he problem of the rolor and polariaation of M'atlf'recl
lipt ia ~everaladditional papen (1869b, 1870). Shortl~· after T~ndall'~ work, Lord Ra~leip (1871. 1881.
1897, 1899) publiahed a aerie~ of papen whirh hearne the ballia for Ill"' rlaMic·al tlleor~ of M'aUerinJ. lie handled the caae of panicle. ver~· much amaller than the wavelenttth of li,hl an•leatablilll~e~l hi~ famo1u• re•
auk thai under theae condition• the IC.'alterinl i11 invenel~ proportional to I he fourth 111•"er of tl~ "·avelenttth.
The fint thorou,tJ treatmenl for a _,herieal partirle of an~ llizr an1l an~ f'14"1·tril·aiiii''OJW'rtieto "a~ ttiu•n b~ Mie {1908). lli11 dution ia an infinite ~ of apherK·ai-IDIMk fuftl·titm .. "itt. amrtlitu•ll- ... wRi•·if'ntM
that are determined from the boundar~ rondition11 at the r.urfac~ of tlw Mfalll"n'. •:a,·h of the.r Mftlwri•·•l·
mode functionll iteelf involvee infinite llet'ietl. 110 that in tile ~IM'raf c·a~ or a lar~ 11ftiM'n' -·ith a I~Hftldfo\. index of refraction the problem of pttin~ nurnerit·al anAwerJI i~ quitf' inmhf'fl. llu"rnr, llw \tit- , .. , .. r atill remaina the fundamental contribution to the aubjet-t.
Since publication or the paper b~ Mie. additittnal inveati~atittnll hau• ''"'II mall!- h~ ·~h~r (IIMJIJ). Gana and Happel (1909), Bromwid1 (19'.20). Ra~ (14121 ), Jobat ( l't.2:i ). lllurMr ( l't.!:i. 11126). Slraltun et al. (1930, 1931, 1941 ), Trinka (1935), Engdhard and FrieM (11137). R~cll- rt al. (IIIII. Ill H. 11115. IIIWt), Ruedy (1941, 1943, 1944 ), LaMer e~ al. (1943, 1946, I ...a). llrillouin (1111:4. Ill U, I«Hfl). L. (;..ld11trin (1945), Sinclair (1947), lloupton et al. (1•)49), Klotabaup anti U111·Lrtt (111111) ancl ot"""- Tiw c .. HD·
plete theory ia ~ven conciael~ b) Stratton (1941) and L. Gold11trin ( 11115 ). 'M1i11 "ill be "''-ir"ed in Set·· tion 3.
It ia not the preaent purpotte to review all thr papet'M clealiDJ "ith llw "'·attf'rinJ ,..,,W..m. lnMtrad, the diiiCIIMffln will be reatricted to OIW partil-ul.r aapet•t u( the fM'ttl.a..m.
With the development of microwave radar duriq Wort.l War II, il "'a11 riMJRII that. fur aullit-irntly
.... 11 wavele~~~iha. rain could produce an appreciable l"l'ho anti 1111batantial attl"nuatittn. 11w f'f'lw• ftlw·
nomenoa. waa important for aeveral reaMJI'Ia. For the uaualtat"twal u~~e olthe f'fjuir-nt. it "a• important
to diatinpieh between atmoapherit- relet-tiona and opn-ational tar~ta. \lnn•o'"· a"' a raeteurollrp·al tool, the obaervance of echoea wall v~ uaeful in mappi~~J the rain areaa and in "'"•"·iiiJ tlleir 1110vernent,., The problem of attenuation waa important to both radar and n-muairalittn. It ito nol ~~trra-iAin,. tlwn,
that tM&e problema l'ftt!ived oon.iderable attention.
12
To invuti~ate theontic.tlly thf. microwan ~?.flection from rain, it ie IM'Ce88ary to aMUme that the
.. ...,pe are randoml~· distributed in Spilce and that the mutual i•ttPraetion bet"een drope ie neglipble. The
fint a1111umption appean to be valid without furt~r comment. "111,. aecond auumption ia based on Trink's
(1935) analysis that for Ra~·leigh 8Cattering ,he mutual intf'raction is negligible for ~phere apaciftl& sreater
than two or three a.,Mre diameter&, plus the fact that in actual rain the average apaeing between drop~ is
many timea this value (Palmer ll949) ). Under these conditions the problem can be divided into two
pilrll: (I) finding the drop-aize distribution and (2) finding the refl«tion from a Bingle drop. 'The drop·
lliu distribution ie all!'lctl)· a meteorological problem; the reflection from a single drop ia a problem in electro
dynamiCII.
The fin1 thorough investigation of the reflection and attenuation effects from rain waa made by Ryde
et al. (1941, 1944, 1945, 1946). llis method was to compute the numerical reaulta for a Binpe drop UNO@
the Mie theory and to 11se thP aASumptiOIIII listed above to mak'l application to the ca~~e of many drope.
The main difticult~· lay in getting the numerical results from the Mie theorv. Despite the tremeadoua
labor involved, however, he computed numerical results for a variety of caaee. For radan ofloapr wave
len«fh, it was found that the Harleigh theory waa adequate, but for radan of Bhorter waveleD«th. a IIIOI'e
euct theor)· was neelled. The work. of R)de is di8CuMed brieRy by L. Goldstein (1945), who baa extended
it somev.hat, Ulltll,; morP detailed mPtPOrolo~(' ... data. Receatly a table or acattering functioaa baa been
prepiir:d by Lowan (Jq4CJ) and these re•ulte have been used by Haddock (1947) in a further contribution to
thi, problet '
In aU caAPII, a major difticulty ha11 Jw,.n the evaluation of the refteetion from a ein«le sphere. It waa
an intereat in thi11 prnb!Pm that prompt4'oi part of the present inveatiBatioD.
In Section 2, tlw c-onmwn abeorption and M-attering pilr&meten are defined. Particular attention ia
~ven to the back-acatt~ri~~~ croee eection, which ia the main aubject of thi1 report.
1be clalllical theory of acattering from a ephere ia reviewed in Section 3. The difticultw in ~tti~
numerical an11wen are discussed, and it ia shown how the computationa may be eimplified by uainc loc· arithmic derivative functions and the recurre~ formulas due to lnfeld (1947). Thi11 method is ueed
to evaluate the back-acattering cro&& sections for water spheres at ~ "" 16.230 t'm in the size ranse from
0.6 < 2.-a/>. < 6.
Ia Section 4 a method for meaauring the baclr.-ecattering from water 11pherell ill deaeribed briefly. Com.
parison ill made of the experimental and the theoretical reaulta.
Aaotber physical problem of interest ia the reftectiGn from a melting snowflake or ice particle. AI a
first approximation, one may comider, aa the mathematit".al model I'OI're8ponding to this phenome-. the
reflection from t.-o concentric spheres with different complex dieleetric facton. Uliq the ume method
•• for a lliup apbere. it ia poeaib1e to determine a ripou formal aolutioa for this acattel'iai- This il doae
in SectionS.
2. CROSS SECTIONS FOR SCATTERING AND ABSORPTION
2.1. TOTAL CROSS SECTIONS
The COMept of acatterin« and ab!'OI'ption parameters is found in many brancbea of phyUCA. The moet
COIIIIIIODly uted panuaeten are tboee related to the total power tcattered and the total power abeorbed by
&n object in the field Of an eJectromagnt'til' ~&\t', The~· may be defined &II folio~ .. : Ul
lr-, • the po"·er denllit~· incident upon the object,
P. • the totalecattered power,
P. • the total abeorbed power,
P, • the total power removed from the incident wave
• (P, + P.).
TIIen tbe total ecatterins CI'OM ~~eetioo ia
the total abeorptioa CI'OM ~~eetioo ia
and the total atteauatioo CI'OM ~eetioo ia
P, Q --· • F,
p, Q.--- Q.+ Q .. F,
2.2. THE BACK--scA1TERING CROSS SECTION
(2.1)
(2.2)
(2.3)
Tile bac:k...catterinc CI'OM ~~eetioo • ia a lumped measure of the abilit~· of a ecatterinc oMtacle to re
radiate eDerl)' iD the directioa of the eouree. Under tbe a.umption of a plane electromapetic wave inci
dent UpoD tM ec&tterer, this parameter call he determined from a knowled~ of the far_..,. N'alterecJ field.
It ia COIIIIDOt1ly defiaed •• followa:
where
F, fT- .f.r'
Jf',
W, • the power denaity ia the plaae wave iaeident upon the ecauerer,
,. • - the power dellllity of the far - ecattend field in the direction or the -~ and
r- the cli&tanee from the ....... terer at which the evaluation or rr·. ia made.
(2.4)
It abould he noted that foe an iaotropic ecatterer, the baclt-ecatterifll CI'OM ~~eetinn • ia equal to the
total ecatterbJc CI'OM aeetion Q, •• defined by Eq. (2.1). In fact, • ma~ be defined in ~neral aa the total
ecatlel'iq CI'OM lleCtion of a fictitioua iaotropic ecatterer which ecattera eners~ in all direc·titHI" with inten..il~
equal to that ecattered directly back toward the aoul'ft b~· the actual ecatteriDJ objf'C-t.
Equation (2.4) may be written in alternative equivalent forme b~ writinc ,. .• anti W ·, PMin! ex1Jiic·itl~.
Thus F, ia pven formaUy by
F • tiP, • tiA.
(!!.;))
where tl..4 ia an element of area located a diatanee r from the ecatterer in the dirertinn of the NIUn't'. In
addition, ti.A. • rW, where tiD il an element of aolid anp meuured from lhe M"<erer. hillf1 thi.._
F, _!tiP, r'tiO
4r w. , ---· F,tiO
(2.6)
(2.7)
where ~P./~ ia e't'aluaaed in the direction of the IOUI'a'.
Since the radiation incident upon the acatterin~ object is .::Mumed to be a plane wave,
E.! If"·--· • 2r. (2.8)
where E. ia the mapitude of the e~trie field in the plane wave and so is the characteristic impediUIC!e of ...-.. u.m, Eq. (2.8), Eq. (2.7) ~~eeomea
, _ S.t,IIP.. (2•9) E.' tiU
At aay poiat in the far sone from the acatterer, the ecattered field is aleo eeeentially a plane wave. There· fore. in temu of the far-zone acattered field E ..
i.· 1117 - IE.l'. w. ~ ~~
Sul.titutiiiJ Eqe. (2.8) and (2.10) into Eq. (2.4) yields
(2.11)
F....Uy, if the lar-soae ecattered field ia referred back to unit rallF- i.e., let E., -= rE.- then Eq. (2.11) .....__ IE., I' .,_...,.E. . (2.12)
AU of tbe.e eqai't'aleat lonna are fouud in the literature.
2.3. RESTRICTIONS ON THE CROSS SECTIONS
Ia the precediac sections. several croee sections have been introduced aa convenient parameten {or iadicatiiiJ the ecatteri~t~ and abeorbiDJ properties of material objects. So lonJ as this ia all that ia implied, ao dU&cultiea .n.e. However, they are often UIOCiated with the arbitrary picture that the Poynti~t~ vector ia a UDique measure of the power ftow throu~h apace. The pitfalls of this illOOI'I'eCt picture are pointed out by Km, (1945). It ebould be noted that althou~ all the crou sectiona defined have the dimellliona of aa .,.._ aoae of tbeee "areu" is in &ener•l equal to the physical area of the object. The value of any one of tbe t::to. eectioDa depends in pneral 011 the waveleDJth, polariaation aad aDJie of iacidenoe of the impiiiJinl radiatioa. u well u oa the dimeneioaa, shape and material of the object.
3. THE THEORY OF THE SCATIERING OF A PLANE WAVE BY A SPHERE
3.1. INTRODUCTION
The paeral problem of a plane wave ecattered by a spherical particle was fint studied in detail by Mie (1908). He treated the problem riproual}· accordinc to claaeical electrodynamics. Briefty, hie method ia u lollowa: The Maxwell field equatiooa are written in apherical ooordinatea. Thae equatiooa are simplified by reeolvi~t~ the electromapetic field into tranavene e~ (TE) and tranavene maptic (TM) wu-. ad particular elementary epherical-wave eolutiona of TE and TM type are found. The complete
l:l
eolution for the ecattered field is then writtf'n formall~ a11 tbr ~urn of all the ele~nt.r~ lll>lutiotu with un· known amplitude coefficients. The incident plane wnr is aleo rxprel!l!ed in terms of a linear combination of the elementary apherical-wavr 10lutiona. Finall~, the initiall~ undetermined coefficient a aft evaluated by applyin~ the boundar~· ronditions at the surface of the sphel'f!. This completes the formaleolution.
The complete aol"ution to this problem ia pven conciwl~ b~ Stratton (1941 ). Jo'ollowi~ llaneen (1935), he developa a art of ortho~onalapherical vector wave fuoction .. each of which i1 a eolution to the vector wave equation. The incident plane wave ia expanded in terma of thear functio11.11, and thr IICAttered field ia expreaard formally aa a llimilar expansion with unknown amplitude coellicient&. Ae in the Mie aolution. the unknown coeflicients aft determined h~· appl~ ins the boundar~· condition• at the 1urface of the apbeft. The method of Stratton ia alao ~iven by L. Goldstein (1945), who ahowa the relation between the coeflidt>nta of Stratton and thoee of .Mie.
In the following aectiona the method of Stratton ie ftviewed followins eeaentiall~· the notation of Gold· stein. The difficultiee of obtainin~ numerieal an&wen for the caar of a •pheft with a complex dielectric'! conatant are dieculled, and it ia shown how lo«aritbmic derivative functiona and the recurftnce formula• due to Inleld (1947) rna~· he used to aimplify the computation&. FinaUy, thi1 method ie used to obtain quantitative reaulte for one apeeial caar invollvins water tpheftl.
3.2. FORMULATION OF THE PROBLEM
Conaider a ephere of radius a, complex dielectric factor E and permeabilit~·,... which ia ieolated in free apace.• The interior of the apheft is called ft~ I; the IW'I'OIIndi~ space is called ftpm 2. The center of the apheft i1 cho.en a& the origin of a rectansular coordinate eyatem. A plane electromaptic wave, with the electric field parallel to the " axi .. ia propa~ated a~ the .z axi1 and strikea the aphere. Periodic time dependence of the form~ is auumed i but in poeral wiD not be written explicitly. MKS unit& are ueed thi'OIIIhout. Quantitiee with bold letten ftpreBent apaet! vector&; accented quantities in roman leuen - e.,., 1, j, 't, ~. i, t - reprewnt uait vecton.
The incident plane wave may be ell.~ ••
where
ia the wave nUIIlbu of apace and
il the cbaraeteriatic velocity ol apace.
• 2r "-c.~~)-).
(3.1)
(3.2)
(3.3)
(3.4)
•'l'bi. i. a ..-ul "- ol the- c--.1 ,......._ treatN by Mie and Strattc. w'-" the _.......,... .....ti- _,. doo h.ve ~ eleetrieal propeniea. Howner, the _._._. ol.autioe i. idnttieal. ·
+ Svat._ -- .-iwl U.. -,.ll.cP. ~ ia _,.na, equatU. writlell hen with ''-of Strata- repa- +i by -i.
16
X
kEI
- ---NT
---z 81 PLANE WAVE
y
11le ortbotonalsphericlll vector wave functiona .re obtained as follows: The ecalar wave equation in ~I coordinate. i1 .oJved by 1eparation of variable. to ~·ield
/, - ~.... P. • (0011 B) a..~ (kr). o•• •n (3.5)
Here, P.• (0011 f) i1 the a~Meiated Le~ndre pol,_ial ofthe lint kind, nth de~ and mth order; a,.fl>(l:r) ia the •pherical Deuel function defined by
a..10 (kr)- (;.)'J.+1(kr) (3.6)
._M(AT) - (;,Jn.+1 (1>(~tr). (3.7)
J.+l(ltr) ia the Deuel functm of the fint kind and half-inte~r order; H.+100 (ltr) ia the Hankel function of
the leCOIId kind and half-intep order. The IUbecripts ~ and o refer to the even and odd • dependenee. The ipberical vector wave functiou are defined as
~~ - v/.. ... . .... (3.8)
m.~ - V X (r t_ ' ... .~ (3.9)
, 1 , a, •• - -V X aa,
0 k o•• Carrym, out the in.tieated operation• ~·ield1
, a ,. eoe 6 1, - -.a.. (lr) P.•(eoel) . ,.. r .- ;w -
+! .a.._,(kr) ~ P.• (eoel) ~,... 8 r af Rn
+kr:,;[n._,(kr)] P.•(eoel):-.f.
17
(3.10)
(3.11)
(3.12)
(3.13)
The incident plane waYe u aow e~:panded in ten111 of the tpherieal veetor wave lunetiaaa. The relatioo betwr.!ln the reetanplar and epberical coordiute 3y1tem1 il readily ~een from F"as. 1. Thu.
1.-.-- .-• -• (liD I eoe•? +COl I COl• I - lin • ~)
je-Jh - .-Jiw _, (lin I lin • 9 + COl I lin • t + eoe • *>· (3.14)
(3.15)
Sinte"te-.- and~-~ are IIOienoidal vec:ton. the~ may be upanded in tern~~ ol aa, •• and •·- able. They 0 0
are fiDite at r - 0; therefore their upalllionl ft41Uire Be.el fuoctioul of the fint kind. Allo. their • de-pendente require~ that m • 1 in the expaaliona. Fiully, compari80U of the ena aad the odd propertift of FAJL (3.12) and (3.13) with EqL (3.14) and (3.15) lhow1 that the formal expall8ioM are
te-•-• - fcc.•-. •• • + c.• a ... •) (3.16) ·-· ... -}tr-1 • J I 4 I ,. - I:(C. mo~. +c. a.a. ). (3.17) ·-·
The t:Oeflicientl are evaluated in the u~~~al wa~ by applyir~~rhe ort ....... lity conditiona. ··or eun1ple, to evaluate C.1, Eq. (3.16) ia multiplied ~ealarly by lin I -., •• 1 and the re1Uitir~~ equation is int~~ted onr I aad ,._ Thil pv• .r: f. f·aa.1_.
1 .-•-• lin I M ~ -.r: r f:CC.1 ta.1•
1 + C.'a.1•1)·aa.1.·'1iniM~. (3.18) I J, •-•
IR
Similarl~-.
J: [ m., .... a.1.' Nn I Jl ~ • 0
!.~ [ m.1•1 ·m.1••
1 !'in I Jl ~ - !:!r(nj'!_ + I :)2 (.;,. IJ) (kr )J: !!n + 1
(." I ., [~~] (- ')• • n(n + 1) J '
c.~ .. -[,.~:~-!,] (-j)"~'
c.•- - [ !!n + 1--J (-j)•- -c.' n(n + 1)
C • [ !!n + I ] ( ')•+ 1 (' , " ""' - n (n + I) - 1 = · • •
TJw,...fc'", tlw inc·iclf'nt 1.tarw ,.·an eapanMion11 a"'
1-;.."' [ :!n + I ] B.=- -I:(-j)• ( 1
(m.,.• -jn .. ,.•·t. r~ •-• n n + )
(3.19)
fur n' ~ n
for ,• • n (3.:!0)
(:i.21)
( :t:!:!)
(:U.'i)
(3.26)
(3.27)
11w induc'f'd ~"tNJdar~ field i11 rnnollruc-trll of t-..·u JNtrt~. one• a1•pl~ in~t in..itlc- tlw IIJ,alf',... and the other apfd~·in!f at all jlliHOtK OUtNde. 'f'hnr I)Mf'tll al'f' "'rilll'n Mil 1''\fNIDI'ionll toimifar to tiH~t~e rur the inc·ident plane "ne IHat ,. il h unJ..no-..n amt:.litucle ... ~flic·wnt~<. TIM' JNtrl a1•tll~ in' uut11icll' 1lw "l"tel'f' i11 refe~d lo &II the ~·all«'n-.1 fM.Icl and ioc indwalfil h~ a IIUt.-riltt "· SirH «' thi .. fwlcl mu111 hr rr~ular at infinit ~ and mullt Ntillf~· thr radialicNJ I'«Niditictll. 1~1 furH'IH"II of tlw ~·cNJcl J..incl ano rrctuirt'«l. Till' ... ·allrrt'«l fM.Id i11 wrillen
E. = 1-~, ~ (-j t [-~(~- + 11
] (a.'m .. ,.• + jl,;o ... ') .~, lltr+ I
(:t2H)
1-~. , . • [ :!n + I J , , . ' , B = - . L( -J) -( + 1- (1,, m.,. -Jn. D .. 1• ).
'"•-• nn )
Till' fwlcl in..Nif' tlw "JIIwrr j,. nofrrrt"tl lo a11 lhr tranllmilll'll fwld ancl it j,. inclic•atl"fl b~ a .... t .... ·rit•t 1. Sinc'f! I hill fwlcl i .. finilr at r - Cl. llt-tw-1 func•licNIII ur 1111' lint kind are rt'«JIIinocl. :\loon. IIIII' fn't' .. , .. ,"t' C'Hnlllalltll a"' "'1"-• ... d h~ the ,,.""1~~~'' ........ tantll of tlte llf'hrre. TitUIIt thr romtlll'' ,.nr numlwr
(:UI)
l'l
replaclel ..._ In Eq. (3.29), t, ia ttw real elective abeolute dieleetrie oonatant and •. i11 1tw real effective
caaductivity. The traMIIIIitted field ia now written
E,- E.~~ -j)~ [,.~:! ~,] (a.'m.a .. 1 + j6.'a.,.')
B,- -E. i:.< -j)• [ 2(n + 11)] (6,.'m.,,.a - ja.i.g.,.•). l'a ••I ft fl +
(:J.J:J)
Equatioaa (3.28) and (3.29) repreeent a formal eolution for the M"attered field. .~II that i" ._teet to
~plete the fermal eolution ia the evaluation of the amplitude ooeftieienta a. • and 6. •. Thili i• dcMW b~ apt.l~
iq the bouDCbry conditiooa at the aurface nf the aphere. The boundar~ OOI'ditionl at r • a are •
The • compoaent of Eq. (3.M) yielda
~ X (E, + E.) - '? X B,
1 X (B, + B,) - ~ X B,.
p,.a.!.CO:I) { ~a"1 (a) + a,.•a,.(l)(a)- a,.'.a,(ll(Na)}
(3.34)
(3.35)
+ . dP,. Ill (roe f) { [Cd. (I) (a)J' + 6.1Cd.. 111 (a)J' _ 6.'(Ncra. <•> (Na)J'} • O. (l.J6) 1
• 1a ltaa
The I ~poaent of Eq. (3.34) yielda
tiP,.1 (roe I) { } • .a, (ll (a) +a,. ;,. ())(a) - a,.'a .. <U (Na)
+ . p,.a (roe I) { (Cd.. Ill (a)J' + 6.'[Cd.. Ill (a))' _ 6.1Ned,. 111 (1\'a)J'} • O. (3.37) J ani 1a kaa
The • ~poaent of Eq. (3.35) yielda
P.• (coa I) { ._ Cl) (a) + 6.•a. <ll (a) _ b.'a. (I) (i'Va) l ~· ~ ~ j
+ .dP,.1 (coal) {(Cd..111 (a))' + a.1Cd..11>(a))' _ a.'(\'Cd,. 111 (Na))' l. O.
J • .,.,_ r,,taa l (3.38)
The I t'JGIIlp.lllftt of Eq. (3.35) yielda
dP,.t (roe I) {._(I) (a) + 6~ '.:, u: (a) _ 6.'.:. 111 (Na) l • ~ · n l
. P.1 {roe f) fla.~,.<•>(a)J' + a.•[a-~,. 11l(a)J' a.'[,'\Cd,. 111 (t'Va)l' [ 0 + 1 an 1 ...a - a·J;rt~ - I • . (3.39)
Equatiooa (3.36), {3.37), (3.38) and (~.39) are all aatiafied if tlw- quntitieto in tlw- brac'f'll I l al'f' idftttirall,·
f111UAI to llei'O. Thit ,-ielda (l.IO)
·• Eq.U.. (S.SS) a.lw ................. witltuat the ....-oe -.nt.o. .a,....,t.oth ...P• I and..._;..,.:!...,. -t ... haH
--... WNt,..
.. ,N-..(I)<N•>Y- .. 1-..•<•>Y- [-..(1)(•))'
M..' .. (J)(Na) - 6.-..•(a) • ._<0 (a)
6,.,N-..ro(Na)f- N6.'(-..•(a)]' • N{-..ro(a)]'.
n-. -.......o.. ......... .,. ..a..t lar the ea • i-ta. Tla-,
.. 00 (Na)(-..ro·(•)Y- ._10 {a}[N-..1l)(Na))' ... - - .. m(Na)[-..Ci(•)f _ ._Ci(•)[N-..ro(N•)J'
._ .. <I)(•)(N-..ro{Na)f- N'._(I)(Na)[-..11)(•))' •- - .. •(•)IN-..M(Na)f- JY'._M(Na)[-..Ci(•)f.
(3.41)
(3.4.2)
(3.43)
(3.44)
(3.45)
Here the pri-. at the ....-. ......... iadieate ..._dadae with l'elpflCtto the .......-• ol the &e.el '-etioa a.w. the llnebta. • • ._ • 2 .. /A; Na • i .. ; N - iJi • (f/tt)t M the .,..p~a iMn ol re· lnctiaa lar the .........
3.3. THE FAR.ZONE nELD AND TOTAL POWER RELATIONS
To.__ the eeattered aDCI U..W power, it il aecmary to bow the r.r....,_ eeattered 6eldi.e., the Wei lar r > & T1lil il lauad by iatiOCiuc:iq the •Y•ptotic .alae lar the ...... Hub! f'uactioD
(3.46)
illto the es.,...._ ,__..... lar the eeattered &eld. Ecp. (3.21) aDCI (3.29), ud -clecbac tenia of biper ..._. &baa (1/r). W1lea tiM il daDe, it M .. that the far....,_ eeattered Wei ill _,...t fora il
E. - B. • 0 (3.47)
E,.- ..... - (L) E.·-jllr I: 211 + 1 ( .. • ~-~ + ,.DJ.l) (101. (3.48) " ·-·•<• + 1) -· •
E,. - - ..... - - (L) E.·-· I: 211 + 1 ( .. • DJ.l + .... ~-~).... (3.49)
lrr ••• 11(11 + 1) • -·
The total Wei at aay pai11t nurul to the lphere il abtaiaed by weetor lldditica ol the iacideat aad eeattencl 6elda. n....
a - a. + •. ; B - B. + B., (S.SO)
The -p~a Poyat~ac Yector COiftJIII oediac to thil total Wei • deUecl by
(3.51)
Tbe--- iBIIicaa. ... -pia CIODjapte. allll '- il the ..apetie ....._, o1.,... (~ • 1/,_). Siaae the lar.-leld ........... the Por•lillt Yector il ndiaL Sal.tit111iac Lt· (3.50) U.to Lt· (3.51) ..... eanyia& oat the weetor ...-1-t yieWI
:!I
S - 'tS, - ~{~ (E;,B,.• - E~B;t*) + ~ (E.,B .. • - E .. B.,*)
+ i (E ;,B .. • + E_.B .. • - E.,IJ,,• - E .. B ,,•) l (:U2)
The -pies Poyntiac Yeetor eorrapoadinc to the iDcident field ia
. ... S; - tS;, -1-2 (E;,B~ • - E.,IJ;t*),
aad that c:orreepoacliJI to the !Cattered field i•
S, - ~S, • ?i (E,,B .. •- E .. B.,*).
The total ecattered power ia obtained b' intev•ti~a~ s .. over the aurface of a la~Je aphere of radiu11 r aad t•kina the real part. Thua,
P,- ~Re}. J.r(E.,B .. •- E .. B.1*)rainltll.. (3.:i5)
The total power Uaorbed by the epl~ere ia obtained b~ intepti~ S, over the aurf~ of • lup llphere •nd takint the aec•tive of the lUI ,-rt. Thua,
P.-- iRe). [ S,.Sainlt114. (3.56)
The total power removed from the incident wne ia equal to the aum of P. and P.. Deaote tbia by P,. Since the iatepal of S;, over a cloeeciiUI'face ia aero, P, can be es,.._d u •n intepal olthe third term in Eq. (3.52). Thu,
P,- P, + P.- - i Ref.rr [ (E,,s .. • + E,,B_,•- E.,B_.•- E,.B;~*) r lin I til,._ (3.57)
The iatep-al for P, ia evaluated u followa: FI"'OII EqL (3.48) and {3.49)
E.,B • _ _! (E.)' E E { [ 2n + 1 ] • [ 2. + 1 ] .. v, .A:r •• 1 •• 1 n(n + 1) m(• + J)
·[ , ,. P.1P.1
6 .. ,. tiP.• tiP.• + .. ,. P.• tiP.• •·•• lia1 1 + • • til til •· • linl •
,. tiP.• P.'] } +6 .... M lial cor• -I (E.)t ,. "' {[ 2n + I ] [ 2ln + I ] [ • tiP.• tiP .. •
E .. B,.• • -;- 1rr .~1 .L:, n(n + I) • •(• + I) • •· •· • -.- •
p' tiP '] } + 6 '8 '* -· -• lin'"' • ... , . .,..
(3.58)
Suhetitute EqL (3.58) ucl (3.59) into i;q. (3.55) ucl reeaU that
J: oW •• - J: liD'••- •. P • ~Re.{ E E [ 2a + 1
]·[ 2a. + 1 ]
' ~ • •• 1 •• 1 11(11 + I) •(• + I)
. [ , ..fP.•P.• + ~.• ~-·) + 6 '6 ,. (P.'P.' + D'.' JP•') .. ._ \ u•• • • • • .,.., • •
+ '6 ,. (~·' lP.' lP.• P.1) .. • ..., • + .... ,
+ 6 ,. (lP.' P.' P.' lP.•)] . } • ..... • ••• + ••• • ... . Uli~t~ the followiq ideatitiee.
aDd
oaepla
By a limilar ~
r( P.• lP.1 + lP.
1 P.')-•• • 0 J, .... • • ... for all " aad •
...,.F..t • P.- ~U E (211 + 1) ( ....... + ... :..•) . .... - ·-·
(3.60)
(3.61)
(3.62)
(3..63)
nitE.' .. P, - - _ u Re E (211 + 1)(-.' + 6.'). C3.64)
......-- ·-· P, aDd P, IDa)' be expre.teed ia tenu of tbe CI'OM lflctioaa defined iD Sectioa 2. Ttu•, llliDc Eq. (2.1),
tbe total eeatteriat erOM eeetioa ie
Q. - :: i:. (211 + 1)( ....... + ....... ). (3.65) .. ·-· aDd. uliat Eq. (2.3), the total•tteauatioa c:ro. eeetion (inelucliac both abeorpLIOD aDd ~a~t...U.,) il
2r • Q, - - u Re E (211 + 1)( .. ' + 6..'). (3.66)
.- ·-· 3.4. THE BACK-SCA1TERING CROSS SECTION
It il DOW ,.-ible to determiae tbe beck-eeatter'iaf CI"'OI eec:tioa de&aed ia Sectioa 2.2. Fn. Eq. (2.9)
(3.67)
Fr.. Eq. (3.55)
~· - ~ (E, s .. • - E .. B,•). (3.61)
23
u-.:E.p. <s.•> ... <s.•>· .-.-. tbaa
._ P.·~-•>- --. w.·<-•>- <-•>•+•["<" + •>]. (3.69)
·-· -· ·-.. • 2 it»-tbat
~·l -~ i I: <-•>•+•(211 + 1)(2. + 1)(•."- 6.")( .... - 6 ... ). (3.70) ... -· ....- ·-··-· . - ~ i i (-1)•+•(2· + 1)(2. + 1)( ... - 6.")( .... - 6 ... ). (3.71)
.- ·-· ... Aa ..._..dYe ...W of cleriYiac ... ..-u.. b the Mek«ll!tteriq CI'Oie IJedioa ia to'* the equiva·
J.t .w.;u. of• ......... ..ay tile fu.- ICattend &eld. Ia riew of the orieatation of the coordiiUite
.,._ ..... <- ,... 1), Eq. (2.11) •
• -.... ,~ .. ,:~/!)I' (3.72) -I E•e:~>l' . -....,.. . &
(3.73)
u.n., &p. (S.48) uc1 (3.8) tapt~~er with Eq. (!.69) Jieldl
IE~L: IE-:!..- (!)IE.e": _!)(- ·r ( .. ·- .. ·)I ·••• •••II
(3.74)
... • -~) l!.(b + ~)(- ·r ( .. ·- ... )I'· (3.75)
Lpatioe (3.75) • w.tical with &f. (3.n).
15. THE SCA1TE.ING AMPLITUDE COEmCIENTS
F.- the preu '"'~it il- daat the n.._tioe of the far....,. eeatteftld &eW aad the ,·ariou•
a...leCiic.l ........ to die ...... 0. of the .catlariaf a.plitaclet?Wiicieatl, .. ' aad 6.',aad the IUIDIU•
a. of the .............. --. Tile ICattend leld .. ,. be repnled M due to the fonecl Oleillatiaa of....,.
Mlic ... electric .......... If tw. poiat of riew ;. tabe. it » ... ,. 10 1ee that the eu.cientl ... are
ueoeiated witla ....._ ofanapetie type. ... diem r•deau ._.are ...a.tecl with Oleilatioaa of electric
type. It » ....W (Eq. (3.12)) dlat the Yeetor waYelaetioe .~ .... 110 rMial caa,_..a. Tberefare. if ... tJ.e. e r•cienu ...... al _. aad tlae r •tmu ... are Ml -. the K leld il ,_..,trulY-ud tbe
BleW .... a radial c-•F t. (See-tAp. (!.21) ad (3.29).) s.iluly. if the eGe.cie•ta •." are aD MrO
ud the oo • ie•la•.' _..--. ... E. leW._ a radial CGaF a•eat ... the B, lielcl » .... y trun-. The ....... ad ellctrie an- of_. ....... .awel'- caW UU~Y- electric (TE) ... tnMY ..
....... ('I'M) t,.,... .......... , . Aa ~~ated ......_ the ... •n.u ...... ._ ... ,. be -a.tecl witla t1ae lonlecl oeeilatioM of...,.UC
... eleetrie-.hipalel. 1P\wr..-tlae._..h .. ••eyappa...-. .. oflhe•t..l ,......__olriln·
.............. of ....... ocan. ........... hy ..... of riln .............. b:v •lilac the
24
denor.tin·._o: .. equal to Zfl'O in'Eqa. (3.44) and (3.45). However, the natural frequenciee are complex while the impreaaed frequent"~ ia alwa~-· real. Therefore, no difficuhiea ari~~e at reaonance.
The problem of the numerical evaluation of the ecatterin8 amplitude coefficient• will now be conaidered. This problem ia 10lved formall~ b~ 1-:qa. (3.44) and (3.45). However, they are quite formidable, and atten• tion ia uauall~ directt'd toward the apecial caaes where cenain aimplifieationa are ~ble. The8e will be reviewed, following Stratton (1941), before the more diftirult vneral problem ia taken up.
For a metal apheft, INa!> l and the aa~·mptotic expreaaions for the Beuellunctiona may be ueed. The) are
Ul(iV ) .., 1 [JV, (n + 1)~] '•
1 0 Na coe 0 - 2 '
[Na~. (ll (Na))' • - sin [ Na - (n ~ 1 )r ]
Subatituting theae npreuiona into F.q11. (3.44) and (3.45) yields
• {.;, (ll (a)} a. • - s,.<i)(a) •
• •- {[a~.u>(a))'}· 6. [a.;, (I) (a)]'
11ae final ~uationa are exart lor a perf~tly eond.,..tin~ •phHP..
(3.76)
(3.77)
(3.78)
(3.79)
For alarp aphere, a>], the al!)mptctic expmw.iona may be used for all the Reaael function& invohed in Jo:qa. (3.44) and (3.45). The reault ia
•: .., ( _ ')·~ [.Un A - N COA A tan MJ, J I+ jN tan M
(3.80)
b." • .. +I~ [Mil A tan M - N Bin A], 1 tan 1U- jN
(3.81)
where \ ... _ _ (11 + I )r . .., = Na _ (n + 1 )r.
• ~ 2 • J~· 2 (3.82)
TI~e~~e et~uation• are induded herauae the~ illu11trate the general oecillatory behavior of the BCatterin~ amplitude roefiKoientll.
}'or a amallaphere, a< ), it ia )JCMBible to pl a good approximation for the ICIItteriqamplitude coellident& b~ subatituting the 11eriea expanlliona of the apherical Beeeel and Hankel functiona into Eqs. (3.44) and (3.45) and keepin~ onl~· a few Ierma of the reaulti~t~ 11e-i~a. This haa been done by L. Goldatein (1945), who correct• the reaulta of Stratton (1941). The llel'iel exfN!nliona of the apherical functions are
.~,.u>(a) • 2"a" E [ ~-J)•(n + m)! ,],.,a ,.., m.(2n +2m+ 1) .
.~.m(a)., 2"a" E [ <_- J)• (n + m)!,J,.,a •·• m.(2n +2m+ 1).
i E [ (2n - 2m)!] or• + 2'a"+l•·• m!(n- m)! ·
(3.83)
(3.84)
Substitutin~ thet!e expaDAiona into Eqs. (3.44) and (3.45) and keepins only a few term ..
i2"' [ n! J [N' - I] '•+a a.: • - 1 (2n + I)! • 2n + 3 a
6 • • _ i2"'[ n! J·[(2n + l)(n + l){N'- I)]· a~•+l • 1 (2n + 1)! (nN' + n + 1)
[ 1 { (2n + 1)[(2n- l)N'- n- II } . +a' + ...
(2n + 3)(2n - l)(nN1 + n + I)
_ i2"'{ n! }'·{(2n + l)(n + l)(N'- 1)} h+l ]· 1 (2n + 1)! (nN' + n + 1) a + · ..
.,. -"'
(3.85)
(3.86)
II a i11eo 11mall that powen of a hisher than the &ixtb may be ~~e~lected, only three ooeflc.hnta are needed:
-j a.• • - (N'- l)a1
45
- 2j[N'- I] { [3(N'- 2)] [2j(N1- 1)] }
"•" • -3- N 2 + 2 a'• 1 - S(N' + 2) a'- 3(N' + 2) a'
[- j(N'- 1}]
b.• • l5(2N' + 3) a•.
(3.87)
(3.88}
(3.89)
Thia is an important caee Iince it applies to the raindrop problem for man~· of the radan in uee. Equations
(3.87), (3.88) and (3.89) are the same ones ueed by Ryde (1941, 1944) if aeeount il taken of the ditference in
definition between the ooelliciente defined by Stratton (ueed here) and thole of Mie (ueed b~- R~de). The
relation between the two eet1 of definitionl ia given by Goldstein {1945):
p •..... - (- J)• j(2n + I )a.•
,_ ..... - (- 1)•+1 j{2rt + I )b.•.
(3.90)
(3.91)
It ehould be noted that if a ie 10 unall that powen of a hiper than the third ma~· be neslected, only the
electric dipole mode ie needed. Thie i1 the caee of Rayleip ~catterinc.
It i• &een that even with the &impli&cationa alorded by the approx.imationll involved in the •pecial tt.aee-.
the numerical evaluation of the ~catterins amplitude ooefticient• i• quite involved. An ex.act evaluation
would be even more 10. Gold1tein (1945) remarka, "An eue~ computation of theee ooefficienta ie out of the
question on account of the lack of ~ablea of Beeaeland Huakel functiona of com .. JL arpment needed here."
The special ca~e~ d.Ueuaeed, howe,er, do not cover all the problema of interest, 10 that il is neceua~· to COO•
lider the problem of pttiot reliable aDIWen in the paeral caee.
Althoup the evaluation of the ~catteri111 amplitude coeftic:aenta il difticult in the pneral caee, it can be
accompliehed. In their 1tudy of the eft'ects of mcteGIOiop:al elemeata on microwave radiation., R~de and
Ryde ( 1945) eet up a schedule for computiat the coef&cienta directly by calcalalinl the values of the 8pherical
Beeeel aad Hankel function~ for the arpmenta Deeded and .W.titutiat tbeaa iato the Mie equivalent of
Eqa. (3.44) aad (3.4.1i). Sincle the work by Ryde and Rycle. Lowaa (1949) hu co.aputed the ooelic:ieata for
various values of a and &ix values of N ~ to waw in the mierowave rep.. 11len are 1tillao
f':Uf'nHn tablea available, howeYer, for the apherieallunctiona of complex arpmeate, aad there ia no pneral rrwthocl for ~1tin1 exact valuea of the coefficient• ell..ept the "brute forft'' method. Ia Section 3.6, IUCh a Jnf'tiKMI "ill lw Jivf'n.
3.6. .X-\ll'-\TIO' OF THE SCATTI-:RI:\G A\fPLITUDE COEFFICIENTS USING LOGARITH. MIC l>l~RIVATIVE FU~CTJO!\S
Tile pneraiiiOiution for the K'atterinK amplitude Mf'tli("if'nl• ia ~ven b~ F.qa. (3.44) and (3.45). 'l'11ew f'qllationa ma~ be rrarranpd aa follow11:
._ -[:..''1(a)] [Na.:. 11){Na)) -N (cu.u>(a)) . l (
[Na:. Ul (Na)]') (''"·(I) (a))') l 6• .:. (I) (a) ((Ncu. ul {Na))') _ (Ia:. <2l (a))') J
[Na:. (II (,\'a)) N [cu.111 (cr)]
(3.92)
(3.93)
It MIOUid be noted that in thia form the equation11 involve the logarithmie derivuivra of [cu. 1'1(a)), [Ncu. U) (Na)]and [cu.") (a)]. Logarilltmic· tkrivative11 of thi11 t~ pe werr ueed by lnfeld ( 1947) in hia stud~ of the apheric-alantf'nna with a r;a1•· and b~ Smith ( I1H8) and Tai (1949) in the l!lucl~ of bic .. lllit·al antenna1o It the notaticm of Smith and Tai i" Ullf'cl, tlw 1~1 clc-nllit~ func·ticmt~ art' clfofinf'fl
S.(.r) == [ • .-:J.+:(.r)],
R.(x) "" [.,JII•+:'~>(x)].
(3.94)
(3.95)
Following Tai (1949), tllf' lc.,:arithmit· df'riuthr .. of tlll'lle Jlt'l!llel dellllit> fu!:.·tiont~ are ckfined a• new func· tiona. •.(s) and p.(.r). Thut~,
S'o (.r) (.z:., (I) (.r)]' cr.(.r)- s.(.r) - (s.:.10 (.r)) ' (3.96)
R'. (.r) [.r.:. (I) (.r) I' p.(x) .. R.(.r} • [s.:. Ill (.r)] . (3.97)
Ia terma of thete funetiona, the ~eatterin& amplitude coefticif'nte are
0 • • _ [.:.
111(a)J·[cr.(a)- N •• (Na)J
• .:.111 (a) p.(a) - Ncr.(Na) (3.98)
6 • __ [.:.111
(a)]·[cr.(Na)- N•.(a)]· • .;.11) (a) cr.(Na) - Np.(a)
(3.99)
Thia form appean to be aimpler aince there are fewer functiona and no derivativea of funetiona involved explicitly. To make the apparent aimplification real. however. it ia neoeeaary to de11101111trate that values for the Josarithmic deri~Vative functiona can be eaail>· obtained. Thia will now be done.
Sinee the followi111 derivatione hold for all the .Beaael denaity functiona. involviq either the Beuel. NeuaUuua • Uaakel functioa.. it is convenient to let
l J.(x)l
Z.{x) - N.(x) .., the ~IN'ral c~Jindrit•al funC'Iicm, H.(x)
I ( ) IY~(x) L- I.__ · h · d · · fi • ~ s • D,.(s) • Inc pnera ....,ant m~e en,·atl'·~ urwtiiNI.
DifFerentiate Eq. (3.101) with respect to x:
IY~(x) - x1Z'•+I(x) + lx- 1Z.+:(x).
Sub.titute the recurrei'IC.'e formula •
Z',.+:(x) • z.-:(x)- [" ~ 1Jz.+~(-") into}:,,; (3.10:\) to rliminatl' tlw drriutin• u( Z:
Divide b~- D.(x): D'.(x) [z._,(x)] "
a.(x) ""----- = --·-- - --· D.(x) z.+:(x) X
•
Thia ia the standard equation uaed to evaluate tiM" lotraritlunit• tleri,athf' fmwtit"'" "IH'n .\' i.- F\"al.
:.!7
n.ICMI)
(J.ICIJ)
(:UU2)
(:i.IOI}
The difficult~· f.'OIIleA when x i11 complex. Equation (:Ut.ltt) f"annot he """'' tiH"n hct·auiW' tahlt-" u(
z.+ 1(x) are not available for x romplex. In thi11 C"a~. Ullf' i11maclr nf tiM" rf'f'ttrwnc"' fnrrmda clew lu lnf,.J,J,
(1947), ,.·hich "ill no" lw deriled. .."'111 1-:q. (3.101 ),
(3.107} Differentiate:
Z'.+:(x) -.%-: [D'.(x)- GJ n .. (x)). (3.108)
DifFerentiate a,ain:
, I ( 3) Z".+&(x) • x-'{D".(x} - _; lY.(x) + .tx~ IJ.Cx)J, (3.109)
1'he 1~1 equatic.n of half·inte~r order i•
rZ".+:(x) + xZ'.+:<x> + [x~- (n + I>~Jz ... :(x)- o. (l.IIO)
SuLMtitutf' F..q11. (3.107), (3.108) ancl (:i.ltJCJ) into E•1· (3.) 10) ami ndlt-t·t l~rm ...
, [n(n + 1) J IY .(x) = --:;;--- - 1 n.(x) · (l.lll)
Thill i• tllf' llllll'lf' a .... ;q. (D. I) or lnfeltl i( n ill replat'f'tl b~ p - •• It ill "'"' Dft'fWar~ Ill~· an .. , ........
lion Cor D'.(x). Sublltitute •;q". (3.10i) and (3.108) into the '"'lllftlln' luno·llla•
• S.. far_,le, '•a-. ..4 T..-u.-• 'llwr:t-ef B...t F......_ pp. 45 and 66.
:!R
and eolwt t.enaa:
Difrerentiate:
IY.(s) • ;u._,(s)- ~D.- 1 (s)- D''.- 1(s)·
SuJ.titutt' Eq. (3.111) into Eq. (3.114) and collect terma:
n n' D'.(%)- :;IY._ 1(s)- :;;D._1(s) + D._ 1(s)·
Now. divide Eq. (3.115) by Eq. (3.113). Thea,
But.
[D'.-J(%)]- • ( )· D ( ) ··-·% ·-·%
,. -- ··-·(%) %
(3.112)
(3.113)
(3.114)
(3.115)
(3.116)
(3.117)
Thia iuhuame u Eq. (0.6) ollnleld ih iueplaced by p - l. It ia 10111etimea ealier to aee Eq. (3.117) if both awaerator and deaomiutor are cleared of fractioaL Tina,
l.(s) • z' + n.d,-•(z) - 111•
II%- z~~._.(z) (3.118)
F.quation (3.118) ci•• the reeurrenee rea.tioa whereby the locarithlllie derivative fuDCtioa of any order can be found if the f'u~~etion of the next lower order ia lnown. It wiD now be lhown that the loweet order can be computed direetl~·· Since ... (s) ia the onl~· function with complex arpment ia Eqa. (3.98) aad (3.99). oaJy thia funetion wiU be COIIIidered:
S.(s) • (%' J1 (s)) • (;J tins, (3.119)
(3.120)
S'e(%) .-.(s) - -- -cot%.
S.(%) (3.121)
Now. if z - r: - jl. thea. · 2e+ 'lliahU
.-.(z) • cot (c - jl) -1111 1 • -.u- 0012c
(3.122)
Jiicl-'-order tenaa may aow be found U1iDJ Eq. (3.118).
It is .een that by uainc Eqa. (3.98) and (3.99) topther with Eqa. {3.106), {3.118) and (3.122), the
-t~ amplitude coefficient• ol any order can be computed exactl~, even in ~~on• whe~ the •pherical
Be.el functions of complex arguments are not tabulated. In the next iM"Ction thi~ mrthod i11 applit-d to the
evaluatioa of the hack-ecatterinc crou eectioa for one particul~r raae involvinJ "ater 11pherel! with 11izea comparable to the wavelength. This caee haa alto been •tudied t!\fJerirMnlall~ and c.'Umpari1011 belween
the theoretical and experimental re~~~lta is made in Section 4.
3.7. NUMERICAL COMPUTATIONS
The method Kiven in Section 3.6 waa ulled to compute lhe IIICIIlterin~ amplitncle ooeflicient11 for water
apbena at A - 16.230 em and 0.6 < a < 6. Tbeee coel&cient• were then ule41 to C'Ompute the back-acatterinc eroea eection, uinc Eq. (3.75). For plottinc purpoeea, it ia convenient lo normalize the back-acatterinc
eroea lflCtion with re11pect to the pometrical eroea eection. II this i1 done, Jo:ct. (3. 7.1) becomes
..!... -.!.I I: (2n + 1) (- 1)• (a..•- 6.•) I'·
... a' •-1 (3.123)
'I'hi. Ia the equation actually ulled in the eomputationa. Since the values of a. • and 6. • clecreaee rapidly
lw n > cr. it is n«e811ry to carry the •ummation only to n - 2cr.
Since the computed values of tbe back-ecatteriq CI'OM eection depend on the index of re&action, it ia
important to have reliable values for tbia quantity. It is recalled that with p.1 • p.1, the complex index of
refnctioa is reblted to the complex dielectric faetol' by
N- (!)'- <t>'- cr.- ir'.>'· (3.124)
where f, is the rebltive dieleetrie factor, and r, and -r', are the real .and imacinary part" of~.. R~de
(1941, 1944) and Gold.tein (1945) determined t by lllinc the Debye (1929) formula,
{3.125)
where f,. ia the rebltive optical dieleetric faetol'; t. ill the relative •••tic dielectric factor; and A. i• the traaaitioa wneleqtb. Tber detenlliMd ~from Eq. (3.125) by uaiq Collie'• (1944) meaaured nluea of (, and r•, lw A • 1.26 em. The validity of U.U., the Debye equatiou for the frequency dependence of~. at eDMtant temperature haa been verified by Suton (1945) and Collie et al. (1948). Slipt difl'ereRCH
hue beea louacl in the experimental valuea of~ but in senenl the qree~~~e~~ot ill pod. Thi• aame method
haa IICCOI'diapy beea ueed in the preeeal reeean:h. The rault is t • 81 - j7.8 at A • 16.230 and temper· • ._aeerwc.
Some of the eomputed valuea for the losarithmic derivative functioaa and -lteriiiJ amplitude roefficieata ue plotted in F"ipo 2-19.
F..- 2 and S abow •.(a) venua a for n - 1, 2, S, ••• , 12. The •alues of ... (a) were comput8 ulinc
Eq. (3.106) with Z.;t:t(a) - laat(a). It ia aeea that •.(a) ia a amooth. DOD~IJ.tiq function. It ap........ +• u (11 + 1)/a wbea a • .,...,...,. Rm. It haa poles at the 111!1'01 of J.H(a). Tbe poles of
... (a), boweftll', do aot cauae the ecatterint amplitude coelicientl to blow up Iince it i• alwa~·· the product J.+t(a)·•.(a) wbieb appean in Eqs. (3.98) IIIII (3.99), and this product haa DO polea.
• • Fie. s. '11M •• ,_.._ far ...a ---··
• Fie. 5. ).......,. ,.. ., ... ,. '-"-·
..... ..... _.. ...... 'i • .,
•
l'lc. 6. n. •• ,_.._far.......... . ••.
••
•
-r. ••• ,
... .. ,.. .... _.. ...... I •'j ,.. _._.. ..... .,
•
FIG... ..... .. ,_..far ...... ····-···
•• r--
•
..
·~
cr~~.. ,
T
I
:u
I I
-
IA/\1'1./\A vvvvvv I
A hilA vvvvv I
~)-•"'......J.,..,... '[ ....... "*,......J ' ' -
I,..,......_...,..(-..,.~
·.;··-~··1 I I
•
I • •; • .....__..,. .. .._,
I •.; • _., ..,..., .. , •• ~,
•
... , .,_
... 1----
~f.J \'
......... . -.. ~·(.,6; I J.,ll .. l)
I '"'• .... _,..,.._,.
I '; ..,......,.,,,. .... ..,,
•
F~r.. 10. n.. •• f.-twa far ca.p1n ._-t .
...
~ .. ._ ='' r_ ·~,_; ·r.·.r. I ... I ....... .,'-....
•·ro. --·r.....
• F.,;. 12. ...... .. ,_.. .............. , II •••
I I I
... '~~AA~ ... _ . ~V'f"'Y-·
.. . l",Pilal r ...... I ,/'''""r},IJ",•I(j~
of--t" ... l Jllfl......,l
I • '• - "tAL ....,.01(\,_,
•·r;. ~" ... '016" .. •'
I All!AA vvvv
.. .. I I _l I . •
• FIG. IJ ........ ,_. ........ _,._-.,._ ••
... I .. ,_: ....
~ ~·r.;K.-..... ,·~~·. 1 I .. I ..... _...,.......,
•· r,. __ .,..._,
..,. . --·"..... --.~.,--
•
F•c. 14. 1M~ _,.1itwle _... ... .,• .ud ,..•.
...
•• •
-u
•• I
. .. ---_.,_, ___ , , , ,'
,/ •• .,., _,, •
•
I I ----___... _,. ---
..' --___ ,.
I • •
-I
,, ,, ----- I
------ I
__ , ,,.~' .,.,
, ,
1 ...... eelllll'•lilllll .... •'• •' ... .,•.
•• • ,.,--........ , ,.,'' .... , -- '" .,~--------------~~-~-~=-----------------~
_, .. I I ···-
•• I
•
ltUL Nat---
IMMt...,. ~IP'T -----
......... - ......... ,,'' ',,
I
-'
--~~',',,, ,_
I -1 •• ~.------.I....----...L.----.....J'------..I..-----J.------.I...J •
•
:n
FIG. IS. 1M_,...., uaplitude eoelieientA, ••' and ••'·
... I
, .... -, , ' , ' , ' , ' •r----~~----·----~--------~,~~
_,..I I ....
... I
\ \ \
--·---- ------.
• I ....... meliei.~a, .. ' IUid ....
:u
...
•
~· •
I I I I ....
.,. •
--T--_, _, ____ _
I I
I I
I
I I
"' ,
}'ll;. II. Tlwt -uf'rinlf .......................... .._ ........ 1 f., •.
•.. ...--....... --..,.....-......,...,
"' ... --............ ,...,. ---
-_,. __ lll&&llii&Mn' MitT -----
0.1,_--~--.,.-.· I --........ NIIT ---
r.- 4 and S lhow the real and imapnary part11 of ,.(a). The values of theee functiona were computed llliq Eq. (3.106) with z.,.,1(a) • H.-:t:1(1)(a). It ia .en that the~· are allo noa-oecillatory. The real ,_a ol ,.(a) a~ - • aa -11/a when a approachea ~. Aa a increa~e~, the real part increa~ea rapidly &o. - • and approachea llei'O. The imapnary part atarta at llei'O and approaches - 1 u a illcrea~n.
r.,.... 6 10 13 lhow the real and imapnary parta of -<Na) for II - 1, 2, ••.• 8. It i. teen that thetle are rapidly oeeilaU.,, hiply damped function&. The real part approachea sero and the imapnary part a,..oeet. l .. Cl iacn:uee.
r....- ., to 191how the lt"atterinr amplitude ooeftieient••·· and •• •, for" - ], 2, ••• ' 7. The pneral oecillatory behuior of the eoelieienta ia teen, aa wellaa the rapid deereaae in amplitude for n > cr.
The theoretie.l normau.ed beckoecaueriaa c:roM-eeetioa curve ia lhown in Fi1. 20, where it ia compared with the ._ ob..u.ed experiMeatally.
4. EXPERIMENTAL DETERMINATION OF THE BACK-SCATTERING FROM WATER SPHERES AND COMPARISON WITH THEORY
4.1. INTRODUCJ'ION
Siace the intnMiuctioa of microwave techniques to the lield of meteorolor,, mueh work hu been done oa the .,.,a&c applieatiaaa to the detection of relectiona from rain and other hydroaleaean. Unfortunately, IliON ol thia work haa been qualitative in nature. Some quantitative IDI!aiUI'nlelltl on rain have been
35
IUde by H. GGW.teia (1945), Manball et aL (1947), J..anciUe and Guan (1448) aad Hooper and Kippax
(1950). CaMi .... the ........ of Yarialll. illYolvecf. &hele meaiUrellletUI ..how ~ apeemenl with ~
theory. It • diliealt. howeY•, to ieolate tbe illdiridual effecla when there are man,· variable.. and it
lllleiM deairable to hne .....e laboratory .......... oa ainp ..,t.eres. Ret"entl~·. thi1 need ha• been at
leut pu1ially 6Ded by the WOI'k ol Klouhaap ud Duckett (1~) on pla~tic aphrrea.Ymulatins raindrope
IIDil tbe war.k of Adela (1950) oa actaal water.,.__. In Section• 4.2 to 4.4. the technique~ u.ed b~· Aden
wiD be cleeerilled hriely, 11D1l the Nlalta obtaiaed wiD be COIDpared wilh &ho-e determined &om theory.
U. THE EXPERIMENTAL PROBLEM
The aperi mat.IIDAIUI'ellletlt of the releetioa of eleetromapetie wne. from individual water ..,._._
iavolvee two illlponut diticulde.: (I} SiDce the tphere ia a very low sain ref;~tor, it ia hard lo obtaill
reliahle....,.. by ordiury paw., .......... aad (2) it ia DOt euy tn maintain a water ephere while
•n..,. aeau are ....... ....._ For tt.. ,...,.... DO direct laboratory meaMJrementa had been under
.._ ~ priar to the wark -tioaed aboYe. Eva DOW, theee difticultiea aJmo.t pn'JC!Iude the --, ol~~~NM~~e~De~~U ia the rep. olpeaaeet iat .... u -i.e., the ~tral repon where water apheres of
actaal raiaclrop Ue are eMDparahle to the wav......... However, it ia poMib1e to perloraa an experiment at
llitbtly loapr waveleattha aad to iac:reue tbe Ue of the water tpberea to •• into the critical aiK repon.
'l'bil ia the expen-t that aetaaDy wu pedocmecl. It ie betined that tbia technique ia uaeful •~ COlli•
,.n.oa hetwea experialeDt aad theory at cme hqaeacy lhould pve a pod indieatioll of the cwrelatioa
to be espected at otb.lnqaeaciea.
.U. THE EXPERIMENTAL TECHNIQUES
The nperimratallllethod aeecl to......_ the badl~t...U., c:10111 teetion wu the ataading-waYe
method of D. Kin« (1948). Thia ID8lbod utili8ee u imap lifteD teelmique. and the atandiq wavea •t
up oa the ~e~ee~~ by the mteractioll of the iaeiclat IIDil reradiated wave. are meaaured •Ions the radiallioe
betweea the acattenr aad the eouree. Thil ID8lbod o&n the advantqa of a •y•tem haviq ablolute
cailntioa. relatively lilllple equiptoeDt aad ............... at low power. n.e main ditadvanup are
that it requira an iaaqe ICftflll whieh il larp. uttifona IIDil ripl. and that obtainiq re~~~lta ia very time
~I· The derivatioa al the fonaulat Deeded IIDil a cliac:uaioa of the approumatio-., •• well u a de·
tailed deecriptioa of the equi.-at .-d. are pYeD by Adea (1950).
Olle of the wont ohetedee to..--,...,.__.. oa water tphaa ia the inability to maintain RICh
.,..._ while ~~~ are beiai takea. Thil obttacle wu OYei'CCIIDe by uans u a container thin .._.,._ical .._.. ro.... ol Stynfoua• IMUIIted oa aluaiaua clilb. Sioce Styrofo.a ha• dielectric
pnpertiN extreaely c:loae to tbaee of air, it had a ...p,ible eSeet oa the ~u. Thua. whea a
...... wu Oed with wata' .... the clilk WM ._.._ iato ita ....-,er plaee i• the imap IICfteD, the effect
wM that ol a hemiaphere ol water expaaed ovw alarp iaap eerea. By itaap theory. the toeUU~e~De~~tl ........ the ...... for. _,.... ....... ia free.,.,..
'-'- COMPARISON OF THE THEO:BETICAL AND EXPERIMENTAL RESULTS
Meaw IDU of the badl«at ..... ~ IICtioa '"" IUde oa 30 wata' epherea in the electrkal abe
...... 0.74 s. s 5.90. Ia r.,. 20. die ....... ol theae IEJ•ea.eau are plotted. topther with thole
......... n.:o-~c...v·· , ....._
•
•
•
THEORETICAL
EXPERIMENTAL
EXPERIMENTAL POINTS ••••••••
37
determioed from theory. The theoretical curve wu obtaiaed by UUJI! the method of losarithmic derivative function• ~ven in Sec:tion 3. It i1 -n that there i1 very ~ a~ment between the uperimental and theoretical re11ult1. For 1pheres very mueh unaller in a than thoR considered here, the re~~~lt1 ahould
be pven to a good approximation by the Rayleip law. For apbera of larpr lise than thoee conllidered here, the raulta should approach the larp-eiae approximation of about 0.64.
5. THE THEORY OF THE SCATTERING OF A PLANE WAVE BY A SPHERE WITH A CONCENTRIC SPHERICAL SHELL
5.1. INTRODUCTION
In the precedin« aectiona, the eeatteriq of eleetromapetie radiation by a linp aphere haa been ooneidered, and ita application to the more paeral problem of llllierowave reflection from rain haa been indicsted. Another phyeical problem of intereat ia the eeatteriftl of llllierowave radiation by raeltins IIDOW and ice
particle.. For purpoeea of analylia, tbia phylieal problem may, to a fint approximation, be replaced b~ the mathematical problem of fiadiac the releetioa from a apbere with a concentrie apherieallhell of dill'erent dieleetrie factor. This latter problem eaa be IOIYed riproualy.
The method of .olution il a direct uteDiioa of that ued Cor the problem of the eeatteri111 from a ainsle apbere. The incident plane wave is expanded in ter1111 of the orthopaalapherical vector wave functions of Strattoa (1941 ). The indiXled aeeondary fielda in the various rep.. are written u llimilar exparuions
with unknown amp6tude coeflicienta. Aa in the preYioul eue, the unknown coeflicienta are determined from the boundary conditioaa. ]o tbia c:aee, however, the matebi111 muat be doae limultaneoual~ over two boundary IUifacee inatead of oae.
5.2. FORMULATION AND SOLUTION OF THE PROBLEM
Coalider a apbere of radiua o, ccmaplex dielectrie factor E. and permeability ,. which ia IUI'I'OIInded by a ipherical ebeU of inner radiuao and outer radiua b with a ccmaplex dieleetrie faetor a and permeability,.. The ftiUitins coalipratioa i1 .......ed to be ieolated in free apace. The interior of tho- lpbere, the interior of the ebeU and the ~ .,.ce are called repooa 1, 2 and 3, rapeetively. Aa in Sectioa 2, the center of the aphere il cboeen u the oripn of a reetanplar coordinate ayatem; the incident eleetromapetic wave il propapted alouJ the • uia, and the eleetrie vector illinearly polarised parallel to the s direction. Thia ia illuatrated in F"JI. 21.
With the conditioaa etated above, the uprelli11111 Cor the iocideot plane wave are euctly the aame u
in Sectioa 3:
:a, -1E. - tE.e-• - E..;l (- j)• [.~: ! !)] (-....• + jA.a.•) (5.1)
B, •1B7 •114 .-/1111 • - E. I: ( - j)• [ 2n + 1 ] (mda1 - jn.a.•).
Ill Dia-l n(n + 1) (5.2)
The indiXled eeeoacluy 6elcl ..... , now be eoaatrueted in three pana. one applyinc in each of the three
Nfiou detiued above. 'l'bftle parta are written u opaaliona limilar to thole for the iaeident wave but
r t------~z
Y AXIS POINTS OUT B·
I .. OINTS OUT ,IIOtl M"Eit
INCIDENT PLANE WAVE
,ROlli M .. Eit
SPHERICAL SHELL;
witll unknown amplitude eoefticieau. The part appl,·ms outlide the lhell i• •1aia referred to •• the BCattered &eld aad i1 iadicated by a aubleript a. The farm of tbie expalllion il the unae •• in EqL (3.28) and (3.29) altboup the value. of a• and 6• are difFerent here:
B. • E..~, (- j)• [,.~:! !)] (a."m.1•1 + j6."Dtl.1 ) (5.3)
B E. ~ ( ')• [ 2n + 1 J (6 a • •) • • - 111 .":
1 - J n(n + l) .~a. - J••"D.I• • (5.4)
B) aaaioc)· with the .-eviou• problem, the field illlide the •phere i• ealled the traaunitted 6eld, and it il indicated by a IUblcript e. A,aia, the formal expalllion1 are the unae •• before (Eqe. (3.32) and (3.33)) with the uadentandiDJ that a.' and 6.' have different valun:
B E' ~ f ')• [ 211 + 1 ] ( t_ I •L I 1) • • ,. :;, ' - J n(n + 1) a.-. •• + l"•lio•• (5.5)
' . ... ] B. • - ~ .~, (- j)• [,.~: -~ !) (6.~a.1
- ja.'D.a.•). (5.6}
llere, 111 - (,a.E,)-& il the comples eharaeteri1tic velocity of the lpbere.
llllide the ipherieal lheU, the tenu involviDJ Deuel function. of both &nt and leCUIICI kimll mUll be retaiaed. The lonaal npaDiioaa may be written
E. • E. .~1 ( - j}• [,.~ : !)] [a.'m.1•1 + A.'mo1•
1 + j(6.'a.a.1 + B.'a.1•1)) (5.7)
B, • - E. E { - j)• [ ~~~ : ! ,] [6.'m.1•1 + ·B.'m.a.• - j{a.'a..,••' + A.'a..,a.1)) (5.8)
.. ••I II II
39
when a.', .A..1, 6.1, B.• are 1lllkDowa amplitude ftl!e._.,~U. ad 111 • u..&)-• ia the comples characterUtic nlocity of the .......
EquatioM (5.3) thNup (5.8) npneeat a lonuldutioa for the iadueeciiiiCOIIduy &eld. AU th.t ia Deedecl to complete the lonul eolutioa ia the enlutioa of the eipt amplitude eod&cieuh. Tbia ie doae by.,.,.. the hoawlary coaditioaeat the-two lllriaeee of dielectric: diecontiuuity. The bouadary condi· tioM at r - • an
txEtl -txE,I (5.9) ·-· ·-· tx B.l -fx B,! (5.10) ·-· ·-· adatr•6an
tx (B, +B.> I -txEtl (5.11) ·-· ·-· tx (B,+B.>I -fxB.I . (5.12) ·-· ·-·
Tbeee lead to two leta olllilllultallflOUS equatioae each iuvolviq four uuknoWDI aelollow1:
[Naea. <I) (N1a)]' 6.' + [Ntcd.m(NIICI))' 6.' + [N~00 (Ntcr)J' B.'. O (5.13) N, N. Nt
- N-,s. ro (N,a) 6.' + N,a,. «•> (Ntcr) 6.1 + N,a,. 00 (N,a) B.'- 0 (5,14)
- [,._(1)(•}]'6.· + 0 6, + !N.n..(I)(N.,))'
• N, 6.' + [N.,..(I)(N.,)]'
N, B.• • (n.<•>(•))' (5.15)
- ._oo(.,)6.' + o 6.' + N,a,. u> (N.,) "·' + N,a,. 00
(N.,) B.'- :..<•>(•} (5.16)
aDd - s.0,(Nacr) ._, +.s.(I)(Nsa) ._, +a. 00 (NtCI) A..• - 0 (5.17)
- (N-.,ro(Naa)]' ._• + [Ntcd. (I) (N,a))' a.1 + [N~ 00 (NIICI)]' A.1 • 0 (5.18)
- .. (I) (•) ._•+O ._• + ._u>(N.,) ... + .. -eN ... > A;.'- a,«l)(P} (5.19)
- [,._oo(.,)Y ._•+O ._• + [N.,..<0 (N.,))' -.! + [N.,...,(N.,)]' A.•- ,,._<•>c,))'. (5.20)
H..-e, a • 2 .. /)., ., - 211'6/).; N 1 - vw; ad N 1 - ~are the co.pln iadicea of relractioa lor the..-... ad llaell, reepec:tinly. AI before. the primee at the equare llraebu indicate Mereatiatioa with r.pect to the ...,_.t of the Be.elluactioa iaaide the brfteu. Tbae two .eh of liaaultaneou. equa· lioDI ... ,. be aolftd b the eipt amplitude ooe6cietah.
To eYaluate the Mek.-atteriq a'OII eeetioa. it illlllllM •• .,. oaly to aolve for the seatteriac amplitude en aS Mllu, ._ • ad 6. •. If thia il doae. aDd if the derintivee of the Be.el cleaeity luactioae are elimiaated by iatrocluciq the lopritlmaic deriuti-.e luactioa., the reeult it:
6.•-- ._co(•} ·[F• + •·<•~1 s.(i)(•) F, + p,.(P)GJ (5.21)
(5.22)
IO
-·~ f't • \',a. 111 (1V.a)cr.(N.cr)).:.(l1(.'\',a).a.'I)(N:!I')P.(.'\':!I')- :.u•(,:!l')cr.(.l\'1•).a.''1(N,a)J
+ "•"· ul (IV,a )).a. nl (N:!I' )cr.(:\'"" ).a. 11' (\':a )p. (.\:a) - :. en (\':a )cr. ( '\':a).a. 111 (NIP )p. (NIP)) (5.23)
(;,- .\·,=.a."1 (.\',cr)•.(/V.a)la."1 (.\'"")a. 1''(.\!:a)- :. 411 (\':o):.•••cv"")l
+ ;v,N,a."' (N.a)):. ol (,'\',a)cr .(\':a ).a. ••I (,'\':!1') - =· cu (.\·:, )z. '"('\':a )p.(N:!Cl) I
f': • 1\'s1a. "1 (N,cr )I.a. " 1 (.\""")cr. (.\.:!1'):. ts• (:\':a )p. (N:a) - .a. 111 (N:a ).r. (N_:cr ).a. ~11 (N:!I' )p.(N~P )I
+ N,N,:. 111 (N1cr)cr.( \ 1cr}l.:. 111 (.\,.a )::.'11 (.\:!l')p. (.\':!1') - :. 111 (N:!I')•. (N,..).:. cs> (N,a)]
Gs • Nsa.n'VY.a)).a.m(.\:cr)cJ.(.\:cr).a.'"'(\:P)- ::.'n(.'V:!I').:.<ai(N:cr)p.(.\':cr))
+ ""•a• ol (,\'•a )~r .(.\.,a )!.a."' (\:I' ).:.'•' (.\',a) - =· ol (N:a ):. "' (.IV:P )).
A~ ~ , a"" j' L (-It(:!,+ l)(fl,:- b.:)
lr •·! 1
(5.24)
(5.25)
(5.26)
(5.27)
•:C.uatiun (5.27). IJJJf'IIH"r "ith E'J"· (5.:!1) th"'"l=h (:l.:!(t). ~'"" tiH" formal l<ltlutiun tu the pntblem of
bad MC'allerinl from a "JJilt're "ith a t'OIH'I'ntrit· "IJilt'rit·.tl ,.Jlt'll. • -"" a t•hf'I·L.. it "'"tuld be noted that
thi11 Mllutiun Mftoultl rf'lhH'I' to that fur a "inj[lr "Jlllt're if a • • anti \ 1 - \: • \'. Sint-e the equation for the 1 ... -Lo~~~·anerinJ ,,__llt'<'tiun j,. tilt' ,..mt' in l1111h , .• ,...,.. thi11 rf'IJUirt'll thf' IW'allf'ring amplitudt' t'Of'flit•ienlll
to Iff' the NOH'. •:,aminatitm of E•t"- (5.:! I I th""'l=h (i.:!(,) rf'\f'al" that I he-~ tlu rechll'f' to E•J"· (3.(J8)
and (:i.«N) uncltor tlH" ••onditiun" "lalc-tl.
-\hli4H1Jfh tiH" •J'J'lit·atiun uf I !Jj,. Jlf'nhlt•m tu JNirlit·ulllr J•h~ ,.j,·lll P"'hlc-m~ ha" lll'f'n intlit·atrcl, the detail~<
o( cllf'IW' •l•tJiit•aticm" "ill nul l11• ;fi\l'n lu•rt•. Tlu·~ 1m•. hu"t'\f'r. lll"in~ ~he-n allPntiun h~ othPr memllf'n;
u( the (;l"f'flll~ ""'" llt·o~~·an·h J)j, j,.iun.
•Sun· tl ...... ,, .. ,,t~·hun uf tl••• •u.u&n·•·rtpl \,J, .• , an•l ~.-rLt1" fl•t:tl) 1 .. ,,. ln~alnl dw .. .,... f!'rl'".,•l•·•,.,.. t~olll"'''" all 11 ...... ~ l'r.fii:M.,.. ... u• , .. .,,.,.a,.., , ... r.nn•·h·r·.
41
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J . ..4,_ l'lty.. ... 21'-,.ATM*,G,.I9U...4T,..__ ... n-:reJ ..... F---.(Ce•HdpUttinnityPr-.'-4a.. ... M-m-.NewYork).
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No. z. No. s.
No. ••
No. 5.
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GEOPHYSICAL RESEARCH PAPERS
leocnpic ellll Noa-IIIOII'apie T..t.ul.aee in the At__,.t-ie Sarr- Leyw, a-. Letuu, ~ ....,... R-.eh Dilwun~ o-a- 1949.
{a...ili.d)
Dil"netio. E--.. U. the P'wlpaptiall el c.-....--.1 w.,. in the At--r•re; N-- A. a.-.. Gtoaph)'lial R-m Direetare~ M..da 1950.
F,, ...... el a--. el Joiat Air F--Y•IMr a.-.u a-1 s-1i11c friele c-dueled 0.... w·illa. ..... Spriltf 1949, ChuiM K A ........ Geoph,.-1 ._. Direetor•ta. M•r 1950.
Ja ...... tiall of Sir•..,._. w·iadll ..... T_,....,_ F- A-*el Plopeptiall StudiN, Au..n P. Cr.y, Ceop~.,-.. R.-eh Direcearete. J- 1950.
Air.CO.,W Fla...a y.,. ia Floetiftlloe, F . ...__ M. E .... A. P. Crery, S.lteta. J. Oliwer, G.apllyMeel R-a. DinetA!n~ Now_._ 1950.
Ph c •· p el the Coafne- ow 1-.1-ic R.-eh <J- 1919), Nitecl lrY Bredlard B. \!..._.. .- Relph J. n...w... Jr., c-p~~,-.. R-a Dlnet.a~. o-tr. 1950.
Pro«tlllillp of the Colla··- oa M- pherie PbJ~b. etlitell by N. C. G-. c.op~~,.R-a. Di,-.., July 1951.
'1"M Diapwlioa el S.r'- W••• ow Multi-LII)'ftWCI M-., N- A. u..ua. Geoph.,.-. a--. DiYW.. A ..... 1951 .
..... M I -· of Stnlollpl-ie o...ity Diootrille .... , ..... s-..,...... T..-...-, L o...-.c •• , .. ,._ ._. om.ia.. o • wl951.
Pw _. .... olthe c-.._ oa l_,._ic...,.. {lilly 1950) Pen A. llllitMIJy N.C. c-_. ...... J. now •••• -· Jr .. c-p~~, ... __.... ()i,--. April 1952.
Prnn•i•r of .... c G ·- - ............. ,... a.ay 1950) Per~ ....... by ....... ... lt ... _. N. C. C.... Geapll71ia8 a-eJt DiYW.. Aprill952.
Pa-t I' p ol P. C .. , I oa .. .._..,. .... ......,, .\....a. .- a.-1 ....,..,., editecl 1tJ ..... J. 0..11•n• Jr.; Cupltylia8 ._... Dmtiaa, May 1952.
A• :1 icn-Pe...__,,.. • .,._...._.,~.._ B.a ... ..... ............. A. c.... Gwpll, ..... __. DiYw.., J.Jy 1951.
