9.6 Spherical Wave Solutions of the Scalar. Chapter 9: Radiating Systems, Multipole Fields and Radiation

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2 Rvw.7 Laa Equao Cya Cooas; Bss Fuos ( L ( R( Q( Z( P Z Z Z Q y Q Q R R R R J (, N ( wh J a N a Bss fuos of h fs a so, svy (s foowg ags. J ( ( N ( 5 Rvw Bss Fuos : If w, h quao fo R as h saa fo of h Bss quao,.7 Laa Equao Cya Cooas; Bss Fuos (ou R R R (.77 wh souos J ( a N (, fo whh w f h Ha fuos: H ( J N ( H ( J ( N ( (.86 H ( J ( N ( a h of Bss fuos (Bss fuos of agay agu I( J( (. ( K ( H ( (. S Jaso. -6, 6 Gashy & Ryh, a Abaow & Sgu fo os of hs sa fuos Sha Wav Souos (ou j ( ( j ( j ( ( ( j ( Fo G. Af, "Mahaa Mhos fo Physss" j ( s, j ( (!! ( ( os, (!! ( ( ( h ( ( [ saa of sh a wavs.] ( ( S Jaso. 6-7 fo fuh os of j,, h, a h Sha Wav Souos (ou Easo of h G fuo : Souo of h G quao ( G(, ( (6.6 s gv by (v S G(, f sa a fo ougog- (6. wav bouay oo. W ay sov (6.6 h sa way as S..9,.. w * G(, g (, Y (, Y (,, sov fo g (, f o a [wh ( ], a h ay bouay oos a,, a. Th su s ( * G (, j ( h ( Y (, Y (, Equag h wo ssos abov fo G (,, w oba ( * j h Y Y ( ( (, (,, (9.98 wh a a, svy, h sa a ag of a. 8

3 Pa I 9.6 Sha Wav Souos (ou Suay of Dffa Equaos a Souos : Sou-f D.E. Laa q. Hho q. Souos y y Casa y,,,.,,,. (S..9 (S. 8. J (,,,.,,,. ya J (S..7 (S. 8.7 (,,,. Y Y(,, j (, (,. sha (Ss..,. (S. 9.6 D.E. wh a G(, ( ( G(, ( o sou b..: G ( b..: ougog wav Souos (G fuos Ss as of G fuo G G [Eq. (6.] Eqs. (.7, (.8, (.68 Eq. ( Pa II 9.6 Sha Wav Souos (ou Suay of Dffa Equaos a Souos : Sou-f D.E. Hho q. Wav Eq. y y,,,. A (, Souos (S. 8. Casa (, J,,,. ( ya J, ( (S. 8.7 sha (, / Y (,, j (, (,. (S. 9.6 ( G (,,, D.E. wh a ( G(, ( ( ( o sou b..: ougog gwav b..: ougog wav Souos ( G [Eq. (6.] G (,,, [Eq. (6.] (G fuos Ss as of G fuo Eq. ( Raao of a Loa Osag Sou Rvw of Ihoogous Wav Equaos a Souos : (6.5 f sa, a A A A J sasfy Lo gaug. (6.6 Bas suu of h hoogous wav quao: f (, (6. Souo of (6. wh ougog-wav b..: (, (, G (,,, f(, (6.5 hoogous souo ( f(, (6.5 wh G (,,, s vaua a (6. h a. s h souo of ( G G (,,, ( ( (6. wh ougog wav b.. 9. Raao of a Loa Osag Sou (ou Usg (6.5 (assu o (6.5 & (6.6, w oba h ga souos fo A a, whh a va fo abay J a. ( (, J, / A (, (6.8,(9. (, ( I ga, h sous, J (, a (,, oa a sa a a a a. Fo sa J( a (, (9. gvs h sa A a Ch. 5 a Ch., sy. ( A ( A ( J (5. ( sou ( ( (.7 Quso: I s sa o. 8 ha (9. s va ov o bouay sufas a s. Why? [S susso bow (6.7 Ch. 6 of us os.]

4 9. Raao of a Loa Osag Sou (ou Fs by Hao Sous : Oy - sous a aa. Raao fo ovg hags a a Ch. a Ch.. H, sa o sous of h fo (as a aa: (, ( (9. J (, J ( Sub. (9. o (9. a ay ou h -gao, w oba A(, A( wh A (, (9. J ( wh. W sha assu ha J( s of A(,.. h sou w o b aff by h fs f hy aa. Ohws, (9. s a ga quao fo A(. 9. Raao of a Loa Osag Sou (ou A s vao of (9.: W sa o hao sous fo h ous. Th, oy (6.6 s qu. A (, A (, J (, (6.6 L J (, J ( a A (, A ( A ( J ( [hoogous Hho wav q.] Th G quao fo h abov quao s G (, ( (6.6 Souo of (6.6 wh ougog wav b.. G (, (6. ( G (, ( (, A J J whh s ( Raao of a Loa Osag Sou (ou Rw (9., A ( J (, (9. H A (vywh (9. Maw qs. gv Z E (ous h sou (9.5 H wh Z 77 (a of f sa,. 97. Thus, gv h sou fuo J (, w ay vaua A ( fo (9. a h oba h fs Ha Efo (9. a (9.5. No ha has b assu fo J, h fo a oh quas whh a ss s of J. No: Th hag sbuo a saa oa a o qu fo h ao of H a E? ( w hy? 5 9. Raao of a Loa Osag Sou (ou Na- F Easo of A( ( (9. J Bfo gog o agba as, w ay ay obsv so ga os of A ( a h sou (. Fo ous h sou a (o, w * a us Y (, Y (,. (.7 S, w hav a. sou * A ( Y (, ( Y (, (9.6 J Th ga (9.6 ys uo offs as (.. Thus, (9.6 shows ha, fo, A ( a b oos o uo ( fs, whh fa off as jus as h sa uo fs, bu wh h. Howv, w w show a ha, fa fo h sou (, A ( bhavs as a ougog sha wav. 6

5 9. Raao of a Loa Osag Sou (ou Fu Easo of A( : W ay fa a A (, whou aoaos, by usg (9.98. Fo ous h sou, w hav,. H, ( a b w ( ( * j ( h ( Y (, Y (, Sub. hs quao o A( (, w oba J ( * J, ( (!! h a ( ( sou ( (! (!!(!!! a a A ( h ( Y (, ( j ( Y (,, (9. wh ( wh a (,, (S Abaow & Sgu, "Haboo of Mahaa Fuos, " Raao of a Loa Osag Sou (ou (9. s a a sso fo A (. W ow assu (.. sou so wavgh. Th, a j ( us o ( j ( (!! (9.88 ( (!! Sub. h ( ( ( a (,, a (9.88 o (9., w oba J, (,, (, Y [ ( ( ( ] a a a A ( (, * J ( Y (, ( s h obao of (9.6 a (9. Jaso. I s va fo a ay ous h sou. Th go ous h sou s ooy v o os (by h ff hysa haas: Th a (sa o: ( Th a (uo o: ( Th fa (aao o: ( 8 Gffhs.. E Do Raao Gffhs E Do Raao: Aoaos Cos wo o hags of +q a qq saag by a sa (. Assu ( a b ss susoa fo. Th su s a osag o: ( q( ˆ qos( ˆ os( ˆ, wh q. Th a oa s (, V(, qos[ ( / ] qos[ ( / ] 9 Aoao #: Ma hs hysa o o a f o. Esa h saao sas by h aw of oss. os ( ( os ( os os[ ( / ] os[ ( os ] os[ ( ]os( os s[ ( ]s( os Aoao #: Th wavgh s uh og ha h o s.

6 Gffhs Th Ra Saa Poa os[ ( / ] os[ ( ]os( os s[ ( ]s( os os[ ( ] s[ ( ] os Th a saa oa s: os os[ ( ] s[ ( ] os ( os (, V os[ ( ] s[ ( ] os ( os os s[ ( os[ ( ] Gffhs Th Ra Saa Poa Aoao #: a h aao o. Th a saa oa s: os V (, s[ ( Th aoaos ( Gffhs Th Ra Vo Poa Gffhs Th Eoag Fs a Poyg Vo Th a vo oa s by h u sy. q I ( ˆ q sˆ / [ /] ˆ J(, q s[ ( A (, / s[ ( ] Ra oas: os V (, s[ ( A (, s[ ( ] ˆ E VV A BA A s E V ( os[ ( ] θˆ s B A ( os[ ( ]ˆ φ s S ( EB ( os[ ( ] ˆ Th oa ow aa s s P S a ( s

7 9. E Do Fs a Raao Rw (: Y (, [ a( a( a ( ] ( A (, * J ( Y (, Ta h [ Y J ] Jy a o by A ( y J J ( ( A A ( J J J y J ( y y, (9.6 gv o obuo baus J wh ( (.8 s oa: J y y J y y (9.6 gvs h o obuo o h souo. I s J va fo a ay ous J J h sou. ( Quso: Why s h o ooo J (s.? 5 9. E Do Fs a Raao (ou Rw (9.6: A ( (9.6 Z Fo (9., a fo (9.5, H A E H H E I h fa o (, (9.8 us o a sha wav H H oo of sou E ZH I (9.9, w s ha E a H a has, a E, H,a a uuay ua. Ths s a ga oy of EM wavs ubou, ufo sa. Gv ay wo of hs quas, w a f h h. (9.8 ( E Do Fs a Raao (ou H (9.8 E I h a o (, (9.8 us o oo H of sou (9. E o ( ( E a H a 9 ou of has avag ow. ( E has h sa saa a as ha of h sa o (., bu wh. ( H ( E E-f gy B-f gy. Qusos : ( Why os E hav h sa f a? ( To oba (9., w hav g a fw s (9.8. Bu so of h g s a s oa h a o? Wha a hy a wha ss a hy oa? 7 P 9. E Do Fs a Raao (ou -avag ow h fa o/u soag * R (9. E H Z (9. Ths vo gvs h o of E,.. h oaao of h aao (s fgu bow. Z y y P oa ow aa I ga,. If, h has a f o, wh, a P y y Z s. (9. Ohws, h o of (h P vay wh, bu P s s gv by (9.. (9. E H o aao a 8

8 9. Mag Do a E Quauo F Rw (: Y (, [ a ( a ( a ( ] A ( (, * J ( Y (, Ta h s [ a ] * A ( ( Y J Y (, ( (,,,. 9 Y(, Y(, s s 8,, 8 os * ( osos ss ( s s os( os os s (.68 9 Thus, 9. Mag Do a E Quauo Fs (ou ( ( ( ( (9. { J J J } A J ( ( ( ( A Q A, quauo aao ag o aao wh ( ( fo a ay (9. ous h sou wh ( J [ag o o]. A gvs h A ag o obuo hough (9. a (9.5 (s.5: H ( ( (9.5 Z ( E ( Mag Do a E Quauo Fs (ou I h fa o (, w hav h sha wav souo: Z P H E ZH P Z o of E I h a o (, ( E a H a 9 ou of has avag ow. H ( H has h sa saa a Z E E as ha of h sa ag o (5.56, bu wh. ( B -f gy E-f gy. Th quauo aao, suss (9.7-(9.5, s o quauo aao oa. H, w oy usa s a aao a by h fgu o h gh. o Coaso bw Sa a T- Cass aos - of E uo fo of uo bw, a B ( : so aso os J, E, a B of h sou sha ( o / q E B haos Fo, a uo sa ( E( aso ( fs a b sg fa. [(.7] o Fo, uo as J ( B ( Tayo ss Q ( ( j j j fs a oa by [(.] of h ows-o ( J ovashg. Th s o - (a a o E o B / ooo fo a soa sou (s.. Ao. h sa f, Q j, a hav h a a - E sa ssos as hos as fo h osog ( ( sha b - haos of h sa ouas, sa uo, bu wh aso ( ( bu wh h (h as J B a quas-sa fs. [(9.98] of. (b fa o EM wavs I - ass, uos a EB, / ga B-fs a (sha EM wavs ag uos a A uo fs /, ga E-fs. av ow vs uhag wh sa.

9 Iu E a Mag Dos 9. C-F La Aa A Quaav Loo a h C - F La Aa : C Tavg wav Egy sou L Tassso R LC osao C-f a aa + a o fa o I h a o, E a B a ay ga by a J, svy ( agy sa f as. I h fa o, E a B a gav hough B a E( EM wavs. 9. C-f La Aa (ou Da Aayss: Th -f a aa s a as of sa s, baus aows h souo of (9. os fo fo ay vau of, whas Ss. 9. a 9., w assu << A ( J (, (9. wh J ( Is ( ( y (9.5 ( / s I / A No: ( J s sy abou J. J( J( J J ( I s h a u oy wh. Quso: : Th aa aas o b a o u. How a h b u fowg o? 5 9. C-f La Aa (ou ( os [ ( ] [ [ ( ( ] 8 [ ( ] = ( f = H, f, w a w os. os / s I A ( (9.5 / os I os os os ( 9.55 s No: os a b g f. Bu os os ( os as a oa obuo o h has ag v a. 6

10 I h fa o, 9. C-f La Aa (ou E H Z (9.6 & 9.9 s A H A A H * Z Z os( os os( fo s a ay os ( os /s, os ( os /s, P R E H H s A ( ZI, ( ZI 8 haf-wav aa ( J P J J (9.57 fu-wav aa ( suoso of haf-wav aas has P os oh ss oh aow ba wh 7 Rw ( C-f La Aa (ou os os os, fo ( s P ZI a ay Lg as ( o aoa o: (.. os ( 8 os 8 os os os P ZI 8 8 ZI os 8 s 5 ( s [va fo ] ( Ths has h sa a as (9., o, whh was v by assug C-f La Aa (ou Raao Rssa a Equva Cu: : J ( Is ( ( y I ( ( y I (a u, ZI ZI 5 8 ZI o P 8 Thus, fo (, P ( s ( s (9.8 P P I R Z a wh R ( a o s ( (9.9 Ra : aao ssa., R s a of fh f fo of a a, s (6.7. 5( ohs [S. -.] I I Equva V u fo V R a a -f u aa sos 9 9. C-f La Aa (ou Pobs:. Th fu-wav aa aao (9.57 a b hough of as h suoso of wo haf-wav aas, o abov h oh, has. Dosa hs by vg P fo h fu-wav aa = by suosg h fs of wo haf-wav aas (ah of gh /, s fgu bow.. If h wo haf-wav aas ob a 8 ou of has, v P aga by h ho of suoso.. Po h aoa agua sbuo of P obs P a. Ea h ff quaavy. aa of gh Ths a ay aa of aa wh o P s gh sg aa fa fo h aa, as of gh os s assu h.

11 9. C-f La Aa (ou Souo o ob : : P of suoso qus ha w a h fs (o h ows of h aas, ah of oa gh. Rw (9.55 os I os os A ( (9.55 s (9.55 as o a sg aa of oa gh (s fg. abov. So h f of ah of h aas hs ob a b oba fo (9.55 by ag (9.55 wh a ssg wh s o h of ah aa (.. by a aa., os os os I A,, (5, s aa wh os a os. os W ay aoa, h oao of (5 by, bu us us h o fo h has ags h oa s., 9. C-f La Aa (ou I s assu ha ah aa hs ob s h haf-wav a, h w s (5 a h suos f of h aas ( has s gv by os( os AA A I os os [ ] ( (6 s os ( os I aa s Fo (, P Z s A aa ZI sa as h fu wav os os s souo (9.57 S Souo o ob : o If h wo haf-wav aas ob a 8 ou of has, w sy a h " " sg (6 wh a " " sg. 9. C-f La Aa (ou Thus, I os os os( os AA A [ ] s s( os os( os I s aa Z Fo (, ( P s A aa ZI 8 s ( os os ( os s os ZI Souo o ob : s s Howo of Cha. 9 Pobs:, 6, 8, 9,, 6, 7,, has o aao ou of has quauo aao Quso: How os a has aay aa wo?

Posterior analysis of the compound truncated Weibull under different loss functions for censored data.

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