SCALAR WAVE EQUATIONS draft 12 nov 2017

Transcription

1 SCLR WV QUTIONS ov 7 Ths hp s o su h us suls o h sl wv quo h s b o h o gl s o Mwll quos. I hols o h o h s opos o h lo-g (..) l b su o o h spl. Th suls ul o usg wv popgo sp oss s h so oos h osllo phoo spl. Th sl wv quo o h.. l s so-o pl vv quo h s l ls sp hoogous. Th s sl wh h s o h Shog quo wh h po o h o o h p. O h oh h h s ol ( o o h spl sl) o h spp p o h soluos h b ob o h ooho ho s (o h.. l) h os g s (Shog). Ths suggss o s ou su wh h plll uo o h oo sp-p sl wv quo. Th usg o wh s o wv hs h g o wvlgh qu wh s h logous o h o wll llow o pp svl logs wh h.. wvs lso o vg o h l wh so ss o l o usg h logous suos h oh (o s h s o h pol b wv hs wll sul l o h s o h v l l h ops s ow s lo ). O h sp-p sl wv quo s s w wll loo o soluo b oo. Th ol psso o h 3D wv quo Cs ll sphl oos wll b gv. W wll h su D ss s ousg o h Cs sp wh h ol po o h l soluo o h D sphl s. I D Cs sp p o h gl soluo o h sp w wll su h ss: h sp- h lo h po pol. W wll h ov o h D Cs sp. H w wll p ou lss o h sp h sp- h lo h po pol. Th suls wll h llu ul us o h opl popgo h hv lu o h l v holog: h popgo o pl wvs lo h Sll lw h gu popgo (h lus h op o o o o) h Dsbu b (DB) holog o wvlgh slo ls os. ll h 3D wol wll b os o h lgh popgo h sp sg o ow sous. I wll l o h osuo o h Khho gl h vluo o h so-ll -l sbuo h s o o h lv us o hg l ls vs.

2 Two ogs o h sl wv quo L us s o h plll oso o h wv quos o h log l o h wvs. W wll NOT ll h wh h s: s ou s s. V W wll ol p h o h.. l h s quo woul hol o oh opo s H H H. Moov: ) Th l os vuu l b h loshp wh s h v (s h suppl os "O logh") ) I ou oo 3) Dlg wh wvs h spso oul vuu s wh s h wvub. Ths s h ll qus hv h s popgo sp. Vuu s o spsv u o.. wvs L us ssu h h log l s ho. Ths s h Th uo () b o s ()() Th -p uo b w s Th puls l o h qu b s os. Th.. l uo s ow ( ) h sp-p uo s soluo o h w quo Ths s: vuu (>) (=) Ths quo os o ) Th lssl quo o h ol g s h su o h h pol p gs V b) h ospo bw phsl qus hl opos h D s gv b p V V I lso s oh s o ospos popos b Pl DBogl bw h qus g ou p h wv qus h p I s po o o h h g quo o h pl logous o h suo o.. wv vuu ls o p Ths s h / s NOT os. Ths pos ou o o h os sg s bw popgo o.. wvs sp. Vuu s spsv u o wvs o ss wh g s os w o ( ) h sp-p uo s soluo o h w quo Ths s: V vuu (V=) (V )

3 Th v ol o h wo wv quos s h sul ou o o s hv logous h oh s. I s po o p h h spso los o obsv h o wvs h logous o h v s h sul o h o / h l gvs o log ls. o h Shog quo V V oo o h 3D soluo. Th sl wv quo h w wll os h gl o o vuu s hoogous l l quo wh pl vvs s h 3D sp. Ths s h o h oo ss o ou ho ps o h vbls (); (); (). Is soluo s gl qu opl o possbl pl o. w so pul ss s possbl o hghl spl h quo s soluo up o so o o l quo. Ths hpps wh s possbl o soluo o h quo h ulls h sp bou oos o o: b C B ; ; pul wh h soluo h o o s uo o ol o o h h vbls. W wll os h h ss: h Cs h ll h sphl os. Th s s b h os po. Th so s l pu uol s. Th h s ul o h sphl wvs h h bss o h Hughs ppl. o ll h o ss ( o oh ss s h pbol l oos h llp oos ) h sg po s o popl pss h gv ss. Lpl opo Cs oos Lpl opo Cll oos s os Lpl opo Sphl oos os s s os s s s s Woul h soluo b possbl b oo w oul w h quo s:

4 oo o h wv quo Cs oos C B C B C B C B vg b BC: C C B B oo o h wv quo Cll oos b b oo o h wv quo Sphl oos s s s o ll ss h l-h s s o h s h pg o o sgl oo. I oh wos h h uos h o h Cs s h s ol pg o h so ol pg o h h ol pg o (h logous o h ll sphl ss). Th gh-h s u s os. Th ol w o hvg os o h l-h s o oo o h 3D sp s h h h l s s os sl. Pls o h hs os NOT h o s Cs s h uo B() s os. I ss h h ospog l s os. Dog h s o h h vbls g h spv w oss s w ol h h su o h squ vlus qus. C C B B W obsv h s possbl h o o wo o h uos BC s os. Wh s NOT possbl s h ll h sulousl os. D ss. L us s loo o soluo h h spl ss o whh (w ssu) h soluo sl os ol p o h s vbl: o h Cs s o h l s o h sphl s. I h s o h s s sll o h oh wo. D Cs soluo Ths s hoogous l o l quo. I h hs wo ll p soluos h gl soluo s l obo o h wo

5 Th gl soluo s o s. Ths s oobl bu s hl g s h l obo bov. D Cll soluo Ths b oobl w s Is soluo s obo o Bssl uos h h gll s Z () wh = p = Z Bus o lg vlus o h l s h quo us o h s l o h s D s h spo soluo us bhv s whh suggss o h sp Bssl uo wh h uos J N J N D Sphl soluo L us p h l-h s Dsp s los sl wh h p ll s h woul ss us ows Bssl uos g J ) w wll spl h h h l ( hs s u pov w loo hl- Bssl uos s uo s soluo o h sphl sl wv wh h s o p o h sgul oos. I s sg o obsv s ow h h squ oulus o h soluo (h pss h s o h. l o h pobbl o pl) s h s h lw o h uo o h s o h l o po hg. Th g o h pol Th D soluos su o hv sgh o wo sg puls o h suls: o phsl wh h op o wvo h oh hl wh h los sl wh h ou lss. ) Wvos

6 ousg o h D Cs s wh w osu h -sp p o h soluo (l us o h.. s) ( ) p p Th sul s h supposo o wo ho uos h osll wh sp wh h o. Th phs u pps h o o oo-quo Ths ss h o s h o-phs = s lo = h ovs og wh h s o h uo oo log sgh l wh sp =/. Wh s ovg og wh h lw? W us b h ou ss s 3D ss o whh w sug so pul o o h.. l. I 3D sp h quo =os s pl ppul o h s h quo = s pl plll o h = pl h ovs log h s. Th sl pls plll o h = o lo s o h oh. Th ospo o h o h os() uo h ss o h wvs. Th wh w ll ow wvos h ov (popg) ows o bws pg o h ho o h sg h quo o phs. I s sg o o h loog h 3D wol h s h s ppul o h popgg wvos os wh h o o h lgh s. W h wvos s h sus ol o h popgo s. Ths bos v o v wh w ov o h D soluo o h sphl oos. H h l quo s spl h wvo quo s wh s g. Bg h l oo hs sbs sph wh us. Th sphl wvs sblg h ul wvs h w obsv howg so h w. Pls obsv h h wv us bus o h / h h l oo h s h ol o h wvos s v o ll h hl o o h lgh s s w l s hlhoo.

7 W wll go b o h op o wvos o s wh g h lss o h o-d ss. )ou I w os h possbl h wvs o wvlgh ss sulousl w s h ppl o supposo. I llows us o l wh pol-ho wvs sug h wvlgh b sl h g up h suls. Ths s quvl o w wh h wvub pss o wvlgh h ospog ho uo hs s ow o. I w os h possbl o suppos ll possbl wvlghs bug o o h o h w w h supposo s Ths s l h o o h ou so. Ths ops wol o possbls pos hos soluos ll bs o h h Nu wos wh h ou ho. W wll svl s pl ls bw wv popgo sps o h ou lss h pobbl hvs h u h Sl Tho o Do. sgh o s ul ool h Khho gl wll b gv wh lg h 3D Cs s. Rlv D Cs soluos Th s o h uo sp wh h o o sgl oss h whol sp ls o h soluo pvousl lul Ths ss h D uo sp w hv gv wvlgh ospog o h wvub o pogssv wv o gssv wv o os l obo o h wo. So o sg ss pl h sp s o o uo.

8 D Cs soluos: Sp I Th s o h sg D ss l wh sp h v. L b lo =. W ssu h os v sbs h hl-sp < whl os v sbs h oh hlsp >. I s o ss o whh s lg h h oh. ogl o h o wvub hs h popgo < whl sbs h wvs >. W us h solv h quo spl o h < h > ss bg o h os s o h s o sp. < b b > Th l soluo qus h h wo uos o =. v o w o ol qu h ou o h soluo uos bu lso o h vv. Th so o l h qu s h h ogl l quo s so-o o whos soluo s upo h ho o wo b oss. Ths s h o jog wo pws lul soluos w o s wo ou oss. I s l h h wo ou oss wll llow o wo o h ou uow oss ppg s b. W bl h oh wo. Ou ho wll b v b phsl osos. I w g sgl wv vllg o - log h posv o o h s s ps b h o h uo. Rhg h = h wv wll b pll s s ow popgg wv log h gv o o h s <. b > pll l s gssv wv I hs s o wv wll vl h > sp log h gv o. I oh wos h wv b wll o b ps (o b =). O ous w g h s s whh sgl wv os o + log h gv o o h s. I h s w woul hv g s l wvs h wv woul b bs (o =). Th gl ss lus h ps o boh og wvs o boh - +. Th wo wvs wlg w o h = oppos o woul b h o s l obuo o h og wvs. I shoul b l h w spl ou su w sl o o h wo spl ss (h s o s) wll owg h h gl suo b ob b supposg h soluos o h s o s sl o. L us h ssu b = lso =. Th ls s bus w hv h possbl o sl b vlu o h ls os. Th u vlu s spl o h plus o h s l wvs s os

9 o h l plu. o hs so w = (h ss o lo ) b = ( ssso ) w < > W sl g < > o whh h vlus = Th ou oos h bo W h g boh h plu lo plu ssso os : Th plu spo s o sgush hos os o h o popul os l o h s lo ssso os R T h spl h squ oulus o h ospog plu os: 4 T R ll bus o h loshp w w ll os s o h o s: 4 T R I s sg o o h h s o h o o h wo ss o h = ( = ) w woul g T R h s: o lo ol ssso. I oh wos: h popgo o h wv h sp. o subl o s o h plu lo o. I ss h wv s ( ls pll) l boh s o > h oppos s. W oul g h hs s h w hv h s lo o h wv vllg o

10 I s o s h ou o lo s h s o h wv og o - o h wv og s p o +. O g w s lgh b o o w (ppul o h o ou D s) h ou o lo wll b h s s o h s b s o w o. Th ol s h sg o h hgs. Bg hs s h h l wv wll hv 8 phs sh s o >. vs wvs Dsp h ul o vsulg suh s o h D.. wvs w shoul os h possbl h h popgo os s g ub o o h wo hl sps. I s o o hos ss whh Quu Mhs hlp: s h s whh V ospos o suo wh h g o h pl s low h h pol V. I s l s s o os h o soluos vlbl s h hs hols o boh h hl-sps. W p h o soluo o s h < hl sp hg pogssv wv og o h l hg h = wh s sul l lso sowh s. I h gh hl-sp > h quo s ow < Is soluo s l obo o pol o l gu b b > Th ou oos h s s bo Loog h uo w s h vgs o lg vlus o. Ths s goo so o sg b =: h hl soluo lso s phsl qu s obvous h plu h s woul qu g. W h o (h s o ho) o su h sol popgo o wv og o h l s o ou ss. W llow o s g = o = b =. < > < > W g R 4 T Ths s g wh h pvous sul wh w ssu h. I s l o h o o h su ( o hs/h ps sus Quu Mhs) o pl how h ol s lo b R= s ogh wh o-ull s ssso o T. Woul bo v lg (v hgh g sp Quu Mhs wh V>> >) lso h s lo o woul bo ull s p lssl hs.

11 D Cs soluos: lo lo s sl o l wh o s oh u. Th os sg s s wh h oh u s h sp (wh = =). L us suppos o hv h sl o hoogous o bsobg l whos v s. L b h hss o h sl. L pl opl wv o u plu qu pg o h sl o l. Is wvlgh vuu wll b whl s h sl wll b Th ospog wvubs wll b spvl wh =. L h l su o h sl b lo = so h s gh su s =.. Bus o los l wv wll pp o h l o h sl whl pu s wv wll popg ows o s gh. Is h sl boh pogssv gssv wv wll s. Dvg h sp o h gos s g. h wvs wll hv h ollowg o: b I II III Th os l o h s lo ssso os R T b R T Th bou oo o ou o h soluos h s vvs hols o boh sus = =. Ths s:

12 b b b b h b w s: b b b b g h s wo quos subg h so wo os ls o b b whh llows o b : b h bus o h h ou oo ls o h o : s os 4 4 Th s ssso o s h: T s s os I oluso: T s Th b pu o o usul o wh w os h pow lo o R o sgl (h s o os l lg so h h gssv wv s go II vshs lg b=). I h s o lso o whh h R

13 I s o subsuo o v h so h 4R R s T Th ob sul s o l lo wh lo ssso opll b h sol v o h u. I s l h h Th o lo o so o vlus. o s wh =3.5 (s o Slo) o hs. 6. O h oh h hs po ol g h shpss o h po sos o T: h hgh h shp T. I o o hv hgh vlus o h lo s usull o wh sp lv ogs sog h v o ul-sl l. Th hol woul bo ubso lhough o possbl. sho w s o os h ol pow lo o R o sgl lso wh popl o pu 4R R W h vl p o h lo ssso o h psso s T wh 4R R pov w b h T s o h lo (wo sus) whl R s o sgl su. I gl wh h wo lo sus o wh ogs o spv pow lo os R R R. R R o hs Th s ool s h ss p. I s v h T s po uo o h pou. L us ll h h s h phsl hss o h lo sl s h wvub s pps s h lo sl. Ths wvub s h vuu lso v pobbl o s vlu s h g u h gs h wv. O h o h qu s h s vwh whh sogl pops o h lo ssso uo T s uo o. T s

14 I hs w h pos o los b op. Now T s po uo o h qu pg o h wo v-sp ps. I s l h sp o h ( o hgh) lv o h wo sus p ssso (T=) ous wh wh s g. I s o qu hs u ssso ospos o h qull sp vlus o whh h s Ths ss h h spo o h s p o h vlu o. Th u o T s hv wh h s uo hvs h vlu. I h s T v o po s h lwh o h ssso ps. I b su o T=.5 (hl u h) whh obvousl s s h s h s Th gu plos T h g 9-96 TH o 6 h S (=3.5) lo wh lo o vlus o Th oubl lo l L us suppos ow o s up s o wo los (V oguo) g o h hss h v. Pov o los llow wh h s spg h wo g sus h ol ssso uo T s h pou o h wo sgl ssso uos T T. T T T s s

15 Th hl lss o h uo s ubso bu gphl sgh b llug. I h ollowg g wo los wh l ss hv b oupl llowg so 5-6% bw h spv vlus o h pou. Th wo sp ssso uos hv slghl po so h wo ps ll o h pogssvl sp o h ghboug os. Th pou s h u h o h qul ss. O o h l pos s o how o sl sp p. I s l h h os pobbl sul wh sul vlus llow o s o ovlp h go o s. Wh sp qu hs o b sl h wo ollowg oos us hol: wh gs o ssl l. Moov lso h o o o h s os s po s o ss h ho o h pl sup. Ths o b vlu b osg s pl sl qu s o h oul bov h loog o h hgh o h s os o s h s lo. I hs s h ssso uo o h s lo s b ho u s s s s T T T L p b h vlu o o whh s sl: s s s p p T p wh w ov o u w o sub g j o o o p :

16 s s s s s j j j j j T p p j p Th ls sp h uss h po o h squ s uo ( o o subo o ulpls o lvs h sul uhg) bgs h gu h g h ll shows h wo l los hv p ssso o ll ps. Now w bl o s h h j-h s p s spl wh sp o o h qu: j j s hgh s s j T j p W ov psso h ls wh qu subsug o j o h quo o h ps: s s T j j j p Ths s h quo o spl uo whos vlop s s T vlop s o h pl s p. Th uo s h s s o h sgl lo bu wh sl o. Ths s h h wh hl u o h vlop s s h h vlop. Th pl osqu s h o v o whh s o suppsso s hghl o h us b s lg s possbl. Wh usg l los hs o b pll hv b spl hgs h v : o s pu s us o oul oo lg hl ool shoul b qu o hv h pop oos. I ollows h o sup s h bgg so h hss. Ths b ob v o l vs b spl og h wh sp o p pllls.

17 Ths woul lso hv h hghl bl sul o so h possbl ps sog v h woul sup bw h wo vs. I h s oo b gl woul ou v opl p plll o h opl s gv b. os D Cs soluos: Co wv W oul ppl suppos h ou lo h s low h go () h ous. Phsll hs b o llowg o hg wh o spl gll sg o h popgo oss ous s h sl o h lv gu. w s o h Sp I s h sg o os s h D ops hs w phsl ss bu h wll g g po s soo s w ov o D ss h hs sl po ol Quu Mhs bg h pso o h so-ll Quu Wll. I hs s w ssu h h popgo os s g ub ous h sl whl hs s l vlu s. L us slghl o ou gos lg h lo o hss b o h og =. W po log wh ou pvous wo bu lso os h h l wvs h sb h soluos gos I III us NOT vg lg s o h og s w p o h sp s o vs wvs. I hs s pls obsv h h pol go I vshs lg s o h og bus ll vlus o h go gv. I ( - ) b II (- ) B III ( ) W spl ou job obsvg h p s wh sp o h og. Ths s h w g h vhg hpps poso s l o wh hpps poso. os( ) I ( - ) II (- ) III ( ) Ths ho sp ou wo b sg o h uow oss. Bus o h s s w ous o h sol =/ o h ou oos: o ulll h h wll uoll hol =-/.

18 os( ) s( ) W g loshp bw h oss os( ) h llows o w h soluo s os( ) I ( - ) os( ) II (- ) os( ) III ( ) O h oh h w o h ls os bu w g os oo o h s o h soluos h ls h wo popgo oss g( ) Ths sul wll b l so ul D. I s sg o obsv h h os bos v hgh. Ths ospos o h quu-hl suo o l Quu Wll wh l hgh pol wlls. Th soluo go III ( h lso go I) b w s os( ) Bg >/ h go h whol po hs gv sg h o lg ls h uo o o. No wv ss ous h wll. Ls h so vlus o gv b os( ) h s o h vlus o. D Cs soluos: Po I (Bgg) Th s o h po s o h upos s. W su h D o h s ow h wo o Wv Mhs s Bloh ho s h o po ovls s h s o g gps h o o b gs h ouo o h ops o Bloh lo o hol so bs o sol s los. w h Bloh ho (98) s ohg ls h h wv-hl logous o h Bgg ho (93) vlop o h sho wvlgh.. wvs h s o -s. Th Bgg ho hs b vlop o 3D po sbuo o h o h ospos o h o suu o sls. I bs h op o Rpol L. W h ssu h h Bloh ho s ow ls s D oulo. W wll ol o s suls. I pl lgh wv ovg h o vls oss po sbuo o v h b po ( ()= (+) ) hs s h h po b pss b s o ou ss V p whos os ow bus h uo s ssu ow gv b V p h h soluo () s o h pou o pl wv po wh h s po o h po : s o h sp uo U() h s

19 p p Wh o s o lul h os (b pug b () () o h wv quo boh pss b h ou pso lulg h vvs loog o h loshp bw os o h l pols) o sovs h h p o h ho o h. I gl ol h s o wll b sgl lg h s h gl h soluo s gv b wv p s o vos w ps s h wv ws popgg hoogous soop sp. Bu o squ o pul vlus o l wh ppohs o h g ulpls o h qu / h ospog o bos lg lg up o qu h gu o wh l. Two hgs us b obsv: ) o s o o s los (o qul) o ol o o hos vlus. I oh wos h ou pso o h wv uo s o o sgl wv p p p o wo wvs o os ) wh wo wvs ps bg h gv o h soluo p p Ths s h h wo wvs o pogssv o gssv wh h s wvlgh. Th h bul up sg wv. I oluso h popgo oss po oulo o h o uss h lo o hos ol wvs h ull h Bgg oo o hos wvs h pogssv popgo s opll bl b l gssv o sg wv buls up. Th popgo go s s Dsbu Rlo h o ops o slo o wvs. I s sg obsv h h s wv o b l s h o s. Th s ospos o h wvlgh h so o w h wvlgh. Th spo o bw ghboug os o s su (hs s h s o ls os) ll oh wvs o h ul o. Ths wll b show lg wh l LD ls os whos spoous sso spu s usull p los o vshs wh w p o h vlu. No wvs vlbl o Bgg (o Bloh) sos o po oulo h ops.

20 Rlv D Cs soluos L us go b o h oo o h soluo o h 3D sl wv quo o h Cs s. Th oo o h soluo llows o w h wv quo s B C B C p wh (h uo s vu b sp obo o h + sgs) Loog h phs w g h wvos h o g pls o h 3D Cs sp. Th oss ow h gol g o opos o vo (h wvvo) whos oulus s s o s ppul o h pl wvos. I s sghow o gv h soluo D Cs sp p wh o w gphl pso o h wvos o wvvo o s opos L us p h ou sus o h pvous ss: h sp h lo h po pol. D Cs soluos: Sp I L us u o h s o os o h D s.

21 W us h solv g h quo spl o h < h > ss bg o h os s o h s o sp. 4 3 < b b b b 4 3 > W shoul b bl o ll wvs h gu o h lss o h vous phs s h soluo. Th ou oos qu h h uo s h s wh ossg h. I oh wos: W wll spl h lulos b usg h s ppoh s o h D s: L b = psg h ol og wv. Th l wv hs o 3 h w s h s wv s vu b h o b ow s. I s v po o obsv wh v h s ou os ss: Ths shoul b o vlu o h vbls lug. Th ol possbl s h. W po sl o h o h soluo s wg h soluo > wh h subsuo Th ou oos ow Th gv I ss h l suo s o h D s wh h sol h s o h ul wvubs w h hv h -opos. Th pul os ou wh w os h gls bw h wvs (whos o s vu b h s lg log h wv-vo ) h ol o h h s h s.

22 Th wv vo hs oulus. Th wv vo hs oulus. Bg = o hs: s s Ths gvs h Sll lw s s I lus h o o h Bws gl s o < (l s s). Th suo sb h pvous s hs o gl oulo. L h o b v o wo hl pls D ss. I wo soluos s o o o h hl sp B h ou oos pos h b B B b b Ths us hol o vlu o h oo. I spl s B b Th osqu s h h soluos ow b w v whou owg h sp psso s B B D QUSTO PUNTO L DISPNS DVONO SSR COMPLTT I DU CPITOLI SGUNTI SRNNO INSRITI NLL VRSION INL DOPO GGIUSTMNTI VOLTI VITR RIPTIZIONI IN OGNI MODO IL PROGRMM PR CONCLUDR L DISPNS UN BRV RGIONMNTO SU UN CSO DI STP INDX COM IL PRCDNT M CON UN POTNZIL PRIODICO LUNGO Y NL SMIPINO DI SINISTR. POI I DU CPITOLI SGUNTI CONCLUDRNNO L DISPNS

23 Gu popgo pp o.4.3 o S.S (Th o) L us os s h bgg s oubl (h boo ll ss wh s o bu h os ll lulo o h sol s o) wh h l l o hss hs v whl h o ls (suppos l h) hv. L us su h popgo o o o h opos o h log l o spll g wh h boo h opo (). Ths opo us b soluo o h wv quo 48 pg.68: Wh o ssus glgbl vo log h svsl o so h: Bo oug l us ll so lv hgs: 4) Th l os vuu l b h loshp wh s h v (s h suppl os "O logh") 5) I ou oo 6) Dlg wh wvs h spso oul vuu s wh s h wvub L us s wh ho o o l ( ) pug = w g: : Ths quo o h spl p o us b solv spl o h og ls wh h v s = wh =. Th soluos wll h b s o o ogh wh h ol vvs wh sp o h bous h s log h o. Whl o h v go w wll loo o osllg soluo h o ls w wll qu h h plu o h l vshs o lg vlus o o o vo g o os.

24 o boh ss w wll po b spo o vbls h s supposg h ()=()B() whh llows o w h pvous quo (vg b B=) s: B B s usul w s h h s s uo o h sol h so o h sol h h s os. Th ol w o solvg h quo s o p h h s sl os. L us ogl s B B wh s os h wll b s h og ls s h v go. Th soluo o h p o quo s è B. L us sl h gv sg o h po so h s h o o pogssv popgo log h s: Now h quo o () suls o b whos soluo s () p bp I > hs gvs osllg soluo ohws h gus o h pols bo l pol s o s o h plu log h o lg s o h v go. Th s o soluo h osllg po s subl o h v go () p bp h bus o h s o h osuo b l w s () os so h os O h o s h o ls w us sl h poll sg o: so h p p wh h oulus o s hs soluo subl o boh h upp h low o l. Bus o h bul- s w loo o h oo o h soluos log h sol =/. L us s o h gl qul pos o. I gvs os p Ths hs possbl soluos o vlu o ol =. I ollows h >: p p lso: os s L oo os h bos:

25 s p os p o h s quo o gs : p os so h: os p os o h so quo w g h oo o g : h s l h oul 56 o h boo. I b hl o g o l o: L us lul h squ o boh ss Th l sul b w s os Th l-h s s h uo os whl o h gh s h s os b h sl wvlgh o h o b h v o h wo ps o ls b h hss o h v go. o l vlu o h os o h gh s h s s o vlus o h h soluo. Usull h lows o h (-o soluo) s h bs o.

26 ll h ollowg oul 57 h boo s spl gv b h o : p os os os

27 l pp o.4.3 o S L us s o g. 34 pg.644 o h gv boo. W loo o h so-ll l h s h sbuo o opl pow wh h opl b o ou o h ls hp lul v lg s (o o lls) wh sp o h pl s o h b gg o h ls s (w os log s u so). W wll us spl gl oulo o h Hughs ppl l s h gu bov wh h pl = ospos o h oupu ls - O h pl h oh oos. L us suppos h w opll ow h sbuo o h l ( ). o h pl. Th Hughs ppl ss h h po o h lgh g su wll bhv s p po sou o whh sphl opl wv popgs h sp. Th o h ol o h g pos pl o h suoug sp wll sul o g (h s o h gl) ll obus o h po sou.. Cosg h opl s h l o h g o h ls l us h gl ppul o h pl ls s h gl plll o hos s ls s // (s oà h quso s l spu b s p u l golo pu l so b gl ss sà l oppo ll golo osì o). o h sou po w wll g sphl wv p h s o h sou po. h obsvo po : Th plu o h ol pubo s h: p whos plu s ( ) wh s lg s op wh h so o h g osg obsvo gos los o h s (hs s >> ) h squ oo h oo s ssll qul o p

28 I h pol w shoul po wh o. s l w h vlu ou o h squ oo obsv h h w squ oo hs h o o + qu h s sll. Ths llows o subsu h oo wh s s o pso L us slghl pul h w psso osg h ls sp h sll ough wh sp o bu h u uh sll h (hs spl s h h b s ol: s us ss b sg h s o h o p) Now llg h os s s // w hv // // p p s s s s Wh w lul h s o h l h s h squ oulus o h pvous psso w g : // p s s Th sul s qu los o h q.(67) o h boo s ll b o opl. I h boo h sol svsl s s os h s lg log h o. Ths s ob b osg h s o h logul pl h s sg boh = = (whh lso pls // s ) h lulg h o o h sul wh sp h s wh boh ull ( h lso // s s ). p os p s s I I So gl s suvv: h gv sg h pol h o h oo o h s h go lso h o o ol log bu h phsl g s h s. Th gph g. 35 l so s ob pug h vlu ½ o h l s usg o h soluo o h l lul h o Co o. Ths pll s h gl. O ous sl psso b ou o //.

29 I s sg o o h h gl psso o h popgg wv s h o // p p s s Th gl pss b o h (b-sol) ou so o h l h oupu. s s o lul h shp o h b sg o sg( s s ) guss sbuo o p b wh b spvl h o h jo s o llps b