Physcs 36: Was Lcur 5 /7/8 Wa Suroson Prncl I s qu a common suaon for wo or mor was o arr a h sam on n sac or o xs oghr along h sam drcon. W wll consdr oday sral moran cass of h combnd ffcs of wo or mor was. L us consdr whr wo was wh dslacmns gn by and ar rsn. Wha s h n dslacmn? Th rncl s ald for lnar PDE s. Th wa quaon s: Suroson Prncl: x If and ar soluons of h abo PDE, hn a b, (whr a and b ar consans) s also a soluon. Ths maks nrfrnc ossbl Pag of 5
Physcs 36: Was Lcur 5 /7/8 Pag of 5 L us dlo h mahmacal formalsm for combnng wo was: Bgnnng wh h wa quaon x L and b soluons. Thn ry h sum of soluons as a ossbl soluon:. W g x x Consruc (was combn n s ) Dsruc (was combn ou of s ) Somhng n bwn and ha h sam frquncy G som wh h sam frquncy
Physcs 36: Was Lcur 5 /7/8 Thrfor, s a soluon. In gnral, w may form a soluon as a lnar combnaon of soluons: Bu suos ( x, ) C ( x, ) c x For soluons,, and, h c s c ( ) LHS RHS So h rncl of suroson s only ald for lnar PDEs (whr h funcon and s dras aar only n frs ordr.). So, Th Addon of Was of h Sam Frquncy As a sarng on, l us say ha w ha wo -D was ha ar harmonc. W can dscrb hs was as: If kx ε, hn w can wr ( x ) sn( ( kx ε )), ( x, ) ( ) sn and x, sn ( ) ( ) o, w ar assumng hy ha h sam frquncy L us suos hs wo was coxs n sac. Th rsulan dsurbanc s h lnar suroson of hs was Thrfor, sn( ) sn( ) ( sn( ) ( ) ( ) sn( )) ( sn( ) ( ) ( ) sn( )) Pag 3 of 5
Physcs 36: Was Lcur 5 /7/8 ( ( ) ( )) sn( ) ( sn( ) sn( )) ( ) ndndn of m Ths can b smlfd f w dfn and ( ) ( ( ) ( )) () ( ) ( sn( ) ( )) () sn sn If w now squar h quaons abo and add sn ( ) sn sn sn sn ( ) (3) If w dd quaon () by () w g an ( ) sn sn Mahmacal Asd: I s ofn connn o mak us of h comlx rrsnaon of was whn dalng wh h suroson of was. Rcall ha can b wrn as: sn sn ( ( kx ε) ) ( ) ( ) whr h wa s h magnary ar of hs quaon. Pag 4 of 5
Physcs 36: Was Lcur 5 /7/8 Smlarly, w can wr: ( ) L ( ) Wha ar and? Facor ou, whch cancls. R Im ( ) R( ) ( ) R ( ) Im( ) ( ) Im anφ R Im ( ) ( ) ( )( ) ( ) ( ) ( ) whch s dncal o Eq. (3), and an ( ) ( ) sn sn W can gnralz hs o was: Pag 5 of 5
Physcs 36: Was Lcur 5 /7/8 Pag 6 of 5 ( ) whr s dfnd as h sourc comlx amlud of h surosonal wa. Th nnsy of h rsulan wa s gn by: ( )( ) ( ) > and sn an W can also us quaons () and () o wr as ( ) sn sn sn Wha dos hs las quaon say? I shows us ha a sngl wa rsuls from h suroson of h orgnal wo. Ths nw wa s harmonc and of h sam frquncy as h orgnal wa, alhough s amlud and has ar dffrn. An moran consqunc of hs s ha w can suroson any numbr of harmonc was hang a gn frquncy, and g a rsulan wa whch s harmonc as wll.
Physcs 36: Was Lcur 5 /7/8 Inrfrnc Th nnsy of a wa s rooronal o h squar of s amlud. Thus from Eq. (3), w s ha h nnsy of rsulng wa s no us h sum of flux dnss of nddual orgnal was, bu ha hr s an addonal rm: ( ) Ths s an nrfrnc rm, whch s a funcon of h dffrnc n has bwn h wo orgnal was. Th crucal facor s δ, whr Whn δ δ, ± π, ± 4π, rsulan amlud s max δ ± π, ± 3π, rsulan amlud s mn Rcall ha -kx ε whr π k. λ So w can wr: π δ x λ ( kx ε ) ( kx ε ) ( x ) ( ε ε ) whr x and x ar h dsancs from h sourcs of h wo was o h on of obsraon and λ s h walngh. L us suos h wo was ar nally n has: ε ε, hn π δ λ ( x x ) If w dfn h ndx of rfracon Pag 7 of 5
Physcs 36: Was Lcur 5 /7/8 λ n, λ walngh n acuum Thn w can wr: π δ n λ ( x x ) δ k Λ hs quany s known as h ocal ah dffrnc (OPD) Λ Dfnons: was for whch ε - ε consan ar cohrn was. Random and Cohrn Sourcs From a rous lcur, w dfn h sourc nnsy as m-arag of h amlud of wa squard: I For wo was, usng Eq. (3) for h rsulan wa, w ha m-arag I I I ( ) If h squar of h was and ha random has ( ncohrn), hn: ( ).. h m-arag wll b zro. Thn, I I I, Or, mor gnrally: I I I Pag 8 of 5
Physcs 36: Was Lcur 5 /7/8 f you ha randomly hasd sourcs of qual amlud and frquncy. (rsulan nnsy arsng from cohrn sourcs s drmnd by h sum of h nddual nnss) [Wha dos hs man? If you ha wo lgh bulbs ha m lgh wh random has, h rsul wll ha an nnsy qual o h sum of h nnss of ach bulb no nrfrnc ffc s obsrd.] If hn - consan cohrn was ( ) Bu ars from and -, so I ars bwn and I I ( ) I II I I ( -) I II If I I, (5) hn I ars bwn 4I consruc nrfrnc and dsruc nrfrnc If you ha sourcs ha ar n has wh and qual amlud, hn I ( )( ) > ( ) I ( ) Pag 9 of 5
Physcs 36: Was Lcur 5 /7/8 f ach amlud s h sam. So hr, h amluds ar addd frs and hn squard o drmn h rsulng nnsy. Sandng Was W alrady cord hs brfly. Hr w assum ha w ha wo harmonc was ralng wh som frquncy and ralng n oos drcons. L us assum h wo was ha qual amlud. Usng comlx rrsnaon ( kx) E E sn (lf) ( kx) E E sn (rgh) E E E E whr h rsulan wa s h ral ar: E ( kx ) ( kx ) E E kx kx ( ) ( kx) ( kx) ( ) Ths s h quaon for a sandng or saonary wa as oosd o a ralng wa. Is rofl dos no mo hrough sac snc s no of h form f(x±). A snasho would look lk a arous ms, sandng was wll look lk snusodal was of arous amluds Whr s kx? (calld nods ) Pag of 5
Physcs 36: Was Lcur 5 /7/8 θ for θ π, 3 π, 5 π, kx (n ½)π n s an ngr o: In h ramn of lasrs (lar n class), w wll fnd ha ha lasr lgh s gnrad n lasr cas, whch ak h form of wo hghly rflcng mrrors surroundng somhng calld a gan mdum. Th lgh n such a cay hn consss of counr-roagang EM was ha form sandng was. I s ycally h cas ha h EM boundary condons a mrror surfacs rqur z nods. Ths mans ha h walngh s rsrcd o hos suord by cay dmnsons o dscr alus. d mλm d, whr m ngr,.. hr s angr of halfwalnghs ha f n h cay lngh. mrror mrror Suroson of Was of Dffrn Frquncy Exc n rar crcumsancs, w nr ncounr suaons whr w ha suroson of was of h sam frquncy. Consdr h suroson of wo was and ( x ) k ( x ) k For smlcy, w assum h was ha h sam amlud and nal has. Th rsulan wa s: Usng h dny [ ( x ) ( k x ) ] k ( β ) ( β ) β Pag of 5
Physcs 36: Was Lcur 5 /7/8 Ths ylds: k km m [( k ) x ( ) ] [( k k ) x ( ) ] (4) k Ths wa can b smlfd by dfnng: ( ) as h arag angular frquncy ( k ) k as h arag roagaon numbr k ( ) m as h modulaon angular frquncy k m ( k ) as h modulaon roagaon numbr k Usng Eq. (4), w g: ( x ) [ kx ] [ k x ], m m or whr m aryng amlud ξ ( x ) ξ ( x, ) [ kx ], ( x ) [ k x ], m m ralng wa wh frquncy whch can b hough of as a m aryng amlud. L us consdr h cas whr and ar larg and. Thn, >> m and x, x, wll ary radly. Th nnsy s ξ ( ) wll chang slowly. Howr, ( ) whr w usd I ξ 4 ( x, ) ( kmx m) [ ( k x ) ] m m sn θ θ θ θ Pag of 5
Physcs 36: Was Lcur 5 /7/8 oc ha I oscllas abou frquncy wh an angular frquncy of m ( - ) ba Ths s a low frquncy nlo modulang a hgh frquncy wa. Tha s, h rsulan wa consss of a hghr frquncy carrr wa modulad by a n funcon. How fas dos h nlo mo? Ths locy s rfrrd o as h grou locy. Pag 3 of 5
Physcs 36: Was Lcur 5 /7/8 W know ha f a waform s of h form ( kx ), hn h locy s. So, k d k k k dk From, k, whr has locy. k Dffrnang, k wh rsc o k, w ha: d g dk d k dk d [o, f h mdum s dsrs, hn dnds on k, so g, ohrws dk and g holds.] If w wr k, hn W can show ha: d g dk d k dk λ dn n d λ g whr n c Proof: d g dk d k dk Pag 4 of 5
Physcs 36: Was Lcur 5 /7/8 d λ d( ) λ ( λ) d dλ Snc c, n d c dn dλ n dλ λ dn λ dn c n dλ n d λ g Pag 5 of 5