Overview. Review Superposition Solution. Review Superposition. Review x and y Swap. Review General Superposition

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1 ylcal aplace Soltos ebay 6 9 aplace Eqato Soltos ylcal Geoety ay aetto Mechacal Egeeg 5B Sea Egeeg Aalyss ebay 6 9 Ovevew evew last class Speposto soltos tocto to aal cooates Atoal soltos of aplace s eqato aal cooates Hoogeos boaes ecto Gaet boay cotos the cyle Hollow cyle obatos of boay cotos Speposto evew Speposto aplace s eqato fo x a y H wth boay cotos show Do ot have hoogeos boay cotos ay cooate ecto Speposto s of sple soltos y H x x E y y x 3 evew Speposto Solto S two soltos wth oe oeo boay x x Solto hee s xy x x y H Solto to ths poble s xy y E y x y H y 4 evew Geeal Speposto a obta solto fo seveal oeo boaes by ceatg a sepaate solto fo each oeo boay Thee-esoal pobles ca have p to have sx sepaate soltos fo xy Swap x a y a swap H a solto fo y y to get solto fo yx xh Set y H y solto fo xh b x to get solto fo x b x 5 evew x a y Swap S two soltos show below x x Swap x a y to get xy fo x x y H xy fo pevosly y x y H E y y 6 ME 5B Egeeg Aalyss

2 ylcal aplace Soltos ebay 6 9 evew x-y Swap Solto x y s xsh y x y B s κ ysh κ x κ H B sh H sh κ H xs x y x x H ys E κ y y [ s xsh y B s κ ysh κ x ] 7 evew ooate Tasfo Swap locato of oe eo boay y H x xy fo pevosly wth x x y x y H xy xh y se S x place of x S x 8 x x y evew y H y Solto x y x y s sh sh H y x y B s xsh H B sh H xs xs S x x x x ylcal aplace Solve aplace s eqato a cyle fo eo boaes at the ses a botto Specfe top boay te solto at 9 What o We Expect? ote slaty to aal ffso eqato s fte at fo both pobles t Sepaato of vaables eslt fo aplace s eqato shol be sla to eslt fo ffso eqato Bessel fcto egefctos ecto What o We Expect? Also sla to aplace eqato fo xy x x H x x y Sepaato of vaables eslt fo aal eqato shol be sla to eslt fo y ectagla cooates Hypebolc se/cose solto fo ME 5B Egeeg Aalyss

3 ylcal aplace Soltos ebay 6 9 ME 5B Egeeg Aalyss 3 3 Sepaato of Vaables opose solto Sepaato of vaables a ODE soltos gve statg pot fo solto D cosh sh B A 4 Boay otos fo all eqes A sh B cosh B te solto at eqes D fo all eqes f so the eos of Solto s s of all egefctos sh 5 Boay oto at aal eqato fo s a St- ovlle poble so we se egefcto expaso fo y boay ego s a p s weght fcto sh sh sh 6 Exaple: a ostat sh sh sh sh sh Sbsttte eqato to geeal solto 7 Exaple: a ostat sh sh sh sh sh sh 8

4 ylcal aplace Soltos ebay 6 9 potat Obsevato solvg DEs by sepaato of vaables val tes have the sae behavo ay eqato ffso a aplace so fa Tes lke /x wth hoogeos boay cotos gve egefctos that ae ses a/o coses Tes lke / [/] / wth hoogeos boay cotos gve Bessel fctos as egefctos 9 Two Boay hages aplace s Eqato two-esoal cylcal ego a Boay cotos Othe boaes: / s fte eo gaet Hee we have hoogeos boay cotos ecto Wat St-ovlle solto ths ecto to get egefcto expasos fo oeo boay at Sepaato of Vaables se to solve eqato ck sepaato of vaables costat to gve se a cose solto A s B cos D Mofe Bessel ctos Mofe Bessel cto lots ν x -ν ν x ν x -ν ν x - Satsfy ofe ffeetal eqato x ν y y x x y x x Eqato above tasfos to y ν y Solto s A ν x B ν x Sce ν ~ x ν ν a ν ae eal 3 x x 4 All becoe fte as x appoaches x x 4 ME 5B Egeeg Aalyss 4

5 ylcal aplace Soltos ebay 6 9 Boay otos fo all eqes A s B cos B te solto at eqes D / fo all eqes / A cos ; o tege s fte at oly f B Solto s s of all egefctos s 5 Boay oto at Eqato fo s a St-ovlle poble; se egefcto expaso s fo boay s [ s ] s 6 Boay oto at Deoato tegal eqato [ s ] eato tegal eqato fo x s cos s cos cos cos s cos 7 Boay oto at eslt fo a costat 4 s [ s ] 4 s 8 Execse Solve pevos poble chagg the boay coto at fo a eo gaet to a eo potetal s fte at ck statg pot fo pevos solto [ As B cos ][ D ] D to elate at B fo / a tege fo 9 3 ME 5B Egeeg Aalyss 5

6 ylcal aplace Soltos ebay 6 9 ME 5B Egeeg Aalyss 6 3 Hollow yle Solto aplace s Eqato two-esoal ego a Boay cotos wth all othe boaes eo: a wll gve St- ovlle solto ecto Expect ofe Bessel fctos ecto sla to pevos poble aot op wthot ego 3 Sepaato of Vaables opose solto sal sepaato of vaables pocess a ODE soltos gve D cos s B A 33 Boay otos fo all eqes A s B cos B fo all eqes As so that tege fo all eqes D so D / Solto s s of all egefctos s 34 Boay oto at Eqato fo s a St-ovlle poble; se egefcto expaso s s/ at s s s 35 Boay oto at Deoato tegal eqato eve o cos cos s cos s cos s s x eato tegal eqato fo 36 fo s s G / fo o oly Defe a G to splfy eslt fo

7 ylcal aplace Soltos ebay 6 9 ME 5B Egeeg Aalyss 7 37 eslts fo G s 4 G s 4 G f Aothe Hollow yle aplace s Eqato two-esoal ego a Boay cotos wth all othe boaes eo: θ aplace s eqato fo cyle wth o agla vaatos 4 Sepaato of Vaables opose solto sal sepaato of vaables pocess a ODE soltos gve D cosh sh B A 4 Boay otos A sh B cosh B a wth fo all eqes D D D Mst have eo eteat eo to avo tval solto D 4 Boay otos Ths s egevale eqato fo Sbsttte to ths eslt to show that egevales epe o as ato / Det

8 ylcal aplace Soltos ebay 6 9 ME 5B Egeeg Aalyss 8 43 g Egevales f / as ato. as ato.5 as ato.9 44 Boay otos Seco ow of atx gves D o D / aal solto s D obe these to get D [ / ] D Mltply by a a sbscpt D 45 Boay otos Defe ew costat: D/ sh D D D Solto s s of all egefctos 46 Boay oto at Eqato fo s a St-ovlle poble; se egefcto expaso at ] [ sh sh sh 47 Solto fo a ostat ] [ sh ] [ sh ] [ ] [ sh sh sh tegals fo aslaw a aege octo of Heat Sols Oxfo Solto fo a ostat sh sh sh sh ovet to esoless fo; we pevosly showe that f / / epes o / / / a /

9 ylcal aplace Soltos ebay 6 9 ossble yle obles a have sol o hollow cyle a se speposto to hale oe tha oe ohoogeos boay a have boay cotos tes of potetal gaets o a lea cobato of the two ctos se fo soltos epe o the ecto fo the hoogeos boay 49 5 ossble yle obles Hoogeos aal boaes gve aal egefcto solto as Bessel fctos a vetcal solto as hypebolc se a cose Hoogeos vetcal boaes gve egefcto solto as ses a coses a aal solto as ofe Bessel fctos o sol cyles wth oa o ae ot peset 5 ME 5B Egeeg Aalyss 9