Overview. Solving PDEs. Solving PDEs II. Midterm Exam. Review Spherical Laplace. The wave equation February 23, ME 501B Engineering Analysis 1
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1 The wave eqao ebay 3 9 aplae Eqao Colso ad The Wave Eqao ay Caeo Mehaal Egeeg 5 Sea Egeeg alyss ebay 3 9 Ovevew evew aeal o dae eeal appoah fo solvg PDEs Ohe deas abo aplae s Eqao Devao ad physal eag of wave eqao Solo of he wave eqao by sepaao of vaables Iodo o he D labe solo of he wave eqao Mde Ea Wedesday Mah 4 Solvg PDEs Pefo eessay opeaos f bodaes o hoogeos Dffso: defe v w v sasfes dffso eqao wh eo boday odos; w sasfes boday aplae eqao se speposo Solo s s of wo o oe solos eah of whh has oly oe oeo boday e eqao appopae oodae syse eagla o yldal 3 Solvg PDEs II e appopae sepaao of vaables solo Wa egefos o epess al o boday odo Use hoogeos boday odos o deee osas solos ad egevales eeal s of all egefos sed o f al o boday odo 4 Mde Ea evew Spheal aplae Wedesday Mah 4 Coves aeal o dffso ad aplae eqaos Ildes aeal p o ad ldg ogh s lee ad hoewok fo Moday Mah Ope book ad oes ldg hoewok solos 5 aplae s eqao a sphee has solos egede polyoals P P ϕ / sφ sφ φ φ P osφ!!!! 6 ME 5 Egeeg alyss
2 The wave eqao ebay 3 9 ME 5 Egeeg alyss 7 evew Hollow Cylde Cosde vaos boday odos Noeo odos o ppe o lowe sfae oly gves essel egefos Noeo odos o e o oe sfae gves se o ose egefos 8 evew Hollow Cylde II aplae s Eqao wo-desoal yldal ego ad o o I I C s [ d I I C s 9 evew Hollow Cylde III aplae s Eqao wo-desoal yldal ego ad o o N Egevales / ad egefo [ sh C Egefo P evew Egevales f / ads ao. ads ao.5 ads ao.9 evew Hollow Cylde IV Egefo epaso P [ [ sh d P C N sh sh P U Solo fo N U a osa evew Colsos ppoah o solvg aplae eqao s sla o ha of dffso eqao Ma dffeee s ha seod deso y o aplae eqao gves losed boday sead of ope boday e Use sepaao of vaables Have egefo solo se/ose essel o ohe oe deso Use egefo epaso o f odo a oe boday
3 The wave eqao ebay 3 9 evew Colsos II Use speposo o solve aplae eqao wh oe ha oe oeo boday ddoal yldal geoey osdeaos Cople essel fos whe adal boday s o egefo solo Ms lde boh ad whe adal oodae does o sa a eo s Veo Calls Ipoa esls fo aplae eqao See oes o veo alls o hapes e ad e eysg fo bakgod deals o gve hee esls ae depede of oodae syse b Caesa sed fo eaples Iode gade ad dvegee whh ae veo/sala fos have eo boday a e ads 3 4 ades ade s a veo we hee Caesa spae whee we have fy Defo of gade Del opeao f f f gad f f j k y j k y gad f s agde ad deo of a gade df/ds ad f s pepedla o le of osa f Physal ades ades of aplae eqao solos ofe popooal o fl es Hea fl ad epeae gade Dffso fl ad ass fao gade Veloy ad veloy poeal deal flow Ce ad eleosa poeal If we have a plo of osa poeal he les pepedla o he poeal ae fl les 5 6 Dvegee Dvegee oves veo v v v y j v k o a sala we as dv v v v y v Defo of dv v v y dvegee Del opeao j k y ass dvegee heoe s veo oal o sfae pog owad dv vdv v d Elosed Vole Sfae 7 8 ME 5 Egeeg alyss 3
4 The wave eqao ebay 3 9 Elosed Vole dv qdv Sfae q d elao o aplae Eqao Eaple of hea fl veo q W/ q s opoe of q oal o sfae d flowg owad Iegad sfae egal q d s hea flow was flowg o hogh fesal aea d Sfae egal gves oal hea flow hogh sfae owad deo Elosed Vole The veo v ay be gade of a sala epeseg a fl: v -k gad Elosed Vole dv vdv dv k gad dv v d Elosed Vole o osa k dv Sfae gad dv dv v d Elosed Vole k Sfae 9 Iepeao of Whe v -k gad s a fl ha s he gade of a sala aplae s eqao fo says ha he e flow of v s eo dv v d k Elosed Vole Sfae Eaple of hs esl show las week esl apples o ay poble ay geoey wh aplae s eqao If Cople Vaable ass Cople aalyss gves sghs o aplae Eqao wo desos os of ople vaable y: f y vy fo eaple f y y y f y ad v y Wha s df/d? Is qe? Cahy-ea odos v v df v v ad he y y d y y Cahy-ea Eaple f y y so y y ad vy y v v ad y y y y Sasfes Cahy-ea odos df v y d df v y d y y 3 Coeo o aplae Eqao Take / of fs Cahy-ea odo ad /y of seod oe ad add he esls o ge aplae eqao v y v y y v y v y y y 4 ME 5 Egeeg alyss 4
5 The wave eqao ebay 3 9 Coeo o aplae Eqao Ipoa esl Take /y of fs Cahy-ea odo ad /y of seod oe ad sba fo aohe aplae eqao v y y v y v y y v y v v y 5 If hee s a fo y ha sasfes a aplae eqao wo desos hee s a assoaed fo vy ha also sasfes aplae s eqao The les of y ad vy ae ally pepedla Typally f s a poeal e.g epeae v s a oespodg fl 6 Ohogoal Solos ddoal esls Show ha he wo solos ad v ae ohogoal a all pos Cosde he gade of eah fo v v j ad v j y y Take he do pod of he gades v v v v v v v j j y y y y j y y 7 Tea aplae eqao solos as ople vaable y vy Cahy heoe fo ople egao shows aplae eqao solos Solos alled hao fos Have a ad o boday If boday s a osa a all pos he solo s he sae osa ego Dhle poble has qe solo eysg seo 8.6 has poofs 8 Nea Poble s o Uqe Cosde he followg poble a ego wh spefed o s bodaes If sasfes he dffeeal eqao ad boday odos y ohe solo pls a osa wll also sasfy he poble odos Colso: a leas pa of he boday s have Dhle o hd kd of boday odo 9 Colsos ppoah o solvg aplae eqao s sla o ha of dffso eqao Ma dffeee s ha seod deso y aplae eqao gves losed boday sead of ope boday e Use sepaao of vaables Have egefo solo se/ose essel o ohe oe deso Use egefo epaso o f odo a oe boday 3 ME 5 Egeeg alyss 5
6 The wave eqao ebay 3 9 Colsos II Use speposo o solve aplae eqao wh oe ha oe oeo boday ddoal yldal geoey osdeaos Cople essel fos whe adal boday s o egefo solo Ms lde boh ad whe adal oodae does o sa a eo s Colsos III esls abo aplae s eqao fo veo aalyss ad ople vaables Whe he gade of he depede vaable sh as T aplae s eqao epeses a fl aplae s eqao says he e oflow s eo The a ad vales of a solo o aplae s eqao o o he boday so a osa boday eas a osa solo have eo boday a e ads 3 3 T Wave Eqao Devao Cosde a sg fed a ad Sg eso T ageal o sg ρ s ass pe legh of sg Seo bewee ad Δ Ne y foe ass es aeleao Δ TΔ T Δs θ Δθ y T sθ ρδ 33 T Wave Eqao Devao Pevos eqao fo veal opoe v l as Δ appoahes eo T Δs θ Δθ T sθ v y ρ Δ Hooal opoe h s elaed o v as v h a θ h dy/d; h s osa Δ TΔ y v y y h h ρ y y h ρ 34 Wave Eqao Sepaao of Vaables Wave pheoea: s wave aplde vayg wh spae ad e s wave speed Ca solve by sal sepaao of vaables ehqe lso have D labe solo wh abay fos ad wh oodaes ad 35 sse XT [ X T T X [ X T X T Dvde by XT esl s T X fo of T X eqal o fo of 36 ME 5 Egeeg alyss 6
7 The wave eqao ebay 3 9 Sepaao of Vaables Woks ssed solo XT gves a fo of eqal a fo of Se ad ae depede boh sdes s eqal a osa fo hs o be e T X T X Choose osa as o splfy lae solo of eslg oday dffeeal eqaos ODE Solve ODEs o e Have sple ODEs wh kow geeal solos d T T T s d d X X d T X os X C s D os [ s os [ C s D os oday Codos ook a ase whee I hs ase X s he solo o a S-ovlle poble X Cs Dos X Cs Dos D X Cs Ms have a ege Egefo s s/ 39 eeal Solo Wh C ad / o solo s [ s os [ C s [ s os [ s eeal solo s of all egefos [ s os s Use egefo epasos o ge al odos 4 Ial Codos [ s os s Spefy f a ad / g a f s g s Ohogoal Egefos Mlply by s/ ad egae f s d a s s d s s d s d Uses ohogoaly fo egefos s s d δ δ s d 4 4 ME 5 Egeeg alyss 7
8 The wave eqao ebay 3 9 ME 5 Egeeg alyss 8 43 Sla esl fo Mlply by s/ ad egae d d d d g s s s s s s a ao lplyg s s d 44 Say: ad d f d d f s s s If g all ae eo d g d d g s s s 45 eeal Solo Sbse ad fo eqaos js fod ad sbse o pevos solo Eae ase whee g so s os s s os 46 eeal Solo fo g o g dees fo s ± y s y s os y s y os s y s os y s y os s y s y s os y s os s s 47 D labe Solo Wave pheoea: s wave aplde vayg wh spae ad e s wave speed D labe solo show below ses abay fos ad wh oodaes ad Poof of solo based o asfog devaves 48 D labe Solo Tasfo eqao fo o sg ad pply asfos o / ad / [ [
9 The wave eqao ebay 3 9 ME 5 Egeeg alyss 9 49 D labe Solo II Coe asfoaos [ [ [ [ [ 5 D labe Solo III Seod devaves sasfy wave eqao [ [ [ [
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