ThewaveEguation1 MultivariableReview2TheTransportEquation3SolvingthewaveEquation

DMultivariableReview

o PartialDifferentiation

f frig afunctionoftwovariables x andy0 fu Howmuchfoeyl changeswhen u changesFfg fy Howmuchfoeyl changeswhen y changes

MultivariableChainRulen sea treat x as a functionof ty yet treaty as a functionof1Question If wegive t a nudge howdoes f faecesyall respondLetSt a change in tSn a change in Nsy a change in y

ObservationSse dateSt

Sy StConclusion

Sf change in f as a response to a change in se andybe s yon yIfdidst Idatstandt og

E Eiht i Ett ySt o dat Ifadfft y t

2 TheTransporteguationPhysicalSetupAwavetravelswitha constantvelocity c and a constantprofileUnit Displacementofwave attheposition x andtime t

GoalSetup a PPE whichdescribessuch a wave

BeginwithsomegeometryTimeslices

U U UUeno is Ueno 2

aUto10

no no Rotc do not zc

t o t I t 2

3DPicture

U

Fitftp

K

KeyObservation Ucr t is constantonthelines

µeU

t

k I

n c tk

U

It

note

no c Y

N KCt Ko t c t

ConclusionUcsc t is constantonthelinesofslope c

ImpliesIf CN t andCreutz lieon the same lineofslope c then

Ucsc t ucxz.tn

BatAllpoints at lyingonthelinepassingthroughcropsofslope c satisfy

u rot at

thereforeluckotctitl constant

PutanotherwayUCKotch t Uckotctz ta for all ti ta

TheTransportEquationBythemultivariablechainrule

ddtucxotct.tl Uncrotctit ddtlxotc.tl 1 UHKo Ct fddt

Cuse NotCt t c Ut Motat t l

dacconstant0

Butthis holdsfor all se and tTherefore ucx.tlmustsatisfythe TransportEquation

fletcun Oforcelloocsectas wate

TheHomogeneousTransportIVPWewishtosolve

CUntUt o woe too t oUcu gcse w Lx Ctw

Initial conditionrecurs_gcse initial waveprofileU

IGY i 2

t

x

Butucx.tl is constanton linessect Kot et

O Ucseotcops gcses for all koThepoint Creo o lies on the linesect Koect

ThereforeUC Kot Ct t g Ro for aCroc1 w t 20

In conclusionafterthesubstitutions seo sa se et

µcmt gcsSolves thetransportequationwithinitialcondition goes

Remarkucx.tl transports goesfromCmo to cnet.tl

TheInhomogeneous Transport IVPNowwewishtosolveCuse Ut fCat wiseLew t ouol.co gcse w Lx Ctw

where fexits is a specifiedfunctionBut

Multivariablechainruleodducnccs.SI d Nta UrcutesisI daysUtcutcss ls as

cureLutes s utcktcs.SIflutes s

FTOCUCKcct.tl Ucsco fotddsucxcc.sisIds ftf Recsslots

AndUcu o GCN

Thereforebythesubstitution n se et

Ucu t Uca ct o f fCutCcs t Ddsgcse ct f fCutCcs t sids

RemarkNow ucn.tl transports andevolvesgoes

se o s feet t

goes god ftflutes sidsComparedto homogeneouscase fcut1 0

se o s feet tgoes goes

DThewaveEguationIn General

A Ucm Nz Nn t waveheightatpositionCnc isentandtime tTheLaplacian

D nytn2zt 1Eren

TheWaveEquation

Du Fer Su _on L Utt Urse O vibratingstringn z UttUrimUmar O vibratingmembranen 3 UttUrim UmmUmm O elasticsolid

RemarksDucati reprensentsdisplacementin somedirection2 D is sometimescalled the D'Alembertian3Wewillfocus on n 1 Higherdimensionsrelyonmultivariable calc

OnedimensionalWaveEquation IVP uNowwewishtosolveUtt Urse O 00Lie too out utcaof guttersucu.oi g.cn wisectoonecko hln oo Lu too Ro se

Heregoes initial waveprofile t ohcm initial wave velocity U

For5791mentat a mightbequickDisplacement at se mightbe slow Ro u

t Oil

SolvingtheWave Equation1 Observe

um t Fn2Then

Utt Unu O t u O3 Let

VCRt UHHH UseCn t U

4By 1 and3E u Fakut un

FewVetVn

5 By 2 and 4Utt Unu o u O VttVa 0

But4the o is the transportequationwith c lGByinitialconditionsofUckittand3 veryhas initial conditions

UCR gcul uneo gca VCRO Utheo Uncle01necko hln her gcel

7 By 5 and6 Vcr.tl solvesthehomogeneoustransportequationse t t O wax Lew t oVIother gch woe Ctw

8 Butthesolution to this transportequation iszit Vcut o h n t gCn t

9 Therefore by 3 and 8 unitssatisfiesUt Use units hex ti g ex t

ButUtUn hue ti g ex El is thenonhomogeneoustransportequationwithc i

10 Byinitial conditionsofucn.ttand 9 ucu.tl satisfies

UntUt hue ti g ex H wax Lew t 0recap gcse w Lx Ctw

Il By 10 ucn.tl isgivenbyUcu t gcutt 1tohee Cs ti s gCu Cs t s Is

tcgcuttitg.cn tis the heads

Examplehtt iii sin iii iii at

necko cosse oo u too

1We observethe initialconditionsUCMo gcse si nu

UtCKol her cosse

2 By 1 anducuitl tcgcuttlg.cn tis Ifutiheads

we concludeUcr E I sincutE since tD t Ibe coscods

I sincutE 1since tD t's sin sett sinex EDsin sett