Energy Density / Energy Flux / Total Energy in 1D. Key Mathematics: density, flux, and the continuity equation.

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1 ecure Phys 375 Eergy Desiy / Eergy Flu / oal Eergy i D Overview ad Moivaio: Fro your sudy of waves i iroducory physics you should be aware ha waves ca raspor eergy fro oe place o aoher cosider he geeraio ad deecio of radio waves, for eaple I he e wo lecures we cosider soe of he deails of he eergy associaed wih wave pheoea o keep i iiially siple, we sar ou wih oe diesioal waves I he e lecure we geeralize he coceps discussed here o hree diesios Key Maheaics: desiy, flu, ad he coiuiy euaio I Desiy, Flu, ad he Coiuiy Euaio e s sar by cosiderig soe uaiy Q ha has associaed wih i a desiy ρ Because we are ieresed i oly oe (spaial) diesio, he desiy associaed wih Q will be so uch Q per ui legh If we he iegrae ha desiy bewee wo pois i space, ad, he we will ge he oal aou of Q bewee he pois ad Maheaically, we wrie his as Q( ) ρ (, ) d (),, We have eplicily icluded ie because Q ay be a dyaic uaiy Ofe, Q is a coserved uaiy ha is, i cao be eiher creaed or desroyed If ha is he case he he chage i Q wihi he regio fro o Q ( ) ρ(, ),, d () us be eual o he e flow of Q io he regio ha is, Q (,, ) j (, ) j( ), (3) where j is he Q curre desiy (or flu) Coveio is ha if j >, he he flow is i he posiive direcio, ad if j <, he he flow is he egaive direcio Noe ha he uis of ρ are [ Q ] ad he uis of j are [ Q] s [ ρ] s j Coulob/s For eaple, if Q represes charge, he [ ρ ] Coulob/ ad [] D M Riffe -- 3/5/3

2 ecure Phys 375 If we ow euae he rhs's of Es () ad (3) we ge (, ) d j(, ) j( ) ρ () Now he rhs of E () ca be wrie as j ( ) ( ) (, ) j j d (5), ad so we ca rewrie E () as (, ) j(, ) ρ d (6) Now, because he liis ad are arbirary, he iegrad us vaish his gives us a ipora relaioship bewee he desiy ad flu, ρ (, ) j(, ) (7) Euaio (7) is kow as he (D) coiuiy euaio Because he desiy ad flu are local uaiies (which eas ha hey ca be defied a each poi i space, as opposed o Q, which is a global uaiy), E (7) is a local saee abou he coservaio of Q II Eergy Desiy ad Flu for D Waves e's ow apply his discussio o he eergy associaed wih D waves ha is, we le Q be he oal eergy associaed wih D waves bewee wo pois ad o be specific, le's hik abou rasverse waves o a srig For his paricular physical syse, where he wave speed c is give by c, where is he esio i he srig ad is he ass desiy (ass per ui legh), he eergy desiy ca be wrie as We do o prove his resul here For is derivaio we refer you o a ierediae echaics e, such Classical Dyaics by Mario ad horo D M Riffe -- 3/5/3

3 ecure Phys 375 D M Riffe -3-3/5/3 ( ), ρ (8) he firs er o he rhs is he kieic eergy desiy ρ while he secod is he poeial eergy desiy ρ So, we have he eergy desiy, bu wha abou he eergy flu j? Well, whaever i is i us saisfy E (7), he coiuiy euaio e's hus calculae ρ ad see wha happes Usig E (8) we have ρ (9) Coparig his wih E (7) we see ha we would like o be able o wrie () as of soe uaiy, which we could he ideify as he flu j o do his we ca ge soe help fro he wave euaio (here we use c for srig waves),, () ad he eualiy of ied parial derivaives,, () o rewrie E (9) as ρ, (3) which ca be copaced as ρ ()

4 ecure Phys 375 We ca hus ideify he eergy flu as j (, ) (5) III Several Eaples e's look a hree eaples: a ravelig wave, a sadig wave, ad wo collidig wave packes I each eaple we cosider he wo eergy desiies ρ ad ρ ad he eergy flu j A ravelig Wave o follow alog a his poi you will eed o go o he class web sie ad brig up he video file Eergy i D ravelig Waveavi, which shows hese ie depede uaiies for he ravelig wave (, ) cos( k ck) (6) Noice ha, as perhaps epeced, ha all uaiies ove o he righ a he wave speed c Furherore, he "wavelegh" of he eergy desiies ad flu is half ha of he displacee For he eergy desiies, his should be obvious fro heir defiiios Furherore, because for his ravelig-wave eaple i is o hard o show ha j cρ, j looks esseially he sae as he oal eergy desiy ρ B Sadig Wave he e video o he web sie, Eergy i D Sadig Waveavi, shows hese sae ie depede uaiies for he sadig wave ( ) si( k) cos( ck) (7), his eaple is a bi ore ieresig Noice ha ow he eergy oscillaes back ad forh bewee kieic ad poeial, which ow have heir (saioary) aia a differe spaial pois A careful eaiaio of he flu shows ha he eergy a a give poi flows oe direcio ad he he oher as i is covered bewee poeial ad kieic C Collidig wave packes he las eaple ca be foud i he video Eergy i D Collidig Pulsesavi he displacee for his wave is give by D M Riffe -- 3/5/3

5 ecure Phys 375 [ ] [ ( k ck) a ] [( ) ] ( ) ( ) ( ) k ck a cos k ck e cos k ck e (8), For os of he ie he wave looks like wo oieracig wave packes (or pulses), each ovig a he speed c bu i differe direcios Noice ha whe he pulses are far apar he behavior of he desiy ad flu is siilar o ha for a ravelig wave, bu as he pulses overlap he desiy ad flu behave i a aer siilar o a sadig wave (which ca, of course, be described as he superposiio of wo ravelig waves) I oal Eergy Now ha we have see soe eaples illusraig he local uaiies ρ (,) ad j (, ), le's cosider he oal eergy associaed wih wave oio I paricular, le's cosider rasverse waves o a srig o he ierval wih he sadard boudary codiios (, ) (, ) I ha case we ca wrie ay wave o he srig as a liear superposiio of sadig waves as (, ) si( ) [ A cos( ω ) B si( ω ) ] π, (9) where ω π c, ad he coefficies A ad B deped upo he iiial codiios Usig E () he oal kieic ad poeial eergies ca be epressed i ers of heir desiies as () ρ (, ) d d, (a) () ρ (, ) d d, (b) e's ow subsiue he geeral for of he displacee o he rhs of i E (9) io E () ad calculae he oal kieic ad poeial eergies For he kieic eergy we have See ecure oes D M Riffe -5-3/5/3

6 ecure Phys 375 π () si( ) ω [ A si( ω ) B cos( ω ) ] si π ( ) ω [ A si( ω ) B cos( ω ) ] d () Noice ha we use differe idices o he wo sus so ha we kow which uaiies go wih each su Swichig he orders of iegraio ad suaio we ca rewrie E () as () { ω ω [ A si( ω ) B cos( ω ) ][ A si( ω ) B cos( ω ) ] si π π ( ) si( ) d} () We ca ow siplify his cosiderably because of he orhogoaliy of he sie fucios ha is, usig π π si( ) ( ) } si d δ, (3) E () becoes Now () ω [ A si( ω ) B cos( ω ) ] () () ω [ A si( ω ) B cos( ω ) ] (5) is siply he kieic eergy coaied i he h oral ode hus E () ca be siply viewed as he su of kieic eergies coaied i all of he oral odes, D M Riffe -6-3/5/3

7 ecure Phys 375 () () (6) Siilarly, sarig wih E (b), oe ca show ha he oal poeial eergy ca be wrie as where () (), (7) () ω [ A cos( ω ) B si( ω ) ] (8) is he poeial eergy coaied i each oral ode Furher, usig Es (5) ad (8) i is o difficul o show ha he oal eergy coaied i each ode, E () () (), is eual o E () ω ( A B ), (9) ad is hus cosa ha he eergy i each oral ode is cosa is due o he fac ha he oral odes do o ierac Why? Because he euaio of oio for each oral ode is idepede of he oher oral odes Noice also ha he eergy is each oral ode is proporioal he suare of he apliude ( ) Fially, because () E is cosa, he oal eergy A B E () E () (3) is also cosa D M Riffe -7-3/5/3

8 ecure Phys 375 Eercises * Eergy Desiy ad Curre for a ravelig Wave Cosider he ravelig wave soluio o he D wave euaio (, ) cos( k ck) (a) Calculae he kieic, poeial ad oal eergy desiies ρ (, ), ρ (, ), ad ρ (,), respecively, ad he eergy curre desiy j (, ) Show ha ρ (, ) ρ (, ) Furher show ha j (, ) cρ(, ) (b) Does he eergy curre flow i he direcio ha you epec? Eplai (c) Show ha he D coiuiy euaio is saisfied by your epressios for ρ (,) ad j (, ) ** Eergy Desiy ad Curre for a Sadig Wave Cosider he sadigwave soluio o he wave euaio for rasverse waves o a srig (, ) si( k) si( ω ) (a) Calculae he kieic ad poeial eergy desiies ρ (, ) ad ρ (, ), respecively, ad he eergy curre desiy j (, ) Epress your aswers usig he paraeers, c, ad ω (b) Show ha he D coiuiy euaio [E (7)] is saisfied for his wave (c) For his wave fid he oal kieic eergy ( ) ad poeial eergy () i a srig of legh o he ierval, assuig ha k π, where is soe posiive ieger Here, use he paraeers,, ad ω o epress your aswers (d) Show ha he oal eergy E ( ) ( ) ( ) is idepede of ie *3 oal Eergy i ibraig Srig (a) As was doe for he kieic eergy i Sec I, show ha he oal poeial eergy coaied i he vibraig srig is give by () ω [ A cos( ω ) B si( ω ) ] (b) Usig Es (5) ad (8) show ha he oal eergy E ( ) i a oral ode is cosa * Cosider he geeric ravelig wave (, ) f ( c) Show ha for his wave (a) ρ ρ ad (b) cρ j D M Riffe -8-3/5/3