Complex plane of Gauss. (a, b)

Transcription

1 FW E:\Excel fles\ch 5 Coplex Oscllatos.doc; page of 4 /4/5; 9: AM Last prted /4/5 9:: AM Carl Fredrch Gauss troduced coplex uers: I the coplex plae the y-axs s the agary axs ad the x-axs s the real axs. Ay coplex uer z ca the e wrtte ters of ts real part plus ts agary part: zˆ = x+ y zˆ = r cos + s = re θ Euler forula (. ( θ θ Oe ca easly prove Euler s forula y expadg the expoetal fucto e x a McLaur power seres, ad the susttutg x wth θ. We ust just use that =-; 3 =- ; 4 =; 5 = y Coplex plae of Gauss θ r a (a, x Coplex uers help us to fd all solutos of a algerac equato le, for exaple: x 3 =-. All we have to do s wrte our uer, real, or coplex, ters of a expoetal: (. 3 π+ π zˆ = = e for =,,,3... the we tae the thrd root of ths equato ad tae all o repettve solutos to accout. ( π+ π ( ( π+ π ( π+ π ( π+ π π 3 3 ˆ 3 3 a e = e z = e = e ; = ˆ 3 π = = ; = z e e cz ˆ = e ( π+ 4π 5π 3 = e 3 ; = we have three uque solutos, whch ca of course also e wrtte trgooetrc ad algerac forats. The sae ca e doe for ay equato of the for

2 FW E:\Excel fles\ch 5 Coplex Oscllatos.doc; page of 4 /4/5; 9: AM Last prted /4/5 9:: AM aa + = zˆ ( θ+ π Frst, wrte a+ ters of a expoetal fucto r ;,,... (.3 r = a + = a+ = ( a+ ( a c θ = arcta a e = (.4 ( θ+ π ( θ π ˆ + z a a e ta radas ˆ z = r e ; =,,,3... = + = + θ = a (.5 exaple = zˆ r = + = θ = 4 = 3 3 : 3 4 ; 3 4 5; arcta (.93+ π 3 3 zˆ = 5 e ; =,, ( zˆ = 5 e =.7e =.7( cos.3 s.3 =.7( = (.93+ π zˆ = 5 e =.7e =.7( = ( ˆ 5 + π z = e =.7e =.7( =.66.5 The coplex cojugate z * (z-star uer of a coplex uer z s otaed y turg all sgs of the agary ut to ther opposte: + ecoes ad vce versa. θ θ zˆ = a + = re ; zˆ = a = re ; (.6 We defe the or (or agtude of a coplex uer as: zˆ = zz ˆˆ = a + = r Ratoalzato of the deoator: * zˆ a β a + a = = = re ; r = = ; β = ta * a+ zz ˆˆ a + a + a (.7 a + β = e a+ a + Soe terestg propertes are a cosequece of Euler s forula (.:

3 FW E:\Excel fles\ch 5 Coplex Oscllatos.doc; page 3 of 4 /4/5; 9: AM Last prted /4/5 9:: AM (.8 ( ( ( ( cosθ + sθ = cosθ sθ = cos θ + s θ DeMovre e = ( ( cosπ ( sπ ( ( ( ( ( ( = ( = = = = = π ( a cosθ + sθ = cos θ s θ + cosθ sθ = cos θ + s θ r ( ( = ; ( = ; ( = ; ( = = ( ( ( ( + r r r r oal coeffcets: = for ad for =! 3 ( 3 ( ( c cosθ + sθ = cos θ + 3cos θsθ + 3cosθ sθ + sθ = cos3θ + s 3θ 3 3 cos θ 3cosθ s θ = cos 3 θ;3cos θ sθ s θ = s 3θ Recall the power expaso of a fucto to a McLaur seres: 3 (.9 ( x ( ( d f ( x ( = f( x = f ; wth f x! dx = whch s the specal case of the Taylor seres, expaso aroud the pot a: (. = ( ( ( ( x a ( ( ; wth d f x f x = f a f ( a x= a! dx The expaso for e x s the: x x x ae = + x+ + + ad!! θ θ θ θ θ θ θ θ e = + θ = cosθ + sθ (. 3! 4! 5! 6! 7! 8! θ ce = cosθ sθ θ θ θ θ e + e e e dcos θ = ad s θ= Oe of the terestg attrutes of coplex expoetal fuctos s ther dervatves. A dervate s sply reduced to a ultplcato y ω, ad a tegrato s reduced to a dvso y ω:

4 FW E:\Excel fles\ch 5 Coplex Oscllatos.doc; page 4 of 4 /4/5; 9: AM Last prted /4/5 9:: AM ˆ( ωt ˆ ωt df t dxe ωt f( t = x ˆ e = = ωxe = ωf( t dt dt (. d fˆ( t = ( ω fˆ( t dt All of ths ecoes very useful whe we wat to solve certa dfferetal equatos, as they occur physcs. Cosder, for exaple, the dfferetal equato of a sprg: Sple Haroc Moto (SHM: Note that s the sprg costat here, ot the wave uer. F = x; s the sprg costat! Newto's secod law apples: d x F = a x = x (.3 dt x+ x= ths s a lear dfferetal equato of the secod order. It has two depedet solutos, whch coe to for the geeral soluto of the d.e. The d.e. requres two tal codtos to tegrate ad to for a uque soluto. We see that cosωt, sωt, ad e ωt, e -ωt are solutos. Let x=acos ωt x = ω x therefore ths s a soluto f ω = (.4 x = Asωt ad As ( ωt + ϕ ; ϕ s called a costat phase shft; phase these are all fuctos whch satsfy the dfferetal equato. Ay lear coato of such fuctos would also e a soluto. If we use real fuctos we for a lear coato of the se ad cose fucto. (.5 x= A cosωt+ B sωt = Cs ( ωt+ ϕ = C'cos ( ωt+ϕ ' If we use coplex solutos we ear d that ultately we eed a real soluto for a real easureet. We therefore use the real part or the agary part of a coplex soluto to descre a physcal stuato, whch rgs us ac to (.5. Whe we pose the tal codtos, the two artrary costats A ad B dsappear.

5 FW E:\Excel fles\ch 5 Coplex Oscllatos.doc; page 5 of 4 /4/5; 9: AM Last prted /4/5 9:: AM We try as geeral solutos oe of the followg: ax(t=acosωt+ Bsωt xt ( = Cs( ωt+ ϕ (.6 ωt ωt czt ˆ( = Ae + Be we use ether the real part of the soluto or the agary part. The tal codtos specfy the locato x(t= ad the speed at v(t= at the te t=. Whe we start wth a fully expaded sprg, ts tal locato s evdetly x(=a ad ts speed s v( =. These stadard tal codtos lead edately to the uque soluto for the oto of the sprg: xt ( = Acos ωtwth ω = (.7 v( t = x = Aωs ωt; at ( = x= ω xt ( We ca fd the sae soluto whe we cosder the total eergy of the sprg: (.8 E = v + x = x + x We ow that the total eergy of a sple sprg wth the force x s coserved. We ca use ths forato to drectly fd the soluto for x ad v: (.9 (. E = x + x For splcty we assue the stadard tal codtos x(=x ad v(= E=E(= x = x + x x = x x x = ( x x dx = ( x x = ω ( x x dt dt dt = ; =± dx ω x x dx ω x x dx dt =± we choose the + sg, ecause we ow ω x x that our aswer ust e ftted to the tal codtos.

6 FW E:\Excel fles\ch 5 Coplex Oscllatos.doc; page 6 of 4 /4/5; 9: AM Last prted /4/5 9:: AM s x x s ω ( x x t = + t t = ω x= xs ( ωt ωt Accordg to the tal codto v(= (. π v(= ωx cos ωt ωt = ωt = ; t= π ut s(x+ = cos x Therefore our soluto s (. x( t = x cosωt It ecoes a t ore challegg whe we also have a dapg ter : af = v( t x (.3 dapg The su of the exteror forces o the sprg s the: F = x x = x or cx + x + x= = x+ x + ω x Nether a cose or a se fucto (aloe wll satsfy the d.e. ths case; a coplex tral soluto however leads to a algerac codto, whch s called the characterstc equato of the dfferetal equato: ωt azt ˆ( = ze ˆ Reeer that dervato eas ultplcato y ω (.4 ω + ω + ω zˆ ( t = ω + ω + ω = quadratc equato ω ± + 4ω ω= 4 = ± ω We dstgush three dfferet possltes, depedg o the value of the radcad, the frst of whch s our ost portat case: (.5 ω= ± ω ; for ω > (.6 ω= ± ω ; ω <

7 FW E:\Excel fles\ch 5 Coplex Oscllatos.doc; page 7 of 4 /4/5; 9: AM Last prted /4/5 9:: AM (.7 sgle soluto: ω (doule root = ; ω = The orgal characterstc equato ca e wrtte as ω = = ω ω = The ost portat (for our purposes here coplex geeral soluto to the dfferetal equato correspods to the case (.5: (.8 ± ω t ± ω t az ˆ = ze ˆ = ze ˆ we ca easly guess at a real soluto x ow (assue that the radcad s postve; reeer the Euler forula: x(t=ae cos t ω = Ae cos ωt wth ω = ω We follow the geeral approach of tryg to solve our equatos wth coplex fuctos, ad whe we have foud a geeral coplex soluto, we tae the real part of that soluto as our physcal soluto.(.8 s a expoetally decreasg cose fucto. We ca rewrte the ew frequecy ω ters of the orgal ω =/, whch we ow call ω a ω = ω = ω = ω = ω = ω 4 (.9 4 ω = ω ; > 4 4 I frst approxato ω =ω. Ths eas that daped oscllatos the sae frequecy apples as log as the uer s sall. 4 To prove ths, t s useful to recall the oal expaso forula (we used ths already wth the relatvstc etc eergy: (.3 r ( x ( r r ( + x = wth = = for ad for = (.3 ( x r ( ( ( + r r r r! + + r x for x

8 FW E:\Excel fles\ch 5 Coplex Oscllatos.doc; page 8 of 4 /4/5; 9: AM Last prted /4/5 9:: AM.5.5(.5 = + = = = = thus (.3 ( x x x ; (.5; ( (.33 ω ω ω = Therefore, frst approxato, ω =ω. Assue that =N/; =.5g; x(=.. Fd a dapg factor, whch would cause the apltude to decrease y 5% perods. (.34 x( t = Ae cos ωt; A=. ;; ω = ω ; ω = = s 4 π ω We are loog for.6=. e ; t = T = = l.5 Ns = =. ; =. π 4 ω ω s π (.35 x(t=ae cos Ae cos wth Fd the te t taes for the apltude to decrease y the factor : t ω = ωt ω = ω.5 = e ; =e ; -l=- A A t (.36 l = t/ Thus, y easurg the te t taes for a sprg oscllato to have ts apltude reduced to half the orgal value, we ca experetally detere the dapg factor. Now, what happes to the eergy? As we have frcto, we do ot expect the total eergy to rea costat. It loses eergy over te. Let us fd out how: de dv dx = v + x dt dt dt dv dx (.37 = x s the d.e. whch we susttute the forula for de dt dt de dx dx = ( x x + x = v dt dt dt

9 FW E:\Excel fles\ch 5 Coplex Oscllatos.doc; page 9 of 4 /4/5; 9: AM Last prted /4/5 9:: AM Ths eas that a daped oscllato does ot coserve eergy ut loses t at the rate of v. A slar calculato ca e doe for the case of the sple pedulu wth the cluso of a dapg costat : The dapg force s proportoal to the velocty of the pedulu: (.38 Fdapg = l θ = s The udaped equato s otaed y oservg that the ass of the pedulu oves crcular oto,.e. ts accelerato s tagetal. It s caused y gsθ, the projecto of g o the tagetal drecto. Sple pedulu oto: (.39 g s θ = aθ gθ = lα = l θ for sall agles θ <. radas After addg the dapg ter we get: (.4 l θ + l θ + gθ = ; l s the legth of the sple pedulu, s the ass, θ s the agle wth the vertcal, ad s the dapg costat. l legth of pedulu θ agle Teso (.4 l g g l θ θ + + lθ = s s ωs; ω ; s lθ l = + + = l = Coparg ths equato wth (.3 we see that: g (.4 ; = ω l s=lθ g We ust therefore ae the approprate susttutos (.35 whch leads to: (.43 t θ(t=ae cos t ω = Ae cos ωt wth ω = ω the apltude A s ow the startg agle θ s= lθ s the arclegth, s = l θ s the tagetal speed, s = l θ s the tagetal accelerato Fd the value for for whch the apltude s cut half. The correspodg te s called the half-lfe t / of the oscllato: l (.44 θ (t=ae A= Ae = t /

10 FW E:\Excel fles\ch 5 Coplex Oscllatos.doc; page of 4 /4/5; 9: AM Last prted /4/5 9:: AM Rears o the specal case (optoal whch the ω = ω =. I ths case our approach yelds oly a sgle soluto to the d.e., aely xe. However, accordg to the theory of dfferetal equatos a lear dfferetal equato of the secod order has two separate solutos, whch coe to a sgle soluto f oe apples the tal codtos. Ths secod soluto s the preset case gve y xte geeral soluto ths case s therefore: (.45 ( t xt ( = x + xt e whch x ad x are costats detered y the tal codtos. I the case of the stadard tal codtos x(=a, v(= ths yelds x = A ad x = A (.46 xt ( = + t Ae t. The Oe ca chec ths y sertg: xt ( = + t Ae = Afor t= (.47 t t x = Ae + Ae tae = tae = for t= Forced Oscllatos: It very ofte happes that a oscllato s drve fro the outsde,.e. that there s a addtoal force appled to the sprg, for exaple, a electrc oscllato wth a dfferet frequecy, whch we ow lael ω f. Ths outsde force ay have the for F cosω f t, wth ω f eg the drvg frequecy. We refer to the ω as the trsc or atural frequecy, the frequecy detered y / the case of the sprg. Oe ca easly see that f ths outsde force s appled log eough, the sprg wll oscllate ore or less wth the ew forced frequecy ω f. Matheatcally we have to solve the followg stuato:

11 FW E:\Excel fles\ch 5 Coplex Oscllatos.doc; page of 4 /4/5; 9: AM Last prted /4/5 9:: AM (.48 F ax + x= cos ω f twe are oly loog for a specal soluto after a log te, whe the forced syste oscllates wth the frequecy ω. We try a ew expoetal soluto ad wrte the rght had sde of equato a ters of a coplex fucto:cosω t e F x=a ˆ e we get ( ωf ω A e e whch eas that A ( ω f ωft ωft ωft + = = Ths eas that we get a ew oscllato wth the apltude: ca = F ( ω ω f appled exteror force. f f ω f t F ( ωf + ω ; ths eas that the apltude s a fucto of the frequecy of the The ew apltude grows to fty as the forced frequecy ω f approaches the atural frequecy ω. Ths s what we call a resoace effect. I realty, there s always a dapg factor /, whch appears the parethess of the deoator (.48 so that A(ω f does ot go to fty ut ca stll grow to extreely large (ad ofte destructve values. If we use the daped equato, we proceed as follows to fd the specal soluto for very large values of t: f t (.49 Set xˆ = Ae ω F ωft ωft F ωft x+ x + x= e ωf + ωf + Ae = e F F ˆ β β A= = re = e ; β = ta a ω ω ω ω ω + ωf where a s the real part ad s the agary part the coplex fucto: (.5 f + f ( f zˆ = ω ωf + ωf = a + a Practce exaple to ratoalze the coplex deoator of a fracto: (.5 = = = = e 4 4 (.974+ π 3+ 4 Arcta π Arcta π e e 5e (.974 π + We ratoalze the coplex apltude accordg to (.7

12 FW E:\Excel fles\ch 5 Coplex Oscllatos.doc; page of 4 /4/5; 9: AM Last prted /4/5 9:: AM (.5 ˆ β A= = e dfferet eag for ; dot cofuse wth, a + a + the ar drag coeffcet!! Aˆ ( ω f = F f + ( ω ωf ω (.53 F xˆ = Aˆ ( ω f e = e ω ( ωft β ( ωft β e f + ( ω ωf β partcular soluto for large t Ths eas that the apltude of the forced oscllato wth dapg s a fucto of the frequecy ω f of the drvg force. If ths frequecy approaches the orgal frequecy of the sprg, the apltude wll approach ts axu value. We call ths effect resoace. The orgal frequecy s referred to as resoace frequecy. As show earler a daped oscllato loses eergy at the rate of de (.54 v dt = Oe ca prove ths easly y cosderg the total eergy ad the usg the dfferetal equato for daped oscllato (see earler (.37: E = x + v; x + x + x= ; x=- ( x + x (.55 de xx = + vv= x v + vx= xx + v( - x x = v dt Ths s evdetly uch easer tha fdg the eergy fro the soluto e cos x = x ω t : Let us use the exaple elow to calculate soe uers for the eergy loss: x = x e cosω t x cosωt (.56 ω = ω (.57 vax = xω Use c for the apltude of your oscllato ad a ass of g. Calculate ω ad choose values for ω t etwee ad π.

13 FW E:\Excel fles\ch 5 Coplex Oscllatos.doc; page 3 of 4 /4/5; 9: AM Last prted /4/5 9:: AM ω ω = = 5.s If =.E-4 we get for the axu eergy loss per secod ( ω 5 v A =..7 =.7 Watts 4 I quatu physcs we fd that the eergy etted y ay oscllator s a ultple of the agular frequecy ad Plac s quatu S -34 The eergy for N oscllators s therefore N ω wth Js (Reder: We have see elsewhere that the agular oetu of the electroc orts atos s also quatzed: l = Ths eas that per secod the oscllator aove ets aout 4x 8 eergy quata. We also fd the result quatu physcs that the al eergy of a oscllator s ot ut ω. Ths s cosstet wth Heseerg s ucertaty relato whch does ot allow ay oject to e totally at rest. It also llustrates the fact that Heseerg s ucertaty relato s ot erely a expresso aout the ucertaty of easureet ut aout the deterate ature of actualty tself. Addtoal fu stuff (optoal Waveuer ad dervatves (optoal: We have see that dervatves are partcularly easy to ota wth coplex expoetal fuctos. It does ot atter whether we tal aout te dervatves, spatal dervatves or partal dervates: f( x = As x; s the waveuer, the uer of coplete wave-cycles the spatal x-drectofttg wth π. The waveuer s for the space coordates what ω s (.58 for the te coordate. (A se or cose wave fucto wth apltude "A" ca e descred as the projecto of crcular oto wth radus "A" o the x or y axs, ω s also the agular frequecy. df = Acos x dx

14 FW E:\Excel fles\ch 5 Coplex Oscllatos.doc; page 4 of 4 /4/5; 9: AM Last prted /4/5 9:: AM π π = ; ω = λ T The wave uer ca ecoe a vector, whe we cosder a se fucto three desos: (.59 s( x x + y y + z z π π π x = ; y = ; z = ; = x, y, z λx λy λz r = xx+ yy+ zz All fuctos f( x ωt +Φ represet lear waves. We dscuss ths the ext chapters 6-8. Ths cludes coplex fuctos whch ca e sply thought of as eg coatos of se ad cose fuctos. ( ˆ(, x ωt Let f x t = Ae fˆ ˆ ˆ f = ω f( x, t ad = fˆ( x, t t x (.6 r ˆ ( ωt Let f (, x y, z, t Ae fˆ (, r t grad fˆ = = (, r t = fˆ (, r t We see that for expoetal fucto the del operator ecoes