Continuous Systems; Waves

Transcription

1 CHAPTER 3 Continuous Systems; Waves 3-. The initial velocities ae zeo and so all of the ν vanish [see Eq. (3.8)]. The given y [see Eq. (3.8a)] µ ae so that A 3πx πx µ sin sin dx Aδ 3 µ 3 A () µ, 3 () The chaacteistic fequency ω 3 is [see Eq. (3.)] and theefoe, 3π τ ω3 (3) ρ 3π τ 3πx qxt (, ) Acos tsin ρ (4) Fo the paticula set of initial conditions used, only one nomal mode is excited. Why? h 435

2 436 CHAPTER 3 The initial conditions ae Because qx, (,) qx, all of the ν vanish. The 3h x, x 3 3h ( x), x 3 qx, () 3 µ ae given y 6h πx 3h πx µ xsin dx+ ( x) sin dx 3 9h π sin (3) π 3 We see that µ fo 3, 6, 9, etc. The displacement function is whee 9 3 h πx πx 4πx qxt (, ) sin cos t sin cos t sin cos t ω + ω π 4 6 ω 4 (4) π τ ω (5) ρ The fequencies ω 3, ω 6, ω 9, etc. ae asent ecause the initial displacement at 3 pevents that point fom eing a node. Thus, none of the hamonics with a node at 3 ae excited. () 3-3. The displacement function is whee qxt, 3 5 sin x cos t sin x cos t sin x cos h π ω π π ω + 5 ω t+ () 3 8 π 9 5 Fo t, π τ ω ρ ω ω qx, π 3 5 sin x π sin x π + sin x + 8h π 9 5 () (3) The figue elow shows the fist tem, the fist two tems, and the fist thee tems of this function. It is evident that the tiangula shape is well epesented y the fist thee tems.

3 CONTINUOUS SYSTEMS; WAVES 437 tem tems 3 tems The time development of q(x,t) is shown elow at intevals of 8 of the fundamental peiod. t, T t 7 T, T 8 8 t 3 T, T 4 4 t 3 5 T, T 8 8 t T 3-4. The coefficients ν ae all zeo and the µ ae given y Eq. (3.8a): 8 πx µ x ( x) sin dx so that π (), even µ 3 (), odd 3 π 3 Since

4 438 CHAPTER 3 the amplitude of the n-th mode is just µ n. The chaacteistic fequencies ae given y Eq. (3.): πx qxt (, ) µ sin cosω t (3) nπ τ ωn (4) ρ 3-5. The initial conditions ae qx, v, x s qxt (, ), othewise The µ ae all zeo and the ν ae given y [see Eq. (3.8)] ( ) ν ω + s s v πx sin dx () fom which 4v π πs sin sin πω, even ν 4v ( ) πs (3) sin, odd πω (Notice that the even modes ae all missing, as expected fom the symmetical natue of the initial conditions.) Now, fom Eq. (3.), π τ ω (4) ρ () and ω ω. Theefoe, Accoding to Eq. (3.5), ν 4v ( ) πs sin, odd (5) πω

5 CONTINUOUS SYSTEMS; WAVES 439 iω t πx qxt (, ) βe sin Theefoe, πx ν sin ω t sin (6) 4v s x 3 s 3 x (, t) sin tsin π sin π sin 3tsin π sin π qx ω ω + (7) πω 9 Notice that some of the odd modes those fo which ( s) sin 3π ae asent The initial conditions ae qx, (,) 4v x x 4 4v 4 x qx x x The velocity at t along the sting, q( x,), is shown in the diagam. () v v The µ ae identically zeo and the ν ae given y: x qx (,) sin dx π ν ω 8v π π sin sin 4 3 πω Oseve that fo 4n, ν is zeo. This happens ecause at t the sting was stuck at 4, and none of the hamonics with modes at that point can e excited. Evaluation of the fist few ν gives ()

6 44 CHAPTER 3 8v ν.44 ν 4 πω 8v.44 8v ν ν 5 4 πω 5 πω.44 8v 8v ν ν πω 6 πω (3) and so, 8v πx πx qxt (, ).44 sin ωt sin + sin ωt sin 4 πω.44 3πx.44 5πx t t sin ω3 sin sin ω5 sin (4) Fom these amplitudes we can find how many d down the fundamental ae the vaious hamonics: Second hamonic: Thid hamonic:.5 log 4.4 d.44 (5).44 7 log 3.3 d.44 These values ae much smalle than those found fo the case of example (3.). Why is this so? (Compae the degee of symmety of the initial conditions in each polem.) (6) 3-7. Since qx, O 3 7 h h 3 7, we know that all of the ν ae zeo and the µ ae given y Eq. (3.8a): The initial condition on qxt (, ) is πx µ qx (,) sin dx ()

7 CONTINUOUS SYSTEMS; WAVES 44 Evaluating the µ we find 7h 3 x, x 3 7 7h 3 4 qx (,) ( x), x 7 7 7h 4 ( x ), x h 4π 3π µ sin sin 3 π 7 7 () (3) Oviously, µ when 4 7 and 3 7 simultaneously ae integes. This will occu when is any multiple of 7 and so we conclude that the modes with fequencies that ae multiples of 7ω will e asent Fo the loaded sting, we have [see Eq. (.5)] Using ρ mdand ( n+ ) d, we have τ π ω sin md ( n + ) () ω d τ π sin ρ ( n+ ) The function ( n + ) τ π sin ρ ( n+ ) () ω ( sin π n + ) τ ( n + ) ρ (3) is plotted in the figue fo n 3, 5, and. Fo compaison, the chaacteistic fequency fo a continuous sting is also plotted: ω π (4) τ ρ

8 44 CHAPTER 3 n ω τ ρ 8 6 Continuous sting n 5 4 n Of couse, the cuves have meaning only at the points fo which is an intege Fom Eq. (3.49), we have: Fom section 3.5, we know that undedamped motion equies: Using () this ecomes D s π τ β ; ω ρ ρ () β < ω D s π τ < 4ρ ρ o ikew ise D D D 4 < 4 4 > ρs π τ ρ π τ s ρ π τ s undedamped citically damped ovedamped The complementay solution to Eq. (3.48) fo undedamped motion can e witten down using Eq. (3.4). The esult is: βt ( t) C e cos ( t ) ηs s ω φ s

9 CONTINUOUS SYSTEMS; WAVES 443 whee ω ω β, ω and β ae as defined in (), and C s and φ s ae aitay constants depending on the initial conditions. The complete solution to Eq. (3.48) is the sum of the paticula and complementay solutions (analogous to Eq. (3.5)): whee Fom Eq. (3.4): Thus η sπ F sin cos s + s πτ D ρ ω + ω ρ ρ βt () t C e cos ( ω t φ ) s s s δ tan s Dω s πτ ρ ρ ω πx qxt (, ) η ( t) sin ( ωt δ ) π F sin cos( t ) ω δ π τ qxt (, ) C exp cos t + sin ρ + ω ω ρ ρ Dt s D x φ ρ 4 ρ ρ πτ D (undedamped) π 3-. Fom Eq. (3.44) the equation fo the diving Fouie coefficient is: sπ x fs () t F( x, t) sin dx If the point x is a node fo nomal coodinate s, then x n whee n is an intege s s (This comes fom the fact that nomal mode s has s-half wavelengths in length.) Fo x n, s sπ x sin sin nπ ; hence fs () t

10 444 CHAPTER 3 Thus, if the sting is diven at an aitay point, none of the nom al modes with nodes at the diving point will e excited. 3-. Fom Eq. (3.44) whee Fxt (, ) is the diving foce, and fs Fxt. (, ) Eq. (3.45) shows that fs () t is the component of (, ) coodinate s. Thus, we desie Fxt (, ) such that sπ x fs () t F( x, t) sin dx () s t is the Fouie coefficient of the Fouie expansion of fo f t s n fo s n Fom the fom of (), we ae led to ty a solution of the fom whee g(t) is a function of t only. Thus (, ) sin n π Fxt gt x nπx sπx fs () t g() t sin sin dx F xt effective in diving nomal Fo n s, the integal is popotional to sin ( n± s) πx ; hence x f t fo s n. s Fo n s, we have th Only the n nomal coodinate will e diven. nπ x fs () t gt () sin dx gt () th Thus, to dive the n hamonic only, nπ x Fxt (, ) gt sin

11 CONTINUOUS SYSTEMS; WAVES The equation to e solved is D s πτ ηs + ηs + ηs () ρ ρ Compae this equation to Eq. (3.35): The solution to Eq. (3.35) is Eq. (3.37): x + βx + ω x βt e A exp( β ω t) + A exp( β ω t) xt Thus, y analogy, the solution to () is Dt D s D s ρ πτ πτ ns () t e A exp t A + exp t 4ρ ρ 4ρ ρ 3-3. Assuming k is eal, while ω and v ae complex, the wave function ecomes ψ i( αt+ iβtkx) ( xt, ) Ae Ae ( αtkx) e βt () whose eal pat is ψ β ( xt, ) Ae t cos( t kx) α () and the wave is damped in time, with damping coefficient β. Fom the elation we otain k ( α β) ( ω (3) v + i k u+ iw) By equating the eal and imaginay pat of this equation we can solve fo α and β in tems of u and w: (4) and kuw α (5) β kw β (6) iku Since we have assumed β to e eal, we choose the solution

12 446 CHAPTER 3 Sustituting this into (5), we have as expected. β kw (7) α ku (8) Then, the phase velocity is otained fom the oscillatoy facto in () y its definition: That is, Reω α V (9) k k V u 3-4. I n I n I n+ V n V n I n+ V n V n+ V n+ Q n Q n C Q n C Q n+ C Q n+ C C Conside the aove cicuit. The cicuit in the at the point etween the and th n th n elements is V. Thus we have n V n inducto is I, and the voltage aove gound Qn C di dt n Vn Vn n We may also wite Q Q C C n n () dqn I n I n+ Diffeentiating () with espect to time and using () gives o din dt () [ I I + I dt C din dt C n n n+ ] [ In In + In+] (3) (4)

13 CONTINUOUS SYSTEMS; WAVES 447 et us define a paamete x which inceases y x in going fom one loop to the next (this will ecome the coodinate x in the continuous case), and let us also define C ; C x x which will ecome the inductance and the capacitance, espectively, pe unit length in the limit x. Fom the aove definitions and (4) ecomes o, din dt I I I (6) + + ( In In) C di n I n dt C Dividing y ( x), and multiplying y ( C ), we find ( In ) ( x ) din C dt But y vitue of the aove definitions, we can now pass to the continuous limit expessed y Then, and fo x, we otain n (, ) I t I x t () ( I( x, t) ) I( x, t) x C t (5) (7) (8) (9) () whee I I () x v t v (3) C 3-5. Conside the wave functions ( ω ) ψ Aexp i tkx ψ Bexp i( ( ω + ω) t ( k+ k) x) () whee ω ω; k k. A and B ae complex constants:

14 448 CHAPTER 3 The supeposition of ψ and ψ is given y ψ ψ + ψ ( φ ) A A exp i a B B exp ( iφ ) ωt kx ωt kx ω k i i exp i ω + t k+ x A exp ( iφa) e + B exp ( iφ) e which can e ewitten as ωt kx φa+ φ ωt kx+ φφa ω k φ i i a φ + ψ exp i w+ t k+ x+ A e + B e () (3) (4) Define t ω x k δ φ φ α a (5) and Theefoe, A e That is, θ is a function of ( ω ) t ( k) i( δ + α) i( δ α) i B e + e θ + Γ (6) ( A B ) Γ + (7) A + B cos θ ( A + B ) B A sin θ ( A + B ) cos sin ( δ + a) ( δ + a) x. Using (6) and (7) (9), we can ewite (4) as (8) (9) and then, ω k φa + φ i ψ Γ exp i ω + t k+ x+ e θ () ω k φa + φ Reψ Γ cos ω + t k+ x cos + θ ω k φa + φ sin ω + t k+ x sin + θ ()

15 CONTINUOUS SYSTEMS; WAVES 449 Fom this expession we see that the wave function is modulated and that the phenomenon of eats occus, ut fo A B, the waves neve eat to zeo amplitude; the minimum amplitude is, fom Eq. (), A B, and the maximum amplitude is A + B. The wave function has the fom shown in the figue. A + B Reψ A B wt kx 3-6. As explained at the end of section 3.6, the wave will e eflected at x x and will then popagate in the x diection We let whee n is an intege. m, j n mj m, j n+ Following the pocedue in Section.9, we wite Assume solutions of the fom Sustituting (3a,) into (a,), we otain fom which we can wite τ F m q q q + q d ( +) n n n n n τ F m q q q + q d ( n+ ) n+ n+ n n+ q n n i( t nkd) () (a) () Ae ω (3a) i[ t ( n ) kd] q Be ω + (3) + τ A Be A Be md ikd ikd ( ) ω + τ B Ae B Ae m d ikd ikd ( ) ω + (4)

16 45 CHAPTER 3 τ τ A ω B cos kd md md τ τ A cos kd+ B ω m d m d The solution to this set of coupled equations is otained y setting the deteminant of the coefficients equal to zeo. We then otain the secula equation (5) Solving fo ω, we find τ τ ω ω τ cos kd md m d mm d (6) ω fom which we find the two solutions ω ω τ 4 + ± + sin kd d m m m m m m d m m m m m m τ sin d m m m m m m τ sin kd kd (7) (8) If m < m, and if we define τ τ ωa, ω, ωc ω a + ω (9) m d m d Then the ω vs. k cuve has the fom shown elow in which two anches appea, the lowe anch eing simila to that fo m m (see Fig. 3-5). ω c ω ω ω a Using (9) we can wite (6) as k π/d ω sin a + () ( ω ω ω ) W ( ω) kd ωω a Fom this expession and the figue aove we see that fo ω > ωc and fo ωa < ω < ω, the wave nume k is complex. If we let k κ +iβ, we then otain fom () ( κ iβ) d κd βd κd βd i κd κd βd βd W( ω) sin sin cosh cos sinh sin cos sinh cosh + + () Equating the eal and imaginay pats, we find

17 CONTINUOUS SYSTEMS; WAVES 45 sin κdcosκdsinh βdcosh βd sin κdcosh βd cos κdsinh βd W( ω) () We have the following possiilities that will satisfy the fist of these equations: a) sin κd, which gives κ. This condition also means that cos κd ; then β is detemined fom the second equation in (): Thus, ω > ω, and κ is puely imaginay in this egion. c ( ω) sinh βd W (3) ) cos κd, which gives κ π/d. Then, sin κd, and cosh βd W( ω) and κ is constant at the value π/d in this egion. c) sinh βd, which gives β. Then, sin κ d W( ω) eal in this egion. Altogethe we have the situation illustated in the diagam.. Thus, ωa < ω < ω,. Thus, ω < ωa o ω < ω < ωc, and κ is π d k κ κ β κ β κ ω a ω ω c ω 3-8. The phase and goup velocities fo the popagation of waves along a loaded sting ae whee V k ( kd ) ω ω sin cd () k kd d ω ω d c () dk cos ( kd ) U k The phase and goup velocities have the fom shown elow. ω ω c sin kd (3) ω c d V(k) V,U U(k) π/d k When k π d, U ut V ω d π c. In this situation, the goup (i.e., the wave envelope) is stationay, ut the wavelets (i.e., the wave stuctue inside the envelope) move fowad with the velocity V.

18 45 CHAPTER The linea mass density of the sting is descied y ρ if x< ; x> ρ ρ > ρ if < x< I II III ρ ρ ρ x Conside the sting to e divided in thee diffeent pats: I fo x <, II fo < x <, and III fo x >. et φ Ae ω i( tkx ) e a wave tain, oscillating with fequency ω, incident fom the left on II. We can wite fo the diffeent zones the coesponding wave functions as follows: ψ I ψ II i( ωt kx) i( ωt+ kx) iωt ikx ikx Ae + Be e Ae Be + i( ωt kx) i( ωt+ kx) iωt ikx ikx Ce + De e Ce + De () ψ III i( ωtkx) Ee Whee k ω, k, V, V V ω τ τ V ρ () ρ and whee τ is the tension in the sting (constant thoughout). To solve the polem we need to state fist the ounday conditions; these will e given y the continuity of the wave function and its deivative at the oundaies x and x. Fo x, we have ψ I ψ ( x ) ψ ( x ) II ψ I II x x x x (3) and fo x, the conditions ae ψ II ψ ( x ) ψ ( x ) III ψ II III x x x x (4) Sustituting ψ as given y () into (3) and (4), we have and A+ B C+ D k (5) A + B C+ D k

19 CONTINUOUS SYSTEMS; WAVES 453 Ce + De Ee ik ik ik k Ce + De Ee k ik ik ik (6) Fom (6) we otain k ik ( k) C + Ee k k ik ( + k) D Ee k (7) Hence, Fom (5) we have k C k + k k e ik D (8) Using (7) and eaanging the aove equation In the same way Fom () and () we otain k k A C + + k k D (9) ( k + k ) ( ) ik A e k k k k k B k + k e + k + k k ik D D () () ( ) ik B k k e A k k e k k ik ( + ) ( ) () On the othe hand, fom (6) and (8) we have which, togethe with () gives E k kd ik k k ( + ) e (3) ik ( + k) E 4kk e A k k e k k ik ( + ) ( ) (4) Since the incident intensity I is popotional to tansmitted intensity is I E, we can wite t A, the eflected intensity is I B, and the total

20 454 CHAPTER 3 B, E I I I t I Sustituting () and (4) into (5), we have, fo the eflected intensity, Fom which I I and fo the tansmitted intensity, we have so that (5) A A ( ) k k e ik i k ( k + k ) e ( k k ) ( k k) ( cos k) (6) I I (7) 4 4 k + k + 6kk k k cos k I t I We oseve that I + I t I, as it must. ( + ) 4kk e ik k i k ( k + k ) e ( k k ) cos (8) 8 kk It I (9) k k k k k k k Fo maximum tansmission we need minimum eflection; that is, the case of est possile tansmission is that in which I I t I () In ode that I, (7) shows that must satisfy the equiement cosk () so that we have m m π π τ, m,,, k ω ρ () The optical analog to the eflection and tansmission of waves on a sting is the ehavio of light waves which ae incident on a medium that consists of two pats of diffeent optical densities (i.e., diffeent indices of efaction). If a lens is given a coating of pecisely the coect thickness of a mateial with the pope index of efaction, thee will e almost no eflected wave.

21 CONTINUOUS SYSTEMS; WAVES I y II We divide the sting into two zones: M I: x < II: x > Then, ψ ψ i( ωt kx) i( ωt+ kx) I Ae + Be i( ωtkx) II Ae () The ounday condition is I ( x ) ( x ) ψ ψ () That is, the sting is continuous at x. But ecause the mass M is attached at x, the deivative of the wave function will not e continuous at this point. The condition on the deivative is otained y integating the wave equation fom x ε to x +ε and then taking the limit ε. Thus, ψ ψ ψ M t x x Sustituting the wave functions fom (), we find II I τ x x II A + B A (4) (3) which can e ewitten as Fom (4) and (6) we otain fom which we wite ik A A B ω MA (5) τ + A B A ( ikτ ω M) ikτ A + B ikτ A B ikτ ω M (6) (7) Define B ω M ω M ikτ A ikτ ω M ω M ikτ (8)

22 456 CHAPTER 3 Then, we can ewite (8) as ω M P tan θ kτ (9) B A ip () + ip And if we sustitute this esult in (4), we otain a elation etween A and A : A A + ip () The eflection coefficient, R B A, will e, fom (), B P tan R A P + + tan θ θ () o, R sin θ (3) and the tansmission coefficient, T A A, will e fom () T A + + tan A P θ (4) o, T cos θ (5) The phase changes fo the eflected and tansmitted waves can e calculated diectly fom () and () if we sustitute iφ B B e B iφ A A e A iφ A A e A (6) Then, and B B i( φ φ A) P B itan ( P) e e (7) A A + P A A i( φa φ ) A itan ( P) e e (8) A A + P

23 CONTINUOUS SYSTEMS; WAVES 457 Hence, the phase changes ae φb φ tan tan ( cot ) A θ P A A ( P) φ φ θ θ tan tan tan (9) 3-. The wave function can e witten as {see Eq. (3.a)] + i( ωtkx) ψ ( xt, ) Ake dk () Since A(k) has a non-vanishing value only in the vicinity of k k, () ecomes Accoding to Eq. (3.3), Theefoe, () can now e expessed as k + k i( ωtkx) ψ ( xt, ) e dk k k ( k k ) ω ω + ω (3) k + k i w k t i t x k ( ω ) ( ω ) (, ) ψ xt e e dk k k ik ( + k)( ω tx) ik ( k)( ω tx) ( ) i k t e e ω ω e i( ω t x) i( ωω k) t i( ω tx) k i( ω tx) k e e e ω t x i (4) and witing the tem in the ackets as a sine, we have () ψ ( xt, ) sin ω t x ( ω tx) k i( ω tk x) e (5) The eal pat of the wave function at t is ( x k) sin Re ψ ( x,) cos kx (6) x If k k, the cosine tem will undego many oscillations in one peiod of the sine tem. That is, the sine tem plays the ole of a slowly vaying amplitude and we have the situation in the figue elow.

24 458 CHAPTER 3 ReΨ(x,) x 3-. a) Using Eq. (3.a), we can wite (fo t ) ψ + ikx x A k e dk (,) + σ ( kk ) ikx B e e dk + ikx σ ( kk) i( kk) x Be e e dk + ikx σu iux Be e e du This integal can e evaluated y completing the squae in the exponent: () + + a x x ax x a e e dx e dx + ax x + a 4a 4a e e dx + a x 4a a e e dx () and letting y x a, we have Using Eq. (E.8c) in Appendix E, we have + + ax x 4a ay e e dx e e dy (3) Theefoe, + ax x π 4a e e dx e (4) a

25 CONTINUOUS SYSTEMS; WAVES 459 π ikx x 4σ ψ ( x,) B e e (5) σ The fom of ψ ( x,) (the wave packet) is Gaussian with a e width of diagam elow. Ψ( x,) 4 σ, as indicated in the σ B σ π σ e B x π σ ) The fequency can e expessed as in Eq. (3.3a): and so, + ( k) ( k k ) ω ω + ω + (6) i( ωtkx) (, ) ψ xt Ak e dk + Ak e iωt+ ω kk tkx + dk i( ωtkx) σ( kk) iω ( kk) t( kk) x Be e e dk Using the same integal as efoe, we find + i( ω tkx) σu i( ω tx) u Be e e du ψ π σ i( ω tk x) ( ω tx) ( xt, ) B e e 4 σ (7) (8) c) Retaining the second-ode tem in the Taylo expansion of ω (k), we have ω( k) ω + ω k k + ω k k + (9) Then, ψ i ω ( k k) t+ ω ( kk) t( kk) x xt e Ake dk i( ωt kx) (, ) + + ω t σ i u i( ωtkx) i( w tx) u Be e e du We notice that if we make the change σ iω t σ, then () ecomes identical to (7). Theefoe, ()

26 46 CHAPTER 3 ψ π σ iw t i( ωtk x) α( xt, ) ( xt, ) B i e () whee α ( xt, ) The e width of the wave packet will now e ( ω t x) σ + iω t 4σ ( ω ) + t () o, w e () t ( ω ) 4σ + t (3) σ () t ω t σ + W 4 e σ (4) In fist ode, W e, shown in the figue aove, does not depend upon the time, ut in second ode, W depends upon t though the expession (4). But, as can e seen fom (8) and (), e the goup velocity is ω, and is the same in oth cases. Thus, the wave packet popagates with velocity ω ut it speads out as a function of time, as illustated elow. Ψ( xt, ) t t t O x