Chapter 1 COUPLED ONE-DIMENSIONAL OSCILLATORS

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1 Chpter COUPLED ONE-DIMENSIONAL OSCILLATORS. Introduction Much of the interesting vibrtionl behvior of periodic systes is reveled by the clssicl oscilltions of chins of sses connected by springs tht obey Hooke s Lw. Recll tht Hooke s Lw siply ens tht when ss connected to spring is displced fro its equilibriu position, there is restoring force, F, directly proportionl to the displceent ψ, i.e. F = ψ. We will see tht the otions of such chins cn be nlyzed using the physics of siple hronic oscilltors nd the concept of norl odes.. Two coupled siple hronic oscilltors We begin by reviewing the siple cse of two sses coupled by Hooke's Lw springs. We wish to find the possible otions of such syste. Lter, we will extend this to liner chins contining finite nd infinite nubers of sses. This syste is odel for other types of coupled oscilltions such s coupled LC circuits, coupled pendulus, etc. The governing equtions for ll systes consisting of two coupled hronic oscilltors cn be put into the se theticl for. This hs the powerful ipliction tht once we hve solved for the behvior of one such syste, we hve effectively solved for ll of the... Equtions of otion Newton s second lw of otion tells us how point ss oves in response to force: r F = dr p dt = d r dt Thus, if we know wht F is, we cn, in principle, solve the differentil eqution to get the trjectory r r (t)!

2 Now, let ψ be the displceent fro equilibriu in the longitudinl direction (long the line connecting the sses) nd pply Hooke's Lw, F = ψ: ψ ψ positive direction We dopt systetic pproch nd define displceents to the right to be positive. Then Force on ss = (force to left) + (force to right) = F = - ψ + ψ - ) ( ψ Force on ss = F = - ψ + ψ - ψ = -ψ - ψ - ) ( ψ Hint: check ech ter for the sign of the displceent! Using our nottion, Newton s Lw is ( ) ( ) ( ) siplicity, ssue tht the sses re equl ( = = ): ψ&& ψ&& F = ψ& &, so let s plug in the forces nd, for + = = + ψ ψ + ψ + ψ We see tht the result is pir of coupled ordinry differentil equtions for ψ nd ψ. They re coupled becuse the second derivtive of ψ or ψ (ccelertion of given ss) depends on both ψ nd ψ. Another wy of expressing this result is the trix eqution

3 ψ&& ψ&& + = + ψ ψ.. Norl odes To proceed further, we ke the iportnt ssuption tht there re prticulr otions of the syste in which both sses oscillte with the se frequency. These specil otions re clled norl odes. We will see s we go long tht for one-diensionl chins there re exctly s ny norl odes s there re sses in the syste. Ech norl ode corresponds to otion with single frequency, but the frequencies of norl odes cn be (n usully re) different. Once we hve identified the otions ssocited with the norl odes nd hve found the norl ode frequencies, we cn express the generl otion of ny ss in ters of superposition of norl odes. To find the norl ode frequencies, ssue tht both sses ove with the se frequency ω : ψ = A e iωt ψ = A e iω t. The coefficients A nd A re coplex. Eventully, we will tke the rel prts to get the ctul otion. The theticl tsk is to find the possible vlues of ω nd the reltionship between the coefficients for ech ω. Note tht for our ssued tie dependence of the displceents, the second derivtives of the displceents (the ccelertions) re proportionl to the displceents theselves: ψ&& ψ&& = ω = ω A e A iωt e iωt = ω = ω If we substitute these expressions for ψ & nd ψ& in the equtions of otion we get the convenient result tht the coupled differentil equtions hve been converted to pir of coupled liner lgebric equtions. In trix for, ω ψ + ψ = ψ + ψ This is n eigenvlue eqution of the for ψ ψ A = ω Łψ ł Łψ ł whose eigenvlues will be the norl ode frequencies (ω-vlues) we seek. ψ ψ 3

4 ..3 Frequencies of norl odes: eigenvlues = To find eigenvlues, we set det( - I) 0 A ω. This yields + ω = 0 Although this ppers to be qurtic eqution, it is relly just qudrtic eqution for ω : + ω = + ω = + ω + ± ± + +ω 4 + = 0 + Thus there re two eigenvlues ( eigenfrequencies ) nd two corresponding norl odes. (We nticipted this with the stteent tht there would be s ny norl odes s there re sses.) Tking the solution with the inus sign we get low frequency ode with frequency ω low = + = This frequency is independent of the spring constnt of the coupling spring! Wht does this suggest bout the otion ssocited with the low frequency norl ode? If we tke the plus sign, we get the frequency of the high frequency ode: ω high = + + = + The high frequency does depend on the coupling. Wht is the ipliction for this ode?..4 Syetries of norl odes: eigenvectors We don t relly hve to guess bout the nture of the otions of the two norl odes. If we find the eigenvectors of the norl odes we will know the reltion between the coefficients A nd A for given ode. This will define the syetry of the otion. 4

5 The procedure is the substitute prticulr eigenfrequency into the trix eqution nd find the reltionship between A nd A (plitudes of the displceents) for tht ode. For the low frequency ode: + + ψ ψ = 0 + ψ ψ = 0 ψ + + ψ = 0 ψ ψ = 0 ψ ψ = 0 so tht ψ = ψ nd therefore A = A. The ode is syetric with the sses oving in phse. Since the sses intin their seprtion in this ode, the frequency is independent of the spring constnt of the coupling spring. If we let A = 0 iϕ = A A e with A 0 rel, we cn express the displceents for the low frequency t iϕ ψ i A0 e ode in the for = e. The rel constnts A iϕ 0 nd ϕ re deterined by the Łψ ł Ł A0 e ł initil conditions. To get the ctul displceents, tke the rel prts: ( cos ϕcos ωt - sinϕ ωt) Re( ψ ) = Re( ψ ) = A0 sin ω = + t iϕ ψ i A0e For the high frequency ode, the se procedure yields = e iϕ ψ Ł -. Ł ł A0e ł This ode is nti-syetric with the sses oving out of phse (equl nd opposite displceents). In this ode, the coupling spring is lterntely stretched nd copressed so the frequency ust depend on the coupling constnt...5 Generl otions: Principle of Superposition We know tht the syste hs eigenfrequencies, ωlow for the syetric ode in which ψ = ψ, nd ω high for the ntisyetric ode in which ψ = -ψ. If we strt the syste with initil conditions corresponding to one of these odes, sy by displcing both sses syetriclly (ntisyetriclly), then the syste will oscillte with single frequency ω ( ω high ). For ore generl initil conditions, both odes re will be excited nd the syste low 5

6 will execute coplex otion corresponding to superposition of the two eigenfrequencies. Applying the Principle of Superposition, we should be ble to express the otions of sses nd s iωt iωbt iωt iωbt ψ = Ae + Be nd ψ = A e + B e where, for siplicity we letω low = ω nd ω high = ωb. The plitudes A, B, A, nd B re coplex constnts to be deterined by the initil conditions. (We will tke the rel prt when we need to.) However, these constnts re not copletely independent of ech other. We know, for exple, tht if B = B = 0 so tht the syste is oscillting only with frequency ω low = ω, then the otion ust be syetric, i.e. ψ = ψ A = A. Siilrly, to get the ntisyetric ode, we need ψ = - ψ B = -B. We cn ssure this by writing iωt iωb t iωt iωb t ψ = Ae + Be nd ψ = Ae - Be Putting in the coplex nture of the coefficients explicitly, we see tht we hve 4 undeterined constnts, the (rel) plitudes A0,B 0 nd the phse fctors, φ, δ. iφ iω t iδ iωbt iφ iωt iδ iωbt ( t) = A e e + B e e ψ ( t) = A e e B e e ψ 0 0 nd 0-0 Now, let s chose prticulr set of initil conditions: ss t the equilibriu position with zero velocity nd ss displced distnce C, lso with zero velocity. Expressing this lgebriclly we hve dψ dt 0 ψ iφ iδ iφ iδ ( 0) = Re( A0e + B0e ) = 0 nd ψ ( 0) = Re( A0e - B0e ) C iφ iδ dψ iφ iδ ( iω A e + iω B e ) = 0 nd = Re( iω A e - iω B e ) 0 = = = Re 0 b 0 0 b 0 dt Rewriting the two equtions for the velocity initil conditions by tking the rel prts of the exponentils, we find - ω A0 sin φ - ωbb0 sin δ = 0 nd - ωa0 sinφ + ωbb0 sinδ = 0 Adding nd subtrcting these two equtions gives - ω A0 sin φ = 0 nd - ω bb0 sinδ = 0 fro which we conclude φ = δ = 0. If we hd forgotten bout the velocity initil condition, it would hve been the se s ssuing A nd B rel, which would hve been OK for this cse. But this cn t be done in generl! Now, go bck to the displceent initil conditions, which now red, A + B = nd A - B C = 0 6

7 whence, A 0 = -B0 = C. So now our equtions for the displceents red These re equivlent to ψ ψ iωt iωbt ( t) = Re[ C( e - e )] = C cos( ωt) - cos( ωbt ) [ ] iωt iωbt ( t) = Re[ C( e + e )] = C cos( ωt ) + cos( ωbt) ψ ψ ( t) ( t) = C sin Ł = C cos Ł [ ] ( ω + ω ) t ( ω - ω ) b sin ł Ł ( ω + ω ) t ( ω - ω ) b cos ł Ł To show this tkes bit of lgebr: use the trig identities sin ( x y ) = sinx cos y cos y sin x in the preceding equtions, crry out the ultipliction, nd introduce the identities x sin x = - cos x nd cos = + cos x. Ł ł Ł ł Bets b b t ł t ł This result shows ( tht the generl otion of the syste consists of high frequency oscilltion ω ) b + ω with frequency ( ωb - ω ) odulted by lower frequency. The effect of the lower frequency (frequency difference) ter is known s bets in nlogy to the throbbing sound tht is herd when two cousticl tones with slightly different frequencies re herd siultneously. The theticl description is identicl superposition of two sines or cosines. The functionl fors of ψ ( t) nd ( t) ωb - ω ψ re shown below for cse where = 0.. ω + ω b 7

8 Fig. X. Displceents ψ ( t) (upper plot) nd ( t) ω ω b b - ω + ω = 0.. ψ (lower plot) for the cse Notice tht the envelopes of the displceents for the two sses re 90 o out of phse. When ss is oscillting with xiu plitude, ss hs iniu plitude nd vice vers. 8

9 Energy is periodiclly exchnged (with frequency hronic oscilltors. ω - ω ) between the two individul b Wek coupling liit Recll the expressions for the two eigenfrequencies: ω low = ω = nd ω high = ω b = + nd recll tht it is the coupling ( ) tht distinguishes the two frequencies. (If = 0, we siply hve two, copletely independent siple hronic oscilltors.) Now, suppose tht the coupling is wek, i.e. <<. Then, using the pproxition + + ε for ε<< we cn write ωhigh = ωb = + = = + Ł + ω ω ł Ł ł The odultion frequency in this liit is to the strength of the coupling. ( ω - ω ) b = ω which is directly proportionl Ł ł Discussion question: Given pir of identicl coupled echnicl oscilltors with known sses, how could you deterine the strength of the coupling by n experient?.3 Mny coupled sses: the ontoic N-ebered chin Lb # is the study of chins of N sses using the CUPS siultions progrs. The siultions studied in Lb # yielded the following results:. An experientlly deterined dispersion reltion (frequency ω versus wve vector k = π/λ) for 5-ss chin.. The dispersion reltion ω(k) is liner t low frequency. 3. There is xiu frequency for the norl odes of the chin. 9

10 4. The norl ode frequencies re repeted for higher wve vectors no new physics for k-vlues beyond the xiu in ω(k). Now let s see how these results follow fro n nlysis of the echnics of the chin using Newton s eqution of otion..3. Equtions of otion We continue with ny sses coupled to their nerest neighbors by Hooke's Lw springs nd find the possible longitudinl otion of such syste. (Coupling with other springs to next neighbors is hoework exple). This syste is odel for other types of coupled oscilltions (trnsverse otion of these sses, coupled LC circuits, pendulus.) n- n n+ Hooke's Lw gives us the force on the n th ss when it is displced distnce ψ n fro its equilibriu position, = -ψ. We re considering only longitudinl otion (long the chin) Fn n in one diension. For the present, we ssue tht ll sses () nd spring constnts () re equl. Lter we will relx this restriction. Then the force on ss n is Fn = n ( ψn ψn ) ( ψn ψ + ) Hint: gin, check ech ter for the sign of the force reltive to tht of the displceent Now, pply Newton s eqution of otion, nerest neighbors: ψ&& ψ&& ψ&& n n = ( ψ = ( ψ n+ = ( ψ F n = n ψ& & nd plug in the forces for ss n nd its n n ψ n+ ψ n ψ n ) ( ψ n ) ( ψ n ) ( ψ ψ n+ n n+ ) ψ ψ n n+ ) ) 0

11 .3. Dispersion Reltion Once gin, we hve set of coupled differentil equtions. The ccelertion of prticulr ss depends not only on its own displceent, but lso those of its neighbors. The route to solution is to ssue, s we did for two coupled oscilltors, tht the syste hs set of norl odes in which ll sses oscillte with the se frequency. For prticulr norl ode with frequency ω, the displceent of ss n cn be expressed s ψ n = A n e iω t. The coefficients A n give the plitudes of the oscilltions for the vrious sses n nd re, in generl, coplex. Note tht the frequency hs no subscript (prticle lbel) ll prticles oscillte with the se frequency in given norl ode. Furtherore, s we sw in in Lb, the plitudes of the prticles oscilltions for the envelope of sinusoid in the norl odes, so we ll further ssue: ψ n = Ae i ( nk δ ) e iωt where n is the prticle s position long the chin, nd k = π is the wve vector. (In onediension, the vector property is expressed only by the sign (±) of k; in two- or λ threediensionl syste, this is true vector, k r ). The wvelength λ is tht of the envelope tht describes the instntneous positions of the sses in tht prticulr ode. Substitute the norl ode solution into Newton s eqution of otion: ω Ae ink iδ e iωt = ( Ae ink iδ e iωt Ae i( n )k iδ e iωt ) Ae ink iδ e iωt Ae i n+ ω ω = eink = e i( n )k + e ink e i( n +)k ω = 4 sin k ω(k) = eik + e ik sin k = ( cosk) ( ( )k iδ e iωt ) Our ssuption of wve-like, oscilltory norl odes hs led us to dispersion reltion ω(k). This function is plotted below for ω x = = nd =.

12 Fig. X. constnts). Dispersion reltion ω(k) for ontoic chin (equl sses nd spring Iportnt fetures of the dispersion reltion:. Infortion is repeted fter k = π. This is clled the Brillouin zone boundry nd corresponds to wvelength of λ =. Sller wvelengths re physiclly eningless s we found in lb.. There is xiu frequency ωx = bove which there re no wve- like norl odes. 3. For sll vlues of k, the dispersion reltion is liner: k = vsk. The proportionlity constnt v s = is just the speed of longitudinl sound (long wvelength wves) long the chin. Here long wvelength ens k << π, or λ >>. The dispersion reltion gives us the reltion between ω nd k for the norl odes, but we hve not yet found the specific frequencies (nd k-vlues) of the odes. These will depend on the nuber of sses in the chin nd the boundry conditions t the ends of the chin.

13 .3.3 Boundry conditions: norl ode frequencies nd wve vectors To find prticulr vlues of ω nd k for the norl odes, we need to specify boundry conditions, nd the totl nuber of sses N. Let us introduce new index q to lbel the frequency ω q nd wve vector k q of the q th norl ode. Then, the dispersion reltion for the ontoic chin becoes ω q =ω x sin k q. Fixed boundry conditions One pproch to finding the norl odes is to pply fixed boundry conditions. This ens tht we fix the ends of the chin t ll ties, i.e. we require the fictitious 0 th nd (N+) th sses to be fixed. In this cse, the norl odes will be stnding wves. ( ) 0 Recll ψ n = Ae i( nk δ) e iωt. The boundry condition i ( k 0-δ = ) i ωt ψ0 Re Ae e = t ll ties requires tht A 0 = Re Ae i (k.0. δ ) = Acos( δ )= 0 which iplies δ = π. Thus, for ny i(kn π/ ) ss n on the chin, A n = Re( Ae )= Acos( kn π / )= Asin kn. These re the wve-like plitudes of the eigenodes. Now, t the other end of the chin, the boundry condition ψ = 0 iplies N + A N + = Asin k q (N +) = 0 k q (N +) = qπ k q = qπ (N +) The ode index q hs vlues, N (N distinct odes) which will give N distinct vlues of k q nd ω q. Now tht we hve k q, we cn find the relted frequencies, ω q, using the dispersion reltion nd our proble is solved. The dispersion reltion, norl ode frequencies nd wve vectors for N = 5 re illustrted in the following plot. 3

14 Figure X. Dispersion reltion for the ontoic chin with norl ode frequencies nd wve vectors for the cse N = 5 with fixed boundry conditions. Iportnt fetures of the norl odes for fixed boundry conditions:. For q < 0, we get no new infortion, since sin(x) = -sin(-x) nd the entire displceent is the se except for phse which cn be bsorbed into the tie dependence.. For q = N +, k N+ = π/ nd A n = Asin ( nπ) = 0 for ll n. This is null ode the displceents of ll sses re zero t ll ties. This k-vlue (π/) defines the Brillouin zone boundry. 3. For q > N +, the frequencies repet those for q N due to the periodicity of the dispersion reltion. 4

15 Periodic boundry conditions Periodic boundry conditions re n lterntive to fixed boundry conditions. Here we hve no requireent on the plitude, but rther on the displceent s whole. We require only tht the otion of the 0 th ss be the se s the otion of the (N+) th, or generlly tht the otion of the n th ss be the se s the otion of the (N+n) th : ψ n =ψ n+ N+ Re Ae i(kn δ ) = Re Ae i(k(n+n +) δ) kn δ = k(n + N +) δ ± πq which siplifies to k q = ± πq (with no requireent on δ). (N +) The norl odes nd corresponding q-vlues for periodic boundry conditions re copred with those for fixed boundry conditions for N = 5 in Figure X. Figure X. Dispersion reltion for the ontoic chin for N = 5 showing norl odes nd ssocited q-vlues for periodic boundry conditions (open points) nd fixed boundry conditions (closed points). Note tht for periodic boundry conditions there re ctully N + = 6 oving sses in the chin nd 6 norl odes (explined below). 5

16 Iportnt fetures of the norl odes for periodic boundry conditions:. We see tht the k spcing hs doubled (q = (N+)/ gives the Brillouin zone boundry). Hve we lost hlf the odes? No! In this cse the different signs of q ARE distinct. Positive nd negtive q vlues correspond to oppositely propgting trveling wves. Thus q = ±, ±,., ±(N+)/ give physiclly distinct odes (now it s cler why the inus sign bove ws dropped). This running wve set of sttes is siply different bsis set fro the stnding wve set. Note tht one generlly sees periodic boundry conditions written s ψ n =ψ n+ N nd not ψ n =ψ n+ N + s written bove. Why? In our proble of N sses with fixed boundry conditions, we relly introduced fictitious 0 th nd (N+) th ss nd de the sttionry. So we relly hd N + unit cells in our proble. With the periodic boundry conditions, we let the 0 th nd N th to (which re relly the se to) prticipte in the otion, so gin we relly hd N + unit cells. 6

17 .4 Ditoic chin Lb # is the study of n N-ebered chin contining lternting sses ( ditoic chin ).4. Equtions of otion Following the procedure we used to nlyze the ontoic chin, we will set up Newton s equtions of otion using Hooke s Lw for the forces. Agin we constrin the otion to onediension long the direction of the chin (longitudinl oscilltions). Ech cell or repeting unit contins two sses, M nd, nd two identicl springs with spring constnt. In the digr below, the drk sses (M) re the lrger, the light ones () re the sller. The distnce fro cell to cell ( lttice constnt ) is, therefore equilibriu spcing between the sses ( intertoic distnce ) is /. The equilibriu position of ss n is x n 0 = n so the ctul position of the ss long the chin is x n = n +ψ n. M n- n M n+ M The force on the hevy ss n is F n = ( ψ n ψ n ) ψ n ψ n+ Fn = Mψ& & n, so we hve Mψ& & ψ ψ ψ ) n = ( n+ n + n For the neighboring light ss n-, the eqution of otion is F ψ & = ψ - ψ + ψ ). n- = & n- ( n- n- n ( ). Newton s lw gives For the ontoic chin, we were ble to obtin the dispersion reltion fro the eqution of otion for single ss in the chin. For the ditoic chin, we hve two distinct equtions becuse M. 7

18 .4. Norl odes nd dispersion reltion We ssue, s usul, tht there exist norl odes of otion in which ll sses oscillte with the se frequency. However, the sll nd lrge sses cn oscillte with different plitudes. Thus we need to write ψ n = Aei ( kn/ ωt ) for M sses. i kn/ ωt ψ n = αae ( ) for sses where α deterines the plitude nd phse of oscilltions reltive to M oscilltions. Once gin note tht the frequency ω hs no subscript n relting it to prticulr ss ll sses ove with se frequency in given ode. The wve vector, k, is deterined by the wvelength of the pttern of displceent long the chin, k = π λ. Now, substitute the ssued displceents for norl ode into the two equtions of otion -i t nd cncel the tie-dependent fctor e ω tht is coon to ll ters. ink/ ink/ i ( ( n-) k/ ink/ i ) ( n+ - Mω e = - e - αe - e - αe ) i( n-) k/ i( n-) k / i( n-) k / i n- - αω e = - αe - e - αe We cn lso cncel the coon fctors e respectively to get k / ( ) ( ) ( ( ) k / ink/ - e ) i( n-) / nd e in the first nd second equtions, ink/ k ( ) ( ) Mω = αe ik/ +αe ik/ αω = e ik/ + e ik/ α which cn be further siplified to get two equtions in the two unknowns α nd ω : Now eliinte α by solving ech eqution for α, Mω = α cos( k / ) αω = cos( k / ) α α = α = Mω cos( k / ) cos ( k / ) ω nd equting the to get the DISPERSION RELATION ω(k): 8

19 cos( k / ) Mω = ω cos( k/ ) 4 cos ( k / ) = ω ( ) Mω ( ) 4 M sin ( k/ ) ω = M + ± M + We cn nticipte tht we will wnt to lbel frequencies nd wve vectors with n index q to identify the norl ode they re ssocited with. Thus, the finl for of the dispersion reltion is ω q = M + ± M + 4 M sin k q / / ( ) / Reeber, frequencies ω q nd wve vectors k q s ssocited with the q th prticulr ss. The sses re identified by their ddress long the chin, n. ode nd not ny A new feture hs eerged in the dispersion reltion of the ditoic chin. Becuse of the ± signs, there re two brnches to the dispersion curve. Ech vlue of the index q hs two frequencies. We will need to distinguish these frequencies. We could do this by introducing n + - dditionl index, sy, by writing ω q nd ω q. However, it is conventionl to lbel the two brnches of the dispersion curve with the following nes: the lower frequency brnch is clled the ACOUSTIC brnch; the higher frequency brnch is clled the OPTIC brnch. There re sound physicl resons for these nes, which we shll discuss shortly. Note the repetition of frequency infortion fter k = π, i.e. beyond the boundry of the Brillouin zone. Wvelengths less thn re physiclly eningless, s we found in Lb #. The two brnches of the dispersion curve re shown in the following figure. They were clculted for the preters: =, M =, = 0.5, = π. 9

20 Figure X. - 0 Acoustic brnch (lower curve) nd optic brnch (upper curve) of the dispersion reltion for ditoic chin with preters =, M =, = 0.5, = π. Iportnt fetures of the dispersion reltion for ditoic chin:. There is xiu frequency for the coustic brnch, nd both xiu nd iniu frequencies for the optic brnch.. Both brnches re periodic in k nd ll possible frequencies cn be obtined with wve vectors in the rnge 0 k π/ (Brillouin zone). 3. Becuse the lttice constnt equls twice the spcing between sses, the Brillouin zone for the ditoic chin is hlf the size of the Brillouin zone for ontoic chin with the se spcing between sses. 4. The coustic brnch is qulittively siilr to the dispersion reltion of ontoic chin nd, in prticulr, is liner for sll vlues of k (or k-vlues tht re integer ultiples of π). 5. At the Brillouin zone boundry, there is gp ( rnge of forbidden frequencies) between the coustic nd optic brnches. The gnitude of the gp depends on the rtio /M nd the gp closes when = M. Now tht we know the frequencies, we cn go bck nd clculte two vlues for α t given k- vlue (one for the coustic odes nd one for the opticl odes). The results re: k k cos cos α = Ł ł nd α o = Ł ł - ω - ωo These expressions for lph re shown below plotted versus k for the se preters used in Fig. X. 0

21 Figure XX. Aplitude rtio α versus k for the coustic odes (centrl curve) nd optic odes (upper nd lower curves) for ditoic chin with preters =, M =, = 0.5, = π. Iportnt fetures of the plitude rtio for ditoic chin:. For the coustic brnch ner t sll k vlues, α. This ens tht sses nd M re oving in phse with equl plitudes. Since the wvelengths re long in this liit (λ >> ), this corresponds to ordinry longitudinl sound wves, hence the ne coustic for this brnch.. For the optic brnch ner k = 0, α o nd is negtive. This ens tht sses nd M re oving out of phse with unequl plitudes. For the preters used for Figs. X nd XX, α o = so tht the sll sses re oscillting out of phse nd with twice the plitude s the lrge sses M. In rel solids where the different sses y lso crry differing ionic electric chrges, such out-of-phse otions couple strongly to electrognetic wves (discussed in section x.x), hence the ne optic for this brnch. 3. For the coustic brnch t the Brillouin zone boundry, α = 0 so tht in this ode, the light sses re not oving t ll nd only the hevy sses M re oscillting. 4. For the optic brnch nd the Brillouin zone boundry, α o so tht in this ode, the hevy sses M re not oving nd only the light sses re oscillting.

22 We sserted previously tht the gp between the optic nd coustic frequencies t the Brillouin zone boundry depends on the rtio /M nd tht the gp vnishes when = M. This is illustrted in Fig. Y where the dispersion reltion is plotted for the cse =, M =.. It cn be seen tht the gp is now very sll copred with tht in Fig. X. The two brnches lost join together to for sets of repeting ontoic dispersion reltions ω 0. Figure Y. k - 0 Acoustic brnch (lower curve) nd optic brnch (upper curve) of the dispersion reltion for ditoic chin with preters =, M =., = 0.5, = π. The plitude rtios α nd α o re shown below in Fig. YY for = nd M =.. The in effect of nerly equl sses is tht α nd α o re lost constnt cross the Brillouin zone except very close to the zone boundry.

23 4 0 - α -4 - k 0 Figure YY. Aplitude rtio α versus k for the coustic odes (centrl curve) nd optic odes (upper nd lower curves) for ditoic chin with preters =, M =., = 0.5, = π. Group ctivity: For the ditoic chin, find the wve vectors, vibrtion frequencies, nd vlues of the constnt α t the following specil points: Group : Long wvelength opticl vibrtion Group : Brillouin zone opticl vibrtion Group 3: Brillouin zone coustic vibrtion Interpret your results in wy tht illustrtes the physicl behvior of the syste. Articulte wht distinguishes n coustic fro n opticl vibrtion. If you finish erly, ove on to nother tsk, swpping roles of tskster, cynic, & recorder. Interpret your results in wy tht illustrtes the physicl behvior of the syste. Articulte wht distinguishes n coustic fro n opticl vibrtion. 3

24 .5 Evnescent wves: forbidden frequencies Fro our study of vrious systes of coupled oscilltors we hve lerned tht the norl ode frequencies re confined to certin rnges, depending on the specific type of syste. For exple, there is xiu frequency for the llowed odes of liner chins of coupled sses (ontoic or ditoic). The se is true for beded string, set of sses connected by continuous string insted of Hooke s Lw springs. In soe cses, there is iniu frequency. Exples re the coupled pendul (hwk ) or optic brnch of ditoic chin (Lb ). The existence of these xiu nd iniu llowed frequencies ens tht there re certin frequency rnges for which no odes exist in coupled systes. Of course, even within the rnges of llowed frequencies, boundry conditions require tht only certin discrete frequencies (eigenfrequencies) re llowed. But these discrete frequencies cn be de rbitrrily close together in syste with lrge nuber of sses. The lws of echnics nd the properties of prticulr syste deterine the frequencies of llowed vibrtionl odes for tht syste. If the syste is excited, sy by displcing the sses fro equilibriu nd then leving it undisturbed, the syste will ove in soe superposition of the llowed norl odes tht depends on the initil conditions. However, there is nothing to prevent us fro driving syste t ny frequency of our choice. In prticulr, we ight choose to drive the syste t frequency tht is not llowed. We could, for exple, oscillte one end of liner chin of sses nd try to excite wves long the chin t forbidden frequency. Wht hppens? Let s consider the exple of ontoic chin. We know tht the dispersion reltion is ω(k) = k sin = ω x sin k We lso know tht depending on the nuber of sses nd the type of boundry conditions, there re norl odes with discrete frequencies in the rnge 0 < ω ωx nd discrete wve vectors in the Brillouin zone 0 < k π /. But now suppose tht we pick driving frequency ω > ω x. Then sin k = ω > nd k ust necessrily be coplex since the sin (x) ω x cnnot exceed for rel x. To see how this goes, first recll tht the sine nd cosine of purely iginry rguents leds to the hyperbolic functions, sinh nd cosh, respectively: e siniθ = e cosiθ = iiθ iiθ - e i + e -iiθ -iiθ e = e = -θ -θ - e i + e + θ + θ e = i + θ - e = coshθ -θ = isinh θ 4

25 Now, letting k = Re ( k)+ i I( k), ω(k) = sin ( Re( k) + i I( k) ) ω x ω(k) = sin Re ( k ) cos ii ( k ) ω x + cos Re ( k ) ω(k) = sin Re ( k ) cosh I ( k ) ω x + icos Re ( k ) The frequency ω ust be rel for ny vlue of k, so cos Re ( k ) = 0 Re( k) = π, 3π... sin ii ( k ) sinh I ( k ) It is sufficient to tke Re(k) = ð/. This ens λ = nd the sses lwys vibrte in ntiphse. Further, ω(k) = sin π ω x cosh I( k) =cosh I( k) I( k) = cosh ω(k ) ω x Now look t displceents gin: i( kn-δ ψ = Ae ) e ψ ψ n n n = = Ae Ae iωt - i πn + in cosh Ł Ł ω ω x ω - ncosh- Ł Ł ωx łł -δ ł ł e iωt i( πn -δ) iωt -βn i( πn -δ) iωt e e = Ae e e where - ω -β β cosh. The rel prt is ψn = Ae n cos ( πn - δ + ωt). Ł ωx ł This describes wve tht is exponentilly dped long the chin with dping length (/β) tht depends on how fr bove ω x we ttept to drive the chin. 5

26 Figure Z. Instntneous plitudes of oscilltions for ontoic chin driven t frequency ω > ωx. The ore strongly dped wve results fro driving frequency tht is further into the forbidden frequency rnge. The horizontl xis represents the distnce long fro the chin fro the point t which it is driven. Discussion questions: Consider set of coupled sses in which the iddle sses re replced by hevier ones. Wht will hppen? (Deonstrte with CUPS session?) Suppose wve propgting with n llowed frequency long chin encounters region in which the frequency of the wve is forbidden. Wht will hppen? Wht hppens to the energy crried by the propgting wve? 6