PEER REVIEW. ... Your future in science will be largely controlled by anonymous letters from your peers. peers. Matt. Corinne

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1 PEER REVIEW 1... Your future in science will be largely controlled by anonymous letters from your peers. Matt peers Corinne

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7 MULTIPLE DRIVNG FREQUENCIES

8 LRC circuit L I = (1/Z)V ext Z must have amplitude & phase I V ext C R Imag I = 1/Z V f V Real

9 V = Re éë V e iwt q( t) = Re éë q e iwt ext 0 û 0 û 9 I(t) = q t é ë ( ) = Re w q 0 e i f q ( +p /2) e iwt û I 0 = wv 0 L é( w 2 - w 2 ) 2 + 4b 2 w 2 ë 0 1/2 û f I = p 2 + arctan -2bw w w 2

10 Y ( w) I = YV ( ) V ( t) = V 0 cos w 0 t w V ( t) = - V 0 cos( 2w 0 t) f I ( w ) V ( t) = V 0 cos( 3w 0 t) w 0 2w 0 3w 0 t

11 V t = V cos w t V cos 2w t + V cos 3w t ext Y ( w) ( ) ( ) ( ) ( ) V ext f I ( w ) t

12 V t = V cos w t V cos 2w t + V cos 3w t ext ( ) ( ) ( ) ( ) Y ( w ) I Y V ext f I ( w ) I ( t) = V 0 Y w0 cos( w 0 t + f ) w0 - V 0 Y 2w0 cos( 2w 0 t + f ) 2w0 + V 0 Y 3w0 cos( 3w 0 t + f ) 3w0 t

13 DRIVING AN OCILLATOR WITH A PERIODIC FORCING FUNCTION THAT IS NOT A PURE SINE V = V e + V e ext iw t i2w t (given) q + q + q = V e + V e 2 iwt i2w t 2 w 2 (Kirchoff) 0 q = q + q 2w 2w = Y V + Y ( w ) 0 w 0 0 ext, w 2w et x, 2w (linear diff eq - superposition) I = I + I I = q I I = + w w w V w L ( 2w ) L w 1/2 ( ( ) ) w0 2w + 4 ( 2w ) 2 2 e 2 w i + arctan w 0 w 1/2 e Ve i w + arctan 2 2 w0 2 iw t 2 ( w) i( 2w ) 2Ve 0 Where are the resonances? t

14 BASIC WAVE CONCEPTS 14

15 Equivalent representations. 15 y(t) = Acos( wt + f) y(t) = B p cos wt ( ) + B q sin( wt) y(t) = Ce i ( wt) -i( wt) + C *e y(t) = Re[ De iwt ] Remember the conversions between A, B, C, D forms - see Main Ch. 1.

16 position y(t) = Acos wt +f ( ) A 1 period T = 2p w = 1 f 16 y A -1 time t = - f w

17 position At a FIXED TIME, y(x) = Acos kx +f ( ) 17 wavelength l = 2p k 1 A y A space time

18 y(x,t) = Acos( wt - kx +f) y(x,t) = B p cos( wt - kx) + B q sin( wt - kx) y(x,t) = Ce i ( wt-kx) -i( wt-kx) + C *e y(x,t) = Re De i wt-kx+f ( ) [ ] Same conversions between A, B, C, D forms as for oscillations - see Main Ch1, Ch9. 18 Other waveforms (e.g.) sawtooth, pulses etc., can be written as superpositions of harmonic waves of different wavelengths and/or frequencies Fourier series and Fourier integrals (transforms)

19 How do these functions arise? PROVIDED w/k = v, a constant, they are solutions to the differential equation: (non-dispersive wave equation) 19 2 x 2 y (x,t) = 1 v 2 2 t 2 y (x,t) This DE results when: Newton s law is applied to a string under tension Kirchoff s law is applied to a coaxial cable Maxwell s equations are applied to source-free media and many other cases

20 Periodic variations in space 20 y(x) = Acos( kx + f) l = 2p k Wave vector, k, dimension: [length -1 ] (wave number is 1/l) Wavelength, l, dimension: [length] Amplitude A, or y 0, dimension: [whatever] Phase, kx+f, dimensionless Phase constant, f, dimensionless

21 Waves - functions of space AND time 21 Looking ahead. We will discuss mostly harmonic waves where variations are sinusoidal. Pulses and non-harmonic waves are just superpositions of harmonic waves Traveling and standing waves Damped, (driven) waves Reflection, transmission, impedance Classical and quantum systems

22 Example: Non dispersive wave equation (a second order linear partial differential equation 22 2 x 2 y (x,t) = 1 v 2 2 t 2 y (x,t) Demonstrate that both standing and traveling waves satisfy this equation (HW) PROVIDED w/k = v. Thus we recognize that v represents the wave velocity. (What does velocity mean for a standing wave?)

23 REFLECTION AND TRANSMISSION 23

24 24 2 y (x,t) = 1 2 y (x,t) x 2 v 2 t 2 What happens when a wave encounters a medium where it propagates with a different velocity?

25 25 Material/field velocity v mat = y/ t dimensions [y. time -1 ] If the material/field velocity is perpendicular to the phase velocity, the wave is transverse. Examples? If the material/field velocity is parallel to the phase velocity, the wave is longitudinal. Examples? Combinations of the above are possible. Examples? Show wave machine

26 Another velocity group velocity There is another way to make something that has the dimensions of a velocity: 26 Group velocity, v group = w/ k, dimensions [length. time -1 ] This describes the propagation of a feature in a wave packet or superposition of waves of different frequencies. We will come back to this concept later.

27 Standing waves - functions of x (only) multiplied by functions of t (only) 27 y(x,t) = Acos kx ( )sin wt ( ) Standing waves are superpositions of traveling waves Also find mixtures of standing waves and traveling waves

28 y(x,t) = Acos( wt - kx +f) y(x,t) = B p cos( wt - kx) + B q sin( wt - kx) y(x,t) = Ce i ( wt-kx) -i( wt-kx) + C *e y(x,t) = Re De i wt-kx+f ( ) [ ] Same conversions between A, B, C, D forms as for oscillations - see Main Ch1, Ch9. 28 Other waveforms (e.g.) sawtooth, pulses etc., can be written as superpositions of harmonic waves of different wavelengths and/or frequencies Fourier series and Fourier integrals (transforms)

29 2 y (x,t) x 2 = 1 v 2 2 y (x,t) v = w k t 2 v v 2 = w 1 = w k 1 k 2 29 Medium changes (i.e v changes) at x = 0 v is one constant for x < 0 and another constant for x > 0. Need piecewise function for y Traveling wave solutions, with wave incident from the left Frequency must be same on both sides (why?), therefore k changes (and l) y (x,t) = Re éae i ( -wt+k 1x) inc ë y (x,t) = Re ébe i ( -wt-k 1x) ref ë y Left (x,t) = y inc (x,t) + y ref (x,t) û û ( ) y trans (x,t) = Re éce i -wt+k 2x ë y Right (x,t) =y trans (x,t) û

30 y left x,t é ë y Left (0,t) = y Right (0,t) A + B = C ( ) ( ) = Re Ae i -wt+k 1x û + Re é Bei -wt-k 1x ë y right ( ) û ( x,t) = Re Ce i -wt+k 2x é ë Rope must be continuous 30 ( ) û y Left x x= 0,t = y Right x x= 0,t Transverse force on vanishingly small rope element (massless) must be zero iak 1 - ibk 1 = ick 2

31 y Left (0,t) = y Right (0,t) ( ) Re é ë Ae i -wt û -wt + Re ébei ë Rope must be continuous ( ) û ( ) -wt = Re écei ë Acos( wt) + Bcos( wt) = C cos( wt) û 31 A + B = C y Left x x= 0,t = y Right x x= 0,t Transverse force on massless rope element is zero ( ) Re é ë ik 1 Ae i -wt ( ) û + Re é ë -ik 1Be i -wt ( ) û = Re é ë ik 2Ce i -wt û k 1 Asin( wt) - k 1 Bsin( wt) = k 2 C sin( wt) k 1 A - k 1 B = k 2 C

32 Force exerted by left side of rope on right side is -T y Left x x=x 0 32 q 1 q 2 T y T x = 0 Force exerted by right side of rope on left side is +T y Right x x=x 0 x iak 1 - ibk 1 = ick 2 Transverse force on vanishingly small rope element (massless) must be zero

33 A + B = C Ak 1 - Bk 1 = Ck 2 Solve simultaneously 33 R y º B A = k 1 - k 2 k 1 + k 2 T y º C A = 2k 1 k 1 + k 2 Displacement reflection and transmission coeffs! y Left (x,t) = e i ( -wt + k 1x) k k 2 e i (-wt -k 1x) k 1 + k 2 y Right (x,t) = 2k 1 e i (-wt +k 2 x) k 1 + k 2

34 Boundary 34 v 1 > v 2 v 2 < v 1 Incident ( )+ reflected ( ) Transmitted ( )

35 Impedance, Z 35 F appl º Z y t Defines Z as the ratio of the applied force to the resulting (material) velocity For rope system: piston applies force at x = 0 producing wave in direction of +ve x ( = tension) -t y x = Z y t

36 Impedance, Z is defined as the ratio of the applied force to the resulting (material) velocity 36 -t y x º Z y t y ( x - vt ) t Þ = -v x -t = -Zv Z = t v = tk w y ( x - vt ) Impedance proportional to k if is constant. Traveling wave eqn Z rope = t v = tm

37 37 ( ) y Left (x,t) = Re e i -wt+k 1 é ë x é y Right (x,t) = Re ê ë û + Re é Z 1 - Z 2 e i -wt-k 1x ê ë Z 1 + Z 2 2Z 1 e i (-wt+k 2x) Z 1 + Z 2 ú û ( ) ú û

38 Pressure/force and displacement 38 t = 0 t = later t = later still Position, x Exercise: Draw y(x,0); p(x,0)

39 y (x,t) F(x,t) = -t x Also obeys the 1-D wave equation it has reflection and transmission coefficients, too! 39 F Left (x,t) = -tik 1 Ae i ( -wt +k 1x) + tik1 Be i ( -wt -k 1x) F Right (x,t) = -tik 2 Ce i ( -wt + k 2 x) R F º ik 1B -ik 1 A = k 2 - k 1 k 1 + k 2 T F º -ik 2C -ik 1 A = 2k 2 k 1 + k 2 Force reflection and transmission coeffs!

40 40 ( ) y Left (x,t) = Re e i -wt+k 1 é ë x é y Right (x,t) = Re ê ë F Left (x,t) = Re e i -wt+k 1 é ë x û + Re é Z 1 - Z 2 e i -wt-k 1x ê ë Z 1 + Z 2 2Z 1 e i (-wt+k 2x) Z 1 + Z 2 ( ) ú û ( ) û + Re é Z 2 - Z 1 e i -wt-k 1x ê ë Z 1 + Z 2 ( ) ú û ú û F Right (x,t) = Re é ê ë 2Z 2 e i (-wt+k 2x) Z 1 + Z 2 ú û

41 Electric circuits: force like voltage; y/ t like current 41 This is where the impedance idea is more familiar. A driving voltage produces a current in a circuit; the proportionality constant is the impedance. The various circuit elements produce currents that are in phase with, ahead of, or behind the driving voltage. There are analogies between the electric and mechanical systems. See Main Ch 10 for a comprehensive listing.

42 42 REFLECTION AND TRANSMISSION - REVIEW Continuity conditions (relationship of positions, forces etc at boundary) Reflection and transmission coefficients for y, Reflection and transmission coefficients for dy dx, Free & fixed boundaries, Phase change at boundary impedance (mechanical and electrical) Mathematical representations of the above