424: Oscillations & Waves

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Delilah Sutton
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1 424: Oscillations & Waves 1

2 PH424: PS 30%; Lab Manuscript 30%; Final 35%; Quizzes/Wrks 5% ~ Jan & Feb 2019 ~ Mon Tue Wed Thu Fri Free motion of an -PS 1a due Lab & oscillator -driven SHM and Discussion: the -Free damped oscillations circuits LCR circuit 7 Simple Harmonic Motion - 4 representations 14 Forced motion & resonances 15 Admittance, Impedance, phase shifts. 16 -PS 2a due Lab Data Workshop. 17 [Math Methods] 11 -PS 1b due Forced motion of a damped oscillator LCR circuit resonance 18 -PS 2b due [Math Methods] 21 Martin Luther King Day 28 Pre-Lab -Wave phenomena, demo lab -Reflection & Transmission 4 [Math Methods] PDEs, wave eqn w/ arb. initial conditions Lab data workshop /presentation 22 [Math Methods] 29 Workshop & discussion: Coax Cable Lab Workshop 5 Wave Propagation Lab data: Peer Review Session Interactive Coax workshop & discovery lab Multiple Driving Frequencies & Superposition 23 [Math Methods] Formal LRC Manuscript Due 30 -PS 4a due -Reflection & Transmission 6 -PS5a due Waves on string simulation (PhET ) Dispersive waves Coax Cable formal abstract due 24 Intro to Wave Mechanics ABCD forms Demo lab. 25 Wave Equation Fourier Solutions -PS3 due PS4b due Wave propagation Labs Data Peer & attenuation Review, Abstract Writing Workshop 7 Wave Energy: kinetic vs. potential ennergy density 8 -PS5b due [one PDE advance problem] Paradigms 424 Review 2

3 Formal Technical Writing One formal lab manuscript is required and one scientific conference abstract with figure. Good technical writing is very similar to writing an essay with sub-headings. We want to hear a convincing scientific story, not a shopping list of everything you did. Check out course web-site (writing tab) We will have data & write-up workshops 3

4 Week 1-2: Oscillations modulations wrt time, f(t) Week 3-5: Waves 1D modulations wrt space, y(x) AND BOTH y(x,t) Superposition of 4 brain neuron activity

5 Are oscillations ubiquitous? Or are they merely a paradigm? White Board Brainstorm: what physical systems have oscillations and/or waves? Superposition of 5 brain neuron activity

6 Are oscillations ubiquitous? Or are they merely a paradigm? PRL 116, (2016) 6

7 2018 Morou & Strickland Chirped Pulse Amplification 10 fs: the atomo laser pulse

8 REPRESENTING SIMPLE HARMONIC MOTION 8

9 amplitude x(t) = Acos( w 0 t + f) phase angle period angular freq T = 2p w 0 = 1 f (cyclic) freq A 1 x 0 position t = - f w 0 -A -1 time determined by initial conditions determined by physical system 9

10 1 Position (cm) x(t) = Acos w 0 t + f ( ) x t 1 2 Velocity (cm/s) ( ) x(t) = Aw 0 cos w 0 t + f + p 2 Acceleration (cm/s 2 ) ( ) x(t) = Aw 0 2 cos w 0 t + f + p v a t t 1 2 time (s) 10

11 These representations of the position of a simple harmonic oscillator as a function of time are all equivalent - there are 2 arbitrary constants in each. Note that A, f, B p and B q are REAL; C and D are COMPLEX. x(t) is real-valued variable in all cases. A: x(t) = Acos( w 0 t +f) B: C: D: x(t) = B p cosw 0 t + B q sinw 0 t x(t) = C exp iw 0 t ( ) + C *exp -iw 0 t ( ) x(t) = Re Dexp iw 0 t ( ) [ ] Engrave these on your soul - and know how to derive the relationships among A & f; B p & B q ; C; and D. 11

12 k m Example: initial conditions x(t) = Acos( w 0 t +f) k m x(t) = B p cosw 0 t + B q sinw 0 t x m = 0.01 kg; k = 36 Nm -1. At t = 0, m is displaced 50mm to the right and is moving to the right at 1.7 ms -1. Express the motion in form A form B 12

13 x(t) = Acos( w 0 t +f) x(t) = 57.5cos( éë 60s -1 ù û t ) mm x(t) = B p cosw 0 t + B q sinw 0 t x(t) = 50mmcos éë 60s -1 ù û t ( ) mmsin ( 60s -1 ù û t) éë B p = Acosf B q = -Asinf A = B p 2 + B q 2 tanf = - B q B p 13

14 k Using complex numbers: initial conditions. Same example as before, but now use the "C" and "D" forms m x(t) = C exp iw 0 t ( ) + C *exp -iw 0 t ( ) k m x(t) = Re Dexp iw 0 t ( ) [ ] x m = 0.01 kg; k = 36 Nm -1. At t = 0, m is displaced 50mm to the right and is moving to the right at 1.7 ms -1. Express the motion in form C form D 14

15 x( t) = x(t) = C exp iw 0 t ( ) + C *exp -iw 0 t ( ) ( 1 1 i60s t i60s t ( i) e + ( i) e )mm x( t) = x(t) = Re Dexp iw 0 t ( ) [ ] Re ( i ) e 1 i60s t mm Acosf = B p = 2Re C [ ] = Re D [ ] Asinf = -B q = 2Im C [ ] = Im D [ ] D = 2C = A tanf = Im [ D ] Re[ D] = Im [ C ] Re[ C] 15

16 Optional Clicker Questions 16

17 A particle executes simple harmonic motion. When the velocity of the particle is a maximum which one of the following gives the correct values of potential energy and acceleration of the particle. (a)potential energy is maximum and acceleration is maximum. (b)potential energy is maximum and acceleration is zero. (c)potential energy is minimum and acceleration is maximum. (d)potential energy is minimum and acceleration is zero. 17

18 A particle executes simple harmonic motion. When the velocity of the particle is a maximum which one of the following gives the correct values of potential energy and acceleration of the particle. (a)potential energy is maximum and acceleration is maximum. (b)potential energy is maximum and acceleration is zero. (c)potential energy is minimum and acceleration is maximum. (d)potential energy is minimum and acceleration is zero. Answer (d). When velocity is maximum displacement is zero so potential energy and acceleration are both zero. 18

19 A mass vibrates on the end of the spring. The mass is replaced with another mass and the frequency of oscillation doubles. The mass was changed by a factor of (a) 1/4 (b) ½ (c) 2 (d) 4 19

20 A mass vibrates on the end of the spring. The mass is replaced with another mass and the frequency of oscillation doubles. The mass was changed by a factor of (a) 1/4 (b) 1/2 (c) 2 (d) 4 Answer (a). Since the frequency has increased the mass must have decreased. Frequency is inversely proportional to the square root of mass, so to double frequency the mass must change by a factor of 1/4. 20

21 A mass vibrates on the end of the spring. The mass is replaced with another mass and the frequency of oscillation doubles. The maximum acceleration of the mass: (a) remains the same. (b) is halved. (c) is doubled. (d) is quadrupled. 21

22 A mass vibrates on the end of the spring. The mass is replaced with another mass and the frequency of oscillation doubles. The maximum acceleration of the mass: (a) remains the same. (b) is halved. (c) is doubled. (d) is quadrupled. Answer (d). Acceleration is proportional to frequency squared. If frequency is doubled than acceleration is quadrupled. 22

23 A particle oscillates on the end of a spring and its position as a function of time is shown below. At the moment when the mass is at the point P it has (a) positive velocity and positive acceleration (b) positive velocity and negative acceleration (c) negative velocity and negative acceleration (d) negative velocity and positive acceleration 23

24 A particle oscillates on the end of a spring and its position as a function of time is shown below. At the moment when the mass is at the point P it has (a) positive velocity and positive acceleration (b) positive velocity and negative acceleration (c) negative velocity and negative acceleration (d) negative velocity and positive acceleration Answer (b). The slope is positive so velocity is positive. Since the slope is getting smaller with time the acceleration is negative. 24

25 Optional Review of Complex Numbers 25

26 Complex numbers z = a + ib z = z e if i = -1 Re z Im z ( ) = a ( ) = b üï ýreal numbers þï Imag b z z = a 2 + b 2 f Real a Argand diagram tanf = b a 26

27 Euler s relation exp if ( ) = cosf + isinf exp iw 0 t ( ) = cos w 0 t ( ) + isin w 0 t ( ) 27

28 Consistency argument z = a + ib z = z e if If these represent the same thing, then the assumed Euler relationship says: a + ib = z cosf + i z sinf Equate real parts: Equate imaginary parts: a = z cosf b = z sinf z = a 2 + b 2 tanf = b a 28

29 x(t) = Re Ae if e iw 0t [ ] [ ] = Re Ae i ( w 0 t+f ) t = T 0 /4 Imag PHASOR p 2 +f f t = 0, T 0, 2T 0 Real w 0 t +f t = t 29

30 Adding complex numbers is easy in rectangular form z = a + ib w = c + id Imag z + w = [ a + c] + i[ b + d] c b a Real d 30

31 Multiplication and division of complex numbers is easy in polar form z = z e if w = w e iq zw = z w e i [ f +q ] Imag z q+f w q f Real 31

32 Another important idea is the COMPLEX CONJUGATE of a complex number. To form the c.c., change i -> -i z = a + ib z* = a - ib z + z* = [ a + a] + i[ b - b] = 2a Imag z = z e if z* = z e -if b z a f Real zz* = z e if z e -if = z 2 The product of a complex number and its complex conjugate is REAL. We say zz* equals mod z squared 32

33 And finally, rationalizing complex numbers, or: what to do when there's an i in the denominator? z = a + ib c + id z = a + ib c + id c - id c - id z = = ac + bd + i bc - ad ( ) c 2 + d 2 ac + bd c 2 + d 2 Re( z) + i bc - ad ( ) c 2 + d 2 Im( z) 33

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

PEER REVIEW. ... Your future in science will be largely controlled by anonymous letters from your peers. peers. Matt. Corinne PEER REVIEW 1... Your future in science will be largely controlled by anonymous letters from your peers. Matt peers Corinne 2 3 4 5 6 MULTIPLE DRIVNG FREQUENCIES LRC circuit L I = (1/Z)V ext Z must have