Understanding Standing Waves and Wave Superposition in Physics

1. Lecture 30 To do : • Chapter 21 • Examine two wave superposition (-wt and +wt) • Examine two wave superposition (-w1t and -w2t) Review for final (Location: CHEM 1351, 7:45 am) Tomorrow: Review session, 2103 CH at 12:05 PM • Last Assignment • HW13, Due Friday, May 7th , 11:59 PM

2. Standing waves • Waves traveling in opposite direction “interfere” with each other. If the conditions are right, same k (2p/l) & w, the superposition generates a standing wave: DRight(x,t)= a sin( kx-wt ) DLeft(x,t)= a sin( kx+wt ) Energy flow in a standing wave is stationary, it “stands” in place. Standing waves have nodes and antinodes Anti-nodes D(x,t)= DL(x,t) + DR(x,t) D(x,t)= 2 a sin(kx) cos(wt) The outer curve is the amplitude function A(x) = ±2 a sin(kx) when wt = pn n = 0,1,2,… k = wave number = 2π/λ Nodes

3. Standing waves on a string • Longest wavelength allowed: ½ of the full wave Fundamental: l/2 = L  l = 2 L Recall v = fl Overtones m > 1

4. Violin, viola, cello, string bass Guitars Ukuleles Mandolins Banjos Vibrating Strings- Superposition Principle D(x,0) Antinode D(0,t)

5. Standing waves in a pipe Open end: Mustbe a displacement antinode (pressure minimum) Closed end: Must be a displacement node (pressure maximum) Blue curves are displacement oscillations. Red curves, pressure. Fundamental: l/2l/2 l/4

6. Standing waves in a pipe

7. Combining Waves Fourier Synthesis

8. DESTRUCTIVEINTERFERENCE CONSTRUCTIVEINTERFERENCE Superposition & Interference • Consider two harmonic waves A and B meet at t=0. • They have same amplitudes and phase, but 2 = 1.15 x 1. • The displacement versus time for each is shown below: A(1t) B(2t) C(t) =A(t)+B(t)

9. A(1t) B(2t) t Tbeat C(t)=A(t)+B(t) Superposition & Interference • Consider A + B,[Recall cos u + cos v = 2 cos((u-v)/2) cos((u+v)/2)] yA(x,t)=A cos(k1x–2p f1t)yB(x,t)=A cos(k2x–2p f2t) Let x=0, y=yA+yB = 2A cos[2p (f1 – f2) t/2] cos[2p (f1 + f2) t/2] and |f1 – f2| ≡ fbeat = = 1 / Tbeat f average≡ (f1 + f2)/2

10. Exercise Superposition • The traces below show beats that occur when two different pairs of waves are added (the time axes are the same). • For which of the two is the difference in frequency of the original waves greater? Pair 1 Pair 2 The frequency difference was the same for both pairs of waves. Need more information.

11. A(1t) B(2t) t Tbeat C(t)=A(t)+B(t) Superposition & Interference • Consider A + B,[Recall cos u + cos v = 2 cos((u-v)/2) cos((u+v)/2)] yA(x,t)=A cos(k1x–2p f1t)yB(x,t)=A cos(k2x–2p f2t) Let x=0, y = yA + yB = 2A cos[2p (f1 – f2)t/2] cos[2p (f1 + f2)t/2] and |f1 – f2| ≡ fbeat = = 1 / Tbeat f average≡ (f1 + f2)/2

12. Review • Final is “semi” cumulative • Early material, more qualitative (i.e., conceptual) • Later material, more quantitative (but will employ major results from early on). • 25-30% will be multiple choice • Remainder will be short answer with the focus on thermodynamics, heat engines, wave motion and wave superposition

13. Exercise Superposition • The traces below show beats that occur when two different pairs of waves are added (the time axes are the same). • For which of the two is the difference in frequency of the original waves greater? Pair 1 Pair 2 The frequency difference was the same for both pairs of waves. Need more information.

14. Organ Pipe Example A 0.9 m organ pipe (open at both ends) is measured to have it’s first harmonic (i.e., its fundamental) at a frequency of 382 Hz. What is the speed of sound (refers to energy transfer) in this pipe? L=0.9 m f = 382 Hzandf l = vwith l = 2 L / m(m = 1) v = 382 x 2(0.9) m  v = 687 m/s

15. Standing Wave Question • What happens to the fundamental frequency of a pipe, if the air (v =300 m/s) is replaced by helium (v = 900 m/s)? Recall: f l = v (A) Increases (B) Same (C) Decreases

16. Chapter 1

17. Important Concepts

18. Chapter 2

19. Chapter 3

20. Chapter 4

21. Chapter 4

22. Chapter 5

23. Chapter 5 & 6

24. Chapter 6 Chapter 7

25. Chapter 7

26. Chapter 7 (Newton’s 3rd Law) & Chapter 8

27. Chapter 9 Chapter 8

28. Chapter 9

29. Chapter 10

30. Chapter 10

31. Chapter 10

32. Chapter 11

33. Chapter 11

34. Chapter 12 and Center of Mass

35. Chapter 12

36. Important Concepts

37. Chapter 12

38. Angular Momentum

39. Hooke’s Law Springs and a Restoring Force • Key fact: w = (k / m)½ is general result where k reflects a constant of the linear restoring force and m is the inertial response (e.g., the “physical pendulum” where w = (k / I)½

40. Simple Harmonic Motion Maximum potential energy Maximum kinetic energy

41. Resonance and damping • Energy transfer is optimal when the driving force varies at the resonant frequency. • Types of motion • Undamped • Underdamped • Critically damped • Overdamped

42. Fluid Flow

43. Density and pressure

44. Response to forces States of Matter and Phase Diagrams

45. Ideal gas equation of state

46. pV diagrams Thermodynamics

47. T can change! Work, Pressure, Volume, Heat

48. Chapter 18

49. Thermal Energy

50. Relationships