1. Chapter 3 Transverse standing waves Resonance and the Overtone Series Mersenne’s laws Longitudinal standing waves Other standing waves and applications http://hyperphysics.phy-astr.gsu.edu/Hbase/waves/string.html http://www.kw.igs.net/~jackord/bp/n2.html www.physics.umd.edu/lecdem/misc/phys102/PH102chap03.ppt

2. Identical waves moving in opposite directions

3. Motion of the spring

4. Motion of point along the spring

5. Stretched strings End effects

6. Standing wave representation

7. Impossible “standing waves”

8. Standing Waves in a Stretched String

9. Stretched String Frequencies and Wavelengths f = v / λ N = 1 λ = 2L f = v/2L = f 1 N = 2 λ = L f = v/L = 2f 1 N = 3 λ = 2L/3 f = 3v/2L = 3f 1 N = 4 λ = L/2 f = 2v/L = 4f 1 N = 5 λ = 2L/5 f = 5v/2L = 5f 1 N = 6 λ = L/3 f = 3v/L = 6f 1

10. Notes of the Overtone Series

11. Mersenne’s Laws f = fundamental frequency L = length of string F = tension in string W = mass per unit length of string

12. Mersenne’s First Law f α 1/L F and W constant If L => 2 L then f => ?? If L => L / 2 then f => ??

13. Mersenne’s First Law f α 1/L F and W constant If L => 2 L then f => f / 2 If L => L / 2 then f => 2 f

14. Mersenne’s Second Law f α √F L and W constant If F => 9 F then f => ?? If F => F / 4 then f => ??

15. Mersenne’s Second Law f α √F L and W constant If F => 9 F then f => 3 f If F => F / 4 then f => f / 2

16. Mersenne’s Third Law f α 1/ √W F and L constant If W => 4 W then f => ?? If W => W / 9 then f => ??

17. Mersenne’s Third Law f α 1/ √W F and L constant If W => 4 W then f => f / 2 If W => W / 9 then f => 3 f

18. Rope Wave Example 110 grams => 3 loops (frequency f) What happens with 990 grams?

19. Rope Wave Example 990 grams => 1 loop (frequency f and wavelength λ) What mass will produce Wavelength λ/2?

20. Rope Wave Example 990 grams => 1 loop (frequency f and wavelength λ) Mass of 250 grams will produce wavelength λ/2.

21. Nanoguitar Silicon strings: length 6 – 12 microns diameter 150 – 200 nanometers http://www.news.cornell.edu/releases/nov03/nemsguitar.ws.html

22. Nanoguitar

23. Longitudinal Standing Waves

24. Kundt’s tube

25. Air Columns End effects

26. “End Effect” Experiment

27. End at right “open” End at right “closed”

28. End configurations for tubes 1. Phase change at open and closed ends

29. End configurations for tubes 2. Node or antinode at open and closed ends

30. Open Tubes

31. “Open” Tube Frequencies and Wavelengths f = v / λ N = 1 λ = 2L o f = v/2L o = f o N = 2 λ = L o f = v/L o = 2f o N = 3 λ = 2L o /3 f = 3v/2L o = 3f o N = 4 λ = 2L o /4 f = 4v/2L o = 4f o N = 5 λ = 2L o /5 f = 5v/2L o = 5f o N = 6 λ = 2L o /6 f = 6v/2L o = 6f o

32. Closed Tubes

33. “Closed” Tube Frequencies and Wavelengths f = v / λ N = 1 λ = 4L c f = v/4L c = f c N = 2 does not exist. N = 3 λ = 4L c /3 f = 3v/4L c = 3f c N = 4 does not exist. N = 5 λ = 4L c /5 f = 5v/4L c = 5f c N = 6 does not exist.

34. Open and Closed Tube Comparison

35. Tube Quiz

36. Fundamental Frequency Open and Closed Tubes If L o = L c what is f o : f c If L o = 2L c what is f o : f c

37. Closed Tube Resonances

38. Applications to Musical Instruments

39. The Flute

40. The Recorder

41. Actual Recorder Finger Hole Positions

42. The Clarinet

44. The Saxophone

45. The Trumpet

46. Various Standing Waves Flame Tube Aluminum Rod Velocity of Sound in Aluminum Chladni Plates (photos)

47. Chladni Plates

48. Violin Body Vibrations

49. Standing Waves in a Membrane Rectangular Membrane Applet Circular Membrane Applet Rectangular Membrane AppletCircular Membrane Applet

50. Beaker Breaker using sound wave resonance Teacup Standing Waves

51. Tacoma Narrows Bridge Collapse

52. The End Any Questions?