1 / 56

Waves

Learn about the formation of standing waves when two waves of the same speed, wavelength, and amplitude travel in opposite directions. Explore resonance and the harmonic series in standing waves on strings and in pipes. Discover how standing waves are formed, their characteristics, and how they can be both useful and harmful.

bootha
Télécharger la présentation

Waves

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Waves Topic 11.1 Standing Waves

  2. Standing Waves • The Formation

  3. When 2 waves of • the same speed • and wavelength • and equal or almost equal amplitudes • travelling in opposite directions • meet, a standing wave is formed

  4. The standing wave is the result of the superposition of the two waves travelling in opposite directions • The main difference between a standing and a travelling wave is that in the case of the standing wave no energy or momentum is transferred down the string

  5. A standing wave is characterised by the fact that there exists a number of points at which the disturbance is always zero • These points are called Nodes • And the amplitudes along the waveform is different • In a travelling wave there are no points where the disturbance is always zero, all the points have the same amplitude.

  6. At the correct frequency a standing wave is formed The frequency is increased until a different standing wave is formed

  7. Resonance • If the frequency of the source of a vibration is exactly equal to the natural frequency of the oscillatory system then the system will resonate • When this occurs the amplitude will get larger and larger • Pushing a swing is an example of resonance • Resonance can be useful and harmful

  8. Airplane wings, engines, bridges, tall buildings are objects that need to be protected against resonance from external vibrations due to wind and other vibrating objects • Soldiers break step when marching over a bridge in case the force which they exert on the bridge starts uncontrollable oscillations of the bridge

  9. Resonance and Standing Waves • Standing Waves on Strings

  10. If you take a wire and stretch it between two points then you can set up a standing wave • The travelling waves are reflected to and fro between the two ends of the wire and interfere to produce the standing wave • This has a node at both ends • and an antinode in the middle • – it is called the fundamental

  11. With this wave the length of the string is equal to half the wave length • L = ½  •  = 2L • As v = f  • Then f = v /  •  f = v / 2L • This is the fundamental frequency of the string (the 1st harmonic)

  12. L =  • This is not the only standing wave that can exist on the string • The next standing wave is This is called the 2ndHarmonic

  13. With this wave the length of the string is equal to the wave length • L =  •  = L • As v = f  • Then f = v /  •  f = v / L • This is the 2nd Harmonic frequency of the string • Notice it is twice the fundamental frequency

  14. L = 3/2  • The next standing wave is This is called the 3rd Harmonic

  15. With this wave the length of the string is equal to 3/2 of the wave length • L =3/2  •  = 2/3L • As v = f  • Then f = v /  •  f = v / 2/3L •  f = 3v / 2L • This is the 3rdHarmonic frequency of the string • Notice it is three times the fundamental frequency

  16. Notice that the only constraint is that the ends of the string are nodes. • In general we find that the wavelengths satisfy  = 2L n Where n = 1,2,3,4……

  17. This is the harmonic series • The fundamental is the dominant vibration and will be the one that the ear will hear above all the others • The harmonics effect the quality of the note • It is for this reason that different musical instruments sounding a note of the same frequency sound different • (it is not the only way though)

  18. Resonance and Standing Waves • Standing Waves in Pipes

  19. Sound standing waves are also formed in pipes • Exactly the same results apply • There are two types of pipes • 1. Open ended • 2. Closed at one end • Nodes exist at closed ends • Antinodes exists at open ends

  20. Open Ended L =  /2 • Fundamental Frequency (1st Harmonic) •  = 2L • As v = f  • Then f = v /  •  f = v / 2L

  21. L =  • 2nd Harmonic •  = L • As v = f  • Then f = v /  •  f = v / L

  22. L = 3/2 • 3rd Harmonic •  = 2/3L • As v = f  • Then f = v /  •  f = v / 2/3L •  f = 3v / 2L

  23. The harmonics are in the same series as the string series • If the fundamental frequency = f • Then the 2nd harmonic is 2f, 3rd is 3f and the 4th is 4f… etc

  24. b) Closed at one End L =  /4 • Fundamental Frequency (1st Harmonic) •  = 4L • As v = f  • Then f = v /  •  f = v / 4L

  25. L = 3/4 • Next Harmonic •  = 4/3L • As v = f  • Then f = v /  •  f = v / 4/3L •  f = 3v / 4L

  26. L = 5/4 • And the next harmonic •  = 4/5L • As v = f  • Then f = v /  •  f = v / 4/5L •  f = 5v / 4L

  27. The harmonics are DIFFERENT to the string and open pipe series • If the fundamental frequency = f • Then there is no 2nd harmonic • The 3rd is 3f • There is no 4th harmonic • The 5th is 5f

  28. Waves Topic 4.3 Standing Waves

  29. Standing Waves • The Formation

  30. When 2 waves of • the same speed • and wavelength • and equal or almost equal amplitudes • travelling in opposite directions • meet, a standing wave is formed

  31. The standing wave is the result of the superposition of the two waves travelling in opposite directions • The main difference between a standing and a travelling wave is that in the case of the standing wave no energy or momentum is transferred down the string

  32. A standing wave is characterised by the fact that there exists a number of points at which the disturbance is always zero • These points are called Nodes • And the amplitudes along the waveform is different • In a travelling wave there are no points where the disturbance is always zero, all the points have the same amplitude.

  33. At the correct frequency a standing wave is formed The frequency is increased until a different standing wave is formed

  34. Resonance • If the frequency of the source of a vibration is exactly equal to the natural frequency of the oscillatory system then the system will resonate • When this occurs the amplitude will get larger and larger • Pushing a swing is an example of resonance • Resonance can be useful and harmful

  35. Airplane wings, engines, bridges, tall buildings are objects that need to be protected against resonance from external vibrations due to wind and other vibrating objects • Soldiers break step when marching over a bridge in case the force which they exert on the bridge starts uncontrollable oscillations of the bridge

  36. Resonance and Standing Waves • Standing Waves on Strings

  37. If you take a wire and stretch it between two points then you can set up a standing wave • The travelling waves are reflected to and fro between the two ends of the wire and interfere to produce the standing wave • This has a node at both ends • and an antinode in the middle • – it is called the fundamental

  38. With this wave the length of the string is equal to half the wave length • L = ½  •  = 2L • As v = f  • Then f = v /  •  f = v / 2L • This is the fundamental frequency of the string (the 1st harmonic)

  39. L =  • This is not the only standing wave that can exist on the string • The next standing wave is This is called the 2ndHarmonic

  40. With this wave the length of the string is equal to the wave length • L =  •  = L • As v = f  • Then f = v /  •  f = v / L • This is the 2nd Harmonic frequency of the string • Notice it is twice the fundamental frequency

  41. L = 3/2  • The next standing wave is This is called the 3rd Harmonic

  42. With this wave the length of the string is equal to 3/2 of the wave length • L =3/2  •  = 2/3L • As v = f  • Then f = v /  •  f = v / 2/3L •  f = 3v / 2L • This is the 3rdHarmonic frequency of the string • Notice it is three times the fundamental frequency

  43. Notice that the only constraint is that the ends of the string are nodes. • In general we find that the wavelengths satisfy  = 2L n Where n = 1,2,3,4……

  44. This is the harmonic series • The fundamental is the dominant vibration and will be the one that the ear will hear above all the others • The harmonics effect the quality of the note • It is for this reason that different musical instruments sounding a note of the same frequency sound different • (it is not the only way though)

  45. Resonance and Standing Waves • Standing Waves in Pipes

  46. Sound standing waves are also formed in pipes • Exactly the same results apply • There are two types of pipes • 1. Open ended • 2. Closed at one end • Nodes exist at closed ends • Antinodes exists at open ends

  47. Open Ended L =  /2 • Fundamental Frequency (1st Harmonic) •  = 2L • As v = f  • Then f = v /  •  f = v / 2L

  48. L =  • 2nd Harmonic •  = L • As v = f  • Then f = v /  •  f = v / L

More Related