1. Standing (or Stationary) Waves
© D Hoult 2007

20. Distance node to node =

21. Distance node to node = half a wavelength
Distance anti-node to anti-node = half a wavelength

23. When a wire under tension is disturbed at a point, waves travel away from the point where the disturbance occurred.

24. When a wire under tension is disturbed at a point, waves travel away from the point where the disturbance occurred.
The waves are then reflected at the fixed ends of the wire. Therefore, we have two waves of the same frequency travelling in opposite sense along the wire.

25. When a wire under tension is disturbed at a point, waves travel away from the point where the disturbance occurred.
The waves are then reflected at the fixed ends of the wire. Therefore, we have two waves of the same frequency travelling in opposite sense along the wire.
Interference occurs between these waves.

26. When a wire under tension is disturbed at a point, waves travel away from the point where the disturbance occurred.
The waves are then reflected at the fixed ends of the wire. Therefore, we have two waves of the same frequency travelling in opposite sense along the wire.
Interference occurs between these waves.
Points where destructive interference occurs are called nodes (or nodal points).
Constructive interference gives anti-nodes.

27. For certain frequencies of disturbance, the nodes and anti-nodes do not move along the wire.
This pattern of stationary nodes and anti-nodes is called a stationary (or standing) wave.

28. For certain frequencies of disturbance, the nodes and anti-nodes do not move along the wire.
This pattern of stationary nodes and anti-nodes is called a stationary (or standing) wave.
In a stationary wave, adjacent points in the medium oscillate in phase but with different amplitudes of oscillation.

29. Any body which is free to move has a natural frequency of oscillation.

30. Any body which is free to move has a natural frequency of oscillation.
If an external force causes a body to oscillate, this force is called the driving (or forcing) oscillation.

31. Any body which is free to move has a natural frequency of oscillation.
If an external force causes a body to oscillate, this force is called the driving (or forcing) oscillation.
In principle a body can be forced to oscillate at any frequency.

32. Any body which is free to move has a natural frequency of oscillation.
If an external force causes a body to oscillate, this force is called the driving (or forcing) oscillation.
In principle a body can be forced to oscillate at any frequency.
However, an oscillating body will give its largest amplitude oscillations when the frequency of the driving oscillation is equal to the natural frequency of the body.

33. Any body which is free to move has a natural frequency of oscillation.
If an external force causes a body to oscillate, this force is called the driving (or forcing) oscillation.
In principle a body can be forced to oscillate at any frequency.
However, an oscillating body will give its largest amplitude oscillations when the frequency of the driving oscillation is equal to the natural frequency of the body.
When this largest amplitude response occurs we say the body is resonating.

34. Resonance of a String under Tension

35. Resonance of a String under Tension
When a wave passes through a point on a string, that point oscillates.

36. Resonance of a String under Tension
When a wave passes through a point on a string, that point oscillates.
If the wave later returns to the same point after a reflection, interference will occur between the existing oscillation and the reflected wave.

37. Resonance of a String under Tension
When a wave passes through a point on a string, that point oscillates.
If the wave later returns to the same point after a reflection, interference will occur between the existing oscillation and the reflected wave.
The interference will be constructive if the “effective distance” moved by the wave is one wavelength.

38. Resonance of a String under Tension
When a wave passes through a point on a string, that point oscillates.
If the wave later returns to the same point after a reflection, interference will occur between the existing oscillation and the reflected wave.
The interference will be constructive if the “effective distance” moved by the wave is one wavelength.
A phase change of 180° makes a wave “look as if” it has travelled an extra half wavelength. more detail

39. Resonance of a String under Tension
When a wave passes through a point on a string, that point oscillates.
If the wave later returns to the same point after a reflection, interference will occur between the existing oscillation and the reflected wave.
The interference will be constructive if the “effective distance” moved by the wave is one wavelength.
A phase change of 180° makes a wave “look as if” it has travelled an extra half wavelength.
So an effective distance of could be due to the wave actually moving half a wavelength and experiencing phase change of p rads.

40. Fundamental Frequency of Resonance (First Harmonic)

41. Fundamental Frequency of Resonance (First Harmonic)
Waves travel towards B (and C) and are reflected with 180° (or p rad) phase change

42. Fundamental Frequency of Resonance (First Harmonic)
Waves travel towards B (and C) and are reflected with 180° (or p rad) phase change
If the “effective distance” travelled by the waves when they return to A is l then resonance occurs

43. Fundamental Frequency of Resonance (First Harmonic)
Waves travel towards B (and C) and are reflected with 180° (or p rad) phase change
If the “effective distance” travelled by the waves when they return to A is l then resonance occurs
A → B → A (or A → C → A ) must be equal to l/2.

44. Fundamental Frequency of Resonance (First Harmonic)
this means that L =

45. Fundamental Frequency of Resonance (First Harmonic)
this means that L = l/2

46. Fundamental Frequency of Resonance (First Harmonic)
this means that L = l/2
speed of waves is given by v =

47. Fundamental Frequency of Resonance (First Harmonic)
this means that L = l/2
speed of waves is given by v = f l

48. Fundamental Frequency of Resonance (First Harmonic)
this means that L = l/2
speed of waves is given by v = f l
therefore the fundamental frequency, fo of vibration of the string is given by

49. Fundamental Frequency of Resonance (First Harmonic)
this means that L = l/2
speed of waves is given by v = f l
therefore the fundamental frequency, fo of vibration of the string is given by v
fo = 2L

50. Fundamental Frequency of Resonance (First Harmonic)
and the distance between adjacent nodes in a stationary wave is half a wavelength

51. Second Harmonic

52. Second Harmonic

53. Second Harmonic

54. Second Harmonic

55. Second Harmonic N N N

56. Second Harmonic N N N
N → N =

57. Second Harmonic N N N
N → N = l/2

58. Second Harmonic N N N
N → N = l/2
L =

59. Second Harmonic N N N
N → N = l/2
L = l

60. Second Harmonic N N N
N → N = l/2
L = l v
f =
therefore L

61. Second Harmonic N N N
N → N = l/2
L = l v
f =
therefore L

62. Second Harmonic N N N
N → N = l/2
L = l v
f = =
therefore L

63. Second Harmonic N N N
N → N = l/2
L = l v
f =
= 2 fo
therefore L

64. Third Harmonic N N N N
L = 3l/2

65. Third Harmonic N N N N
L = 3l/2
therefore f = 3 fo

66. A wire/string under tension will resonate at

67. A wire/string under tension will resonate atv
fo = 2L

68. v fo = 2L A wire/string under tension will resonate at
and integral multiples of this fundamental frequency

69. v fo = 2L f = n fo A wire/string under tension will resonate at
and integral multiples of this fundamental frequency
f = n fo

70. The speed of waves on a string under tension depends on

71. The speed of waves on a string under tension depends on
the tension, Te

72. The speed of waves on a string under tension depends on
the tension, Te
the mass per unit length of the string, µ

73. The speed of waves on a string under tension depends on
the tension, Te
the mass per unit length of the string, µ
We might therefore suggest that

74. Te v = µ The speed of waves on a string under tension depends on
the tension, Te
the mass per unit length of the string, µ
We might therefore suggest that Te
v =

75. Te v = µ The speed of waves on a string under tension depends on
the tension, Te
the mass per unit length of the string, µ
We might therefore suggest that Te
v =
However, consideration of the units of the quantities leads us to suggest

76. Te v = µ The speed of waves on a string under tension depends on
the tension, Te
the mass per unit length of the string, µ
We might therefore suggest that Te
v =
However, consideration of the units of the quantities leads us to suggest

77. This relation has been confirmed by experiment.