waves topic 11.1 standing waves. v the formation
TRANSCRIPT
![Page 1: Waves Topic 11.1 Standing Waves. v The Formation](https://reader035.vdocuments.us/reader035/viewer/2022062322/56649f4e5503460f94c70084/html5/thumbnails/1.jpg)
WavesTopic 11.1
Standing Waves
![Page 2: Waves Topic 11.1 Standing Waves. v The Formation](https://reader035.vdocuments.us/reader035/viewer/2022062322/56649f4e5503460f94c70084/html5/thumbnails/2.jpg)
Standing Waves
The Formation
![Page 3: Waves Topic 11.1 Standing Waves. v The Formation](https://reader035.vdocuments.us/reader035/viewer/2022062322/56649f4e5503460f94c70084/html5/thumbnails/3.jpg)
• 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
![Page 4: Waves Topic 11.1 Standing Waves. v The Formation](https://reader035.vdocuments.us/reader035/viewer/2022062322/56649f4e5503460f94c70084/html5/thumbnails/4.jpg)
• 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
![Page 5: Waves Topic 11.1 Standing Waves. v The Formation](https://reader035.vdocuments.us/reader035/viewer/2022062322/56649f4e5503460f94c70084/html5/thumbnails/5.jpg)
• 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.
![Page 6: Waves Topic 11.1 Standing Waves. v The Formation](https://reader035.vdocuments.us/reader035/viewer/2022062322/56649f4e5503460f94c70084/html5/thumbnails/6.jpg)
At the correct frequencya standing wave is formed
The frequency is increaseduntil a different standing wave is formed
![Page 7: Waves Topic 11.1 Standing Waves. v The Formation](https://reader035.vdocuments.us/reader035/viewer/2022062322/56649f4e5503460f94c70084/html5/thumbnails/7.jpg)
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
![Page 8: Waves Topic 11.1 Standing Waves. v The Formation](https://reader035.vdocuments.us/reader035/viewer/2022062322/56649f4e5503460f94c70084/html5/thumbnails/8.jpg)
• 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
![Page 9: Waves Topic 11.1 Standing Waves. v The Formation](https://reader035.vdocuments.us/reader035/viewer/2022062322/56649f4e5503460f94c70084/html5/thumbnails/9.jpg)
Resonance and Standing Waves
1. Standing Waves on Strings
![Page 10: Waves Topic 11.1 Standing Waves. v The Formation](https://reader035.vdocuments.us/reader035/viewer/2022062322/56649f4e5503460f94c70084/html5/thumbnails/10.jpg)
![Page 11: Waves Topic 11.1 Standing Waves. v The Formation](https://reader035.vdocuments.us/reader035/viewer/2022062322/56649f4e5503460f94c70084/html5/thumbnails/11.jpg)
• 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
![Page 12: Waves Topic 11.1 Standing Waves. v The Formation](https://reader035.vdocuments.us/reader035/viewer/2022062322/56649f4e5503460f94c70084/html5/thumbnails/12.jpg)
• 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)
![Page 13: Waves Topic 11.1 Standing Waves. v The Formation](https://reader035.vdocuments.us/reader035/viewer/2022062322/56649f4e5503460f94c70084/html5/thumbnails/13.jpg)
• This is not the only standing wave that can exist on the string
• The next standing wave is
This is called the 2nd Harmonic
L =
![Page 14: Waves Topic 11.1 Standing Waves. v The Formation](https://reader035.vdocuments.us/reader035/viewer/2022062322/56649f4e5503460f94c70084/html5/thumbnails/14.jpg)
• 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
![Page 15: Waves Topic 11.1 Standing Waves. v The Formation](https://reader035.vdocuments.us/reader035/viewer/2022062322/56649f4e5503460f94c70084/html5/thumbnails/15.jpg)
• The next standing wave is
L = 3/2
This is called the 3rd Harmonic
![Page 16: Waves Topic 11.1 Standing Waves. v The Formation](https://reader035.vdocuments.us/reader035/viewer/2022062322/56649f4e5503460f94c70084/html5/thumbnails/16.jpg)
• 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 3rd Harmonic frequency of the string• Notice it is three times the fundamental frequency
![Page 17: Waves Topic 11.1 Standing Waves. v The Formation](https://reader035.vdocuments.us/reader035/viewer/2022062322/56649f4e5503460f94c70084/html5/thumbnails/17.jpg)
• 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……
![Page 18: Waves Topic 11.1 Standing Waves. v The Formation](https://reader035.vdocuments.us/reader035/viewer/2022062322/56649f4e5503460f94c70084/html5/thumbnails/18.jpg)
• 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)
![Page 19: Waves Topic 11.1 Standing Waves. v The Formation](https://reader035.vdocuments.us/reader035/viewer/2022062322/56649f4e5503460f94c70084/html5/thumbnails/19.jpg)
Resonance and Standing Waves
2. Standing Waves in Pipes
![Page 20: Waves Topic 11.1 Standing Waves. v The Formation](https://reader035.vdocuments.us/reader035/viewer/2022062322/56649f4e5503460f94c70084/html5/thumbnails/20.jpg)
• 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
![Page 21: Waves Topic 11.1 Standing Waves. v The Formation](https://reader035.vdocuments.us/reader035/viewer/2022062322/56649f4e5503460f94c70084/html5/thumbnails/21.jpg)
a) Open Ended
• Fundamental Frequency (1st Harmonic)
L = /2
= 2L
•As v = f •Then f = v / f = v / 2L
![Page 22: Waves Topic 11.1 Standing Waves. v The Formation](https://reader035.vdocuments.us/reader035/viewer/2022062322/56649f4e5503460f94c70084/html5/thumbnails/22.jpg)
• 2nd Harmonic
= L
•As v = f •Then f = v / f = v / L
L =
![Page 23: Waves Topic 11.1 Standing Waves. v The Formation](https://reader035.vdocuments.us/reader035/viewer/2022062322/56649f4e5503460f94c70084/html5/thumbnails/23.jpg)
• 3rd Harmonic
= 2/3L
•As v = f •Then f = v /
f = v / 2/3L f = 3v / 2L
L = 3/2
![Page 24: Waves Topic 11.1 Standing Waves. v The Formation](https://reader035.vdocuments.us/reader035/viewer/2022062322/56649f4e5503460f94c70084/html5/thumbnails/24.jpg)
• 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
![Page 25: Waves Topic 11.1 Standing Waves. v The Formation](https://reader035.vdocuments.us/reader035/viewer/2022062322/56649f4e5503460f94c70084/html5/thumbnails/25.jpg)
b) Closed at one End
• Fundamental Frequency (1st Harmonic)
L = /4
= 4L
•As v = f •Then f = v / f = v / 4L
![Page 26: Waves Topic 11.1 Standing Waves. v The Formation](https://reader035.vdocuments.us/reader035/viewer/2022062322/56649f4e5503460f94c70084/html5/thumbnails/26.jpg)
• Next Harmonic
= 4/3L
•As v = f •Then f = v /
f = v / 4/3L f = 3v / 4L
L = 3/4
![Page 27: Waves Topic 11.1 Standing Waves. v The Formation](https://reader035.vdocuments.us/reader035/viewer/2022062322/56649f4e5503460f94c70084/html5/thumbnails/27.jpg)
• And the next harmonic
= 4/5L
•As v = f •Then f = v /
f = v / 4/5L f = 5v / 4L
L = 5/4
![Page 28: Waves Topic 11.1 Standing Waves. v The Formation](https://reader035.vdocuments.us/reader035/viewer/2022062322/56649f4e5503460f94c70084/html5/thumbnails/28.jpg)
• 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
![Page 29: Waves Topic 11.1 Standing Waves. v The Formation](https://reader035.vdocuments.us/reader035/viewer/2022062322/56649f4e5503460f94c70084/html5/thumbnails/29.jpg)
WavesTopic 4.3
Standing Waves
![Page 30: Waves Topic 11.1 Standing Waves. v The Formation](https://reader035.vdocuments.us/reader035/viewer/2022062322/56649f4e5503460f94c70084/html5/thumbnails/30.jpg)
Standing Waves
The Formation
![Page 31: Waves Topic 11.1 Standing Waves. v The Formation](https://reader035.vdocuments.us/reader035/viewer/2022062322/56649f4e5503460f94c70084/html5/thumbnails/31.jpg)
• 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
![Page 32: Waves Topic 11.1 Standing Waves. v The Formation](https://reader035.vdocuments.us/reader035/viewer/2022062322/56649f4e5503460f94c70084/html5/thumbnails/32.jpg)
• 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
![Page 33: Waves Topic 11.1 Standing Waves. v The Formation](https://reader035.vdocuments.us/reader035/viewer/2022062322/56649f4e5503460f94c70084/html5/thumbnails/33.jpg)
• 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.
![Page 34: Waves Topic 11.1 Standing Waves. v The Formation](https://reader035.vdocuments.us/reader035/viewer/2022062322/56649f4e5503460f94c70084/html5/thumbnails/34.jpg)
At the correct frequencya standing wave is formed
The frequency is increaseduntil a different standing wave is formed
![Page 35: Waves Topic 11.1 Standing Waves. v The Formation](https://reader035.vdocuments.us/reader035/viewer/2022062322/56649f4e5503460f94c70084/html5/thumbnails/35.jpg)
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
![Page 36: Waves Topic 11.1 Standing Waves. v The Formation](https://reader035.vdocuments.us/reader035/viewer/2022062322/56649f4e5503460f94c70084/html5/thumbnails/36.jpg)
• 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
![Page 37: Waves Topic 11.1 Standing Waves. v The Formation](https://reader035.vdocuments.us/reader035/viewer/2022062322/56649f4e5503460f94c70084/html5/thumbnails/37.jpg)
Resonance and Standing Waves
1. Standing Waves on Strings
![Page 38: Waves Topic 11.1 Standing Waves. v The Formation](https://reader035.vdocuments.us/reader035/viewer/2022062322/56649f4e5503460f94c70084/html5/thumbnails/38.jpg)
![Page 39: Waves Topic 11.1 Standing Waves. v The Formation](https://reader035.vdocuments.us/reader035/viewer/2022062322/56649f4e5503460f94c70084/html5/thumbnails/39.jpg)
• 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
![Page 40: Waves Topic 11.1 Standing Waves. v The Formation](https://reader035.vdocuments.us/reader035/viewer/2022062322/56649f4e5503460f94c70084/html5/thumbnails/40.jpg)
• 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)
![Page 41: Waves Topic 11.1 Standing Waves. v The Formation](https://reader035.vdocuments.us/reader035/viewer/2022062322/56649f4e5503460f94c70084/html5/thumbnails/41.jpg)
• This is not the only standing wave that can exist on the string
• The next standing wave is
This is called the 2nd Harmonic
L =
![Page 42: Waves Topic 11.1 Standing Waves. v The Formation](https://reader035.vdocuments.us/reader035/viewer/2022062322/56649f4e5503460f94c70084/html5/thumbnails/42.jpg)
• 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
![Page 43: Waves Topic 11.1 Standing Waves. v The Formation](https://reader035.vdocuments.us/reader035/viewer/2022062322/56649f4e5503460f94c70084/html5/thumbnails/43.jpg)
• The next standing wave is
L = 3/2
This is called the 3rd Harmonic
![Page 44: Waves Topic 11.1 Standing Waves. v The Formation](https://reader035.vdocuments.us/reader035/viewer/2022062322/56649f4e5503460f94c70084/html5/thumbnails/44.jpg)
• 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 3rd Harmonic frequency of the string• Notice it is three times the fundamental frequency
![Page 45: Waves Topic 11.1 Standing Waves. v The Formation](https://reader035.vdocuments.us/reader035/viewer/2022062322/56649f4e5503460f94c70084/html5/thumbnails/45.jpg)
• 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……
![Page 46: Waves Topic 11.1 Standing Waves. v The Formation](https://reader035.vdocuments.us/reader035/viewer/2022062322/56649f4e5503460f94c70084/html5/thumbnails/46.jpg)
• 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)
![Page 47: Waves Topic 11.1 Standing Waves. v The Formation](https://reader035.vdocuments.us/reader035/viewer/2022062322/56649f4e5503460f94c70084/html5/thumbnails/47.jpg)
Resonance and Standing Waves
2. Standing Waves in Pipes
![Page 48: Waves Topic 11.1 Standing Waves. v The Formation](https://reader035.vdocuments.us/reader035/viewer/2022062322/56649f4e5503460f94c70084/html5/thumbnails/48.jpg)
• 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
![Page 49: Waves Topic 11.1 Standing Waves. v The Formation](https://reader035.vdocuments.us/reader035/viewer/2022062322/56649f4e5503460f94c70084/html5/thumbnails/49.jpg)
a) Open Ended
• Fundamental Frequency (1st Harmonic)
L = /2
= 2L
•As v = f •Then f = v / f = v / 2L
![Page 50: Waves Topic 11.1 Standing Waves. v The Formation](https://reader035.vdocuments.us/reader035/viewer/2022062322/56649f4e5503460f94c70084/html5/thumbnails/50.jpg)
• 2nd Harmonic
= L
•As v = f •Then f = v / f = v / L
L =
![Page 51: Waves Topic 11.1 Standing Waves. v The Formation](https://reader035.vdocuments.us/reader035/viewer/2022062322/56649f4e5503460f94c70084/html5/thumbnails/51.jpg)
• 3rd Harmonic
= 2/3L
•As v = f •Then f = v /
f = v / 2/3L f = 3v / 2L
L = 3/2
![Page 52: Waves Topic 11.1 Standing Waves. v The Formation](https://reader035.vdocuments.us/reader035/viewer/2022062322/56649f4e5503460f94c70084/html5/thumbnails/52.jpg)
• 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
![Page 53: Waves Topic 11.1 Standing Waves. v The Formation](https://reader035.vdocuments.us/reader035/viewer/2022062322/56649f4e5503460f94c70084/html5/thumbnails/53.jpg)
b) Closed at one End
• Fundamental Frequency (1st Harmonic)
L = /4
= 4L
•As v = f •Then f = v / f = v / 4L
![Page 54: Waves Topic 11.1 Standing Waves. v The Formation](https://reader035.vdocuments.us/reader035/viewer/2022062322/56649f4e5503460f94c70084/html5/thumbnails/54.jpg)
• Next Harmonic
= 4/3L
•As v = f •Then f = v /
f = v / 4/3L f = 3v / 4L
L = 3/4
![Page 55: Waves Topic 11.1 Standing Waves. v The Formation](https://reader035.vdocuments.us/reader035/viewer/2022062322/56649f4e5503460f94c70084/html5/thumbnails/55.jpg)
• And the next harmonic
= 4/5L
•As v = f •Then f = v /
f = v / 4/5L f = 5v / 4L
L = 5/4
![Page 56: Waves Topic 11.1 Standing Waves. v The Formation](https://reader035.vdocuments.us/reader035/viewer/2022062322/56649f4e5503460f94c70084/html5/thumbnails/56.jpg)
• 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