Waves Topic 11 1 Standing Waves Standing Waves
- Slides: 56
Waves Topic 11. 1 Standing Waves
Standing Waves v. The Formation
• 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
• 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
• 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.
At the correct frequency a standing wave is formed The frequency is increased until a different standing wave is formed
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
• 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
Resonance and Standing Waves 1. Standing Waves on Strings
• 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
• With this wave the length of the string is equal to half the wave length • L=½ • = 2 L • As v = f • Then f = v / • f = v / 2 L • This is the fundamental frequency of the string (the 1 st harmonic)
• This is not the only standing wave that can exist on the string • The next standing wave is L= This is called the 2 nd Harmonic
• 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 2 nd Harmonic frequency of the string • Notice it is twice the fundamental frequency
• The next standing wave is L = 3/2 This is called the 3 rd Harmonic
• With this wave the length of the string is equal to 3/2 of the wave length • L =3/2 • = 2/3 L • As v = f • Then f = v / • f = v / 2/3 L • f = 3 v / 2 L • This is the 3 rd Harmonic frequency of the string • Notice it is three times the fundamental frequency
• Notice that the only constraint is that the ends of the string are nodes. • In general we find that the wavelengths satisfy = 2 L n Where n = 1, 2, 3, 4……
• 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)
Resonance and Standing Waves 2. Standing Waves in Pipes
• 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
a) Open Ended • Fundamental Frequency (1 st Harmonic) L = /2 • = 2 L • As v = f • Then f = v / • f = v / 2 L
• 2 nd Harmonic L= • = L • As v = f • Then f = v / • f = v / L
• 3 rd Harmonic L = 3/2 • = 2/3 L • As v = f • Then f = v / • f = v / 2/3 L • f = 3 v / 2 L
• The harmonics are in the same series as the string series • If the fundamental frequency = f • Then the 2 nd harmonic is 2 f, 3 rd is 3 f and the 4 th is 4 f… etc
b) Closed at one End • Fundamental Frequency (1 st Harmonic) L = /4 • = 4 L • As v = f • Then f = v / • f = v / 4 L
• Next Harmonic L = 3/4 • = 4/3 L • As v = f • Then f = v / • f = v / 4/3 L • f = 3 v / 4 L
• And the next harmonic L = 5/4 • = 4/5 L • As v = f • Then f = v / • f = v / 4/5 L • f = 5 v / 4 L
• The harmonics are DIFFERENT to the string and open pipe series • If the fundamental frequency = f • Then there is no 2 nd harmonic • The 3 rd is 3 f • There is no 4 th harmonic • The 5 th is 5 f
Waves Topic 4. 3 Standing Waves
Standing Waves v. The Formation
• 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
• 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
• 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.
At the correct frequency a standing wave is formed The frequency is increased until a different standing wave is formed
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
• 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
Resonance and Standing Waves 1. Standing Waves on Strings
• 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
• With this wave the length of the string is equal to half the wave length • L=½ • = 2 L • As v = f • Then f = v / • f = v / 2 L • This is the fundamental frequency of the string (the 1 st harmonic)
• This is not the only standing wave that can exist on the string • The next standing wave is L= This is called the 2 nd Harmonic
• 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 2 nd Harmonic frequency of the string • Notice it is twice the fundamental frequency
• The next standing wave is L = 3/2 This is called the 3 rd Harmonic
• With this wave the length of the string is equal to 3/2 of the wave length • L =3/2 • = 2/3 L • As v = f • Then f = v / • f = v / 2/3 L • f = 3 v / 2 L • This is the 3 rd Harmonic frequency of the string • Notice it is three times the fundamental frequency
• Notice that the only constraint is that the ends of the string are nodes. • In general we find that the wavelengths satisfy = 2 L n Where n = 1, 2, 3, 4……
• 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)
Resonance and Standing Waves 2. Standing Waves in Pipes
• 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
a) Open Ended • Fundamental Frequency (1 st Harmonic) L = /2 • = 2 L • As v = f • Then f = v / • f = v / 2 L
• 2 nd Harmonic L= • = L • As v = f • Then f = v / • f = v / L
• 3 rd Harmonic L = 3/2 • = 2/3 L • As v = f • Then f = v / • f = v / 2/3 L • f = 3 v / 2 L
• The harmonics are in the same series as the string series • If the fundamental frequency = f • Then the 2 nd harmonic is 2 f, 3 rd is 3 f and the 4 th is 4 f… etc
b) Closed at one End • Fundamental Frequency (1 st Harmonic) L = /4 • = 4 L • As v = f • Then f = v / • f = v / 4 L
• Next Harmonic L = 3/4 • = 4/3 L • As v = f • Then f = v / • f = v / 4/3 L • f = 3 v / 4 L
• And the next harmonic L = 5/4 • = 4/5 L • As v = f • Then f = v / • f = v / 4/5 L • f = 5 v / 4 L
• The harmonics are DIFFERENT to the string and open pipe series • If the fundamental frequency = f • Then there is no 2 nd harmonic • The 3 rd is 3 f • There is no 4 th harmonic • The 5 th is 5 f
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