## Standing Waves on a String

Here we are looking at the first through the third harmonicof a standing wave on a stringĀ

Here we are looking at the first through the third harmonicof a standing wave on a stringĀ

Standing Wave Description | |||
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Calculations | Decsription | Picture | |

Nodes | Destructive Interferenc | Here the incoming crest of the the wave is meeting with the trough of the reflection of the previous wave, thereby creating a Node. | |

Anti- Nodes | Constructive Interference | Here the incoming crest of the the wave is meeting with the crest of the reflection of the previous wave, thereby creating an Anti-Node | |

Note that in all Cases the Length of the String is Constant The speed of a wave in a given medium is also Constant and is given by: $V_{wave}=f\lambda$ In General Frequency is given by: $f=\frac{V_{wave}}{\lambda}$ |
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First Harmonic or Fundemental Frequency | Wavelength Calculations: $\lambda _{1}=2L$ |
Lowest possible frequency. This is usually the frequency that is played on stringed instruments. We are seeing half a wave, with an anti-node in the center and nods at either end | |

Frequency Calculations: $f_{1}=\frac{V_{wave}}{2L}$ |
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Second Harmonic | Wavelength Calculations:
$\lambda _{2}=L$ or $L=\lambda_{2} $ |
Here we have the second possible frequency and we are seeing a full wavelength. Notice that we now have two anti-nodes and we have added a node in the centre for a total of three nodes. | |

Frequency Calculations: $f_{2}=\frac{V_{wave}}{L}$ |
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Third Harmonic | Wavelength Calculations:
$\lambda_{3}=\frac{2L}{3}$ or $L=\frac{3}{2}\lambda_{3} $ |
This is the third possilble frequency and we are seeing a wavelength and a half. There are now three anti-nodes and we have nodes at either end plus two in the middle | |

Frequency Calculations: $f_{3}=\frac{3V_{wave}}{2L}$ |
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General Equations for Frequency | $f_{n}=\frac{nV_{wave}}{2L}$ or $f_{n}=nf_{1}$ |
n = 1, 2, 3, 4, etc this is the harmonic number. | |

General Equations for Wavelength | $\lambda_{n}=\frac{2L}{n}$ | n = 1, 2, 3, 4, etc this is the harmonic number. |