## Wave Speed on a Stretched String

14.3 Wave Speed on a Stretched String

### Learning Objectives

By the end of this section, you will be able to:

- Determine the factors that affect the speed of a wave on a string
- Write a mathematical expression for the speed of a wave on a string and generalize these concepts for other media

### Wave Speed on a String under Tension

**Practice!**

Practice 14.3.1 |
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What physical factor or factors affect the speed of a wave traveling in a string? |

(a) only the tension force in the string |

(b) both the tension force in the string and the mass density of the string |

(c) only the mass density of the string |

(d) only the amplitude of the string’s displacement from equilibrium |

(e) both the tension force in the string and the amplitude of the string’s displacement from equilibrium |

The physical factors that affect the speed of a wave traveling in a string are **both the tension force in the string and the mass density of the string**. The speed of a wave in a string depends on the magnitude of the tension, *F _{T}*, in the string and the mass density, µ, of the string. The speed of the wave can be calculated with .

Practice 14.3.2 |
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The six strings of a guitar are the same length and under nearly the same tension, but they have different thicknesses. On which string do waves travel the fastest? |

(a) The thickest string |

(b) The thinnest string |

(c) The wave speed is the same on all strings. |

Practice 14.3.3 |
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If you double the wavelength λ of a wave on a string, what happens to the wave speed v and the wave frequency f? |

(a) v is doubled and f is doubled. |

(b) v is doubled and f is unchanged. |

(c) v is unchanged and f is halved. |

(d) v is unchanged and f is doubled. |

(e) v is halved and f is unchanged. |

Practice 14.3.4 |
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Consider a light string like the one shown in the figure. If you pluck on the vertical part of the string, you can produce a wave in the string that will propagate along the string with speed v. What would happen to the speed of the wave if you replaced the hanging block with one that is twice as heavy? |

(a) The speed of the wave would increase by a factor of 2. |

(b) The speed of the wave would increase by a factor of . |

(c) The speed of the wave would decrease by a factor of . |

(d) The speed of the wave would decrease by a factor of 2. |

(e) The speed of the wave would not change. |

If you replaced the hanging block with one that is twice as heavy, **the speed of the wave would increase by a factor of **. The speed of a wave in a string depends on the magnitude of the tension, *F _{T}*, in the string and the mass density, µ, of the string. The speed of the wave can be calculated with . If you hang a block from the string that is twice as heavy, the tension in the string will double in magnitude. The calculation for the new wave speed using the heavier block is , which is faster than the original speed by a factor of .

Practice 14.3.5 |
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Consider a light string like the one shown in the figure. If you pluck on the vertical part of the string, you can produce a wave in the string that will propagate along the string with speed v. What would happen to the speed of the wave if you replaced the string with one that has two times the mass density? |

(a) The speed of the wave would increase by a factor of 2. |

(b) The speed of the wave would increase by a factor of . |

(c) The speed of the wave would decrease by a factor of . |

(d) The speed of the wave would decrease by a factor of 2. |

(e) The speed of the wave would not change. |

If you replaced the string with one that has two times the mass density, **the speed of the wave would decrease by a factor of **. The speed of a wave in a string depends on the magnitude of the tension, *F _{T}*, in the string and the mass density, µ, of the string. The speed of the wave can be calculated with . If you use a string that has twice the mass density of the original string, then the mass density becomes 2µ. The calculation for the new wave speed using the heavier string is , which is slower than the original speed by a factor of .

**Discuss!**

A heavy uniform rope hangs from the ceiling, and a small amplitude transverse wave is started by jiggling the rope at the bottom. As the wave travels up the rope, will the wave speed increase, decrease, or remain constant?

Three waves are traveling along identical strings. Wave B has twice the amplitude of the other two. Wave C has 1/2 the wavelength than A or B.

(a) Which wave goes fastest?

(b) Which wave, A or C, has the higher frequency?