## Standing Waves and Resonance

14.6 Standing Waves and Resonance

### Learning Objectives

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

- Describe standing waves and explain how they are produced
- Describe the modes of a standing wave on a string
- Provide examples of standing waves beyond the waves on a string

**Practice!**

Practice 14.6.1 |
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A string of length L is rigidly held at both ends. The string is plucked – like you might strum a guitar string – and a standing wave is formed in the string. What is true about the motion of the particles located at a node on this standing wave? |

(a) The particles at a node oscillate back and forth with maximum amplitude perpendicular to the length of the string. |

(b) The particles at a node do not move at all. |

(c) The particles at a node move back and forth with maximum amplitude parallel to the length of the string. |

(d) The particles at a node move with a constant speed in the direction of the wave. |

For a standing wave, **the particles at a node do not move at all**. In a standing wave, the wave pattern stays in the same position on the string and the amplitude of the wave fluctuates. A node is defined as a point that does not move.

For a standing wave on a string, the antinodes are defined as points that move perpendicular to the length of the string with maximum amplitude. The particles at an antinode move in simple harmonic motion about the equilibrium position of the string.

Practice 14.6.2 |
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A string of length L is rigidly held at both ends. The string is plucked – like you might strum a guitar string – and a standing wave is formed in the string. What is true about the motion of the particles located at an antinode on this standing wave? |

(a) The particles at an antinode do not move at all. |

(b) The particles at an antinode move back and forth with maximum amplitude parallel to the length of the string. |

(c) The particles at an antinode oscillate back and forth with maximum amplitude perpendicular to the length of the string. |

(d) The particles at an antinode move with a constant speed in the direction of the wave. |

For a standing wave, **the particles at an antinode oscillate back and forth with maximum amplitude perpendicular to the length of the string**. In a standing wave, the wave pattern stays in the same position on the string and the amplitude of the wave fluctuates. For a standing wave on a string, the antinodes are defined as points that move perpendicular to the length of the string with maximum amplitude. The particles at an antinode move in simple harmonic motion about the equilibrium position of the string.

A node is defined as a point that does not move.

Practice 14.6.3 |
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A string of length L is rigidly held at both ends. The string is plucked – like you might strum a guitar string – and a standing wave is formed in the string. The figure shows a time exposure photograph of this standing wave. What is the wavelength? |

(a) L |

(b) 2L |

(c) L/2 |

(d) 3L/2 |

(e) 2L/3 |

The wavelength of the standing wave shown is equal to **2 L**. This standing wave pattern exhibits two nodes and one antinode. The distance between two successive nodes is equal to half a wavelength. So this string of length

*L*contains half a wavelength: L = /2 and therefore = 2

*L*.

Practice 14.6.4 |
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A string of length L is rigidly held at both ends. The string is plucked – like you might strum a guitar string – and a standing wave is formed in the string. The figure shows a time exposure photograph of this standing wave. What is the wavelength? |

(a) L |

(b) 2L |

(c) L/2 |

(d) 3L/2 |

(e) 2L/3 |

The wavelength of the standing wave shown is equal to ** L**. This standing wave pattern exhibits three nodes and two antinodes. The distance between two successive nodes is equal to half a wavelength. So this string of length

*L*contains two halves of a wavelength: L = 2/2 and therefore =

*L*.

Practice 14.6.5 |
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A string of length L is rigidly held at both ends. The string is plucked – like you might strum a guitar string – and a standing wave is formed in the string. The figure shows a time exposure photograph of this standing wave. What is the wavelength? |

(a) L |

(b) 2L |

(c) L/2 |

(d) 3L/2 |

(e) 2L/3 |

The wavelength of the standing wave shown is equal to **2 L/3**. This standing wave pattern exhibits four nodes and three antinodes. The distance between two successive nodes is equal to half a wavelength. So this string of length

*L*contains three halves of a wavelength: L = 3/2 and therefore = 2

*L*/3.

Practice 14.6.6 |
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A string of length L is rigidly held at both ends. The string is plucked – like you might strum a guitar string – and a standing wave is formed in the string. The figure shows a time exposure photograph of this standing wave. What is the distance between successive antinodes on this standing wave? |

(a) /2, where is the wavelength of the standing wave |

(b) , where is the wavelength of the standing wave |

(c) 3/2, where is the wavelength of the standing wave |

(d) 2, where is the wavelength of the standing wave |

(e) 2/3, where is the wavelength of the standing wave |

The distance between successive antinodes on this standing wave is ** /2, where is the wavelength of the standing wave**. The distance between two successive nodes is equal to half a wavelength and the distance between two successive antinodes is also equal to half a wavelength. This is true for any standing wave.

Practice 14.6.7 |
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A string of length L is rigidly held at both ends. The string is plucked – like you might strum a guitar string – and a standing wave is formed in the string. The fundamental frequency for this standing wave is where F is the tension in the string and µ is the mass density of the string. What is the frequency of the third harmonic, or second overtone?_{T} |

(a) f_{1} |

(b) 2f_{1} |

(c) 3f_{1} |

(d) f_{1}/2 |

(e) f_{1}/3 |

The frequency of the third harmonic, or second overtone, is **3 f_{1}**. The frequencies of standing waves, which are also called harmonics, are determined by multiplying the fundamental frequency,

*f*

_{1}, by the number of antinodes,

*n*:

*f*=

_{n}*nf*

_{1}. The third harmonic contains three antinodes, which gives

*n*= 3.

Practice 14.6.8 |
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In the apparatus shown in the figure, tension is created in the wire by the weight of the hanging block. A standing wave with frequency f is produced in the vibrating portion of the wire. What will happen to the frequency of the standing wave in the wire if a mass with half the weight is used? |

(a) The frequency of the standing wave would increase by a factor of . |

(b) The frequency of the standing wave would increase by a factor of 2. |

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

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

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

If a weight with half the mass was used, **the frequency of the standing wave would decrease by a factor of **. The frequency of the standing wave depends directly on the wave speed, *v*, according to *f* = *v*/2*L*, where *L* is the length of the wire. The wave speed is calculated with so the wave speed will change with changing tension in the wire. Using a lighter block will reduce the tension in the wire by a factor of 2. Therefore the wave speed will decrease by according to: .

Practice 14.6.9 |
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While a string is vibrating, you gently touch the midpoint of the string to ensure that the string does not vibrate at that point. The lowest-frequency standing wave that could be present on the string vibrates at |

(a) the fundamental frequency. |

(b) twice the fundamental frequency. |

(c) three times the fundamental frequency. |

(d) four times the fundamental frequency. |

(e) There is not enough information given to decide. |

**Discuss!**

Middle C on a piano has a fundamental frequency of 262 Hz, and the first A above middle C has a fundamental frequency of 440 Hz.

(a) Calculate the frequencies of the next two harmonics of the C string.

(b) If the A and C strings have the same linear mass density μ and length L, determine the ratio of tensions in the two strings.