PhysicsLAB Resource Lesson
Resonance in Pipes

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Every object, substance, has a natural frequency at which it is "willing" to vibrate. When an external agent applies a forced vibration that matches this natural frequency, the object begins to vibrate with ever increasing amplitude, or resonate.
  • For a swing, that natural frequency depends on its length, T = 2π√(L/g). If the swing is pushed at a frequency which either matches the swing's natural frequency or is a sub-multiple of that natural frequency, then the swing's amplitude builds, and we say that it is in resonance.
  • When air is blown across the top of a soda bottle, standing waves are set up in the air column inside the bottle and the bottle "sings." 
  • When two identical tuning forks are placed side by side, the vibrations of one fork can force the second fork to begin to vibrate or resonate.
 
 
Open-Open Pipes
 
Pipes can either be open on both ends or on only one end. The open ends act as free-end reflectors (producing antinodes) and the closed ends act as fixed-end reflectors (producing nodes). Let's start our investigation with a pipe open at both ends, for example, a flute.
 
Notice in the animation that both ends always remained open or "free" to move, that is they are antinodes. A summary of the first three harmonics for an open-open pipe are shown below.

Refer to the following information for the next seven questions.

Open-Open Pipe Waveforms
fundamental frequency
1st harmonic
fo
1st overtone
2nd harmonic
f1 = 2fo
2nd overtone
3rd harmonic
f2 = 3fo

Note that the frequency subscript matches the order of the overtone, NOT the order of the harmonic.

 Given a pipe open on both ends is 1.0 meter long. What is the wavelength of the lowest frequency which causes it to resonate?

 If the air temperature is 20ºC, what is the speed of sound in the pipe?

 What is the fundamental frequency of this pipe?

 In this same 1-meter pipe, what would be the frequency of the 4th harmonic if the temperature remains constant?

 If the temperature remains constant but the pipe is cut in half, what would be the new frequency of the fourth harmonic?

 Could the 0.5-meter pipe resonate at a frequency of 1543.5 hz?

 Could the original 1-meter pipe resonate at a frequency of 1543.5 hz?

 
Open-Closed Pipes
 
Now let's examine a thumb piano, an instrument which is open at one end and closed at the other.
 
Notice in the animation that while the open end always remained "free" to move, the clamped end always remains "fixed." The free-end is an antinode while the fixed-end is a node. A summary of the first three harmonics for an open-closed vibrating system are shown below.
 
Observe that there are no "even harmonics" among the resonance states of this type of vibrating system. This stems from the fact that the fundamental frequency is a half-loop or ¼λ. Since every overtone represents the addition of a complete loop, which contains two half-loops, we can never add just one more half-loop. Thus, we cannot generate even harmonics.

Refer to the following information for the next four questions.

Open-Closed Waveforms
fundamental frequency
1st harmonic
fo
1st overtone
3rd harmonic
f1 = 3fo
2nd overtone
5th harmonic
f2 = 5fo

Note that the frequency subscript matches the order of the overtone, NOT the order of the harmonic.

 Given a pipe open on one end and closed on the other is 1.0 meter long. What is the wavelength of the lowest frequency which causes it to resonate?

 What is the lowest frequency to resonate in the pipe if the air temperature is 20ºC?

 What would be the next lowest frequency to resonate in the pipe if the temperature remains constant?

 What are the frequency and wavelength of the 2nd overtone, or 5th harmonic?




 
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