Musical instruments

# 31 Review and Homework: Standing waves and resonance

### Important Terms

• forced vibration
• natural frequency
• frequency response
• resonance
• standing wave
• mode
• node
• antinode
• open string
• string length
• displacement node
• pressure node
• end correction
• Helmholtz resonator

### Review Questions

1. What is resonance? When does it occur?
2. What is the relationship between standing waves and resonance?
3. Give an example of amplification of a sound that is an example of resonance. Explain why your choice is an example of resonance.
4. Give an example of amplification of a sounds that is not an example of resonance. Explain why your choice is not an example of resonance.
5. A bare tuning fork is not very loud when it is struck. You can make the sound louder by touching the base of the tuning fork to a table (or a wall). Describe what’s happening. Is this an example of resonance? Describe the evidence that supports your answer. Identify any object(s) that are resonating.
6. In a Mythbusters video, a singer uses his voice to shatter a wine glass. Before trying to shatter the glass, the singer taps the glass. Discuss the physics behind this important first step. Use at least two technical terms from the reading in your explanation.
7. The sound of an unknown instrument is recorded. The FFT of a short sample (about 0.1 second) of the instrument’s sound shows peaks at 500 Hz,1600 Hz, 4200 Hz. The FFT does not extend beyond 5000 Hz- so there may be other overtones (above 5000 Hz) in the sound, but that information is not known. Could this sound have been produced by a string instrument? Is it possible to tell from the information provided? Discuss the relevant issues. Don’t forget to support any claims you make with appropriate evidence and logic.
8. Different strings on the guitar produce different pitches. While they are all the same length, the strings are made of differing thicknesses and (in some cases) different materials.
• Do all the (open) strings have the same fundamental wavelength? Explain.
• Do waves travel at the same speed on all the strings? Explain,
• Do all the (open) strings have the same fundamental frequency? Explain based on your answers to the previous two questions- do not use your experience as a listener/string player to justify your answer.
9. A violin player plays the A-string and hears a pitch of 426 Hz. The violinist uses the peg in the tuning box to tighten the string slightly and hears a new pitch of 440 Hz (which makes orchestra’s conductor happier).
• What happens to the fundamental wavelength of the open string? Explain.
• What happens to the speed of waves on the string? Explain.
10. A guitar player plays the (upper) E-string and hears a pitch of 660 Hz. The guitarist presses the E-string firmly against the fingerboard at a location about 1 cm from the end of the string.
• What happens to the fundamental wavelength? Explain.
• What happens to the tension of the string? Explain.
• What happens to the speed of waves on the string? Explain.
11. Pan pipes (also called Pan flutes) are an ancient instrument that many cultures produced. They consist of an array of pipes of different lengths strapped together. The pipes are stopped at one end and sound is produced by blowing across the open end. What effect would each of the following changes have on the pitches produced by a set of Pan pipes? Explain each of your answers.
• make each pipe shorter
• change the material of the pipes from steel to copper
• drill a hole in the end of the tube
• cool the air in the room

### Numericals

1. Tim and Mary are playing with a spring on the floor. The two ends of the spring are 2.4 meters apart. When Mary moves her hand back and forth at 3 times per second, a “one football” pattern is established on the spring.
• What is the wavelength of the “one football” pattern?
• What is the speed (in meters/sec) of waves on this slinky?
• How many times per second should Mary move her hand up and down to establish a pattern with 2 “footballs”?
2. Each string on the cello is the same length (70.0 cm). When the cello’s G-string is played (and the cellist is not touching the string), it produces a fundamental frequency of 100 Hz. If the cellist pushes the string down firmly against the fingerboard, she can effectively shorten the length of the string by any amount she wants. Suppose she shortens the string by 12 cm. What is the new fundamental frequency (in Hz)?
3. The fundamental frequency of the open cello C-string is 100 Hz. What other frequencies would you expect to see in an FFT graph of the sound of the cello’s open C-string?
4. One of the pipes in a set of Pan pipes is 12.4 cm long. Find the fundamental frequency (in Hz) produced by this pipe. Pan pipes are an example of an open-closed air column. Assume speed of sound in air is 345 m/s. (Not sure what Pan pipes are? Check out the wikipedia article).
5. Suppose you twirled a singing pipe over your head, starting slowly at first and gradually twirl it faster and faster.  (A singing pipe is a flexible hose open at both ends). You start slowly, twirling it faster and faster and notice that it produces only specific pitches. The lowest frequency you produce is 600 Hz, and the pipe also produces pitches at 800 Hz, 1000 Hz, 1200 Hz and 1400 Hz before your arm gives out.
• What is the fundamental frequency of this tube? (Hint: The lowest pitch this tube can produce is not the fundamental).
• How long is this tube?
6. pipes (also called Pan flutes) are an ancient instrument that many cultures produced. They consist of an array of pipes of different lengths strapped together. The pipes are stopped at one end and sound is produced by blowing across the open end. A sound of one of the pipes from a Pan pipe is recorded and found to have a fundamental frequency of 330 Hz. What other frequencies would you expect to see on an FFT of this sound?
7. A BoomWhacker toy is a hollow plastic tube that makes pitched sound when you strike it. (It is an example of an open-open resonating air column). Recording of one BoomWhacker is made and analyzed. It is found that the fundamental frequency is 440 Hz. What other frequencies are likely to be on the FFT? Suppose you placed a cap on one end of the BoomWhacker. What would the new fundamental be? What would other frequencies would the BoomWhacker produce now?
8. Pan pipes (also called Pan flutes) are an ancient instrument that many cultures produced. They consist of an array of pipes of different lengths strapped together. The pipes are stopped at one end and sound is produced by blowing across the open end. One of the tubes is 105 mm (or 10.5 cm) long. What wavelengths of sound can this instrument produce?
9. Find the wavelength of each standing wave described below. Briefly show your work- this could be as little as one appropriate, well labeled picture.
• A string 60 cm long that has three loops in it
• A string 40 cm long vibrating at its fundamental frequency
• A standing wave in an air column 90 cm long with three nodes and three antinodes
• A standing wave in an air column 90 cm long with three nodes and two antinodes
• The entire bottom surface of a tube is filled with light cork particles. Different frequencies are played near one end of the tube. Nothing happens until a 500 Hz tone is played into the tube when the sound gets very loud and the cork collects into piles that 16 cm apart
10. A BoomWhacker toy is a hollow plastic tube that makes pitched sound when you strike it. Each tube comes with a removable cap, called an Octavator. One of the BoomWhackers is 46.5 cm long without the cap and 48.0 cm long with the cap on.
• What fundamental frequency will this BW produce without the cap?
• List the frequencies of at least two overtones that this pipe produces without the cap.
• What fundamental frequency will this BW produce if you put the cap on?
• List the frequencies of at least two overtones that this pipe produces with the cap on.