Buggé: Waves 4

Can you hear what I hear? 4.1 Observation Experiment There are many different types of materials that can produce sound in the room. Try them out and be careful to record your observations, especially about, the type of sound made (a specific description), how fast you are trying to produce the sound and anything interesting you see when the sound is being made. 4.2 Observe and explain Observe the series of sketches at the right. The pulses travel at speed v along the string of length L. Based on your analysis and using these two quantities, decide the time interval T1 (called the period for the first vibration) between upward pulls of the hand so that each upward pull supports or enhances the pulse that has just returned to and reflected from your hand after one trip down the string and back. The frequency of these upward pulls is f1 = 1/T1. The vibration that is produced is called the fundamental or first harmonic standing wave vibration, and its frequency is labeled f1.

 

Buggé: Waves 4

4.3 Observe and explain Observe the series of sketches at the right. The pulses travel at speed v along the string of length L. Note that as the hand is initiating an upward pulse, as in (a), an earlier pulse has reflected from the right and is starting its return to the hand. Based on your analysis and the quantities v and L, decide the time interval T2 between upward pulls of the hand so that each upward pull supports or enhances this second vibration. The frequency of this vibration is f2 = 1/T2. This is called the second harmonic standing wave vibration and its frequency is labeled f2. Note in the last sketch, N is called a node (a point where the string does not vibrate); A is called an antinode (a point where the string has its maximum amplitude vibration). 4.4 Observe and find a pattern Tie one end of a rope (or one end of a long, tightly wound spring) securely to a post. Hold the other end in your hand and vibrate it up and down at different frequencies. At most frequencies the rope responds little to your efforts. However, at special frequencies big amplitude vibrations occur (the figure shows four of these vibrations). If you videotape the process and view the video frame by frame, you find that the second vibration is twice the frequency of the first, the third is three times the frequency of the first, and so forth. (a) Write expressions for the frequency of the other standing wave vibrations—there are many more possible than shown in the figure. (b) Write expressions for the wavelengths of the observed and of other standing wave vibrations.

 

 

Buggé: Waves 4

A standing wave is the result of the superposition of waves reflecting back and forth between the ends of a string or pipe of length L. In a standing wave: (a) All points in the medium vibrate in a repeating pattern about the equilibrium position—the disturbance is not traveling. (b) The disturbance of different points in the medium have different amplitudes. Points that have a zero amplitude and do not vibrate at all are called nodes; points that vibrate with the largest amplitude are called antinodes. Frequencies of standing waves on a string of the length L:

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fn = n( v2L) where n = 1, 2, 3, …n

where v =

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Fstring tension

m /L is the speed of the disturbance on the string.

Frequencies of standing waves in an open pipe (open at both ends):

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fn = n( v2L) where n = 1, 2, 3, … n

where v is the speed of sound in the air in the pipe and L is its length. Frequencies of standing waves in a closed pipe (open at one end and closed at the other):

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fn = n( v4L) where n = 1, 3, 5, 7, . . . . n

where v is the speed of sound in the air in the pipe and L is its length. 4.5 Predict and test The A string on a violin vibrates at 440 Hz (called concert A). The string is 0.33 m long. (a) Find the speed of a pulse on the string. (b) Predict the frequency of the string’s vibration if you change the length of the string to 0.22 m and pluck it again.

Buggé: Waves 4

4.6 Predict and test Return to the Whirly Tube ™. We express the fundamental frequencies of vibration of a tube open at both ends the same as the expression for a string (fn = n v/2L, where v is the speed of sound in the air inside the tube, L is the length of the tube, and n is an integer 1, 2, 3…). You have a Whirly Tube™ that is 0.86 m long and is open at both ends. When swung in the air it produces a sound of a particular frequency. If swung faster, the frequency is higher. You can get about three distinct frequency sounds. (recall, vsoundinair = 340 m/s) (a) Use the expression for the standing wave frequencies in tubes to predict the frequencies of the Whirly Tube(TM). 4.7 Explain Wind instruments such as trumpets, flutes, clarinets, and the pipes in organs consist of columns of air inside tubes. They also have opening and closing valves or slides, in the case of a trombone. (a) Explain how the valves and slides allow a musician to change the frequency of sound that these instruments produce. (b) Whistle and try to change the frequency of sound produced. How did you do it? Is the reason consistent with the explanation you provided in part (a)? 4.8 Represent and reason Read the descriptions of situations and answer the questions that follow. Situation I: A 0.50-m-long string vibrates in three segments, with a frequency 240 Hz. Situation II: A 0.68-m pipe is open at both ends. The speed of sound is 340 m/s. (a) What is the fundamental frequency of the string? (b) What is the speed of a wave on this string? (c) What is the fundamental frequency of the pipe? (d) What is the fundamental frequency of the pipe when one end is closed? 4.9 Regular problem A banjo G string is 0.69 m long and has a fundamental frequency of 392 Hz. (a) Determine the speed of a wave on the string. (b) Identify three other frequencies at which the string can vibrate. (c) How far from the end of the banjo string should a fret be placed so that when the string is pressed against the fret, the fundamental frequency is increased to 490 Hz?