Resonance and Harmonics Problems and Questions

  1. Why are troops often told to “break step” when crossing a bridge (they shouldn’t march in unison)?
  2. In order to push someone progressively higher on a swing, it isn’t necessary to exactly match the natural frequency of the swing when you push. Suppose that it takes 3.6 seconds (frequency = 0.278 hz) for the swing to make one complete cycle. What other periods could be used to increase the amplitude of the swing? Explain.
  3. The sitar, a stringed Indian musical instrument has on set of strings mounted above an identical set of “sympathetic strings” that vibrate but are never physically plucked by the player. Using appropriate terminology, how does the sitar work?
  4. Draw three simple diagrams showing the fundamental frequency of a vibrating guitar string as well as the third and fifth harmonic frequencies (2nd and 4th overtones).

Waves in Strings

(n = number of antinodes, or just which harmonic; F = tension; = mass/length)

Waves in Pipes

Open-end tube: (n = 1,2,3…; both ends are antinodes)

Closed-end tube: (n = 1,3,5,7…; closed end is a node, open end is an antinode)

Beat FrequencyOther Equations

  1. Jimi Hendrix needs to find the first three harmonics in a 0.750 meter guitar string if the string has a mass per unit length of 2.00x10-3 kg/m and the tension is 67.0 N.
  2. Suppose that Jimi wants to tune his guitar so that a second harmonic of 256 hz is produced by the same guitar string. To what tension should the string be changed to achieve this frequency? Does this tension make sense?
  3. Suppose that another string on Jimi’s guitar is heavier, with a value of 7.95 x10-3 kg/m. How does this affect the frequency of vibration? Calculate the first harmonic if the heavier string using the same length and tension as in the first problem.
  4. A pipe that is open at both ends (“open pipe”) is 2.46 m long. Determine the frequencies of the first three harmonics if the pipe is open both ends. Use 345 m/s as the speed of sound.
  5. What are the three lowest possible frequencies in a closed-end tube with a length of 2.46 meters?
  6. One day while walking along the Caribbean coast in Jamaica, Bob Marley found a discarded cardboard wrapping paper tube. When he held a vibrating 256 hz tuning fork to one end, it resonated in the first harmonic. If the temperature was a balmy 36oC in Jamaica that day, how long was the tube? Hint: first determine the speed of sound, and then determine the length of the tube.
  7. While tuning a piano, Giovanni discovers that the G above middle C is vibrating with a higher frequency than his G tuning fork, which vibrates at 392 hz. He plays the piano key and tuning fork at the same time and hears a beat frequency of 2 hz. What is the frequency of the G on the piano?
  8. The same note played on a guitar and a clarinet (or any other instrument) sounds distinctly different. What term is used represents this difference and what causes the difference (in terms of the sound waves, not the instruments themselves)?
  9. While tuning her guitar, Marjorie hears a beat frequency of 3 beats per second when listening simultaneously to a tuning fork and a plucked guitar string. When she tightens the string, the beat frequency increases to 5 beats per second. What should Marjorie do and why?

Answers: (5) 122 hz, 244 hz, 366 hz; (6) 73.7 N, yes, it is tighter so the frequency is higher;(7) 61.2 hz, the frequency will be lower; (8) 70.0, 140, 210 hz; (9) 35.0 hz , 105 hz, 175 hz; (10) 0.690 m; (11) 394 hz.