The Physics
Opus in profectus

Standing Waves

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  1. Write something.
  2. Write something else.
  3. Schumann Resonances
    The ionosphere is a layer in the Earth's upper atmosphere where a large portion of the atoms and molecules have been ionized by exposure to the ultraviolet radiation of the sun. With so many charged particles free to roam around, the ionosphere is a reasonably good conductor of electricity. The surface of the Earth is also a reasonably good conductor. This should be somewhat obvious since 70% of the Earth's surface is covered in saltwater, which will short out electrical equipment as everyone knows, and the remaining 30% is exposed rock or soil, the stuff that electrical circuits are grounded to. The layer of atmosphere in between these two conductors is ordinary, non ionized air, which is transparent to radio waves. For extremely low frequency (ELF) radiation, the gap between the Earth and its ionosphere acts as a spherical wave guide — a kind of racetrack for radio waves. Lightning and other natural phenomena generate ELF waves at all sorts of different frequencies. Those frequencies that are just right will travel around the Earth, meet themselves in phase, and form standing waves. The set of frequencies that will do this are known as the Schumann resonances in honor of Winfried Schumann (1888–1974), the scientist who predicted their existence in 1952.
    1. Complete the following table…
      Schumann resonances
      harmonic λ (km) ƒpredicted (Hz) ƒobserved (Hz) Δƒ/ƒobserved
      first 7.8
      second 14
      third 20
      fourth 26
      fifth 33
      sixth 39
      seventh 45
    2. Do the predicted Schumann resonances agree with the observed values to a reasonable degree? Account for any significant discrepancies.
  4. Write something completely different.


  1. A common form of hearing loss is associated with resonance in the ear canal. When this happens, there is reduced sensitivity to sounds around 4,000 Hz (since this frequency is consistently louder than all the others).
    1. What fraction of a wavelength fits in the ear canal while it is resonating at its fundamental frequency? (Recall that the ear canal starts at the opening in the outer ear and ends at the eardrum.)
    2. From this information determine the length of the ear canal in the average human. (Assume that the speed of sound in the air in the ear canal is about 348 m/s.)
  2. A 1.00 m vertical tube is partially submerged in water. The height of the tube above the water can be adjusted to any value from 0 m (the tube is completely submerged) to 1 m (the bottom of the tube is just touching the top of the water). A vibrating tuning fork is held just over the open top end of the tube. Pick one of the notes from the table on the right and answer the following questions.
    1. Determine the wavelength of the sound emitted from the tuning fork in air at room temperature (vsound = 343 m/s).
    2. How long is ¼, ½, ¾, 1¼, 1½, 1¾ of the wavelength you calculated in part a.?
    3. At what heights will resonance occur? (Just highlight the answers from part b. that satisfy this condition.)
  3. Three instruments in an orchestra are playing the same note — a concert A of 440 Hz. For each instrument decide whether it has two fixed ends, two free ends, or one fixed and one free end. Then state the frequencies of all the harmonics up to the seventh. The instruments are…
    1. a clarinet
    2. a flute
    3. a violin
  4. The saxophone was invented by the Belgian instrument maker Adolphe Sax (1814–94) in the Nineteenth Century. I have decided to invent my own instrument called the elertphone. The elertphone is designed to play only one note, concert a which has a frequency of 440 Hz, and is always played when the air is 0 ℃. The elertphone is a tube that is open at both ends like an organ pipe or a drinking straw. Determine the length of a properly designed elertphone.


  1. resonance-tube.txt
    A tuning fork was held over a half closed tube, the length of which was adjusted until the sound from the tuning fork was at its loudest. Use this data to determine the speed of sound in air at room temperature.
    The columns in this data set are as follows:
    1. Note of tuning fork (scientific scale except where indicated)
    2. Frequency in hertz
    3. Length of tube in meters
  2. vibrating-string.txt
    A one meter piece of ordinary string was connected to a variable oscillator that was fixed at both ends. The oscillator was dialed through different frequencies of vibration until transverse standing waves formed in the string. A photogate was then used to time the period of vibration since the oscillator was not calibrated in any way. Use this data to determine the speed of transverse waves in the string.
    The columns in this data set are as follows:
    1. Number of antinodes (or the number of the harmonic)
    2. Period of oscillation in seconds