The Physics
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Opus in profectus

Standing Waves

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Problems

practice

  1. Tuning forks
    note pitch (Hz)
    C4 256
    D4 288
    E4 320
    F4 341
    G4 384
    A4 427
    B4 480
    C5 512
    D5 576
    E5 640
    F5 682
    G5 768
    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.)

    To see a similar experiment using a shorter tube and, more importantly, to hear what resonance sounds like watch the video below.

    youtu.be/YpCA9CVsVos

  2. In contemporary Western music, the most common type of clarinet is the soprano clarinet. The most common type of flute is the concert flute. Some characteristics of the two instruments are summarised in the table below. The two instruments have almost the same length, but noticeably different frequency ranges. The purpose of this exercise is to explain why that is the case.
    Woodwind instruments compared Source: Vienna Symphonic Library
    soprano
    clarinet    		 concert
    flute
    key B♭ C
    highest
    concert pitch    		 G6
    1567.98 Hz    		 D7
    2349.32 Hz
    lowest
    concert pitch    		 D3
    146.83 Hz    		 B3
    246.94 Hz
    length 66 cm 68 cm
    inner diameter
    (bore)    		 12.7 mm 19 mm
    1. How does the
    2. How is this problem supposed to work? I forgot.
  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 electric 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) fpredicted (Hz) fobserved (Hz) ∆f/fobserved (%)
      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. A seiche (pronounced "saysh") is a type of standing wave that occurs in bays, lakes, and seas. Seiches typically form when strong winds push a surge of water toward one side of a large body of water. This raises the water level on one side and lowers it on another. When the wind stops, the surge sloshes back until it hits the other side where it is reflected. The seiche may then continue to oscillate back and forth for several hours or days, depending on the size of the body of water. Seiches are sometimes confused with tides, but those are caused by the gravitational interaction of the Earth with the Moon (and to a lesser extent, the Earth with the Sun). The animation below shows a cross-sectional representation of a seiche oscillating at its fundamental frequency in a cartoon body of water.

    Cartoon animation of a seiche

    Lake Erie on the border between the US and Canada is a convenient place to observe seiches. The long axis of the lake is nearly aligned with the direction of the prevailing winds, which makes seiches relatively common, and the lake is surrounded by monitoring stations, which makes accessing water level data relatively easy. The map below shows the location of seven of these monitoring stations on the American shore of Lake Erie.

    Map of Lake Erie with water level monitoring stations labeled

    On Monday 31 December 2018 (New Year's Eve), a noticeable seiche formed in Lake Erie. Water level displacement data from the seven stations shown on the map above were plotted on the graph below. Although the seiche lasted for six days, the graph below only shows data for two whole days and a few hours before and after. The data after Thursday morning are a little too messy for an introductory physics textbook.

    Water level displacement (mm) vs. time (days, hours)

    Answer the following questions about the seiche described in the text, animation, map, and graph above. Start by answering some questions that are about standing waves.

    1. Which two monitoring stations are closest to the one node of the seiche? Explain your reasoning.
    2. Which two monitoring stations are closest to each of the two antinodes of the seiche? Explain your reasoning.
    3. When it comes to seiches, is Lake Erie more like a system that is fixed at both ends, free at both ends, or fixed at one end and free at the other. Explain your reasoning.
    4. How long is one wavelength of a seiche oscillating in its fundamental mode in Lake Erie compared to the length of the lake itself?

    Determine some temporal characteristics of the seiche. State all times to the nearest hour (or day and hour).

    1. At what time did the seiche start? How did you identify this time?
    2. How many half-cycles of the seiche are shown in the graph of water level displacement?
    3. At what time did the last half-cycle you counted in the previous question end? How did you identify this time?
    4. What is the period of the seiche?
    5. If the seiche dissipated after six days, how many complete cycles were there?

    Compare your results to those predicted by theory. The fundamental period of a seiche can be estimated using Merian's equation, named after the Swiss mathematician Johann Rudolf Merian (1797–1871). If you have been doing this problem right, the period computed from this equation and the value computed from the data shown in the graph should be equal to two significant digits. Merian's equation works quite well for Lake Erie.

    T =  2ℓ
    √gh

    Where

    T =  period of the fundamental mode of oscillation
    ℓ =  effective length of the body of water
    h =  average depth of the body of water
    g =  acceleration due to gravity
    1. What does Professor Google say the values of ℓ, h, and g are for Lake Erie?
    2. What is the fundamental period of a seiche in Lake Erie according to Merian's equation and the values you found online?

    Some final questions on waves in general and standing waves in particular.

    1. What is the wave speed of this seiche in Lake Erie?
    2. The water level graph shows evidence of higher harmonics in the lake. How does the graph show this?
    3. What are the periods of the 2nd, 3rd, and 4th harmonics?

numerical

  1. Here's an animation of a wave.

    Animation of a standingling wave

    Determine its…
    1. period in seconds
    2. frequency in hertz
    3. frequency in millihertz
  2. 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.)
  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 (1814–1894) in the 19th 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 °C. 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.

statistical

  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
  3. speed-of-sound.txt
    A tuning fork was held over a tube that was partially submerged in water. The height of the tube above the waterline 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)
    2. Frequency in hertz
    3. Height of the tube above the waterline in millimeters