The Physics
Opus in profectus

Periodic Waves

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  1. The graph below is a record of sea level heights recorded at Hanimaadhoo, Maldives during the tsunami of 26 December 2004. The data were filtered to eliminate the normal tidal fluctuations, so what you are seeing is the increase in sea level due to the tsunami. (Data source: University of Hawaii Sea Level Center)

    Line graph

    Natural phenomena are normally noisy (in the statistical sense) but from 10:50 to 12:05 local time the changes in sea level at Hanimaadhoo were most nearly periodic. During this time interval determine the tsunami's mean…
    1. amplitude
    2. period
    The speed of a tsunami varies with depth. In the open ocean they normally move as fast as a commercial jet airplane (about 250 m/s or 900 kph) but slow down to the speed of a car on a neighborhood street when they reach the shallow waters of the shore (about 15 m/s or 55 kph). Given these speeds, determine the mean wavelength of the segment of the tsunami that arrived in Hanimaadhoo between 10:50 and 12:05 when they were…
    1. in deep water
    2. near the shore
    One final question.
    1. How would the amplitude of a tsunami near shore compare to the amplitude of the same wave in the open ocean? Explain your reasoning.
  2. Each of the six strings on an acoustic guitar is 80 cm long and generates a wave that's twice the length of the string when picked or strummed. Determine the wave speed of each string given the following tuning.
    1. 082.41 Hz
    2. 110.00 Hz
    3. 146.83 Hz
    4. 196.00 Hz
    5. 246.94 Hz
    6. 329.63 Hz
  3. A viola string is 36 cm long and plays a certain note when the finger is not resting on the string. How far from the bridge should a finger be placed so as to produce a note that is…
    1. an octave higher (double the original frequency)?
    2. two octaves higher (quadruple the original frequency)?
    3. a fifth higher (32 the original frequency)?
    4. a fourth higher (43 the original frequency)?

    Viola with a left hand on the fingerboard

  4. tsunamis.txt
    The greatest recorded earthquake (magnitude of 9.5) occurred on 22 May 1960 in Chile. The second largest earthquake (magnitude 9.2) occurred on 27 March 1964 during the Christian Holiday of Good Friday, which is why it is also known as the Good Friday Earthquake. A large earthquake (magnitude 8.8) occurred in Chile on 27 February 2010 that grabbed my attention and motivated me to write this problem. All three earthquakes generated tsunamis for which I was able to find useful data.

    Tsunamis are sometimes called "tidal waves" but this name is misleading. Tsunamis, which are seismic in origin, and tides, which are caused by the gravitational pull of the Moon and Sun, are completely unrelated. The word tsunami is derived from the Japanese phrase "harbor wave" (津波) since tsunamis are most intense in harbors where the underlying terrain focuses their energy. The term "tidal wave" is somewhat appropriate since the waves generated by earthquakes result in long period waves that sometimes look like the changes in water depth caused by the tides.

    The accompanying tab-delimited text file provides the following data for the tsunamis associated with the three earthquakes described above. (Source: National Centers for Environmental Information)

    1. Location of town, harbor, or facility
    2. Region (state, province, or country)
    3. Transit time in minutes after the earthquake began
    4. Distance from the epicenter in kilometers measured along a great circle (the shortest path on the surface of a sphere)

    Use this information to determine the speed of a tsunami in…

    1. km/min
    2. km/hr
    3. m/s
    4. and mph if you live in the United States


  1. Here's an animation of a wave.

    Animation of a traveling wave

    Determine its…
    1. period in seconds
    2. frequency in hertz
    3. frequency in millihertz
  2. Sprinting speed is related to frequency and stride length in much the same way that wave speed is related to frequency and wavelength. Use this analogy to analyze and compare the world record setting Jamaican sprinter Usain Bolt to the average of his competitors in three of his best performances ever.
    1. Use the data given and complete the table below. (Report all computed values with 2 decimal places of precision.)
    2. What factor in Mr. Bolt's running style seems to give him an advantage?
    Kinematic parameters in three men's 100 meter races
    Usain Bolt Beijing
    time (s) 9.69 9.58 9.63
    speed (m/s)
    stride fre­quency (Hz) 4.24
    stride length (m) 2.47
    number of strides 41.40
    Adapted from Maćkała and Mero, 2013
    average of the rest Beijing
    time (s) 9.96 9.91 9.86
    speed (m/s)
    stride fre­quency (Hz) 4.55
    stride length (m) 2.23
    number of strides 44.45
  3. Gravitational waves are disturbances in the gravitational field that propagate at the speed of light and are formed whenever objects with mass accelerate.
    1. A black hole is the remnant of a very massive star that has collapsed to the point where nothing, not even light, can escape it. The first direct detection of gravitational waves was the merger of two black holes in a distant galaxy by the Laser Interferometer Gravitational-Wave Observatory in 2015. LIGO's two detectors simultaneously observed an 8 cycle, 0.2 s gravitational chirp that swept upward in frequency and amplitude from 35 to 250 Hz as the black holes spiraled into one another — something like a giant toilet flushing. Given this information, determine…
      1. the shortest wavelength detected in the chirp
      2. the longest wavelength detected in the chirp
      3. the average frequency of the chirp
      4. Why isn't the answer to the previous question equal to the average of the initial and final frequencies of the chirp?
    2. A pulsar is a highly magnetized, burned out star (a bit more massive than the Sun) that has collapsed down to the size of a city, is made up almost entirely of neutrons, and is spinning as fast as a kitchen blender. The North American Nanohertz Observatory of Gravitational Waves (NANOGrav) is a consortium of scientists whose goal is to detect gravitational waves with frequencies of about 10 nanohertz using precisely timed millisecond pulsars as clocks. Given this information, determine…
      1. the frequency of a 1.5578 ms pulsar
      2. the period of a 10 nHz gravitational wave in…
        1. seconds
        2. years
      3. What kind of astronomical phenomena might emit ~10 nHz gravitational waves?
  4. Gravitational waves from some orbiting astronomical objects (like the Earth orbiting the Sun) have frequencies that are best stated with nanohertz as the unit (1 nHz = 10−9 Hz or one billionth of a hertz).
    1. What is the frequency of the gravitational waves produced when the Earth orbits the Sun in nanohertz?
    2. What is the period of a nanohertz gravitational wave in years?
    Explain the reasoning behind your answers to these two questions. "I googled it" is not a form of reasoning.