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Opus in profectus

The Nature of Waves

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Practice

practice problem 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).

Natural phenomena are normally very 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 and
  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.

solution

  1. The amplitude is open to some interpretation. The crests seem to average around +90 cm between 10:50 and 12:05, so 90 cm would be an acceptable answer. The highest crest appears to be +100 cm, so 100 cm would also be an acceptable answer. The troughs seem to average −60 cm. If we split the difference between +90 cm and −60 cm we'd get 75 cm for an amplitude. The deepest trough looks like its −70 cm. If we split the difference between +100 cm and −70 cm we'd get 87.5 cm for an amplitude. Any one of these answers is reasonable as far as I'm concerned. Recall that the problem said that real physical data is statistically noisy. This makes it impossible to point to an exact answer. Exact isn't a scientific concept, anyway. It's a word that really belongs to mathematics.
  2. There are 3.5 wave cycles from 10:50 to 12:05 (75 minutes or 4,500 seconds). Divide time by number to get period — the time for one cycle. We'll do it once with each unit just because neither unit seems better than the other right now.
     
    T =  t  =  75 min  = 21.4 min
    n 3.5
    T =  t  =  4,500 s  = 1,290 s
    n 3.5
  3. Start with the wave speed equation (wave speed in terms of period) and solve it for wavelength.
     
    v =  λ  ⇒  λ = vT
    T

    Put numbers in. Get answer out. I think I'll use the SI values (meters and seconds, not kilometers or hours or minutes).

    λ = vT
    λ = (250 m/s)(1,290 s)
    λ = 321,000 m
  4. Tsunamis have long periods and long wavelengths. This would be hard to perceive in the open ocean. The changes would be too gradual to see or feel.

  5. Repeat the previous problem with one number different — the slower wavespeed near shore.
     
    λ = vT
    λ = (15 m/s)(1,290 s)
    λ = 19,300 m
  6. Near shore wavelength are much shorter, but they're still pretty long. Note that the period hasn't changed. That value is set by the source of the wave — an earthquake in this case.

  7. The amplitude and wavelength of a wave in the ocean together describe something like a volume. This volume can't change. Squash the tsunami horizontally and it grows vertically. The amplitude increases as the wave moves toward the shore.

practice problem 2

Write something else.

solution

Answer it.

practice problem 3

Write something different.

solution

Answer it.

practice problem 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.

  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

Source: National Oceanic and Atmospheric Administration (NOAA)

solution

Plot distance on the y axis and time on the x axis so that the slope of the best fit line will be the speed.

It appears that the average speed of a tsunami in the pacific is…

  1. 11 km/min
  2. 660 km/h
  3. 180 m/s
  4. 410 mph