Localized Wave Packets

We now add more than two plane waves together.  We will also ignore the time dependent component of the wave.

The original wave is cos(pi*x), in which case k = pi.  The other buttons add more components to the original wave.  For instance, the "Add 0.1*k button" plots in dark blue the function cos(pi*x)+cos(1.1*pi*x)+cos(0.9*pi*x).  The light blue plot is the original wave.  There is a change in scale on the x-axis for the last two buttons.

Student Exercises:

  1. Why does the wave envelope repeat at x = 0, and 20 for the second and third buttons?
  2. Why is your answer to Exercise 1 no longer valid for the last button?
  3. What would be necessary for a wave envelope to repeat at  infinity?  This would mean that the wave is localized around the origin.  What would the amplitude of the wave be in this case?  Is it necessary that all the components have unity amplitude?

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