Problems for Intermediate Methods in Theoretical Physics Edward F. Redish

Normal modes on a beaded string

In class, we have derived the normal modes on a continuous elastic string starting from the continuous wave equation. We also derived the wave equation for a string consisting of a finite number of massive beads connected by massless springs. When we took the limit of the beads large and the size of the springs small, we recovered the continuous wave equation. However, there are some interesting peculiarities to the finite bead model that have interested implications for the study of solids. Our finite bead model is actually a better approximation for vibrations in a solid than is the continuous wave equation if you are at a scale where you have to consider the fact that it is made up of atoms. To look at some of these properties, download and run the program StWave.exe. (You will also need to download the file egavga.bgi and put it in the same directory with StWave.) Information about the program is available at Info Page but it should be fairly obvious how to run it.)

Set the program StWave to show 20 beads connected by springs stretched between walls a distance L (= 20 meters) apart. If you ask for the n-th mode, it sets the initial conditions to be

y(x,0) = A sin(nπx/L

velocity = 0.

(a) Sketch the first 4 modes (labeled "harmonics" in the program). Do they look like the modes on the continuous string? What do the 20th and 21st mode look like? Explain why they look this way from considering the initial conditions.

(b) Explore the other modes and then look at modes 22 through 42. Particularly note what happens in modes 40, 41, and 42. What's going on? Explain. (The blocks of 21 modes corresponds to what is called a "Brillouin zone" in solid state physics.)

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