Homework 7, due 1 November 2005.

[goodwine]

Note: The example problems from class are worked out in full detail in the "Wave equation" section of the course notes. The details included in the notes, which are in a slightly different order than from class, may be helpful for this homework. If you want to include animations in the course notes, select "Enhanced PDF" at the bottom of the list of options. This option is good for viewing on a computer, but not for printing. Don't select "Enhanced PDF" if you are only going to print it.

1. Solve the wave equation for the case where the tension, mass per unit length and length of the string are given by
   • 
   and the initial shape and velocity profile are given by
   • 
   respectively.
   1. Plot the shape of the first three modes.
   2. Plot the shape of the sum of enough modes for a sufficient number of different times to adequately illustrate the nature of the solution.
   3. If everything is kept the same, but the tension is doubled, what happens to the nature of the response?
   4. If the mass per unit length is doubled, what happens to the nature of the response?
   5. If the length is doubled, what happens to the nature of the response?
2. Solve the wave equation for the same string as in the previous problem where the initial shape and velocity profiles are given by
   • 
   1. Plot the shape of the first three modes.
   2. Plot the shape of the sum of enough modes for a sufficient number of different times to adequately illustrate the nature of the solution.
   3. Would this response and the response from the first problem sound the same? Explain your reasoning.

[esabelha]

Professor,

In problems 1a and 2., you say to plot the shape of the first three modes. Does this mean that you only want the f(x) and g(x) terms plotted for the first three n's, or is the mode the u(x,0) term?

[goodwine]

Professor,

In problems 1a and 2., you say to plot the shape of the first three modes. Does this mean that you only want the f(x) and g(x) terms plotted for the first three n's, or is the mode the u(x,0) term?

It's basically the same thing. The first three terms from the f(x) and g(x) series, which are the same as the first three terms in the u(x,0) and d/dt u(x,0) series.

It's not just sin(n pi x/L) and cos(n pi x/L), make sure you have them multiplied by the right coeffficients (and a_n and b_n's).