Question 1 Consider the vibrating string problem for a string of length L, whose ends are held fixed. Suppose that that the string is released from rest with an initial displacement 1.25- I O.8L f(x, t = 0) = 5 20.81 (a) Perform a separation of variables approach and show that the eigen- functions are of the form Un (x) ~ sin(kr) cos(Wnt) (b) By computing the inner product between the eigenfunctions at t = 0 and the initial displacement f(x), obtain the coefficients necessary for the solution that satisfies the initial and boundary conditions. an L sin(kne)f (x)d. (c) Hence show that the solution is y(r, t) = > 12.5 hein2,2 sin(0.8na) sin L | cos nact LQuestion 2 Generate some plots of the above solution at different times. You can use a spreadsheet for this, but a short bit of code (e.g. MATLAB or Octave, or Python) will be much quicker and easier. Specifically, assume L = 1 and c = 1. Then take 21 points from 0 to 1 (steps of 0.05) and compute the function for the first 4 terms in the series when t = 0. This is the initial condition. Use more points and eigen-modes if your code allows this. Repeat the above process to produce two plots at later times: t = 0.5 and t = 1.0. BONUS: create an animation of the wave moving through the domain for several time units