​​ ​ It is given that the vibration of a uniform string is modelled by the wave equation 2 2 u c2 u , where u(x,t) is the displacement of the string and c is a non-zero constant, t2 x2 c=3 . Assume that both ends of the string are fixed. The initial position of the string is set by:  2x 0  x  1 ux, 0 2   2x  2 1  x  1 2 The string is also set into motion from its initial position with an initial velocity: u(x, 0)  t  0 0 x  1  1 x 1  2 2 where is any non-zero constant. Follow the steps below and use the method of separation of variables u(x,t) F (x)G(t) to find the displacement u (x, t) . (a) Sketch the initial position of the string.[2 marks] (b) Write down the fixed-end boundary conditions at x 0 and x 1 for the displacement u(x,t), and then deduce the boundary conditions for the function F (x) . [2 marks] (c) Explain in your own words what is the physical process behind the mathematical relation given by u(x, 0) . [2 marks] (d) Using the method of separation of variables, find two constant-coefficient ordinary differential equations. [2 marks] (e) Given a separation constant k=0, discuss the general solution of the problem. [4 marks] (f) Given a separation constant k>0, discuss the general solution of the problem. [5 marks] (g) Given a separation constant k=-p2 <0 i. Show all steps to find the eigen-function Fn (x) . ii. Given the eigen-values kn , find Gn (t) . Hint: Use the notation Cn and Dn for the iii. Write a general form of the un (x, t) solution of the problem. iv. Use the superposition principle to obtain a general solution of the problem. v. Find Cn . t constants involved in the definition of Gn (t) . (h) Find the series form for the subsequent time-dependent displacement vi. Find Dn. [27 marks] uniform string and write down the first six non-zero terms of the solution. [6 marks] u(x, t) of the (i) Using the results of (h) and also a mathematical software of your choice, represent graphically u(x, t) for x [0,1] , for 1 and at five different times t (1 / 6,1 / 3,1 / 2, 2, 3) or any other five interesting times. Comment on the physical behaviour of your string at your chosen times and on long term. [10 marks] [2+2+2+2+4+5+27+6+10=60 marks] As this is a long proof, working neatly and presenting your work as a small project is essential for obtaining good marks. Highlight your steps, your results for each part. The mathematical software used to graphically represent your solutions can be: Mathematica, Matlab, or any other online graphical software. You have to specify exactly what you have used, web address, include photos of your screens or codes, etc. The graphs should be printed images. Include your own code at the end of the assignment. Plots done by hand are not going to be marked. The learning outcome of this assignment is understanding using computer software of the behaviour of your string in time.