Q3 A guitar string of length 1 meter is secured to the x-axis, at x = 0 and x = 1, for all time t. The string vibrates and the displacement is represented by u(x,t). u(x,t) is then governed by the equations and conditions as below:
∂²u(x,t)/∂t² = ∂²u(x,t)/∂x², 0 < x < 1, t > 0,
Boundary conditions: u(0,t) = u(1,t) = 0, t > 0
Initial conditions: u(x,0) = sin 3πx, 0 ≤ x ≤ 1
∂u(x,0)/∂t = 8 sin 4πx + 6 sin 6πx, 0 ≤ x ≤ 1
(a) By showing all possible cases, use separable method to find the general solution, u(x,t), of the vibrating guitar string problem. (17 marks)
(b) Show that the solution of the wave equation with the given initial and boundary conditions is:
u(x,t) = cos 3πt sin 3πx + (2/π) sin 4πt sin 4πx + (1/π) sin 6πt sin 6πx.