The one dimensional wave equation describing the vibration of a string about the x-axis is given by ∂²U/∂x² = 1/c² ∂²U/∂t² where U(x, t) is the displacement of the string about the x-axis at position x and time t .
A string is stretched between two fixed points at a distance l apart. Motion is started by displacing the string in the form U(x, t = 0) = a sin(π x/l).
Using the separation of variables method where U(x, t) = X(x)T(t), express the
(i) boundary conditions for X(x = 0) and X(x = l).
(ii) initial conditions U(x, t = 0) and ∂U(x, t = 0)/∂t.
(iii) Solve for U(x, t).
You may assume that ∫₀ᵃ sin(nπ/a x) sin(mπ/a x) dx = { a/2, n = m; 0, n ≠ m