Problem: Consider the initial boundary value problem
{
utt = c²uxx, t > 0
u(x, 0) = f(x),
ut(x, 0) = g(x),
for u(x, t) â â, where f and g are two given twice differentiable functions. Show that this problem is solved by the d'Alembert solution
u(x, t) = 1/2 [f(x + ct) + f(x - ct)] + 1/2c [G(x + ct) - G(x - ct)].
Where G is an antiderivative of g. Sketch the solution for t = 0, 1, 2, 3, where f(x) and g(x) are as given below.
a)
f(x) = { 1 - x², |x| < 1
{ 0, |x| > 1, g(x) = 0
b)
f(x) = 0, g(x) = { sin Ďx, |x| < 1
{ 0, |x| > 1