Consider the wave equation

Question

Consider the wave equation $\frac{d^2}{dt^2} u(x,t)= \frac{d^2}{dx^2} u(x,t)$ -$a\leq x \leq a$ ,$t \geq 0$ subject to the initial conditions $u_t (x,0)= 0$, $u(x,0)=x$.
Find a solution using the d’Alembert procedure.

Answer 1

Just seckilling this problem by using D’Alembert’s formula:

$u(x,t)=\dfrac{x+t+x-t}{2}+\dfrac{1}{2}\int_{x-t}^{x+t}0~ds=x$

Note that this solution suitable for $x,t\in\mathbb{C}$ , not only suitable for $-a\leq x\leq a$ and $t\geq0$ .