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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.

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Presumably, you have some notes that tell you what the d'Alembert procedure is, and how to use it, so why not tell us what you know about the problem, how you start out, where you get stuck, and so on? –  Gerry Myerson Dec 10 '12 at 6:25
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1 Answer

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 $t\geq0$ .

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I believe you need to solve using the reflection principle, which means your soltution is valid in the triangular region given by $$T = \left\{(x,t) \in \mathbb{R}\,\big|\, t > 0\,,\,t < x+a,\,t<a-x\right\}$$ and then it has to be extended using the Rhombus Theorem with the reflection condition $f \to -f$ in the boundaries. –  Pragabhava Dec 10 '12 at 17:41
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