From Evans' PDE Book - we want to solve the 1D Wave Equation on the half-space
$$u_{tt}-u_{xx} = 0 \text{ in } \mathbb{R}_+ \times (0, \infty)$$ $$u = g, u_t = h \text{ on } \mathbb{R}_+ \times \{t = 0\}$$ $$u = 0 \text{ on } \{x = 0\} \times (0, \infty)$$
Where $g$ and $h$ are given with $g(0) = h(0) = 0$.
It then goes on to perform an odd extension, and then you can apply D'alemberts Formula for the Wave Equation on the whole line.
My question is - why can't I use an even extension? Where does this break down if I do that? The calculations all seem to work fine (can do the extension and write down a formula after apply D'alemberts).