Wave equation with boundary condition

Question

Given the wave equation $u_{xx}=\frac{1}{v^2}u_{tt}$ (for $v=const$) with the boundary condition $u(0,t)=0$ for all $t\in \mathbb R$. There exists a simply proof that $u=u(x,t)$ must be: $$u(x,t)=f(x+vt)-f(x-vt)\ ?$$ I only know to prove that (without the boundary condition) $u(x,t)=f(x+vt)+g(x-vt)$ for some $f,g$ smooth functions.

Answer 1

Impose the boundary condition on $u(x,t)=f(x+v\,t)+g(x-v\,t)$ to get $$ f(v\,t)+g(-v\,t)=0. $$ Letting $v\,t=z$ we see that $$ g(z)=-f(-z). $$