Please, do not upload selfless :D
I give example of the proper solution to one of the problems.
\begin{align}
&u_{tt}-c^2u_{xx}=0,\\
&u|_{t=0}=g(x),\quad u_t|_{t=0}=h(x)
\end{align}
with
\begin{equation}
g(x)=0, \qquad
h(x)=\left\{\begin{aligned}
&1 &&|x| < 1,\\
&0 && |x| \ge 1.
\end{aligned}\right.
\end{equation}
Observe that $g$, $h$ are even with respect to $x$. Then $u$ is even with respect to $x$. Since $g=0$ we conclude that $u$ is odd with respect to $t$. So basically we need to consider only $x>0,t>0$. Here teal and orange are characteristics passing through the ends of the segment $(-1,1)$ on $\{t=0\}$
(http://www.math.toronto.edu/courses/apm346h1/20159/Forum/P2.3.2.svg)