We have the following for a semi-infinite interval $$ u_{tt}-c^2u_{xx} = 0 \quad \quad 0 < c < \infty $$ Initial conditions are piecewise defined. $$ u(x,0) = \begin{cases} 0, & 0<x<2 \\ 1, & 2<x<3 \\ 0,& x>3 \end{cases} $$ $$ u_t(x,0) = 0 $$ We have boundary conditions: $$ u_x(0,t) = 0 $$ $\textbf{My attempt}$ at this is as follows: We use D'Alembert solution: $$u(x,t) = F(x-ct) + G(x+ct) $$ We note that we have a problem with zero initial velocity. This means that if $u(x,0) = f(x)$ then we have that: $$ u(x,t) = \frac{1}{2}f(x-ct) + \frac{1}{2} f(x+ct)$$ We note $f(x)$ is given piecewise but it only takes on positive arguments. $f(x-ct)$ has negative arguments though. We summarize this as follows
$$ \begin{array}{|c|c|c|c|} \hline \mbox{Interval} & F(x) = \frac{1}{2}f(x-ct) & G(x) = \frac{1}{2}f(x+ct)&u(x,t)\\ \hline 0<x<ct & ? & 0 & ? + 0 \\ \hline ct<x < 2-ct & 0 & 0 & 0\\ \hline 2-ct<x<2+ct & 0 & \frac{1}{2}(1)&\frac{1}{2}\\ \hline 2+ct <x<3-ct & \frac{1}{2}(1) & \frac{1}{2}(1)&1\\ \hline 3-ct < x< 3+ct & \frac{1}{2}(1) & 0&\frac{1}{2} \\ \hline x > 3+ct & 0 & 0&0 \\ \hline \end{array} $$ So solving for $x \in (ct, \infty)$ is not a problem. One will simply ass up the row to get $u(x,t)$.\ $\textbf{The Problem}$ arises there where the $?$ has been placed as $f(x)$ takes a negative argument. We try and solve for it using our boundary conditions. $$ u(x,t)_x = \frac{dF(x-ct)}{d(x-ct)} + \frac{G(x+ct)}{d(x+ct)} $$ $$ \therefore u(0,t)_x = \frac{dF(ct)}{d(ct)} + \frac{G(ct)}{d(ct)} = 0 $$ $$ -\frac{1}{c}\frac{dF(-ct)}{d(t)} + \frac{1}{c}\frac{G(ct)}{d(t)} = 0 $$ $$\therefore \frac{dF(-ct)}{d(t)} = \frac{G(ct)}{d(t)} $$ $$ \therefore F(-ct) = G(ct) + k$$ Let $z$ denote a negative argument. Then we have $$ F(z) = G(-z) + k $$ $$\therefore F(x-ct) = G(ct-x) + k$$ Now from the table, we have that $G(ct-x) = 0, \quad 0<x<ct$. So should we assume $F(x-ct) = k$ for this interval. There is no way of determining $k$ right? I think so. Thus the first interval has solution $u(x,t) = k$ where $k$ is an undetermined constant.
$\textbf{Question:}$ please help me undertand whether what I have done is indeed correct. If it is wrong, please help me to correct it. If it is correct and you feel I should be using a better method, please help me understand that method please. Thank you very much for your time.
P.S. this question is from here