I am studying for my PDEs final and this is one of the practice questions from Strauss and I don't understand how to solve it.
Plot $u$ vs $x$ for the following problem on the half-line $$u_{tt} = c^2 u_{xx}$$ $$u_x(0,t) = 0$$ $$u(x,0) = 0$$ $$u_t(x,0) = \begin{cases} V & x\in(a, 2a) \\ 0 & x \notin (a,2a)\end{cases}$$
for $t = 0, a/c, 3a/2c, 2a/c, 3a/c$
The method that I was told was good to follow was to split the $x$ vs $t$ plot into regions or zones and use that plot to find the limits of integration for the general solution.
But I don't know how to obtain the lines the divide the plane or where to put my solution for some specific $t$. Say for $t = a/c$ we can mark a horizontal line that will give us our solution, but I don't know how to find $a/c$ on the vertical $t$ axis with relation to the other lines that divide the plane. I don't understand the solutions online. Any help is appreciated and sorry for the bad drawing.