Consider a taut string of length $10$ with wave speed $c = 1$ (in suitable units), with a fixed end at $x = −5$ and a free end at $x = 5$. The deflection of the string is denoted $u(t, x)$ for $−5 \leq x\leq 5$. At $t = 0$, the string is stationary with $u(0, x) = 0$ for $\lvert x\rvert > 1$, $u(0, x) = 2(1+x)$ for $−1 < x ≤ 0$, and $u(0, x) = 2(1−x)$ for $0 < x ≤ 1$. Determine the shape of the string at $t = 0, t = 2, t = 5,$ and $t = 10$, i.e. sketch graphs of $u(t, x)$ for $−5 ≤ x ≤ 5$ and $t\in \{0, 2, 5, 10\}$.
The general solution to the wave equation with zero initial velocity is $u(t,x)=\frac{1}{2}(f(x+t)+f(x-t))$, where $f(x)=u(0,x)$. The correct answers are shown below. I understand the graphs of $t=0,2,10$, the graph essentially splits in half and moves left and right (initially the the height is $2$). My question is how does the height of the graph go from $1$ at $t=2$ back to a height of $2$ at $t=5$? Should it not remain at $1$? I understand its shape though.
