I seem to remember watching a video where someone took a rope which was fixed at one end and gave it a jerk (or maybe more than one), and then did something to the end he was holding (fixed it somehow), so the pulse he created resulted in a displacement of the rope which was constant. This was explained as being the result of the waves bouncing back and forth between the fixed ends, and somehow interfering constructively to result in a time-constant displacement of the rope.
I've looked online and can't find the video, and my memory is hazy enough for me to distrust it on this matter. I am wondering if what I am describing is even possible, and if so, under what conditions. More formally:
The general solution to the wave equation in one dimension was given by d'Alembert as $$u(x,t) = f_1(x-ct) + f_2(x+ct),$$ where $f_1$ models disturbance traveling in the positive $x$ direction and $f_2$ in the negative $x$ direction. Is there any selection of initial conditions $u(x,0)$ and $u_t(x,0)$ ($u_t$ is the partial of $u$ w.r.t. time $t$) as well as boundary conditions for the string (e.g. $u(0,t) = 0$ for all $t$), and selection of $f_1, f_2$ such, for some $\tau \geq 0$ and any real number $r$ we have
$$u(x,\tau) = u(x,\tau+r) \neq 0~?$$