Please let me know if my title or notation should be changed to something more precise.
In the study of waves, I am considering eigenfunction expansions of waves on a finite interval $0 \leq x \leq L$ as products
$$y_n(x,t) = C_n \sin(kx)\sin(\omega t)$$ with a dispersion relation, $\omega = f(k)$, and where $C_n$ is the amplitude that is determined from initial conditions $y(x,0) = y_0(x)$.
Suppose $y_0(x)$ is piecewise constant on $[0,L]$, and consider the solution to our equation, $y(x,t)$. Sometimes, there exist future times $t_i >0$ where $y(x,t_i)$ is again piecewise constant, but not equal to $y_0(x)$.
For example, consider the equation on $0 \leq x \leq L$, $$\frac{\partial^2 \Psi}{\partial x^2} = i \frac{1}{c^2} \frac{\partial \Psi}{\partial t}$$ This has dispersion relation $\omega = c^2 k^2$. Playing around with solutions to this equation, one can see that a piecewise constant initial condition will produce piecewise constant profiles at certain later times. For example, if $$y_0(x)= \begin{cases} 0 & x\leq \frac{L}{4} \\ 1 & \frac{L}{4} \leq x\leq \frac{L}{2}\\ 0 & \frac{L}{2} \leq x \leq L \end{cases}$$
we see at later times some of the following solution profiles:
The solutions here are not exactly piecewise constant because I came across this accidentally, so could only approximate the correct times. And, of course, we see Gibbs oscillations, which prevents the solution from ever being truly constant. However, I hope the phenomenon I am describing is clear.
I have several questions here:
(1) Why does this occur??
(2) For what dispersion relations $\omega = f(k)$ does this occur?
(3) Given that $y_0(x)$ is piecewise constant, suppose there exists a time $t_1 > 0$ such that $y(t_1,x)$ is also piecewise constant. What will be the properties of $y(t_1, x)$? Is there a relationship between $\int_0^L y(t_1,x) dx$ and $\int_0^L y_0(x) dx$?
(4) Given that $y_0(x)$ is piecewise constant, suppose there exists a time $t_1 > 0$ such that $y(t_1,x)$ is also piecewise constant. Does this guarantee that there will exist infinitely many further times $t_2 < t_3 < ...$ with $y(x,t_i)$ piecewise constant? If so, what are these times?
I appreciate any and all help on this question!