Wave on a string: fixed vs free-end reflection

What you are seeing: two parallel strings each simulating the 1D wave equation $y_{tt} = c^2 y_{xx}$ at $c = 1$ on a length-4 grid. A Gaussian pulse is launched moving rightward. The top string has fixed ends ($y = 0$); the bottom has free ends ($y_x = 0$). Watch what happens when the pulse reaches the far edge.

Fixed-end reflection inverts the pulse: an upward bump returns as downward. Free-end reflection preserves sign: an upward bump returns upward. This is analogous to optical reflection off media of higher vs lower refractive index.