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.

Figure 1. Wave-on-string reflections. Method: three-point FD stencil with

c \Delta t / \Delta x = 0.5

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WHAT TO TRY

Launch the pulse and watch it hit the end: a fixed end reflects it inverted, a crest returning as a trough, while a free end reflects it upright. The boundary condition flips the sign or not.
Compare the two strings side by side, same pulse and speed but opposite end conditions, the cleanest way to see why a guitar string and an open organ pipe differ.
The reflected pulse keeps its shape and speed: the wave equation is linear and non-dispersive, so information bounces without smearing.