Click here for explanation and discussion
A pulse traveling down a string and reflected from a fixed end will be reversed (up to down), but reflected from a free end it will stay up.
This can be understood by imagining a string twice as long, of the same thickness and tension, but with equal pulses having opposite velocities coming in from the two ends simultaneously and passing through each other in the middle. In this linear system, as an up pulse passes through a down pulse, the middle of the long string never moves!
So, the left-hand half of the double length string, which satisfies the same identical equation of motion as the string with the fixed end (same tension, same density), and has the same “boundary condition” of never moving at the center point, behaves in exactly the same way as the complete shorter string.
By a string having a "free" end, we mean the end is free to move vertically, but the string is still under tension. This is achieved by having the string end attached to a ring of negligible mass free to move up and down on a frictionless vertical pole. The string tension can have no vertical component at the end, or the ring would accelerate at a huge rate, so the string must end horizontally at all times.
Two identical up pulses meeting symmetrically on the longer string will at all times add to zero slope at the central point, so one-half of the longer string will in this case behave exactly like this shorter string with a free end.
All this is fully discussed, with some enlightening pictures, in my lecture here.
Also available as an animated spreadsheet here.
Code by Casey Bowler