We argued above that waves have to reflect of of the ends of stretched strings because of energy conservation. This is true independent of whether the end is fixed or free - in neither case can the string do work on the wall or rod to which it is affixed. However, the behavior of the reflected wave is different in the two cases.
Suppose a wave pulse is incident on the fixed end of a string. One way to ``discover'' a wave solution that apparently conserves energy is to imagine that the string continues through the barrier. At the same time the pulse hits the barrier, an identical pulse hits the barrier from the other, ``imaginary'' side.
Since the two pulses are identical, energy will clearly be conserved. The one going from left to right will transmit its energy onto the imaginary string beyond at the same rate energy appears going from right to left from the imaginary string.
However, we still have two choices to consider. The wave from the imaginary string could be right side up the same as the incident wave or upside down. Energy is conserved either way!
If the right side up wave (left to right) encounters an upside down wave (right to left) they will always be opposite at the barrier, and when superposed they will cancel at the barrier. This corresponds to a fixed string. On the other hand, if a right side up wave encounters a right side up wave, they will add at the barrier with opposite slope. There will be a maximum at the barrier with zero slope - just what is needed for a free string.
From this we deduce the general rule that wave pulses invert when reflected from a fixed boundary (string fixed at one end) and reflect right side up from a free boundary.
When two strings of different weight (mass density) are connected, wave pulses on one string are both transmitted onto the other and are generally partially reflected from the boundary. Computing the transmitted and reflected waves is straightforward but beyond the scope of this class (it starts to involve real math and studies of boundary conditions). However, the following qualitative properties of the transmitted and reflected waves should be learned: