At a fixed end, the string's transverse displacement is zero. Consider the general traveling wave solution to the wave equation:
. If the string is fixed at , then and
, which indicates that displacement traveling waves reflect from a fixed end with an inversion (or a reflection coefficient of -1).
Since the displacement at a rigid termination is always zero, the physical velocity must also be zero for all time. Therefore, traveling-wave components of velocity will also reflect with a reflection coefficient of -1 at such a boundary.
Force and velocity traveling-wave components are related by the wave impedance as
. At a rigid terminiation,
(from above). Thus, force wave components can be related at a rigid termination as:
From this, we see that traveling-wave components of force are related by a reflection coefficient of +1 at a rigid termination.
At a free end,
because no transverse force is possible. At such a boundary, traveling-wave components of force must reflect with a coefficient of -1. To determine the reflection coefficient for displacement waves, we first note that force waves are proportional to the string slope. Differentiation of our general traveling-wave solution by leads to (see this link):
After integrating this expression with respect to time, we have
at , indicating that displacement traveling waves reflect with a coefficient of +1.