A fixed boundary is one in which the wave experiences no displacement along the boundary when it hits. The wave is reflected back, and because the boundary does not allow any displacement, the up and down motion of the wave is inverted. The speed and the wavelength of the reflected wave will be the same as the original wave. The amplitude will be somewhat less, depending upon how much energy was absorbed by the boundary material.
A free boundary allows the wave to experience some displacement as it hits the boundary. The point along the boundary at which the wave hits moves with the curve of the wave. The boundary blocks further forward motion of the wave but allows continued movement following the curve of the wave. The wave is bounced back away from the boundary, but there is no change in polarity. Like the fixed boundary case, the velocity and wavelength of the reflected wave will be the same.
For the boundary to be free, it must be frictionless and have no mass because either condition would exert force upon the incoming wave. Because of this restriction, there is no change of amplitude in the reflected wave because there is no inertial force upon the wave from the boundary and no drag force. The slope of the wave at the point of contact with the boundary must be 0, so not all wave equations are possible in a system with a free boundary.
Sometimes a wave will contact a boundary that is neither totally free nor totally fixed. In these cases, part of the wave may be transmitted past the barrier, with reduced amplitude and width but the same polarity. The rest of the force of the wave will be reflected back from the barrier, with reduced amplitude and reversed polarity.