from pulling the string through the hole in the
wall at x=L. The ring, however, does not interfere with the motion
of the string in the y-direction and thus leads to a free
boundary condition. This condition is also termed open because,
again in the case of sound, such a condition can be achieved in a pipe
with an open end which allows the air to move freely in and out of the
pipe.
Figure 3: Free boundary condition: (a) physical realization; (b) free
body diagram for ring
To derive the mathematical form of this type of boundary condition, we
again consider the motion of the end of the string. In this case, it
attaches to the massless ring, and so to determine the laws of motion
for this end, we consider the free-body diagram of the ring, as in
Figure 3b. The only forces acting on the ring come from
the contact with the frictionless pole and with the string. Because
the pole is frictionless, the force from the pole is a pure normal
force N. The force from the string is the tension force which,
again, acts along the tangent direction to the string with components
determined by (7,11),
and 
, where the
partial derivative is evaluated at the location of the ring, which we
here call 
. Because
the ring has mass m=0, we find for the x-component of Newton's law
and therefore conclude that the normal force acting on the ring is just force on the right with which we apply the tension to the string being transmitted along the full length of the string. From the y-component we learn,
that the solution must approach a free end with zero slope (as in the sketch in Figure 3a.) at all times t
