I considered the frictionless tube solution and decided that
it was a cheat which was beyond the bounds of acceptability.
The table will exert a horizontal component of force on the
rope, and the solution I seek would include that force.
To see that the horizontal component is present (Donald knows
this already, which is why he suggested the frictionless tube
in the first place) one need merely consider what the rope is
doing at the midpoint of its slide off the table. The portion
of the rope which is still on the tabletop will be moving
horizontally, thust the rope has a horizontal component of
momentum. It is easy to see, then, that there must have been
a horizontal force acting on the rope.
One benefit of installing that frictionless tube in the model
was that it allowed me to show my students the neat derivation
of wave motion in a string under tension that I call the glass
tube method. If the rope is moving just at the wave velocity
through the tube then the tube exerts *no* force on the rope.
The problem makes a good object for one-on-one teaching (or
one-on-two, as was the case here) but otherwise it is a rotten
problem to pose because it is ill-formed.