What is the Uniqueness Theorem and its Application in Physics?

[JaredPM]

Could someone give me an applied math example of the uniqueness theorem in the physical sciences (physics, chemistry, biology)? Because I am not sure of its application. I understand that there is an interval (x,y)~intial conditions.

 

[HallsofIvy]

Here is a classic example- suppose you have a wire attached to a fixed point on a wall and pulled taut. You "snap" the wire at some point, sending a single upward "bulge" toward the wall. It will hit the wall and "bounce" back. Does it come back upward as the initial wave or reversed and downward?

To answer that, imagine that the wire extends on the other side of the wall (so there is no "wall") and there is an "image" of the wave on the other side. Both waves are, initially, moving toward the wall. In order that the point where the wired, and its image, meet the wall does not move the two waves must cancel, not add. That means that the "image" wave, that started on the other side of the wall, but continues through to this side, must be reversed.

Because those two waves, on the single wire, give a valid solution to the wave equation, that satisifies all the initial and boundary conditions, and such solutions are unique, that answers the original question- the wave must bounce back from the fixed point on the wall reversed.