Can a standing wave form on a string with both end open I am fascinated with an idea of an standing wave forming on a string with both end open. 
If we assume two identical waves coming in of an infinitely long string then for a short period of time, they will meet and form a standing wave, my question is would that standing wave last for a considerable amount of time like they do when both ends are tied. What would the expression of duration for that if possible?
Thanks a lot.
 A: Yes, it most certainly can.  It's much easier to visualize if you consider a length of flexible steel or plastic.  You can shake it a bit, then toss it in the air so it's not constrained, and it will (if properly initiated) vibrate at a resonant wavelength.
I think the confusion most people will get from your question is that everyday string is "floppy," i.e. no elastic rebound unless it's under tension.  And there's no way to apply tension without constraining an end.  Or maybe I should say it's very difficult: you'd have to hold the ends with some sort of clamp running on a high-bandwidth feedback loop so that it can maintain tension while simultaneously moving up and down to match the standing wave's current amplitude at the end of the string.   
A: A "standing wave" is not a real wave - it is simply our observing the superposition of two waves - one traveling to the left and one traveling to the right. If they have the same amplitude and propagate at the same velocity there will be stationary points on the string. This is true regardless of whether the ends of the string are "open" or "closed".
Usually we fix the end of the string for two reasons: 1) we need to hold the string to be able to apply tension; 2) at the fixed point the wave is reflected (and inverted) so from one traveling wave we end up creating waves traveling both left and right.
But if you create these waves by other means their superposition will look like a standing wave, yes.
A: Somehow you have to impose rigidity on a flexible object.  I think this could be done by turning the string into a long thin magnet. The ends would then repel each other. You could put the string in orbit and then perturb it to look at the waves. I think it is intuitively obvious that standing waves would be possible.  I don't have the maths to prove it.
