Estimating the maximum possible frequency of standing waves on a fixed string

Question

I would like to understand if/how I can estimate an upper limit on the frequency of (standing) waves that can be generated on a string fixed at both ends. 

As a first step, the properties of the medium (material of the string) will determine this. In addition to the rigidity, what other properties of the medium, specifically? And how (mathematically) might rigidity (and any properties) affect this? I would expect that for a given amount of energy imparted to the string, the maximum frequency is inversely proportional to the rigidity. But is this a linear relation, or is there a power law? 

Additionally I would like to be able to generalize the question to the upper limit of the frequency / lower bound to the wavelength in a (quantized) medium. How might one proceed with this case? 

Finally, if more energy than can be held in a wave of the max frequency is imparted, how might the excess energy get dissipated(?)?

Comments

The basics are well documented... There are nicer pages, like this one, with quick answers in a context : Strings, standing waves and harmonics . Note that there is not any new constant. To go further, it depends the approximations you are able to accept. Ask the Oleg A. Godin articles, ie on arxiv ie Anomalous transparency of water-air interface for low-frequency sound . And then, please reformulate the question. Explain particularly what you do mean exactly by quantized because here it is very broad :)