If you put these new values into the original formula then you should get the new frequency but with and unknown m in it. If your string length increases by l (lower case L) the new L is L'=L+ l and the new m' = m L/(L+ l). The problem is that stretching the string will alter both, so length appears twice in the formula. I assume you can measure the start length L and the extension but how about m? The grey boxes give the wave velocity and then the fundamental frequency, in terms of Length (L) and Mass per Unit length (m). There are many possible links but this one may help. The formula requires you to know a lot about the string but, if you already have a known frequency for the given length, you don't need all the data. How frequency will change if I stretch a string by certain amount (0.14 mm in my case)?
