Waves on a string and frequency

[Physgeek64]

So I was reading our lecture notes about generalising the motion of a string by considering it as a system of N masses, and one of the arguments was that no matter where you are on the string, each mass element will oscillate with the same frequency. This makes intuitive sense for an ideal string with no friction, but is there a way to mathematically prove this/ does it only apply to ideal strings, or can it be extended to realistic strings?

Many thanks :)

 

[Svein]

https://en.wikipedia.org/wiki/Wave_equation

 

[Physgeek64]

https://en.wikipedia.org/wiki/Wave_equation

I appreciate the reply, and I've had a scan through, but unless I've missed something I don't think it quite answers my question. It talks about eigenmodes, but not really about why this occurs.

Thank you though :)