This is a very valid question, and I'd like to know the answer too!
To clarify what I think is being asked; if you have waves on string with a free end, for example, then the relationship between the frequency f of the waves, their wavelength w, and their speed v is of course v = fw. If f is increased, this equation tells us that several things could happen;
- the speed could stay the same, meaning that the wavelength would get shorter
- the wavelength could stay the same, meaning that the speed would get larger
- some more complicated combination, whereby both speed and wavelength alter
It's only the first that actually happens. Why? I've not been able to find an adequate explanation. I've seen explanations in specific cases (it's clear for light as it always travels at the same speed, there's an explanation for strings connceted with tension, etc.), but I've not seen an argument that deals with the general case. I wonder if it's a consequence of the conservation of energy.