Increasing wave speed on a tring

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To increase wave speed on a string i would have to decrease the frequency and the string thickness correct? the string thickness would not make an impact correct?
 
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The wave velocity v along a string is equal to

v = (T0/ρ)½ meters per second

where T0 is string tension in Newtons, and ρ is string mass per unit length (Kg/m). (Note that the units are correct). If you decrease the string thickness without changing tension, the velocity increases. The string thickness (i.e., mass per unit length) is important.

Bob S