Energy and Power of a wave traveling along a string

[DrunkApple]

Homework Statement

Two wires are made of the same material but
the second wire has twice the diameter and
twice the length of the first wire. When the
two wires are stretched, and the tension in
the second wire is also twice the tension in the
first wire, the fundamental frequency of the
first wire is 500 Hz.
What is the fundamental frequency of the
second wire?
Answer in units of Hz

Homework Equations

$l_{2}$ = 2$l_{1}$
$r_{2}$ = 2$r_{1}$
$τ_{2}$ = 2$τ_{1}$
$f_{1}$ = 500 Hz

The Attempt at a Solution

From the help of others, I got the answer but there is one confusing part.
He said that $\mu_{2}$ = 4$\mu_{1}$, but obviously that's not what I got in the below equations. $v_{2}$ = $\sqrt{.5}v_{1}$ So it was Would you explain how this is so?

$v_{2}$ = $\sqrt{\frac{τ_{2}}{\mu_{2}}}$
= $\sqrt{\frac{2τ_{1}}{m_{2}/2l_{1}}}$
= $\sqrt{\frac{4τ_{1}l_{1}}{m_{2}}}$
= 2$v_{1}$

 

[jbsteele]

f_{2} = \frac{v_{2}}{2l_{2}} = \frac{2v_{1}}{4l_{1}} = \frac{f_{1}}{2} = 250 Hz