Speed of waves in a nonuniform wire

MECP

1. The problem statement, all variables and given/known data
A wire that is 7.75 m long and has a uniform density of 10.0 g/cm^3 is pulled to a tension of Ft=376 N. The wire, however, does not have a uniform thickness; rather, it varies uniformly from an initial radius of 1.0 mm to a final radius of 3.3 mm where is it attached to a wall. If you send a wave pulse down the length of the string, how long does it take to reach the wall?


2. Relevant equations
Volume=∫∏(0.001 + 2.97e-4 X)^2 dx
Alternately, Volume= (∏h/3)*(a^2 +ab + b^2) where a= 3.3 mm, b= 1 mm, h=7.75 m (This is the volume of a conical frustrum)
Velocity= √(Ft/μ)
μ=mass/length


3. The attempt at a solution

Using the equations for volume, I found that Vol=1.234*10^-4 m^3. Then I multiplied by 10,000 kg/m^3 to get the mass of the wire. I divided that number by 7.75 and then divided 376 by that number, then took the square root. (Using the velocity function). I found the velocity to be 48.62 m/s. Then I divided 7.75 m by 48.62 m/s to get the time, 0.159 seconds.


My question is- is it really that simple? I couldn't find much information on nonuniform wires. Does the velocity of the wave stay constant throughout the whole wire?