How do I find the normal modes of massless string w/ masses?

[ChickenTarm]

Homework Statement

So, a string with length L and a mass of M is given tension T. Find the frequencies of the smallest three modes of transverse motion. Then compare with a massless string with the same tension and length, but there are 3 masses of M/3 equally spaced. So this is problem #1
http://www.physics.purdue.edu/~jones105/phys42200_Spring2013/Assignment_5_Spring2013.pdf

Homework Equations

ν * λ = velocity
velocity = sqrt(T * L / M)
ν_n = nν₁ n = 1, 2, 3
ν₁ = √(T/(4ML))

The Attempt at a Solution

I tried using coupled oscillators and the equation for finding the frequencies.
ω_q=2ω₀|sin(q/2)|
q = nπ/(N+1) where n is the index and N is the number of particles
This does not give me the correct answer.
The correct answer is: .84ν₁, 1.55ν₁, and 2.04ν₁.

 

[TSny]

Hello. Welcome to PF!

What did you use for ω_o?

 

[ChickenTarm]

Hello. Welcome to PF!

What did you use for ω_o?

ω₀² = T/((M/3)(L/4))

 

[TSny]

OK. That should give the correct answer. You'll have to show your detailed calculations in order to find the mistake.