# Eigenvalues of a string with fixed ends and a mass in the middle

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1. Apr 6, 2016

### Frank93

Hi there. First of all, sorry for my bad english. I ´m trying to solve next exercise, from Vibrations and Waves in Continuos Mechanical Systems (Hagedorn, DasGupta): Determine the eigenfrequencies and mode-shapes of transverse vibration of a taut string with fixed ends and a discrete mass in the middle.

I set next wave equation:

T * ∂^2 Y/ ∂^2 X = ( ρ + m*δ(x-L/2) ) * ∂^2 Y/ ∂^2 t

where
T: tension, ρ: lineal density of the string, m: mass of the particle in the middle of the string, δ: Dirac delta, L: lenght of the string, and the boundary conditions are Y(0,t)=0 and Y(L,t)=0.

I dont know how to continue this, because to determine the eigenvalues, I need the wave equation, but it isnt the usual Y(x,t) = [A*Cos(k*x)+B*Sin(k*x)]*[C*Cos(w*t)+D*Sin(w*t)]. How can I solve this differntial equation? Or another way to solve this?

2. Apr 8, 2016

3. Apr 8, 2016

### Frank93

Thank you drvrm!!!