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Homework Help: Eigenvalues of a string with fixed ends and a mass in the middle

  1. Apr 6, 2016 #1
    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

    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 don`t know how to continue this, because to determine the eigenvalues, I need the wave equation, but it isn`t 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. jcsd
  3. Apr 8, 2016 #2
  4. Apr 8, 2016 #3
    Thank you drvrm!!!
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