Vibrating string displacement, partial differential problem

  1. 1. The problem statement, all variables and given/known data
    A damped vibrating string of length 1, that satisfies
    u_tt = u_xx - ([tex]\beta[/tex])u_t

    with the boundary conditions:
    u(0,t)=0
    u(1,t)=0

    initial conditions:

    u(x,0)=f(x)
    u_t(x,t)=0

    solve for u(x,t) if [tex]\beta[/tex]^2 < 4Pi^2


    3. The attempt at a solution

    if u(x,t)=F(x)G(t)

    So by using partial differential equations, I got:

    G''+[tex]\beta[/tex]G'+GP^2=0
    and
    F''+FP^2=0

    I solved for F(x) with B.Cs and got:
    F(x)=C*Sin(P*x), where C is a const.

    when I tried to solve for G(t), I got a long equation with 2 constants in there.
    If I try to solve for u(x,t) by using the F(x)G(t), I will have something with three unknow constants.

    Am I on the right track?
    I'm not sure how/when to apply I.Cs and [tex]\beta[/tex] to solve for u(x,t).


    Thank you very much!
     
  2. jcsd
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