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Math Help - wave partial differential eqn

  1. #1
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    wave partial differential eqn

    Can someone help me solve this: the vibrating string partial differential equation if an external force per unit mass proportional to displacement acts on the string.

    So I have uxx - ku = (1/a^2)utt

    How do I solve this? I've only seen it without external forces (here Pauls Online Notes : Differential Equations - Vibrating String). I've just started studying PDE's.

    Thanks for your help.
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  2. #2
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    Quote Originally Posted by PvtBillPilgrim View Post
    Can someone help me solve this: the vibrating string partial differential equation if an external force per unit mass proportional to displacement acts on the string.

    So I have uxx - ku = (1/a^2)utt

    How do I solve this? I've only seen it without external forces (here Pauls Online Notes : Differential Equations - Vibrating String). I've just started studying PDE's.

    Thanks for your help.
    Do you have boundary and initial conditions to go with this PDE?
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  3. #3
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    Quote Originally Posted by danny arrigo View Post
    Do you have boundary and initial conditions to go with this PDE?
    That's what I was wondering.

    I assume that the vibrating string has fixed endpoints.

    u(0,t) = u(L,t) = 0

    But you need some initial conditions as well to determine all arbitrary constants.
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    Quote Originally Posted by Rincewind View Post
    That's what I was wondering.

    I assume that the vibrating string has fixed endpoints.

    u(0,t) = u(L,t) = 0

    But you need some initial conditions as well to determine all arbitrary constants.
    It could also have a free endpoint where u_x(0,t) =0,\;\; \text{or}\;\;\;u_x(L,t) = 0.
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  5. #5
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    Quote Originally Posted by danny arrigo View Post
    It could also have a free endpoint where u_x(0,t) =0,\;\; \text{or}\;\;\;u_x(L,t) = 0.
    Yup. Anything is possible. I wonder if PvtBillPilgrim will clarify.
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  6. #6
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    Quote Originally Posted by Rincewind View Post
    That's what I was wondering.

    I assume that the vibrating string has fixed endpoints.

    u(0,t) = u(L,t) = 0

    But you need some initial conditions as well to determine all arbitrary constants.
    I want these conditions.
    Basically, u(x,0)=f(x)
    du/dt(x,0)=g(x) where this is a partial derivative
    u(0,t)=0=u(L,t)
    How does the -ku play into the final solution? It's solved in the website I gave in the original post, but with no external forces.
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  7. #7
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    Quote Originally Posted by PvtBillPilgrim View Post
    I want these conditions.
    Basically, u(x,0)=f(x)
    du/dt(x,0)=g(x) where this is a partial derivative
    u(0,t)=0=u(L,t)
    How does the -ku play into the final solution? It's solved in the website I gave in the original post, but with no external forces.
    Use the usual separation of variables u = T(t) X(x) so your equation separates

    \frac{T''}{a^2 T} = \frac{X'' - k X}{X} = c

    From the second

    X'' - k X = c X or X'' + \omega^2 X = 0 so c = - k - \omega^2. I'm thinking you can take it from here.
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