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Math Help - 3D Partial Differential Equation

  1. #1
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    3D Partial Differential Equation

    A rectangular plate of sides a and b has its edges fixed in the xy plane and set into transverse vibration. If the initial displacement is f(x,y) and the initial velocity is g(x,y), find the displacement u(x,y,t).
    Does anybody know how to solve this partial differential equation?
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  2. #2
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    Quote Originally Posted by BenWong View Post
    A rectangular plate of sides a and b has its edges fixed in the xy plane and set into transverse vibration. If the initial displacement is f(x,y) and the initial velocity is g(x,y), find the displacement u(x,y,t).
    Does anybody know how to solve this partial differential equation?
    It helps if you give us the PDE. Is it

    u_{tt} = c^2 \left( u_{xx} + u_{yy}\right) or u_{tt} + c^2 \left( u_{xxxx} + 2 u_{xxyy}  + u_{yyyy}\right)=0\;?
    Last edited by Jester; March 14th 2009 at 04:09 PM. Reason: changed PDE
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  3. #3
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    I believe it's the first one you had, although I'm not sure. The textbook I'm reading just has the physical setting.
    Let's assume the first.
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  4. #4
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    Assume the usual separation of variables u = T(t)X(x)Y(y). Substitute and separate into 3 ODE, i.e.
    \frac{1}{c^2} \frac{T''}{T} = \frac{X''}{X} + \frac{Y''}{Y}

    so

    \frac{X''}{X} = \lambda_1, \frac{Y''}{Y} = \lambda_2 , and \frac{1}{c^2} \frac{T''}{T} =\lambda_1 + \lambda_2

    Now, bring in the BC's for the first two ODE's giving (details omitted)

    X = c_1 \sin \frac{ \pi\, x }{a},\;\;\;Y = c_2 \sin \frac{ \pi\, y}{b}

    which leads to

    \frac{1}{c^2} \frac{T''}{T} = - \left( \frac{\pi}{a} \right)^2 - \left( \frac{\pi}{b} \right)^2

    which you can solve (again leading to sin's and cos's). Then match your initial condition.
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