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Math Help - Wave Equation using seperation of variables

  1. #1
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    Wave Equation using seperation of variables

    Hello i was wondering if someone could help me out and show me how to solve the following heat wave problem.We dont have a book for this class and i couldnt make it to my last class so i dont knoow how to go about this question

    Solve the wave equation using seperation of variables and show thte the solution reduces to D'Alembert's solution

    \mu_{tt} = c^2\mu_{xx}
    \mu(0,t) = 0
    \mu(L,t) = 0
    \mu(x,0) = f(x)
    \mu_t(x,0) = 0
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  2. #2
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    Quote Originally Posted by flaming View Post
    Hello i was wondering if someone could help me out and show me how to solve the following heat wave problem.We dont have a book for this class and i couldnt make it to my last class so i dont knoow how to go about this question

    Solve the wave equation using seperation of variables and show thte the solution reduces to D'Alembert's solution

    \mu_{tt} = c^2\mu_{xx}
    \mu(0,t) = 0
    \mu(L,t) = 0
    \mu(x,0) = f(x)
    \mu_t(x,0) = 0
    I would feel better if you would at least show that you know what "separation of variables" is!

    Assume \mu(x,t)= X(x)T(t). Then \mu_{tt}= XT^{''} and \mu_{xx}= X^{''} T so the equation becomes XT^{''} = c^2X^{''} T. Dividing on both sides of the equation by XT gives \frac{T^{''} }{T}= c^2\frac{X^{''} }{X}. Since the left side depends only on t and the right side only on x, to be equal they must each equal a constant.

    That gives two ordinary equations:
    c^2\frac{X^{''} }{X}= \lambda
    or
    c^2X^{''} = \lambda X
    and
    \frac{T^{''} }{T}= \lambda
    or
    T^{''} = \lambda T.
    You should be able to find the general solution to each of those, depending on \lambda of course.
    What must \lambda be in order that X(0)= 0 and X(L)= 0?

    I have a feeling I have repeated what your text book says!
    Last edited by mr fantastic; March 28th 2009 at 01:48 PM. Reason: Added a missing latex tag. Replaced " with ^{''}
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