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Homework Help: About Separation of Variables for the Laplace Equation

  1. Sep 7, 2009 #1
    1. The problem statement, all variables and given/known data

    This is a try for the solution of Laplace Equation. We have to calculate the potential distribution in a cylinder coordinate. However, there is a step really bring us trouble. Please go to the detail. You can either read it in the related URL, or in my PDF attachment..
    The uncompleted solution is:

    2. Relevant equations

    The method on the book is that:

    3. The attempt at a solution

    I really do not know what the basis of above equation is. Why can we get (2) from (1)? Does anyone give me any advice?
    Thanks in advance.



    Attached Files:

  2. jcsd
  3. Sep 8, 2009 #2


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    You do exactly what they say you should do:

    [tex]U_0=\sum_{m=0}^{\infty}A_m\sinh\left(\frac{P_m h}{a}\right)J_0\left(\frac{P_m r}{a}\right)[/tex]

    [tex]\implies\int_0^a U_0 J_0\left(\frac{P_n r}{a}\right)rdr=\int_0^a \left[\sum_{m=0}^{\infty}A_m\sinh\left(\frac{P_m h}{a}\right)J_0\left(\frac{P_m r}{a}\right)\right] J_0\left(\frac{P_n r}{a}\right)rdr=\sum_{m=0}^{\infty}A_m\sinh\left(\frac{P_m h}{a}\right)\left[\int_0^aJ_0\left(\frac{P_m r}{a}\right) J_0\left(\frac{P_n r}{a}\right)rdr\right][/tex]

    What does the orthoganality condition tell you about the integral on the RHS?
  4. Sep 8, 2009 #3
    Wo, Thanks for your reply so soon!

    I understood your means.

    About the orthogonality condition, actually, there is one of the charactrestics of Bessel function, isn't it?

    we have:

    [tex]\int _0^{\alpha }J_0\left(\frac{P_mr}{\alpha }\right)J_0\left(\frac{P_nr}{\alpha }\right)rdr=0[/tex] if [tex]m\neq n[/tex]

    where [tex]P_m[/tex] is the solution of Bessel Function [tex]J_0(x)=0[/tex]


  5. Sep 8, 2009 #4


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    Right, so the only non-zero term in the sum

    [tex]\sum_{m=0}^{\infty}A_m\sinh\left(\frac{P_m h}{a}\right)\left[\int_0^aJ_0\left(\frac{P_m r}{a}\right) J_0\left(\frac{P_n r}{a}\right)rdr\right][/tex]

    will be the [itex]m=n[/itex] term.

    [tex]\implies\int_0^a U_0 J_0\left(\frac{P_m r}{a}\right)rdr=A_m\sinh\left(\frac{P_m h}{a}\right)\int_0^a\left[J_0\left(\frac{P_m r}{a}\right)\right]^2 rdr[/tex]
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