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Thread: Please help in expansion of Helmholtz Equation in Spherical coordingaes.

  1. #1
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    Please help in expansion of Helmholtz Equation in Spherical coordingaes.

    The book gave
    $\displaystyle \int_{0}^{a}\int_{0}^{2\pi}\int_{0}^{\pi} j_{n} (\lambda_{n,j}r) j_{n'} (\lambda_{n',j'}r) Y_{n,m}(\theta,\phi)\overline{Y}_{n',m'}(\theta,\p hi) \sin\theta \;dr d\theta\; d\phi=\frac{a^{3}}{2}j^2_{n+1}(\alpha_{n+\frac{1}{ 2},j})$
    For $\displaystyle (n=n')$, $\displaystyle (j=j')$ and $\displaystyle (m= m')$

    I got only
    $\displaystyle \int_{0}^{a}\int_{0}^{2\pi}\int_{0}^{\pi} j_{n} (\lambda_{n,j}r) j_{n'} (\lambda_{n',j'}r) Y_{n,m}(\theta,\phi)\overline{Y}_{n',m'}(\theta,\p hi) \sin\theta \;dr d\theta\;d\phi=\frac{a^{2}}{2}j^2_{n+1}(\alpha_{(n +\frac{1}{2},j)})$



    Helmholtz equation: $\displaystyle \nabla^2 u(r,\theta,\phi)=-k u(r,\theta,\phi)$ Where $\displaystyle u_{n,m}(r,\theta,\phi)=R_{n}(r)Y_{n,m}(\theta,\phi )$

    Where $\displaystyle Y_{n,m}(\theta,\phi)=\sqrt{\frac{(2n+1)(n-m)!}{4\pi(n+m)!}} P_{n}^{m}(\cos\theta)e^{jm\phi}$ is the Spherical Harmonics.

    And $\displaystyle R_{n}(r)=j_{n}(\lambda_{n,j} r)=\sqrt{\frac{\pi}{2\lambda_{n,j} r}} J_{n+\frac{1}{2}}(\lambda_{n,j} r)$ (1) is the Spherical Bessel function.


    Orthogonal properties stated that

    For $\displaystyle 0\leq\; r \leq \;a $ where $\displaystyle R(0)$ is finite and $\displaystyle R(a)=0$:

    $\displaystyle R(a)=0\Rightarrow\; \lambda{n,j}=\frac{\alpha_{n,j}}{a}$

    $\displaystyle \int_{0}^{2\pi}\int_{0}^{\pi}Y_{n,m}(\theta,\phi)\ overline{Y}_{n',m'}(\theta,\phi) \sin\theta \;d\theta \;d\phi=0$ For $\displaystyle (n\neq \;n')$ and $\displaystyle (m\neq \;m')$

    $\displaystyle \int_{0}^{2\pi}\int_{0}^{\pi}Y_{n,m}(\theta,\phi)\ overline{Y}_{n',m'}(\theta,\phi) \sin\theta \;d\theta\; d\phi=1 $ For $\displaystyle (n=n')$ and $\displaystyle (m= m')$(2)

    And $\displaystyle \int_{0}^{a} r J_n^2(\lambda_{n,j} r)dr=\frac {a^{2}}{2}J_{n+1}^2(\alpha_{n,j})$ (3) where $\displaystyle \alpha_{n,j}$ is the j zero of the Bessel function.


    Here is my work:
    For $\displaystyle (n=n')$, $\displaystyle (j=j')$ and $\displaystyle (m= m')$

    $\displaystyle \int_{0}^{a}\int_{0}^{2\pi}\int_{0}^{\pi} j_{n} (\lambda_{n,j}r) j_{n'} (\lambda_{n',j'}r) Y_{n,m}(\theta,\phi)\overline{Y}_{n',m'}(\theta,\p hi) \sin\theta \;dr d\theta\; d\phi=\int_{0}^{a} j_{n}^{2} (\lambda_{n,j}r)dr\;\int_{0}^{2\pi}\int_{0}^{\pi}| Y_{n,m}(\theta,\phi)|^{2}\sin\theta \;d\theta \;d\phi=\int_{0}^{a} j_{n}^{2} (\lambda_{n,j}r)dr$

    As the two have different independent variables and from (2), $\displaystyle \int_{0}^{2\pi}\int_{0}^{\pi}|Y_{n,m}(\theta,\phi) |^{2}\sin\theta \;d\theta \;d\phi=1$

    Using (1), (3)
    $\displaystyle \int_{0}^{a}\int_{0}^{2\pi}\int_{0}^{\pi} j_{n} (\lambda_{(n,j)}r) j_{n'} (\lambda_{n',j'}r) Y_{n,m}(\theta,\phi)\overline{Y}_{n',m'}(\theta,\p hi) \sin\theta \;dr d\theta \;d\phi = \int_{0}^{a} j_{n}^{2} (\lambda_{(n,j)}r)rdr= \frac{\pi}{2\lambda_{(n,j)}}\int_{0}^{a} J_{n+\frac{1}{2}}^{2}(\lambda_{(n,j)}r)rd r$

    $\displaystyle R(a)=0\;\Rightarrow\;\lambda_{(n,j)}=\frac{\alpha_ {(n+\frac{1}{2},j)}}{a}$ as $\displaystyle \alpha_{(n+\frac{1}{2},j)}$ is the $\displaystyle j^{th}$ zero of $\displaystyle J_{n+\frac{1}{2}}(\lambda_{(n,j)})$

    $\displaystyle \frac{\pi}{2\lambda_{(n,j)}}\int_{0}^{a} J_{n+\frac{1}{2}}^{2}(\lambda_{(n,j)}r)rd r= \frac{\pi}{2\lambda_{(n,j)}} \frac{a^2}{2}J_{n+\frac{3}{2}}(\alpha_{(n+\frac{1} {2},j)})=\frac{a^{2}}{2}j^2_{n+1}(\alpha_{(n+\frac {1}{2},j)})$

    I am missing an $\displaystyle a$. Please help

    Thanks
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  2. #2
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    Re: Please help in expansion of Helmholtz Equation in Spherical coordingaes.

    This is about proof of orthogonality of spherical bessel functions
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  3. #3
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    Re: Please help in expansion of Helmholtz Equation in Spherical coordingaes.

    I already find proof that for $\displaystyle 0\leq r \leq a $ where $\displaystyle R(0)$ is finite and $\displaystyle R(a)=0$:

    $\displaystyle \int_{0}^{a}\int_{0}^{2\pi}\int_{0}^{\pi} j_{n} (\lambda_{(n,j)}r) j_{n'} (\lambda_{n',j'}r) Y_{n,m}(\theta,\phi)\overline{Y}_{n',m'}(\theta,\p hi) \sin\theta \;dr d\theta d\phi = \int_{0}^{a} j_{n}^{2} (\lambda_{(n,j)}r)rdr$



    So all I need to proof is

    $\displaystyle \int_{0}^{a} j_{n}^{2} (\lambda_{(n,j)}r)rdr=\frac{a^{3}}{2}j^2_{n+1}(\al pha_{(n+\frac{1}{2},j)})$




    But as in the last post:

    $\displaystyle \int_{0}^{a} j_{n}^{2} (\lambda_{(n,j)}r)rdr= \frac{\pi}{2\lambda_{(n,j)}}\int_{0}^{a} J_{n+\frac{1}{2}}^{2}(\lambda_{(n,j)}r)rd r$

    $\displaystyle R(a)=0\;\Rightarrow\;\lambda_{(n,j)}=\frac{\alpha_ {(n+\frac{1}{2},j)}}{a}$ as $\displaystyle \alpha_{(n+\frac{1}{2},j)}$ is the $\displaystyle j^{th}$ zero of $\displaystyle J_{n+\frac{1}{2}}(\lambda_{(n,j)})$

    $\displaystyle \frac{\pi}{2\lambda_{(n,j)}}\int_{0}^{a} J_{n+\frac{1}{2}}^{2}(\lambda_{(n,j)}r)rd r= \frac{\pi}{2\lambda_{(n,j)}} \frac{a^2}{2}J_{n+\frac{3}{2}}(\alpha_{(n+\frac{1} {2},j)})=\frac{a^{2}}{2}j^2_{n+1}(\alpha_{(n+\frac {1}{2},j)})$

    I am missing the $\displaystyle a$. This time, it's a lot simpler and more focus, please help me on this.

    Thanks
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