Results 1 to 3 of 3

Math Help - Please help in expansion of Helmholtz Equation in Spherical coordingaes.

  1. #1
    Junior Member
    Joined
    Jul 2013
    From
    Arizona
    Posts
    44
    Thanks
    2

    Please help in expansion of Helmholtz Equation in Spherical coordingaes.

    The book gave
    \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 (n=n'), (j=j') and (m= m')

    I got only
    \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: \nabla^2 u(r,\theta,\phi)=-k u(r,\theta,\phi) Where u_{n,m}(r,\theta,\phi)=R_{n}(r)Y_{n,m}(\theta,\phi  )

    Where 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 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 0\leq\; r \leq \;a where R(0) is finite and R(a)=0:

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

    \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 (n\neq \;n') and (m\neq \;m')

    \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 (n=n') and (m= m')(2)

    And \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 \alpha_{n,j} is the j zero of the Bessel function.


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

    \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), \int_{0}^{2\pi}\int_{0}^{\pi}|Y_{n,m}(\theta,\phi)  |^{2}\sin\theta \;d\theta \;d\phi=1

    Using (1), (3)
    \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

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

     \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 a. Please help

    Thanks
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Junior Member
    Joined
    Jul 2013
    From
    Arizona
    Posts
    44
    Thanks
    2

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

    This is about proof of orthogonality of spherical bessel functions
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member
    Joined
    Jul 2013
    From
    Arizona
    Posts
    44
    Thanks
    2

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

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

    \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

    \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:

    \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

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

     \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 a. This time, it's a lot simpler and more focus, please help me on this.

    Thanks
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Helmholtz eq. in cylindrical disk (+boundary value)
    Posted in the Differential Equations Forum
    Replies: 1
    Last Post: July 13th 2012, 10:08 AM
  2. [SOLVED] Spherical Equation
    Posted in the Calculus Forum
    Replies: 1
    Last Post: July 8th 2011, 07:38 AM
  3. Wave Equation In Spherical Coordinates
    Posted in the Differential Equations Forum
    Replies: 1
    Last Post: May 4th 2011, 01:37 PM
  4. Homogeneous Helmholtz Equation with Variable Coefficient
    Posted in the Differential Equations Forum
    Replies: 3
    Last Post: June 21st 2010, 07:31 PM
  5. Replies: 5
    Last Post: October 15th 2009, 04:33 PM

Search Tags


/mathhelpforum @mathhelpforum