1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

2D Green's Function - Bessel function equivalence

  1. Jan 16, 2015 #1

    RUber

    User Avatar
    Homework Helper

    1. The problem statement, all variables and given/known data
    This is not a homework problem per se, but I have been working on it for a few days, and cannot make the logical connection, so here it is:
    -- The problem is to show that
    ##\frac{1}{4\pi} \int_{-\infty}^{\infty} \frac{ e^{-\sqrt{\xi ^2 + \alpha^2 } |y-y'| + i \xi (x-x') }}{\sqrt{\xi ^2 + \alpha^2 }} d\xi= \frac i4 H_0^{(1)}(i \alpha R) = \frac 1{2\pi} K_0(\alpha R) ##
    Where ##R=\sqrt{(x-x')^2 + (y-y')^2 } ##.
    ##\alpha## is the constant wavenumber of a time-harmonic wave.
    This is part of the nitty-gritty explanation of the 2D free space Green's function for waves.
    Because the answer is a Bessel function, I expect there to be a change to cylindrical coordinates, which I attempt in section 3.

    2. Relevant equations
    I have a claim that the equation above is equivalent to
    ##i \int_{C(\phi)} e ^{ i k r \cos \beta } d\beta ##
    with ##C(\phi)## defined by:
    ## C(\phi) = \left\{ \begin{array}{l l} \displaystyle
    x=-|\phi| & y \text{ from } i \infty \text{ to } 0 \\
    y=0 & x \text{ from } -|\phi| \text{ to } \pi - |\phi|\\
    x= \pi - |\phi| & y \text{ from } 0 \text{ to } -i \infty
    \end{array} \right. ##

    Referring to Gradshteyn and Ryzhik, \cite{Gradshteyn2000}, this functional form is equivalent to $\rmi \pi H_0^1(kr)$.

    3. The attempt at a solution
    Attempting to change to polar coordinates centered at (x',y') using ## x = R\cos\theta, y = R\sin\theta## gives:
    ##\frac{1}{4\pi} \int_{-\infty}^{\infty} \frac{ e^{-\sqrt{\xi ^2 + \alpha^2 } R\sin\theta + i \xi R\cos\theta }}{\sqrt{\xi ^2 + \alpha^2 }} d\xi ## for ##\theta \in [0, \pi]##
    and
    ##\frac{1}{4\pi} \int_{-\infty}^{\infty} \frac{ e^{\sqrt{\xi ^2 + \alpha^2 } R\sin\theta + i \xi R\cos\theta }}{\sqrt{\xi ^2 + \alpha^2 }} d\xi ## for ##\theta \in [ -\pi,0]##.
    The reference I have simply says that with an appropriate subsitition, the equivalence can be seen.
    I cannot see what change of variables to make from here. I wonder if perhaps there is a more intuitive form of the Bessel K function that might help me see the connection.
    Thank you to anyone who might be able to point me in the right direction.
     
  2. jcsd
  3. Jan 21, 2015 #2
    Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: 2D Green's Function - Bessel function equivalence
  1. Green Function (Replies: 1)

  2. A green function (Replies: 5)

  3. Green Function (Replies: 3)

  4. Green's Function (Replies: 13)

Loading...