Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Two dimensional Poisson's equation, Green's function technique

  1. Aug 4, 2010 #1
    While considering perturbed gravitational potential of incompressible fluid in rectangular configuration, I encountered two dimensional Poisson's equation including the step function.
    I want to solve this equation

    [tex] \left( \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial z^2} \right) \psi(x, z) = [ \theta( z - ( a + \epsilon \cos(kx) ) } ) - \theta( z - a ) ] + [ \theta( z - ( - a - \epsilon \cos(kx) ) ) - \theta( z - (- a) ) ] [/tex]

    [tex]a[/tex] is the height from [tex]z=0[/tex] plane and [tex]\epsilon [/tex] is a small number much smaller than [tex]a[/tex].
    The source term is periodic in x direction with wavenumber [tex] k [/tex] and has a reflection symmetry.
    Hence I expect [tex]\psi[/tex] would be also periodic in x-direction and be an even function about z=0 plane.

    Do I have to use green's technique here to solve Poisson's equation involving periodic load?
    Can it be reduced to Helmholtz equation in one dimension like [tex] \psi^{\prime \prime} - k^2 \psi = ... [/tex] ?

    Any help would be greatly appreciated.

    Thank you~
    Last edited: Aug 4, 2010
  2. jcsd
  3. Aug 4, 2010 #2
    What if I simplify the problem?

    [tex] \left( \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial z^2} \right) \psi(x, z) = \theta(z - ( a + \epsilon \cos(kx) ) } [/tex]

    If I can solve the above one then the superposed solution can be obtained.

    help me. T.T
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook