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

[omyojj]

Hi,
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

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

$$a$$ is the height from $$z=0$$ plane and $$\epsilon$$ is a small number much smaller than $$a$$.
The source term is periodic in x direction with wavenumber $$k$$ and has a reflection symmetry.
Hence I expect $$\psi$$ 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 $$\psi^{\prime \prime} - k^2 \psi = ...$$ ?

Any help would be greatly appreciated.

Thank you~

 

[omyojj]

Ok..
What if I simplify the problem?

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

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

help me. T.T