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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~

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

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