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1. The problem statement, all variables and given/known data

The basic equation of the scalar Kirchhoff diffraction theory is:

[tex]\psi(\vec r_1) = \frac{1}{4\pi}\int_{S_2}\left[\frac{e^{ikr}}{r}\nabla \psi(\vec r_2) - \psi(\vec r_2)\nabla (\frac{e^{ikr}}{r})\right] \cdot d\vec S_2[/tex]

where [itex]\psi[/itex] satisfies the homogeneous (three-dimensional) Helmholtz equation and [itex]r = |\vec r_1 - \vec r_2|[/itex]. Derive the above equation, assuming that [itex]\vec r_1[/itex] is interior to the closed surface [itex]S_2[/itex]

2. Relevant equations

Helmholtz Equation

[tex](\nabla^2 + k^2)\psi = 0[/tex]

Modified Helmholtz Equation

[tex](\nabla^2 - k^2)\psi = 0[/tex]

Has Green's Function

[tex]G(\vec r_1, \vec r_2) = \frac{\mathrm{exp}(-k|\vec r_1 - \vec r_2|)}{4\pi|\vec r_1 - \vec r_2|}[/tex]

Green's Function of three-dimensional Laplacian ([itex]\nabla^2[/itex]) is proportional to

[tex]G(\vec r_1, \vec r_2) \propto \int \mathrm{exp}(i\vec k\cdot(\vec r_1 - \vec r_2))\frac{d^3k}{k^2}[/tex]

3. The attempt at a solution

We have [itex]\psi[/itex] as a solution of the equation

[tex](\nabla_1{}^2 + k^2)\psi(\vec r_1) = (\nabla_1{}^2 - k^2)\psi(\vec r_1) + 2k^2\psi(\vec r_1) = 0[/tex]

or

[tex]\nabla_1{}^2 \psi(\vec r_1) = -k^2\psi(\vec r_1)[/tex]

Which gives that

[tex](\nabla_1{}^2 - k^2)\psi(\vec r_1) = -2k^2\psi(\vec r_1) = 2\nabla_1{}^2 \psi(\vec r_1)[/tex]

So we can write the solution in terms of the Green's function for the modified Helmholtz Equation

[tex]\psi(\vec r_1) & = & \int_{V_2}G(\vec r_1, \vec r_2)2\nabla_1{}^2\psi(\vec r_1) dV_2[/tex]

And then I don't know how to take it from there to the solution. It doesn't look like it should be too hard, but I don't know how to do it.

Any help would be greatly appreciated.

Thanks in advance,

Devin

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# Homework Help: Green's Functions

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