PDE Helmholtz eq. in 3D + boundary value)

[marqushogas]

Hi!This is a quite sophisticated problem, but it’s interesting and challenging!

Consider the following case: Let’s say we have a 3-dimensional disk with a radius $r_{2}$ and a thickness $d$ (so it actually is a cylinder with a quite short height compared to radius). We’re interested in solving the (complex) vectorfield $E_{z}$ directed in the $\hat{z}$ direction for this disk. The PDE for this field is:

$\nabla^2 E_{z}+k \sigma E_{z}=0$

where $\sigma\geq0$ and $k$ is a pure imaginary number, with a real part 0 and a negative imaginary part. This disk has cylindrical rotation symmetry so $E_{z}$ does not depend on $\phi$. If we choose our cylindrical coordinate system so that the z-axis passes through the center and that z=0 at one of the circular planes of the disk; then the boundary values on the disk are the following:

1. $E_{z}(\rho,z=0)=0$ for all $\rho\in[0,r_{2}]$.
2. $\int_0^{r_{2}}E_{z}(\rho,z' )\rho\,d\rho=0$ for all $z' \in(0,d)$.
3. $\int_{r_{1}}^{r_{2}}E_{z}(\rho,z=d)\rho\,d\rho=-I/\sigma$, where $0 \leq r_{1} \leq r_{2}$ is a constant and $I$ is a complex constant.
4. $E_{z}(r' ,z=d)=0$ for all $r' \in[r_{0},r_{1}]$, where $r_{0}$ is a constant such that $0 \leq r_{0} \leq r_{1} \leq r_{2}$.
5. $\int_0^{r_{0}}E_{z}(\rho,z=d)\rho d \rho=I/\sigma$.
6. Obviously $E_z$ must also be finite for all points in the disk.

I would be very thankful for any insight or idea on how to solve this problem (full solutions not necessary acquired!). So if you can help me in any way I owe you a huge amount of thankfulness and respect!

(I have posted this problem in the classical physics forum as well because the background to the problem is in electromagnetism, but the problem here is mainly mathematical)

 

[HallsofIvy]

It looks to me like a "separation of variables" problem. Look for a solution of the form $E(z, r, \theta)= Z(z)R(r)\Theta(/theta)$

Once you do that, you will probably find that the equations in z and $\theta$ are relatively simple "constant coefficients" equations and that the equation in r is a "Bessel" equation.

 

[marqushogas]

Thanks, I have tried separation of variables but I'm not able to make the solution satisfy the boundary conditions and the sums are becoming quite hard to evaluate I think. (if you've done this would you like to show me?)

In more detail I arrived with one factor that is a function of $\rho$ alone and another that is a function $z$ alone (in this problem we does not have a dependence of $\phi$). In fact I got:

$P_{C}(\rho) = A_{C} J_{0}(j\sqrt{C}\rho)$

$Z_{C}(z) = B_{C} e^{z \sqrt{-k\sigma-C}}+D_{C} e^{-z \sqrt{-k\sigma-C}}$

($A, B, D$ being constants yet to be determined)

and $C$ is the (arbitrary) separation constant. To get a more general solution you can sum a linear combination of $P_{C}(\rho)Z_{C}(z)$ over all $C$, so that

$E_{z} = \sum_{C}\alpha_{C}P_{C}(\rho)Z_{C}(z)$

Where $\alpha_{C}$ is some "constant" depending on $C$. I have tried to choose $C = -k\sigma +n^2$ where $n$ is a positive integer so that the sum would actually look like a Fourier series (easy to evaluate). But then I can't make it satisfy the boundary condition! Any idea of how to choose $C$ in a smart way?? Or something else?

Big thanks!