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Math Help - Helmholtz eq. in cylindrical disk (+boundary value)

  1. #1
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    Helmholtz eq. in cylindrical disk (+boundary value)

    Hi! This is a quite sophisticated problem, but its fun and interesting!

    Consider the following case: Lets 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). Were 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)\rhod\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!
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  2. #2
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    Re: Helmholtz eq. in cylindrical disk (+boundary value)

    Sorry! In boundary value 3. it should be -I/\sigma. (but I don't think that changes things too much).
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