Helmholtz eq. in cylindrical disk (+boundary value)
Hi! This is a quite sophisticated problem, but it’s fun and interesting!
Consider the following case: Let’s say we have a 3-dimensional disk with a radius and a thickness (so it actually is a cylinder with a quite short height compared to radius). We’re interested in solving the (complex) vectorfield directed in the direction for this disk. The PDE for this field is:
where and is a pure imaginary number, with a real part 0 and a negative imaginary part. This disk has cylindrical rotation symmetry so does not depend on . 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:
- for all .
- for all .
- , where is a constant and is a complex constant.
- for all , where is a constant such that .
- Obviously 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!
Re: Helmholtz eq. in cylindrical disk (+boundary value)
Sorry! In boundary value 3. it should be . (but I don't think that changes things too much).