3D Laplace solution in Cylindrical Coordinates For a Hollow Cylindrical Tube

[jkthejetplane]

Homework Statement

Find the general series solution for laplace in cylindrical coordinates

Relevant Equations

for this i have always used (s,phi,z)

Here is the initial problem and my attempt at getting Laplace solution. I get lost near the end and after some research, ended up with the Bessel equation and function. I don't completely understand what this is or even if this i the direction I go in.
This is a supplemental thing that I want to nail down for review to get my brain up to speed again for this semester

 

[pasmith]

You are missing a term from your final equation.

The solution must be periodic in \phi, so the dependence will be \Phi&#039;&#039; = -n^2\Phi. You do then have Z&#039;&#039; = CZ, but at this point there's no reason to believe that C \geq 0. So your radial dependence satisfies <br /> s^2 S&#039;&#039; + s S&#039; + (Cs^2 - n^2)S = 0. Setting k = |C|^{1/2} and x = ks turns this into <br /> x^2 \frac{d^2S}{dx^2} + x \frac{dS}{dx} + (\operatorname{sgn}(C) x^2 - n^2)S = 0 which is the Bessel equation if C &gt; 0 and the modified Bessel equation if C &lt; 0. If C = 0 then the dependence on z is Az + B and the radial dependence is s^\alpha where \alpha depends on n.

The Bessel functions are oscillatory with amplitude decaying to zero as x \to \infty. The Bessel function of the first kind ,J_n, is bounded at the origin with J_0(0) = 1 and J_n(0) = 0 for n \geq 1. The Bessel function of the second kind, Y_n, blows up at the origin. The modified Bessel functions are monotonic and positive with the modified function of the first kind, I_n, being bounded at the origin with I_0(0) = 1 and I_n(0) = 0 for n \geq 1 and increasing without limit as x \to \infty, and the modified function of the second kind, K_n, blowing up at the origin and decaying to 0 as x \to \infty.