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

In summary, the conversation discusses the process of finding a Laplace solution and the use of the Bessel equation and function. It also mentions the properties of the Bessel and modified Bessel functions, including their periodicity, oscillatory behavior, and different behaviors at the origin and infinity. The importance of a missing term in the final equation is also emphasized.
  • #1
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

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  • #2
You are missing a term from your final equation.

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

The Bessel functions are oscillatory with amplitude decaying to zero as [itex]x \to \infty[/itex]. The Bessel function of the first kind ,[itex]J_n[/itex], is bounded at the origin with [itex]J_0(0) = 1[/itex] and [itex]J_n(0) = 0[/itex] for [itex]n \geq 1[/itex]. The Bessel function of the second kind, [itex]Y_n[/itex], blows up at the origin. The modified Bessel functions are monotonic and positive with the modified function of the first kind, [itex]I_n[/itex], being bounded at the origin with [itex]I_0(0) = 1[/itex] and [itex]I_n(0) = 0[/itex] for [itex]n \geq 1[/itex] and increasing without limit as [itex]x \to \infty[/itex], and the modified function of the second kind, [itex]K_n[/itex], blowing up at the origin and decaying to 0 as [itex]x \to \infty[/itex].
 
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What is the 3D Laplace solution in Cylindrical Coordinates for a Hollow Cylindrical Tube?

The 3D Laplace solution in Cylindrical Coordinates for a Hollow Cylindrical Tube is a mathematical expression that describes the electric potential or gravitational potential inside and outside of a hollow cylindrical tube in three-dimensional space. It is derived using the Laplace equation, which is a partial differential equation that describes the variation of a scalar field in space.

What are Cylindrical Coordinates?

Cylindrical coordinates are a type of coordinate system used to describe points in three-dimensional space. They consist of a radial distance, an azimuthal angle, and a vertical distance from a reference point. In the context of a hollow cylindrical tube, the radial distance would represent the distance from the center of the tube, the azimuthal angle would represent the angle around the tube, and the vertical distance would represent the height along the tube.

Why is the 3D Laplace solution in Cylindrical Coordinates important?

The 3D Laplace solution in Cylindrical Coordinates is important because it allows us to accurately describe the electric or gravitational potential inside and outside of a hollow cylindrical tube. This is useful in many scientific and engineering applications, such as in the design of electrical circuits or in the study of fluid flow in pipes.

How is the 3D Laplace solution in Cylindrical Coordinates derived?

The 3D Laplace solution in Cylindrical Coordinates is derived by solving the Laplace equation in cylindrical coordinates. This involves using mathematical techniques such as separation of variables and boundary conditions to obtain a solution that satisfies the equation and the given boundary conditions.

What are some real-world applications of the 3D Laplace solution in Cylindrical Coordinates?

The 3D Laplace solution in Cylindrical Coordinates has many real-world applications, such as in the design of electronic circuits, the study of fluid flow in pipes and channels, and the analysis of electromagnetic fields in cylindrical structures. It is also used in the study of heat transfer and diffusion processes in cylindrical systems.

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