Question

By writing Laplace's equation in cylindrical coordinates \(r, \theta,\) and \(z\) and then assuming that the solution is axially symmetric (no dependence on \(\theta),\) we obtain the equation $$ u_{r r}+(1 / r) u_{r}+u_{z z}=0 $$ Assuming that \(u(r, z)=R(r) Z(z),\) show that \(R\) and \(Z\) satisfy the equations $$ r R^{n}+R^{\prime}+\lambda^{2} r R=0, \quad Z^{\prime \prime}-\lambda^{2} Z=0 $$ The equation for \(R\) is Bessel's equation of order zero with independent variable \(\lambda\).

Short answer

Answer: The equations for R and Z are as follows: For R: $$ r R^{''}(r) + R^{'}(r) + \lambda^2 r R(r) = 0 $$ For Z: $$ Z^{''}(z) - \lambda^2 Z(z) = 0 $$

Step-by-step solution

Separate the variables in Laplace's equation

Let us assume that the solution u(r, z) = R(r)Z(z). To do this, plug the assumed solution into the given Laplace's equation in cylindrical coordinates: $$ u_{rr} + \frac{1}{r} u_r + u_{zz} = \left(R''(r)Z(z) + \frac{1}{r}R'(r)Z(z) + R(r)Z''(z)\right) = 0 $$

Divide by RZ

Divide the entire equation by \(R(r)Z(z)\) to separate the variables: $$ \frac{R''(r)}{R(r)} + \frac{1}{r}\frac{R'(r)}{R(r)} + \frac{Z''(z)}{Z(z)} = 0 $$

Combine the R terms and Z terms separately

Now, rearrange the equation such that the terms with R are on the left side and the terms with Z are on the right side: $$ \frac{R''(r)}{R(r)} + \frac{1}{r}\frac{R'(r)}{R(r)} = -\frac{Z''(z)}{Z(z)} $$ Both sides of the equation must be equal to a constant (since they depend on different variables \(r\) and \(z\)). We denote this constant as \(\lambda^2\): $$ \frac{R''(r)}{R(r)} + \frac{1}{r}\frac{R'(r)}{R(r)} = \lambda^2 $$ $$ -\frac{Z''(z)}{Z(z)} = \lambda^2 $$

Analyze the equations for R and Z separately

Now, we deal with the R and Z equations separately. For the R equation: $$ \frac{R''(r)}{R(r)} + \frac{1}{r}\frac{R'(r)}{R(r)} = \lambda^2 $$ Multiplying both sides by \(R(r)\) and simplifying results in: $$ r R^{''}(r) + R^{'}(r) + \lambda^2 r R(r) = 0 $$ For the Z equation: $$ -\frac{Z''(z)}{Z(z)} = \lambda^2 $$ Multiplying both sides by \(Z(z)\) and simplifying results in: $$ Z^{''}(z) - \lambda^2 Z(z) = 0 $$

Final Result

The equations for both \(R\) and \(Z\) are: $$ r R^{''}(r) + R^{'}(r) + \lambda^2 r R(r) = 0 $$ $$ Z^{''}(z) - \lambda^2 Z(z) = 0 $$ The R equation is a Bessel's equation of order zero with an independent variable of \(\lambda\). We have successfully derived the equations for \(R\) and \(Z\).

Key concepts

Bessel's Equation

Bessel's equations arise naturally when solving partial differential equations (PDEs) in cylindrical coordinates using the method of separation of variables. Specifically, these equations are encountered in problems that have radial symmetry, such as heat conduction in a cylindrical object or the vibrations of a circular drumhead.

The general form of Bessel's equation is given by:
\[ x^2 y'' + x y' + (x^2 - n^2) y = 0 \]
where \( n \) is the order of the Bessel's equation and \( y \) is a function of \( x \). The solutions to this equation are known as Bessel functions; they are denoted as \( J_n(x) \) for Bessel functions of the first kind and \( Y_n(x) \) for the second kind.

In the context of Laplace's equation in cylindrical coordinates, when we look for a solution that is axially symmetric (meaning the solution does not change with the angle \( \theta \)), we end up with a Bessel's equation of zero order. Here, the radial part of the solution, denoted as \( R(r) \), must satisfy this special form of Bessel's equation:
\[ r R''(r) + R'(r) + \lambda^2 r R(r) = 0 \]
This equation has significant applications in physics and engineering, particularly in the study of wave propagation and static potentials.

Separation of Variables

Separation of variables is a widely used method for solving PDEs, where the solution is written as the product of functions, each depending on a single coordinate. The key idea is to reduce a complex PDE into simpler, ordinary differential equations (ODEs) that are more manageable.

In our Laplace's equation example, the method begins by assuming the solution can be expressed as \( u(r, z) = R(r)Z(z) \), where \( R \) is a function of radius \( r \) alone and \( Z \) is a function of \( z \) alone. By substituting this into the original PDE and separating the variables we obtain:\
\[ \frac{R''(r)}{R(r)} + \frac{1}{r}\frac{R'(r)}{R(r)} = -\frac{Z''(z)}{Z(z)} = \lambda^2 \]
Potentially, separation of variables can simplify the original PDE problem into ODE problems, as seen in the exercise. This is particularly helpful because ODEs have a well-established set of solution techniques.

Axially Symmetric Solutions

Axially symmetric solutions are those that do not change along a particular axis of symmetry—in this case, the \( \theta \) axis in cylindrical coordinates. This type of symmetry is key in simplifying the Laplace equation for problems such as flow around a cylinder, electric potential around a wire, or the temperature distribution in a long, circular pipe.

By assuming axial symmetry, we neglect any derivatives with respect to \( \theta \), reducing the three-dimensional problem to a two-dimensional one. The solution thus depends only on \( r \) and \( z \), which is more straightforward to analyze. The assumption of axial symmetry leads directly to a form of Bessel's equation, facilitating the integration process and allowing for closed-form solutions in many cases. This simplification is instrumental in solving many practical problems that exhibit cylindrical symmetry.

Partial Differential Equations

Partial Differential Equations (PDEs) are equations that involve rates of change with respect to multiple variables. They are fundamental in describing various phenomena such as physics, engineering, and finance.

Laplace's equation, \( abla^2 u = 0 \), is a classic PDE known for describing the behavior of potentials in electrostatics, gravitation, and fluid flow, among other areas. The solutions to Laplace's equation are known as harmonic functions and exhibit unique properties, such as having no local maxima or minima within a domain.

When we work with problems with a cylindrical geometry, it is natural to use cylindrical coordinates and rewrite Laplace's equation accordingly. This is where the knowledge of transforming PDEs into different coordinate systems becomes crucial for finding solutions that align with the physical symmetry of the problem at hand. The ability to solve PDEs by methods like separation of variables and understanding their solutions' behavior, such as axially symmetric solutions, plays a crucial role in the modeling and simulation of real-world systems.