Question

The wave equation in polar coordinates is $$ u_{r r}+(1 / r) u_{r}+\left(1 / r^{2}\right) u_{\theta \theta}=a^{-2} u_{t t} $$ Show that if \(u(r, \theta, t)=R(r) \Theta(\theta) T(t),\) then \(R, \Theta,\) and \(T\) satisfy the ordinary differential equations $$ \begin{aligned} r^{2} R^{\prime \prime}+r R^{\prime}+\left(\lambda^{2} r^{2}-n^{2}\right) R &=0 \\ \Theta^{\prime \prime}+n^{2} \Theta &=0 \\\ T^{\prime \prime}+\lambda^{2} a^{2} T &=0 \end{aligned} $$

Short answer

Question: Show that substituting the expression \(u(r, \theta, t) = R(r) \Theta(\theta) T(t)\) into the wave equation in polar coordinates results in a set of ordinary differential equations for the functions \(R\), \(\Theta\), and \(T\). Answer: After substitution and separating the variables, we obtained the following ordinary differential equations: 1. \(r^{2} R^{\prime \prime}(r)+r R^{\prime}(r)+\left(\lambda^{2} r^{2}-n^{2}\right) R(r) = 0\) 2. \(\Theta^{\prime \prime}(\theta)+n^{2} \Theta(\theta) =0\) 3. \(T^{\prime \prime}(t)+\lambda^{2} a^{2} T(t) = 0\)

Step-by-step solution

Substitute the expression for \(u\) into the wave equation

Replace \(u(r, \theta, t)\) in the given wave equation with \(R(r) \Theta(\theta) T(t)\) and obtain: $$\left[R''(r) + \frac{1}{r}R'(r) + \frac{1}{r^2} \Theta''(\theta)\right] \Theta(\theta)T(t) = a^{-2} R(r) \Theta(\theta) T''(t).$$

Divide both sides by \(R \Theta T\)

Divide the equation obtained in Step 1 by the product of \(R(r)\), \(\Theta(\theta)\), and \(T(t)\): $$\frac{R''(r) + \frac{1}{r}R'(r) + \frac{1}{r^2}\Theta''(\theta) \Theta(\theta) }{R(r) \Theta(\theta)} = \frac{a^{-2} T''(t)}{T(t)}.$$

Separate the variables

Since the left-hand side of the equation depends only on \(r\) and \(\theta\), and the right-hand side depends only on \(t\), we can set each side equal to a constant \(\lambda^2\). Thus, we have two equations: $$\frac{R''(r) + \frac{1}{r}R'(r)}{R(r)} - \frac{n^{2}}{r^{2}} = \lambda^2,$$ $$\frac{1}{\Theta(\theta)} \Theta''(\theta) = -n^2,$$ where \(n\) is an integer to maintain the periodicity of the angular function, and $$\frac{a^{-2} T''(t)}{T(t)} = \lambda^2.$$ Now we can write three separate ordinary differential equations for \(R(r)\), \(\Theta(\theta)\), and \(T(t)\):

Write the ordinary differential equations

From the separated variables equations in Step 3, we obtain the three ordinary differential equations: $$ \begin{aligned} r^{2} R^{\prime \prime}(r)+r R^{\prime}(r)+\left(\lambda^{2} r^{2}-n^{2}\right) R(r) &=0, \\ \Theta^{\prime \prime}(\theta)+n^{2} \Theta(\theta) &=0, \\\ T^{\prime \prime}(t)+\lambda^{2} a^{2} T(t) &=0. \end{aligned} $$ These are the required ordinary differential equations for \(R\), \(\Theta\), and \(T\).

Key concepts

Polar Coordinates

Polar coordinates are a way of representing points in a plane using a distance and an angle. Instead of using traditional x and y coordinates, polar coordinates employ:

• \( r \) - the radial distance from a fixed point, known as the pole (equivalent to the origin in Cartesian coordinates).
• \( \theta \) - the angular coordinate, which is the angle from a fixed direction (typically the positive x-axis).

The main advantage of polar coordinates is that they simplify problems involving symmetry around a central point. In many cases, such as in this wave equation problem, the circular symmetry makes polar coordinates a more natural choice than Cartesian coordinates. This makes equations easier to solve and visualize. Converting between polar and Cartesian coordinates involves the relationships: \( x = r \cos(\theta) \) and \( y = r \sin(\theta) \). This transformation is crucial when solving problems with circular or spherical symmetry.

Ordinary Differential Equations

Ordinary Differential Equations (ODEs) are equations involving functions and their derivatives. An ODE relates some function of one variable to its derivatives and is distinguished from partial differential equations, which involve multiple variables. In the wave equation example, the ODEs for \( R(r) \), \( \Theta(\theta) \), and \( T(t) \) focus on solutions that are only functions of their respective single variables.
The process of solving these involves:

• Finding the roots or solutions that satisfy the given equation with respect to their variable.
• In the case of our wave equation, manipulating the separated ordinary differential equations to find general solutions (often involving specific functions like trigonometric functions for angular solutions).

Understanding the behavior of these equations is useful in many fields, such as physics and engineering, for modeling oscillations, waves, and other dynamic processes. By solving the ODEs for \( R(r) \), \( \Theta(\theta) \), and \( T(t) \), one can understand how the components of the wave behave individually.

Variable Separation

Variable separation is a powerful technique used to solve differential equations, where variables are separated into different parts of an equation to be solved independently. This approach works particularly well with problems that can be expressed as products of single-variable functions.
The key idea is to manipulate equations such that each side of the equation depends on a different variable. This allows us to equate each part to a constant, leading to simpler ordinary differential equations.

• In our wave equation exercise, expressing the unknown function as \( u(r, \theta, t) = R(r) \Theta(\theta) T(t) \) enables separation of variables because each function is only dependent on its specified variable.
• The result is a set of ODEs for each function (\( R(r) \), \( \Theta(\theta) \), \( T(t) \)), where techniques for solving ODEs can be applied separately to each part.

This method simplifies complex PDEs by transforming them into several easier-to-manage ODEs, allowing for more straightforward solutions.

Partial Differential Equations

Partial Differential Equations (PDEs) involve multiple independent variables and their partial derivatives. They are essential for modeling various phenomena in physics, engineering, and other sciences. The wave equation given in polar coordinates is a PDE because the function \( u \) depends on more than one variable (\( r, \theta, \) and \( t \)).
Key characteristics of PDEs include:

• PDEs can describe distributions over a spatial domain and their evolution over time.
• The wave equation is a second-order PDE, meaning the highest derivative order involved is two.

The method of separation of variables, as shown in the solution, is a common strategy to tackle PDEs by reducing them into simpler ODEs. Solving the ODEs provides insights into the entire system's behavior, demonstrating how different variables influence the solution. Understanding PDEs helps model waves, heat flow, and other dynamic systems in a more comprehensive way than ODEs alone.