Question

The motion of a circular elastic membrane, such as a drumhead, is governed by the two-dimensional wave equation in polar coordinates $$ u_{r r}+(1 / r) u_{r}+\left(1 / r^{2}\right) u_{\theta \theta}=a^{-2} u_{t t} $$ Assuming that \(u(r, \theta, t)=R(r) \Theta(\theta) T(t),\) find ordinary differential equations satisfied by \(R(r), \Theta(\theta),\) and \(T(t) .\)

Short answer

Question: Determine the ordinary differential equations satisfied by \(R(r)\), \(\Theta(\theta)\), and \(T(t)\) for the given wave equation \(u(r,\theta,t)=R(r)\Theta(\theta)T(t)\) in polar coordinates. Answer: The ODEs satisfied by \(R(r)\), \(\Theta(\theta)\), and \(T(t)\) are 1. For the radial function: $$ R''(r) + \frac{1}{r}R'(r) + k^2 R(r) = 0 $$ 2. For the angular function: $$ \Theta''(\theta) + m^2 \Theta(\theta) = 0 $$ 3. And for the time function: $$ T''(t) + a^2 k^2 T(t) = 0 $$

Step-by-step solution

Apply Separation of Variables to Wave Equation

We are given the form \(u(r,\theta,t) = R(r) \Theta(\theta) T(t)\). First, we'll plug this into the wave equation and simplify it. $$ R''(r)\Theta(\theta)T(t) + (1/r) R'(r)\Theta(\theta)T(t) + (1/r^2) R(r)\Theta''(\theta)T(t) = a^{-2}R(r)\Theta(\theta)T''(t) $$

Divide by \(R(r)\Theta(\theta)T(t)\)

To separate the variables, we need to divide the entire equation by \(R(r)\Theta(\theta)T(t)\). This will give us a left-hand side that is dependent only on \(r\) and \(\theta\) and a right-hand side that is dependent only on \(t\). $$ \frac{R''(r)}{R(r)} + \frac{(1/r)R'(r)}{R(r)} + \frac{(1/r^2)\Theta''(\theta)}{\Theta(\theta)} = \frac{a^{-2}T''(t)}{T(t)} $$

Set Each Side to Constant

Since the left-hand side of the equation is dependent only on \(r\) and \(\theta\), and the right-hand side is dependent only on \(t\), each side must be equal to a constant, which we will call \(-k^2\) to simplify future steps. $$ \frac{R''(r)}{R(r)} + \frac{(1/r)R'(r)}{R(r)} + \frac{(1/r^2)\Theta''(\theta)}{\Theta(\theta)} = -k^2 = \frac{a^{-2}T''(t)}{T(t)} $$

Separate the Radial and Angular Functions

From the equation above, we have: $$ \frac{R''(r)}{R(r)} + \frac{(1/r)R'(r)}{R(r)} + \frac{(1/r^2)\Theta''(\theta)}{\Theta(\theta)} = -k^2 $$ Rearranging the equation, we can isolate the terms involving \(r\): $$ \frac{R''(r)}{R(r)} + \frac{(1/r)R'(r)}{R(r)} = -k^2 - \frac{(1/r^2)\Theta''(\theta)}{\Theta(\theta)} $$ Since we have separated the \(r\) and \(\theta\) variables, the terms on the left side involving \(r\) must equal a constant, and the terms on the right side involving \(\theta\) must equal another constant. We will set the left side equal to \(-k^2\) and the right side equal to \(-m^2\): $$ \frac{R''(r)}{R(r)} + \frac{(1/r)R'(r)}{R(r)} = -k^2 \\ -k^2 - \frac{(1/r^2)\Theta''(\theta)}{\Theta(\theta)} = -m^2 $$

Solve for \(R(r), \Theta(\theta)\), and \(T(t)\) ODEs

Now, we have separate ordinary differential equations for the radial, angular, and time functions: 1. For the radial function: $$ R''(r) + \frac{1}{r}R'(r) + k^2 R(r) = 0 $$ 2. For the angular function: $$ \Theta''(\theta) + m^2 \Theta(\theta) = 0 $$ 3. And for the time function: $$ T''(t) + a^2 k^2 T(t) = 0 $$ These are the ordinary differential equations satisfied by \(R(r)\), \(\Theta(\theta)\), and \(T(t)\).

Key concepts

Separation of Variables

The concept of separation of variables is a crucial technique in solving partial differential equations, like the wave equation given for the motion of a circular elastic membrane. By assuming a solution of the form \(u(r, \theta, t) = R(r) \Theta(\theta) T(t)\), we are implying that the solution can be broken down into a product of functions, each dependent on only one of the variables: \(r\), \(\theta\), or \(t\).
This method simplifies the original equation by reducing a complex partial differential equation (PDE) into simpler ordinary differential equations (ODEs) that are easier to solve.

• The first step involves assuming the product solution and substituting it into the given PDE.
• Next, the equation is rewritten such that each side of the equation depends on different groups of variables.
• By dividing by \(R(r)\Theta(\theta)T(t)\), it allows us to set each group to a constant, leading to separate ODEs.

This systematic approach is advantageous for problems defined in polar coordinates, where radial and angular symmetries allow the application of this method efficiently. It simplifies solving multi-variable problems by treating them separately, offering a powerful analytical tool in mathematical physics.

Ordinary Differential Equations

In solving the wave equation via separation of variables, ordinary differential equations (ODEs) for each separate function were obtained. An ODE is an equation containing a function and its derivatives, typically focusing on functions of a single variable. They're used to describe the behavior of dynamic systems and are fundamental in modeling natural phenomena.
Let's look at the ODEs derived:

• Radial function \(R(r)\): The ODE \(R''(r) + \frac{1}{r}R'(r) + k^2 R(r) = 0\) describes how the radial component changes with respect to \(r\). This equation accounts for radial symmetry and is importantly structured due to the polar coordinate system.


• Angular function \(\Theta(\theta)\): The equation \(\Theta''(\theta) + m^2 \Theta(\theta) = 0\) involves second-derivatives reflecting periodic or oscillatory behavior, common in angular treatments.


• Time function \(T(t)\): The ODE \(T''(t) + a^2 k^2 T(t) = 0\) is typical of wave-like phenomena, with solutions usually involving sinusoidal functions in terms of \(t\).

Each of these ODEs can be solved individually using methods appropriate for linear second-order differential equations, often involving characteristic equations or exploiting known solutions like sine and cosine for oscillatory systems.

Polar Coordinates

The use of polar coordinates in the wave equation for a circular elastic membrane is ideal due to the inherent circular symmetry of the problem. Unlike Cartesian coordinates, polar coordinates are described by a pair \((r, \theta)\), where \(r\) is the radius from a fixed point known as the pole (or origin), and \(\theta\) is the angular coordinate (or azimuth) which measures the angle from a fixed reference direction.
Polar coordinates simplify the description of problems involving circular or spherical regions. In the wave equation,

• The radial part \(r\) allows observing how the distance from the center affects the behavior of the wave.


• The angular part \(\theta\) considers the variation around the circumference, which is essential for waves propagating in a rotatory symmetry.


• The wave equation in polar coordinates incorporates specific terms like \(\frac{1}{r}\) and \(\frac{1}{r^2}\), reflecting the divergence and convergence in circular domains.

This coordinate system enables handling and finding solutions more naturally fitting the physical scenario, especially useful in engineering and physics problems involving circular boundaries or rotational symmetries.