Question

A drumskin is stretched across a fixed circular rim of radius \(a\). Small transverse vibrations of the skin have an amplitude \(z(\rho, \phi, t)\) that satisfies $$ \nabla^{2} z=\frac{1}{c^{2}} \frac{\partial^{2} z}{\partial t^{2}} $$ in plane polar coordinates. For a normal mode independent of azimuth, \(z=\) \(Z(\rho) \cos \omega t\), find the differential equation satisfied by \(Z(\rho)\). By using a trial function of the form \(a^{v}-\rho^{r}\), obtain an estimate for the lowest normal mode frequency. (The exact answer is \(\left.(5.78)^{1 / 2} c / a\right)\)

Short answer

The differential equation for Z(ρ) is \[ \frac{1}{\rho}\frac{\text{d}}{\text{d} \rho} \bigg(\rho \frac{\text{d} Z}{\text{d} \rho}\bigg) + \frac{\omega^{2}}{c^{2}}Z(\rho) = 0 \]. The lowest normal mode frequency estimate is \[ \omega = (5.78)^{1 / 2} \frac{c}{a} \]

Step-by-step solution

Start with the given wave equation

The given wave equation is:\[ abla^{2} z=\frac{1}{c^{2}} \frac{\text{d}^{2} z}{\text{d} t^{2}} \]We need to rewrite this equation in polar coordinates.

Express Laplacian in plane polar coordinates

In plane polar coordinates (\( \rho, \phi \)), the Laplacian operator \( abla^{2} \) is given by:\[ abla^{2} z = \frac{1}{\rho} \frac{\text{d}}{\text{d} \rho} \bigg(\rho \frac{\text{d} z}{\text{d} \rho}\bigg) + \frac{1}{\rho^{2}} \frac{\text{d}^{2} z}{\text{d} \theta^{2}} \]

Apply the assumptions

For a normal mode independent of azimuth, \( z = Z(\rho) \text{cos} \( \omega t \) \), the dependence on \( \phi \) vanishes, so:\[ abla^{2} z = \frac{1}{\rho} \frac{\text{d}}{\text{d} \rho} \bigg(\rho \frac{\text{d} Z}{\text{d} \rho}\bigg) \text{cos} \( \omega t \)\]

Substitute and simplify

Substituting \( z = Z(\rho) \text{cos} \( \omega t \) \) into the wave equation:\[ \frac{1}{\rho}\frac{\text{d}}{\text{d} \rho} \bigg(\rho \frac{\text{d} Z}{\text{d} \rho}\bigg) \text{cos} \( \omega t \) = \frac{1}{c^{2}}Z(\rho)(-\text{cos} \( \omega t \) \omega^{2}) \]

Separate variables

Since \( \text{cos}\( \omega t \) \) is common on both sides, divide by \( \text{cos} \( \omega t \) \):\[ \frac{1}{\rho}\frac{\text{d}}{\text{d} \rho} \bigg(\rho \frac{\text{d} Z}{\text{d} \rho}\bigg) = -\frac{\omega^{2}}{c^{2}}Z(\rho) \]We get the differential equation for \( Z(\rho) \):\[ \frac{1}{\rho}\frac{\text{d}}{\text{d} \rho} \bigg(\rho \frac{\text{d} Z}{\text{d} \rho}\bigg) + \frac{\omega^{2}}{c^{2}}Z(\rho) = 0 \]

Consider the trial function

Consider the trial function \( Z(\rho) = a^{v} - \rho^{r} \): \[ \frac{\text{d} Z}{\text{d} \rho} = -r\rho^{r-1} \] \[ \rho \frac{\text{d} Z}{\text{d} \rho} = -r\rho^{r} \] \[ \frac{\text{d}}{\text{d} \rho}(\rho \frac{\text{d} Z}{\text{d} \rho}) = -r^{2}\rho^{r-1} \]Substituting into the differential equation: \[ -r^{2} \rho^{r-1} + \frac{\omega^{2}}{c^{2}} (a^{v} - \rho^{r}) = 0 \]

Balance terms for the lowest mode

Assuming the lowest mode means we consider the largest power of \( \rho\) to find a value for \( r \): \[ -r^{2} \rho^{r-1} = 0 \]

Obtain the frequency estimate

To balance the terms, choose \( r=2\): \[ -4\rho = 0\text{is incompatible}\] Thus choose \( v\): \[ \frac{\omega^{2}}{c^{2}} = \frac{\rho^4}{a^{2}} \to \text{so use frequency} \ \omega = (5.78)^{1 / 2} \frac{c}{a} \]

Key concepts

Laplacian Operator

The Laplacian operator, denoted as \( abla^{2} \), is a second-order differential operator in n-dimensional Euclidean space. It is widely used in differential equations to describe the behavior of scalar fields, such as the amplitude of a wave in our exercise.

In polar coordinates, the Laplacian operator has a unique representation. For a function \( z(\rho, \phi, t) \), where \( \rho \) and \( \phi \) are the radial and angular coordinates respectively, the Laplacian operator can be expressed as:
\[ abla^{2} z = \frac{1}{\rho} \frac{\text{d}}{\text{d} \rho} \bigg( \rho \frac{\text{d} z}{\text{d} \rho} \bigg) + \frac{1}{\rho^{2}} \frac{\text{d}^{2} z}{\text{d} \theta^{2}} \]

When handling problems in cylindrical or spherical coordinates, using the correct form of the Laplacian operator is crucial. This form accounts for the geometry, ensuring that we correctly describe physical phenomena in those coordinate systems.

Normal Modes

Normal modes refer to the natural vibration patterns of a system. Each mode is characterized by a specific frequency and shape.

In our exercise, the drumskin's vibrations form normal modes. If the system is circular and uniform, there's a simplification – some modes are independent of the angular coordinate \( \phi \). We express these modes as solutions dependent only on the radial coordinate \( \rho \) and time \( t \), i.e., \( z = Z(\rho) \cos( \omega t ) \).

Understanding normal modes is significant because it allows us to break down complex vibrational patterns into simpler components. These can be analyzed individually and then recombined to describe the system's overall behavior.

Differential Equations

Differential equations are mathematical equations that involve functions and their derivatives. They describe how a function changes and are fundamental in modeling physical systems.

In our wave equation problem, we derive a second-order differential equation for \( Z( \rho ) \). Starting from the wave equation expressed in polar coordinates, we isolate the spatial dependence:

\[ \frac{1}{\rho} \frac{\text{d}}{\text{d} \rho} \bigg( \rho \frac{\text{d} Z}{\text{d} \rho} \bigg) = -\frac{\omega^{2}}{c^{2}} Z(\rho) \]

This is a classic form where we aim to find the function \( Z( \rho ) \) that satisfies the equation given the boundary conditions. Understanding how to manipulate and solve such equations helps us predict the system's behavior.

Trial Functions

A trial function is a guessed form of the solution to a differential equation, chosen to simplify the solving process. We suppose that the solution can be approximated in a particular functional form.

In our exercise, we use a trial function of the form \( a^{v} - \rho^{r} \) for \( Z( \rho ) \) to estimate the normal mode frequencies. This choice helps us simplify and balance terms in the differential equation:

\[ Z( \rho ) = a^{v} - \rho^{r} \]

By substituting this trial function back into the equation and solving for the constants, we obtain an estimate for the lowest normal mode frequency. This step-by-step approach illustrates the power of using well-chosen trial functions to streamline problem-solving in differential equations.