The vibration

Vibration

Vibration refers to mechanical oscillations about an equilibrium point. The oscillations may be periodic such as the motion of a pendulum or random such as the movement of a tire on a gravel road.Vibration is occasionally "desirable"...

s of an idealized circular drum

Drum

The drum is a member of the percussion group of musical instruments, which is technically classified as the membranophones. Drums consist of at least one membrane, called a drumhead or drum skin, that is stretched over a shell and struck, either directly with the player's hands, or with a...

, essentially an elastic membrane

Acoustic membrane

An acoustic membrane is a thin layer that vibrates and are used in acoustics to produce or transfer sound, such as a drum, microphone, or loudspeaker....

 of uniform thickness attached to a rigid circular frame, are solutions of the wave equation

Wave equation

The wave equation is an important second-order linear partial differential equation for the description of waves – as they occur in physics – such as sound waves, light waves and water waves. It arises in fields like acoustics, electromagnetics, and fluid dynamics...

 with zero boundary conditions.

There exist infinitely many ways in which a drum can vibrate, depending on the shape of the drum at some initial time and the rate of change

Derivative

In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...

 of the shape of the drum at the initial time. Using separation of variables

Separation of variables

In mathematics, separation of variables is any of several methods for solving ordinary and partial differential equations, in which algebra allows one to rewrite an equation so that each of two variables occurs on a different side of the equation....

, it is possible to find a collection of "simple" vibration modes, and it can be proved that any arbitrarily complex vibration of a drum can be decomposed as a series

Series (mathematics)

A series is the sum of the terms of a sequence. Finite sequences and series have defined first and last terms, whereas infinite sequences and series continue indefinitely....

 of the simpler vibrations (analogously to the Fourier series

Fourier series

In mathematics, a Fourier series decomposes periodic functions or periodic signals into the sum of a set of simple oscillating functions, namely sines and cosines...

).

Motivation

Analyzing the vibrating drum problem explains percussion instruments such as drum

Drum

The drum is a member of the percussion group of musical instruments, which is technically classified as the membranophones. Drums consist of at least one membrane, called a drumhead or drum skin, that is stretched over a shell and struck, either directly with the player's hands, or with a...

s and timpani

Timpani

Timpani, or kettledrums, are musical instruments in the percussion family. A type of drum, they consist of a skin called a head stretched over a large bowl traditionally made of copper. They are played by striking the head with a specialized drum stick called a timpani stick or timpani mallet...

. However, there is also a biological application in the working of the eardrum

Eardrum

The eardrum, or tympanic membrane, is a thin membrane that separates the external ear from the middle ear in humans and other tetrapods. Its function is to transmit sound from the air to the ossicles inside the middle ear. The malleus bone bridges the gap between the eardrum and the other ossicles...

. From an educational point of view the modes of a two-dimensional object are a convenient way to visually demonstrate the meaning of modes, nodes, antinodes and even quantum number

Quantum number

Quantum numbers describe values of conserved quantities in the dynamics of the quantum system. Perhaps the most peculiar aspect of quantum mechanics is the quantization of observable quantities. This is distinguished from classical mechanics where the values can range continuously...

s. These concepts are important to the understanding of the structure of the atom.

The problem

Consider an open disk  of radius centered at the origin, which will represent the "still" drum shape. At any time the height of the drum shape at a point in measured from the "still" drum shape will be denoted by which can take both positive and negative values. Let denote the boundary

Boundary (topology)

In topology and mathematics in general, the boundary of a subset S of a topological space X is the set of points which can be approached both from S and from the outside of S. More precisely, it is the set of points in the closure of S, not belonging to the interior of S. An element of the boundary...

 of that is, the circle of radius centered at the origin, which represents the rigid frame to which the drum is attached.

The mathematical equation that governs the vibration of the drum is the wave equation with zero boundary conditions,

Here, is a positive constant, which gives the "speed" of vibration.

Due to the circular geometry of , it will be convenient to use cylindrical coordinates, Then, the above equations are written as

The radially symmetric case

We will first study the possible modes of vibration of a circular drum that are radially symmetric. Then, the function does not depend on the angle and the wave equation simplifies to


We will look for solutions in separated variables, Substituting this in the equation above and dividing both sides by yields

The left-hand side of this equality does not depend on and the right-hand side does not depend on it follows that both sides must equal to some constant We get separate equations for and :

The equation for has solutions which exponentially grow or decay for are linear or constant for and are periodic for Physically it is expected that a solution to the problem of a vibrating drum will be oscillatory in time, and this leaves only the third case, when Then, is a linear combination of sine and cosine functions,

Turning to the equation for with the observation that all solutions of this second-order differential equation are a linear combination of Bessel function

Bessel function

In mathematics, Bessel functions, first defined by the mathematician Daniel Bernoulli and generalized by Friedrich Bessel, are canonical solutions y of Bessel's differential equation:...

s of order 0,


The Bessel function is unbounded for which results in an unphysical solution to the vibrating drum problem, so the constant must be null. We will also assume as otherwise this constant can be absorbed later into the constants and coming from It follows that

The requirement that height be zero on the boundary of the drum results in the condition

The Bessel function has an infinite number of positive roots,

We get that for so

Therefore, the radially symmetric solutions of the vibrating drum problem that can be represented in separated variables are

where

The general case

The general case, when can also depend on the angle is treated similarly. We assume a solution in separated variables,

Substituting this into the wave equation and separating the variables, gives

where is a constant. As before, from the equation for it follows that with and

From the equation

we obtain, by multiplying both sides by and separating variables, that

and

for some constant Since is periodic, with period being an angular variable, it follows that

where and and are some constants. This also implies

Going back to the equation for its solution is a linear combination of Bessel function

Bessel function

In mathematics, Bessel functions, first defined by the mathematician Daniel Bernoulli and generalized by Friedrich Bessel, are canonical solutions y of Bessel's differential equation:...

s and With a similar argument as in the previous section, we arrive at

where with the -th positive root of

We showed that all solutions in separated variables of the vibrating drum problem are of the form

for

Animations of several vibration modes

A number of modes are shown below together with their quantum numbers. The analogous wave functions of the hydrogen atom are also indicated as well as the associated angular frequency .