Vibrations of a circular membrane

A two-dimensional elastic membrane under tension can support transverse vibrations.

The properties of an idealized drumhead can be modeled by the vibrations of a circular membrane of uniform thickness, attached to a rigid frame.

Due to the phenomenon of resonance, at certain vibration frequencies, its resonant frequencies, the membrane can store vibrational energy, the surface moving in a characteristic pattern of standing waves.

A membrane has an infinite number of these normal modes, starting with a lowest frequency one called the fundamental frequency.

The vibrations of the membrane are given by the solutions of the two-dimensional wave equation with Dirichlet boundary conditions which represent the constraint of the frame.

It can be shown that any arbitrarily complex vibration of the membrane can be decomposed into a possibly infinite series of the membrane's normal modes.

This is analogous to the decomposition of a time signal into a Fourier series.

The study of vibrations on drums led mathematicians to pose a famous mathematical problem on whether the shape of a drum can be heard, with an answer (it cannot) being given in 1992 in the two-dimensional setting.

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 numbers.

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

centered at the origin, which will represent the "still" drum head shape.

the height of the drum head shape at a point

measured from the "still" drum head shape will be denoted by

centered at the origin, which represents the rigid frame to which the drum head is attached.

The mathematical equation that governs the vibration of the drum head is the wave equation with zero boundary conditions, Due to the circular geometry of

is a positive constant, which gives the speed at which transverse vibration waves propagate in the membrane.

In terms of the physical parameters, the wave speed, c, is given by where

We will first study the possible modes of vibration of a circular drum head that are axisymmetric.

and the wave equation simplifies to We will look for solutions in separated variables,

yields The left-hand side of this equality does not depend on

Physically it is expected that a solution to the problem of a vibrating drum head will be oscillatory in time, and this leaves only the third case,

is a linear combination of sine and cosine functions, Turning to the equation for

which results in an unphysical solution to the vibrating drum head problem, so the constant

be zero on the boundary of the drum head results in the condition The Bessel function

of the vibrating drum head problem that can be represented in separated variables are where

its solution is a linear combination of Bessel functions

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

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

The analogous wave functions of the hydrogen atom are also indicated as well as the associated angular frequencies

can easily be computed using the following Python code with the scipy library:[1]