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. This is called a normal mode. A membrane has an infinite number of these normal modes, starting with a lowest frequency one called the fundamental mode.
There exist infinitely many ways in which a membrane can vibrate, each depending on the shape of the membrane at some initial time, and the transverse velocity of each point on the membrane at that time. 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.
Analyzing the vibrating drum head problem explains percussion instruments such as drums and timpani. However, there is also a biological application in the working of the eardrum. 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.
Consider an open disk of radius centered at the origin, which will represent the "still" drum head shape. At any time the height of the drum head shape at a point in measured from the "still" drum head shape will be denoted by which can take both positive and negative values. Let denote the boundary of that is, the circle of radius 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 , it will be convenient to use cylindrical coordinates, Then, the above equations are written as
Here, 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 , is the radial membrane resultant at the membrane boundary (), , is the membrane thickness, and is the membrane density. If the membrane has uniform tension, the uniform tension force at a given radius, may be written
where is the membrane resultant in the azimuthal direction.
The radially symmetric case
We will first study the possible modes of vibration of a circular drum head 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 head 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 functions of order 0, since this is a special case of Bessel's differential equation:
The Bessel function is unbounded for which results in an unphysical solution to the vibrating drum head 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 head 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 head problem that can be represented in separated variables are
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
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 functions 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 head problem are of the form
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 .
- H. Asmar, Nakhle (2005). Partial differential equations with Fourier series and boundary value problems. Upper Saddle River, N.J.: Pearson Prentice Hall. p. 198. ISBN 0-13-148096-0.