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.
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.
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.
We will first study the possible modes of vibration of a circular drum head that are axisymmetric. 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 be 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, so we choose for convenience. 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 axisymmetric solutions of the vibrating drum head problem that can be represented in separated variables are
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 frequencies . The values of are the roots of the Bessel function . This is deduced from the boundary condition which yields .
Mode (1s) with
Mode (2s) with
Mode (3s) with
Mode (2p) with
Mode (3p) with
Mode U12 (3p) V2
Mode (4p) with
Mode (3d) with
Mode (4d) with
Mode (5d) with
More values of can easily be computed using the following Python code with the scipy library:
fromscipyimportspecialasscm=0# order of the Bessel function (i.e. angular mode for the circular membrane)nz=3# desired number of rootsalpha_mn=sc.jn_zeros(m,nz)# outputs nz zeros of Jm