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- Drum vibration mode12.gif -
One of the possible modes of vibration of an idealized circular drum head (mode u_{12} with the notation below). Other possible modes are shown at the bottom of the article.

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.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.

Practical significance

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.

The problem

Consider an open disk Omega of radius a centered at the origin, which will represent the “still” drum head shape. At any time t, the height of the drum head shape at a point (x, y) in Omega measured from the “still” drum head shape will be denoted by u(x, y, t), which can take both positive and negative values. Let partial Omega denote the boundary of Omega, that is, the circle of radius a 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,

frac{partial^2 u}{partial t^2} = c^2 left(frac{partial^2 u}{partial x^2}+frac{partial^2 u}{partial y^2}right) text{ for }(x, y) in Omega ,

u = 0text{ on }partial Omega.,

Due to the circular geometry of Omega, it will be convenient to use cylindrical coordinates, (r, theta, z). Then, the above equations are written as

frac{partial^2 u}{partial t^2} = c^2 left(frac{partial^2 u}{partial r^2}+frac {1}{r}frac{partial u}{partial r}+frac{1}{r^2}frac{partial^2 u}{partial theta^2}right) text{ for } 0 le r < a, 0 le theta le 2pi,

u = 0text{ for } r=a.,

Here, c 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

c = sqrt{frac{N_{rr}^*}{rho h}}

where N_{rr}^*, is the radial membrane resultant at the membrane boundary ( r = a), h, is the membrane thickness, and rho is the membrane density. If the membrane has uniform tension, the uniform tension force at a given radius, r may be written

F = rN^{r}_{rr}=rN^{r}_{thetatheta}

where N^{r}_{thetatheta} = N^{r}_{rr} is the membrane resultant in the azimuthal direction.

The axisymmetric case

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

frac{partial^2 u}{partial t^2} = c^2 left(frac{partial^2 u}{partial r^2}+frac {1}{r}frac{partial u}{partial r}right) .

We will look for solutions in separated variables, u(r, t) = R(r)T(t). Substituting this in the equation above and dividing both sides by c^2R(r)T(t) yields

frac{T(t)}{c^2T(t)} = frac{1}{R(r)}left(R(r) + frac{1}{r}R’(r)right).

The left-hand side of this equality does not depend on r, and the right-hand side does not depend on t, it follows that both sides must be equal to some constant K. We get separate equations for T(t) and R(r):

T’’(t) = Kc^2T(t) , rR’’(r)+R’(r)-KrR(r)=0.,

The equation for T(t) has solutions which exponentially grow or decay for K>0, are linear or constant for K=0 and are periodic for K0 and

T(t)=Acos clambda t + Bsin c lambda t.,

From the equation

frac{R(r)}{R(r)}+frac{R’(r)}{rR(r)} + frac{Theta(theta)}{r^2Theta(theta)}=-lambda^2

we obtain, by multiplying both sides by r^2 and separating variables, that

lambda^2r^2+frac{r^2R’’(r)}{R(r)}+frac{rR’(r)}{R(r)}=L

and

-frac{Theta’’(theta)}{Theta(theta)}=L,

for some constant L. Since Theta(theta) is periodic, with period 2pi, theta being an angular variable, it follows that

Theta(theta)=Ccos mtheta + D sin mtheta,,

where m=0, 1, dots and C and D are some constants. This also implies L=m^2.Going back to the equation for R(r), its solution is a linear combination of Bessel functions J_m and Y_m. With a similar argument as in the previous section, we arrive at

R(r) = J_m(lambda_{mn}r),, m=0, 1, dots, n=1, 2, dots,

where lambda_{mn}=alpha_{mn}/a, with alpha_{mn} the n-th positive root of J_m.We showed that all solutions in separated variables of the vibrating drum head problem are of the form

u_{mn}(r, theta, t) = left(Acos clambda_{mn} t + Bsin clambda_{mn} tright)J_mleft(lambda_{mn} rright)(Ccos mtheta + D sin mtheta)

for m=0, 1, dots, n=1, 2, dots

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 frequencies omega_{mn}=lambda_{mn}c=dfrac{alpha_{mn}}{a}c=alpha_{mn}c/a. The values of alpha_{mn} are the roots of the Bessel function J_m. This is deduced from the boundary condition forall theta in [0,2pi], forall t, u_{mn}(r=a, theta, t) = 0 which yields J_m(lambda_{mn}a) = J_m(alpha_{mn}) = 0.Image:Drum vibration mode01.gif|Mode u_{01} (1s) with alpha_{01}=2.40483Image:Drum vibration mode02.gif|Mode u_{02} (2s) with alpha_{02}=5.52008Image:Drum vibration mode03.gif|Mode u_{03} (3s) with alpha_{03}=8.65373File:Drum vibration mode11.gif|Mode u_{11} (2p) with alpha_{11}=3.83171File:Drum vibration mode12.gif|Mode u_{12} (3p) with alpha_{12}=7.01559File:Drum vibration mode13.gif|Mode u_{13} (4p) with alpha_{13}=10.1735 Image:Drum vibration mode21.gif|Mode u_{21} (3d) with alpha_{21}=5.13562Image:Drum vibration mode22.gif|Mode u_{22} (4d) with alpha_{22}=8.41724Image:Drum vibration mode23.gif|Mode u_{23} (5d) with alpha_{23}=11.6198More values of alpha_{mn} can easily be computed using the following Python code with the scipy library:SciPy user guide on Bessel functionsfrom scipy import special as scm = 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

References

BOOK, H. Asmar, Nakhle, Partial differential equations with Fourier series and boundary value problems, 2005, Pearson Prentice Hall, Upper Saddle River, N.J., 0-13-148096-0, 198,

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Vibrations of a circular membrane

Practical significance

The problem

The axisymmetric case

Animations of several vibration modes

See also

References