Vibrational Modes Of A Drum
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A two-dimensional elastic membrane under tension can support transverse vibrations. The properties of an idealized
drumhead A drumhead or drum skin is a membrane stretched over one or both of the open ends of a drum. The drumhead is struck with sticks, mallets, or hands, so that it vibrates and the sound resonates through the drum. Additionally outside of percus ...
can be modeled by the vibrations of a circular membrane of uniform thickness, attached to a rigid frame. Due to the phenomenon of
resonance Resonance describes the phenomenon of increased amplitude that occurs when the frequency of an applied periodic force (or a Fourier component of it) is equal or close to a natural frequency of the system on which it acts. When an oscillatin ...
, at certain vibration
frequencies Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from ''angular frequency''. Frequency is measured in hertz (Hz) which is eq ...
, its
resonant frequencies Resonance describes the phenomenon of increased amplitude that occurs when the frequency of an applied periodic force (or a Fourier component of it) is equal or close to a natural frequency of the system on which it acts. When an oscillati ...
, the membrane can store vibrational energy, the surface moving in a characteristic pattern of
standing wave In physics, a standing wave, also known as a stationary wave, is a wave that oscillates in time but whose peak amplitude profile does not move in space. The peak amplitude of the wave oscillations at any point in space is constant with respect ...
s. This is called a
normal mode A normal mode of a dynamical system is a pattern of motion in which all parts of the system move sinusoidally with the same frequency and with a fixed phase relation. The free motion described by the normal modes takes place at fixed frequencies. ...
. A membrane has an infinite number of these normal modes, starting with a lowest frequency one called the
fundamental mode A normal mode of a dynamical system is a pattern of motion in which all parts of the system move sinusoidally with the same frequency and with a fixed phase relation. The free motion described by the normal modes takes place at fixed frequencies ...
. 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 The (two-way) wave equation is a second-order linear partial differential equation for the description of waves or standing wave fields — as they occur in classical physics — such as mechanical waves (e.g. water waves, sound waves and s ...
with
Dirichlet boundary conditions In the mathematical study of differential equations, the Dirichlet (or first-type) boundary condition is a type of boundary condition, named after Peter Gustav Lejeune Dirichlet (1805–1859). When imposed on an ordinary or a partial differential ...
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 Series may refer to: People with the name * Caroline Series (born 1951), English mathematician, daughter of George Series * George Series (1920–1995), English physicist Arts, entertainment, and media Music * Series, the ordered sets used i ...
of the membrane's normal modes. This is analogous to the decomposition of a time signal into a
Fourier series A Fourier series () is a summation of harmonically related sinusoidal functions, also known as components or harmonics. The result of the summation is a periodic function whose functional form is determined by the choices of cycle length (or ''p ...
. 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 being given in 1992 in the two-dimensional setting.


Motivation

Analyzing the vibrating drum head problem explains percussion instruments such as
drum The drum is a member of the percussion group of musical instruments. In the Hornbostel-Sachs classification system, it is a membranophone. Drums consist of at least one membrane, called a drumhead or drum skin, that is stretched over a she ...
s and
timpani Timpani (; ) or kettledrums (also informally called timps) are musical instruments in the percussion family. A type of drum categorised as a hemispherical drum, they consist of a membrane called a head stretched over a large bowl traditionall ...
. However, there is also a biological application in the working of the
eardrum In the anatomy of humans and various other tetrapods, the eardrum, also called the tympanic membrane or myringa, is a thin, cone-shaped membrane that separates the external ear The outer ear, external ear, or auris externa is the extern ...
. 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 In quantum physics and chemistry, quantum numbers describe values of conserved quantities in the dynamics of a quantum system. Quantum numbers correspond to eigenvalues of operators that commute with the Hamiltonian—quantities that can be kno ...
s. These concepts are important to the understanding of the structure of the atom.


The problem

Consider an
open disk In geometry, a disk (also spelled disc). is the region in a plane bounded by a circle. A disk is said to be ''closed'' if it contains the circle that constitutes its boundary, and ''open'' if it does not. For a radius, r, an open disk is usua ...
\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 Boundary or Boundaries may refer to: * Border, in political geography Entertainment *Boundaries (2016 film), ''Boundaries'' (2016 film), a 2016 Canadian film *Boundaries (2018 film), ''Boundaries'' (2018 film), a 2018 American-Canadian road trip ...
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 = c^2 \left(\frac+\frac\right) \text(x, y) \in \Omega \, : u = 0\text\partial \Omega.\, Due to the circular geometry of \Omega, it will be convenient to use
cylindrical coordinates A cylindrical coordinate system is a three-dimensional coordinate system that specifies point positions by the distance from a chosen reference axis ''(axis L in the image opposite)'', the direction from the axis relative to a chosen reference di ...
, (r, \theta, z). Then, the above equations are written as :\frac = c^2 \left(\frac+\frac \frac+\frac\frac\right) \text 0 \le r < a, 0 \le \theta \le 2\pi\, : u = 0\text 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 where N_^*, 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^_=rN^_ where N^_ = N^_ 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 Rotational symmetry, also known as radial symmetry in geometry, is the property a shape has when it looks the same after some rotation by a partial turn. An object's degree of rotational symmetry is the number of distinct orientations in which i ...
. Then, the function u does not depend on the angle \theta, and the wave equation simplifies to :\frac = c^2 \left(\frac+\frac \frac\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 = \frac\left(R''(r) + \fracR'(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 K<0. 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, K<0, so we choose K=-\lambda^2 for convenience. Then, T(t) is a linear combination of sine and cosine functions, : T(t)=A\cos c\lambda t + B\sin c \lambda t.\, Turning to the equation for R(r), with the observation that K=-\lambda^2, all solutions of this second-order differential equation are a linear combination of
Bessel function Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions of Bessel's differential equation x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0 for an arbitrary ...
s of order 0, since this is a special case of
Bessel's differential equation Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions of Bessel's differential equation x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0 for an arbitrary ...
: :R(r) = c_1 J_0(\lambda r)+ c_2 Y_0(\lambda r).\, The Bessel function Y_0 is unbounded for r\to 0, which results in an unphysical solution to the vibrating drum head problem, so the constant c_2 must be null. We will also assume c_1=1, as otherwise this constant can be absorbed later into the constants A and B coming from T(t). It follows that : R(r) = J_0(\lambda r). The requirement that height u be zero on the boundary of the drum head results in the condition : R(a) = J_0(\lambda a) = 0. The Bessel function J_0 has an infinite number of positive roots, : 0< \alpha_ < \alpha_ < \cdots We get that \lambda a=\alpha_, for n=1, 2, \dots, so : R(r) = J_0\left(\fracr\right). Therefore, the axisymmetric solutions u of the vibrating drum head problem that can be represented in separated variables are : u_(r, t) = \left(A\cos c\lambda_ t + B\sin c\lambda_ t\right)J_0\left(\lambda_ r\right)\textn=1, 2, \dots, \, where \lambda_ = \alpha_/a.


The general case

The general case, when u can also depend on the angle \theta, is treated similarly. We assume a solution in separated variables, : u(r, \theta, t) = R(r)\Theta(\theta)T(t).\, Substituting this into the wave equation and separating the variables, gives : \frac = \frac+\frac + \frac=K where K is a constant. As before, from the equation for T(t) it follows that K=-\lambda^2 with \lambda>0 and : T(t)=A\cos c\lambda t + B\sin c \lambda t.\, From the equation : \frac+\frac + \frac=-\lambda^2 we obtain, by multiplying both sides by r^2 and separating variables, that : \lambda^2r^2+\frac+\frac=L and : -\frac=L, for some constant L. Since \Theta(\theta) is periodic, with period 2\pi, \theta being an angular variable, it follows that : \Theta(\theta)=C\cos m\theta + D \sin m\theta,\, 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 function Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions of Bessel's differential equation x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0 for an arbitrary ...
s J_m and Y_m. With a similar argument as in the previous section, we arrive at : R(r) = J_m(\lambda_r),\, m=0, 1, \dots, n=1, 2, \dots, where \lambda_=\alpha_/a, with \alpha_ 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_(r, \theta, t) = \left(A\cos c\lambda_ t + B\sin c\lambda_ t\right)J_m\left(\lambda_ r\right)(C\cos m\theta + D \sin m\theta) 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_=\lambda_c=\dfracc=\alpha_c/a. The values of \alpha_ are the roots of the Bessel function J_m. This is deduced from the boundary condition \forall \theta \in ,2\pi \forall t, \ u_(r=a, \theta, t) = 0 which yields J_m(\lambda_a) = J_m(\alpha_) = 0. Image:Drum vibration mode01.gif, Mode u_ (1s) with \alpha_=2.40483 Image:Drum vibration mode02.gif, Mode u_ (2s) with \alpha_=5.52008 Image:Drum vibration mode03.gif, Mode u_ (3s) with \alpha_=8.65373 Image:Drum vibration mode11.gif, Mode u_ (2p) with \alpha_=3.83171 Image:Drum vibration mode12.gif, Mode u_ (3p) with \alpha_=7.01559 Image:Drum vibration mode13.gif, Mode u_ (4p) with \alpha_=10.1735 Image:Drum vibration mode21.gif, Mode u_ (3d) with \alpha_=5.13562 Image:Drum vibration mode22.gif, Mode u_ (4d) with \alpha_=8.41724 Image:Drum vibration mode23.gif, Mode u_ (5d) with \alpha_=11.6198 More values of \alpha_ can easily be computed using the following Python code with the scipy library:SciPy user guide on Bessel functions
/ref> from scipy import special as sc m = 0 # order of the Bessel function (i.e. angular mode for the circular membrane) nz = 3 # desired number of roots alpha_mn = sc.jn_zeros(m, nz) # outputs nz zeros of Jm


See also

*
Vibrating string A vibration in a strings (music), string is a wave. Acoustic resonance#Resonance of a string, Resonance causes a vibrating string to produce a sound with constant frequency, i.e. constant pitch (music), pitch. If the length or tension of the strin ...
, the one-dimensional case *
Chladni patterns Ernst Florens Friedrich Chladni (, , ; 30 November 1756 – 3 April 1827) was a German physicist and musician. His most important work, for which he is sometimes labeled as the father of acoustics, included research on vibrating plates an ...
, an early description of a related phenomenon, in particular with musical instruments; see also
cymatics Cymatics (from grc, κῦμα, translit=kyma, translation=wave) is a subset of modal vibrational phenomena. The term was coined by Hans Jenny (1904-1972), a Swiss follower of the philosophical school known as anthroposophy. Typically the surf ...
*
Hearing the shape of a drum To hear the shape of a drum is to infer information about the shape of the drumhead from the sound it makes, i.e., from the list of overtones, via the use of mathematical theory. "Can One Hear the Shape of a Drum?" is the title of a 1966 article ...
, characterising the modes with respect to the shape of the membrane *
Atomic orbital In atomic theory and quantum mechanics, an atomic orbital is a function describing the location and wave-like behavior of an electron in an atom. This function can be used to calculate the probability of finding any electron of an atom in any spe ...
, a related quantum-mechanical and three-dimensional problem


References

* {{Cite book , author=H. Asmar, Nakhle , title=Partial differential equations with Fourier series and boundary value problems , year=2005 , publisher=Pearson Prentice Hall , location=Upper Saddle River, N.J. , isbn=0-13-148096-0 , page=198 Partial differential equations Mechanical vibrations Drumming