- Vibrations of a circular drum
drum(mode with the notation below). Other modes are shown at the bottom of the article.]
vibrations of an idealized circular drum, essentially an elastic membraneof uniform thickness attached to a rigid circular frame, are solutions of the wave equationwith zero boundary conditions.
There exist infinitely many ways in which a drum can vibrate, depending on the shape of the drum at some initial time and the rate of change of the shape of the drum at the initial time. Using
separation of variables, it is possible to find a collection of "simple" vibration modes, and it can be proved that any arbitrarily complex vibration of a drum can be decomposed as a linear combination of the simpler vibrations.
The most obvious relevance of the vibrating drum problem is to the analysis of certain percussion instruments like a
drumor a 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 are all concepts that students who are first exposed to the structure of the atom need to become familiar with.
open diskof radius centered at the origin, which will represent the "still" drum shape. At any time the height of the drum shape at a point in measured from the "still" drum 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 is attached.
The mathematical equation that governs the vibration of the drum is the wave equation with zero boundary conditions,
Here, is a positive constant, which gives the "speed" of vibration.
Due to the circular geometry, it will be convenient to use
polar coordinates, and Then, the above equations are written as
The radially symmetric case
We will first study the possible modes of vibration of a circular drum 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 will be oscillatory in time, and this leaves only the third case, when for some number 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,
The Bessel function is unbounded for which results in an unphysical solution to the vibrating drum 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 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 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 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.
Hearing the shape of a drum
Wikimedia Foundation. 2010.
Look at other dictionaries:
Drum — For other uses, see Drum (disambiguation). The drum is a member of the percussion group of musical instruments, which is technically classified as the membranophones. Drums consist of at least one membrane, called a drumhead or drum skin, that … Wikipedia
Normal mode — For other types of mode, see Mode (disambiguation). Vibration of a single normal mode of a circular disc with a pinned boundary condition along the entire outer edge. See other modes. A normal mode of an oscillating system is a pattern of motio … Wikipedia
Vibrating string — A vibration in a string is a wave. Usually a vibrating string produces a sound whose frequency in most cases is constant. Therefore, since frequency characterizes the pitch, the sound produced is a constant note.Vibrating strings are the basis of … Wikipedia
Bessel function — In mathematics, Bessel functions, first defined by the mathematician Daniel Bernoulli and generalized by Friedrich Bessel, are canonical solutions y(x) of Bessel s differential equation: for an arbitrary real or complex number α (the order of the … Wikipedia
Ernst Chladni — Chladni redirects here. For the lunar crater, see Chladni (crater). Ernst Chladni Ernst Chladni Born … Wikipedia
Wave equation — Not to be confused with Wave function. The wave equation is an important second order linear partial differential equation for the description of waves – as they occur in physics – such as sound waves, light waves and water waves. It arises in… … Wikipedia
List of mathematics articles (V) — NOTOC Vac Vacuous truth Vague topology Valence of average numbers Valentin Vornicu Validity (statistics) Valuation (algebra) Valuation (logic) Valuation (mathematics) Valuation (measure theory) Valuation of options Valuation ring Valuative… … Wikipedia
Mode shape — ] In the study of vibration in engineering, a mode shape describes the expected curvature (or displacement) of a surface vibrating at a particular mode. To determine the vibration of a system, the mode shape is multiplied by a function that… … Wikipedia
Membrane (selective barrier) — Schematic of size based membrane exclusion A membrane is a layer of material which serves as a selective barrier between two phases and remains impermeable to specific particles, molecules, or substances when exposed to the action of a driving… … Wikipedia
Vibration — For the soul music group, see The Vibrations. For the machining context, see Machining vibrations. For the albums, see Vibrations (Roy Ayers album) and Vibrations (The Three Sounds album). Classical mechanics … Wikipedia