Usuario:Felipebm/Vibraciones de un tambor circular

Uno de los posibles modos de vibración de un tambor circular ideal (modo con la notación mostrada más abajo). Otros modos son mostrados al final del artículo.

The vibrations of an idealized circular drum, essentially an elastic membrane of uniform thickness attached to a rigid circular frame, are solutions of the wave equation with 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 series of the simpler vibrations (analogously to the Fourier series).

Motivation

editar

The most obvious relevance of the vibrating drum problem is to the analysis of certain 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

editar

Consider an open disk   of 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

editar

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   (Note that this   actually plays the role of a wavevector, which is often denoted by  ). 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

 

where  

The general case

editar

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

 

and

 

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

 

for  

Animations of several vibration modes

editar

A number of modes are shown below together with their quantum numbers. The analogous wave functions of the hydrogen atom are also indicated.

See also

editar

References

editar
  • H. Asmar, Nakhle (2005). Partial differential equations with Fourier series and boundary value problems. Upper Saddle River, N.J.: Pearson Prentice Hall. pp. page 198. ISBN 0-13-148096-0.