Module for the Vibrating Drum, a Wave Equation in 2D

Numerical Methods for O. D. E.  using Mathematica. (c) John H. Mathews, 2003

Background.

    The two dimensional wave equation is   ,  

in rectangular coordinates it is   ,  

and in polar coordinates it is   .

     Consider a drum head that a flexible circular membrane of radius .  Assume that it is struck in the center and this produces radial vibrations only where the displacement depends only on time and distance from the center.  Then   satisfies the D.E.
      
    .

Example 1.  Consider a drum head of radius . For convenience, choose the parameter  . The method of separation of variables permits us to use the substitution  .  Use this substitution and obtain the D.E.

    .  
    
Solve this D.E. and plot the solution over the interval  .

Example 2.  We want to familiarize ourselves with Bessel functions.
Use Mathematica to differentiate and  verify that the function  
            
is a solution to the
            D.E.   .  
Use known identities for Bessel functions to simplify the final computation  

Example 3.  In Example 1, the boundary condition for the D.E. is  ,  i.e. the drum head has radius  .
Thus the parameter    must be chosen to be a root of the Bessel function.
The zeros do not have a simple formula. However it is known that they are "close to" multiples of  .  
Verify this and find the first five zeros.

Example 4.  Plot the functions   is the n-th root of  .
Since we will be considering a drum of unit radius, plot   over the interval  .

Background.

    The solution we are seeking in Example 1 is   where the boundary condition   requires that  ,  hence  . Therefore the fundamental solutions to the wave equation for the drum head is  

    ,  for  n = 1,2,3.  

Example 5.  The initial displacement for a fundamental solution is  .  
Plot the functions for  n = 1,2,3.  

The first fundamental solution vibrates up and down throughout the entire disk of radius 1.

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