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 c. Assume that it is struck in the center and this produces radial vibrations only where the displacement depends only on time t and distance r from the center. Then u(t,r) satisfies the D.E. .

Computer Lab Work.
 

Exercise 1. Consider a drum head of radius c = 1. For convenience, choose the parameter a = 1. The method of separation of variables permits us to use the substitution . Use this substitution and obtain the D.E. . Then solve this D.E. and plot the solution over the interval -15 <= r <= 15.

The solution we require is the one involving the function BesselJ[0,r w].
For illustration, set w = 1 and graph y = BesselJ[0,r].

Exercise 2 (a). We want to familiarize ourselves with Bessel functions.
Use Mathematica to differentiate and verify that the function f[x]= BesselJ[0,x] is a solution to the D.E. .

2 (b). Use known identities for Bessel functions to simplify the computation

Exercise 3. The boundary condition for the D.E. is R[1]=0, i.e. the drum head has radius c = 1. Thus the parameter w 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.

Multiples of will be sufficiently close to be starting values for Mathematica's FindRoot subroutine. The first root is

We can put all five of them in an array called "roots." Then redraw the graph with horizontal axis ticks at the integers.

Exercise 4. Plot the functions is the i-th root of BesselJ[0,r].

Conclusion. The solution we were seeking in Exercise 1 was where the boundary condition R[1]=0 requires that , hence . Therefore the fundamental solutions to the wave equation for the drum head is , for n=1,2,3, ...

Exercise 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.

The second fundamental solution has a circle of radius 0.435651 as a node where there is no vibration and it moves up and down in opposite directions on the inside and outside of this circle.

The third fundamental solution has a two circular nodes of radius 0.277895 and 0.637884 where there is no vibration and it moves up and down in opposite directions between circles.

Solutions.

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(c) John H. Mathews, 1998