Final Observations and Conclusions

We here draw some observations of a physical nature. The wave equation we are considering here can be applied to actual physical situations only for small oscillations. This restriction has not been followed in the general case. We can fix this problem by introducing a small multiplicative constant which depends on the normal mode and on the membrane radius, in such way that the maximum amplitude is much smaller than the radius. This has effect only when the option constrained is used in the animations.

The oscillation frequency of the animations does not necessarily correspond to the physical frequency . It can be changed by clicking the acceleration button.[It seems that the frame rate cannot be set up in the animation command. It must be adjusted after the animation has been launched in the animation window by pressing the acceleration button.] To calibrate the frame rate we have to choose the units, give numerical values for the parameters and use the function to find the true physical frequency. From this value we adjust the frame rate to the physical value .

In the sections describing the normal modes, we have animated just one of them, without considering the degenerated ones. All of them are similar. If the degeneracy is in the variable, they differ from a rotation of (for the circular and ringlike membrane). If the degeneracy is in the variable, they differ by a time translation; the modes with the term are at the horizontal position at , while the modes with the term are at maximum amplitude position at .

The main conclusion of this work is that the computer algebra systems provide powerful pedagogical tools to visualize and to aid in understanding important characteristics of the membrane problem. It can be used in the classroom to help clarify some aspects of the problem, and the worksheets can be given to students for their own analysis at home. This work can be extended to include the vibration of strings, rods and plates. The animations printed here can be reproduced in the Maple system once one has copied the packages available in the membrane web pages. With the actual animation, one can examine the dynamic processes of the vibration. One can change the physical parameters and choose the initial conditions. The Fourier coefficients and the wave equation solution are found automatically. The calculations are performed analytically as far as possible. If it fails to find closed solutions, it is done numerically.

We have used these worksheets in courses of Mathematical Methods for Physics and Engineering at Pontifícia Universidade Católica of Rio de Janeiro in Brazil for more than one hundred students. We were able to analyze problems which before could not be studied in undergraduate courses because the number of calculations involved, such as the oscillation of rods with many kinds of boundary conditions. With the aid of computer algebra systems it is possible to analyze more realistic problems. Since these systems have many limitations, the teacher must introduce them carefully, but we highly recommend the use of Maple to analyze the oscillation of strings, membranes, rods and plates.