Computer algebra takes on the vibrating-membrane problem

The vibrating membrane is a classic problem of mathematical physics. It provides a particularly good example of how a computer algebra system (CAS) can serve as a pedagogic tool for scientific analysis. In this article, we focus on the oscillation of a circular membrane. We concentrate on the pedagogical aspects and on the part of the theory in which a CAS can be the most helpful. Maple is an appropriate environment to analyze the membrane problem. We used the Maple system's main aspects together to solve the problem. We used the algebraic aspect to develop the problem step by step from the wave equation. We used the graphical aspect to animate the normal modes and the movement caused by initial conditions. We used the numerical aspect to find the zeros of the Bessel functions, to perform numerical integration when needed and to plot the results. The programming aspect let us make packages containing tools to analyze the problem. With these tools, we can change physical parameters and choose many kinds of initial conditions.