ABSTRACT
The Helmholtz equation plus the boundary condition form an eigenvalue problem, a three dimensional Sturm Liouville problem. This chapter considers a circular drumhead for which it is appropriate to use the solutions of the Helmholtz equation in cylindrical coordinates. Superposing the normal modes enables us to satisfy initial conditions on the displacement and velocity of the drumhead. The time dependence in the heat conduction equation and the wave equation adds another dimension compared to Laplace's equation, and after separation of the time dependence, leads to the Helmholtz equation. With suitable boundary conditions the Helmholtz equation becomes an eigenvalue problem with a complete set of eigen-functions un and eigenvalues kn2. These are the normal modes of the problem. The eigenvalue problem is exactly the same as for the diffusion equation.
