A brief history of the membrane problem

The analysis of the movement of membranes is very old[Kline]. In 1764 Euler published a paper[Euler] in which he obtained the wave equation for the vertical displacement of circular and rectangular membranes. The radial part of the circular membrane is described by a Bessel differential function of integer order. It was the first time in which the Bessel function of arbitrary order had been analyzed[Watson]. Euler found solutions for these equations using the method of separating variables which correspond to the normal modes of oscillation. The general solution as a normal mode superposition was obtained much later, only after Fourier's work.

Fourier had problems publishing his paper, finished in 1807, on heat conduction, in which he stated that any function can be expanded in series of sines and cossines. The members of the Science Academy of Paris did not at first accept Fourier's new ideas, arguing lack of rigor. In 1819 the paper was finally published[Fourier].

Poisson applied immediatly Fourier's method to many physical problems, including the membrane problem [Poisson ]. This was the first successful solution for this problem. His analysis of the rectangular membrane and circular membrane with axial movements was very complete. The problem was further developed by Clebsch[Clebsch] in the circular case, including the effects of stiffness and rotatory inertia. This problem has been considered by many other researchers: Lamé, Mathieu, Rayleigh and others. It has important applications in the theory of sound[Morse][Rayleigh].

We mention some reasons why the analysis of the movement of membranes, of many shapes and with different boundary conditions, became a classic problem of Mathematical Physics. This problem provides a good example of the use of Fourier expansion, which follows the method of separation of variables for linear partial differential equations. Depending on the membrane shape, many kinds of special functions, like Bessel functions for circular membranes and Mathieu functions for eliptic membranes, are used. The membrane movement is an example of the Sturm-Liouville eigenvalue problem[Courant]. It can be used to illustrate some general theorems such as those involving the nodal lines of eigenfunctions of second-order auto-adjoint differential equations.