ABSTRACT
Daniel Bernoulli determined the normal modes of a hanging chain in 1732. Given its importance, there are many derivations for the model governing the small transverse vibrations of a one-dimensional elastic continuum, such as a string on a piano, violin or guitar. The derivation of the wave equation just given follows directly from first principles, Newton's laws and conservation of mass. It is based on the principle of least action which states that the action, the integral over any time interval during the motion of the kinetic energy minus the potential energy, must be stationary when compared to all possible (virtual) motions of the physical system. This chapter suggests how shooting methods developed earlier to treat Sturm-Liouville eigenvalue problems can be extended to higher order self-adjoint problems. The eigenvalues and eigenfunctions associated with the vibrations of a bar subject to the usual boundary conditions behave qualitatively like those of Sturm-Liouville problems.
