Eigenfunction orthogonality for one-dimensional acoustic systems with interior or end point conditions

Some common exercises presented in introductory acoustics courses and texts illustrate solutions involving eigenvalues and eigenfunctions. Challenging extensions of these, even for one-dimensional (1D) systems, might involve a mass or spring loading the acoustic medium at an end point or at an interior point. These problems might be extended further by requiring that some given function be expanded in a series of the eigenfunctions, but such extended problems may lead to unexpected complications in regard to eigenfunction orthogonality. In this paper, Sturm-Liouville theory is used to develop a systematic method for predetermining eigenfunction orthogonality for 1D systems loaded at end points or interior points or having properties that change with jump discontinuities.