On the mathematical conditions for the existence of periodic fluctuations in non-uniform media

In many areas of mathematical physics where one is interested in the propagation of waves through non-uniform media, it is often assumed that periodic excitations result in periodic responses. This assumption is examined by rigorously investigating the existence of periodic solutions of linear hyperbolic differential equations whose coefficients vary with position and whose solution must satisfy periodic boundary or source data. It is shown that the nature of the coefficients of the undifferentiated terms of the differential system is crucial in determining whether or not the solution is periodic. In physical applications, these coefficients usually depend on the gradients of media properties as well as on the media properties themselves. In particular, it is shown that for a general hyperbolic system of two equations in one space dimension, the solution is not periodic. Moreover, this can remain true even if the media gradients are assumed small. However, if the media gradients vanish, or if they vanish except for a bounded region of space, the solution is shown to be periodic for a large enough time. Furthermore, if these gradients vanish asymptotically at large distances, then the disturbances will be asymptotically periodic for increasing time. Special attention is given to the propagation of infinitesimal pressure disturbances through non-uniform steady flows of a lossless fluid.