In some cases the reflection functions associated with the discontinuities in conical bores are growing exponentials [J. Martinez and J. Agul-lo, J. Acoust. Soc. Am. 84, 1613--1619 (1988)]. It is shown that discontinuities can be modeled by a Sturm--Liouville system using a pressurelike quantity as the dependent variable. For the subset of discontinuities exhibiting the growing exponential reflection function, the Sturm--Liouville potential function is an energy well. This well is shown to support exactly one trapped energy mode which corresponds to the growing exponential. It is shown that in the region surrounding the discontinuity for these systems, traveling Fourier components taken together with their reflected waves do not constitute a complete set and that the trapped mode is required to complete the set. On the other hand, for systems which do not exhibit the growing exponential, the Sturm--Liouville potential is an energy barrier with no trapped modes, and the Fourier components compose a complete set within the conical regions. Furthermore, a change of dependent variable can be used to go from a Sturm--Liouville description involving an energy well to one involving a barrier, thus eliminating the trapped mode.