The propagation of whistler energy in slight enhancements or depressions of ionization aligned with the geomagnetic field has been studied earlier in the two limiting cases of a discontinuous boundary or very slowly varying density gradients. These two approaches have yielded different results, particularly for the trapping of energy in enhancements of ionization. Therefore a study of the reflection and transmission for the full range of gradients is desirable. In this paper, full-wave methods are applied to bell shaped ducts of arbitrary scale size. The results indicate criteria for applicability of the two simplifying limiting cases. This phase of the study is further extended by examination of a coupling theory. For a variety of magnetospheric conditions the cross-coupling terms between characteristically different waves are negligible; this leads to almost perfect reflection and substantiates the use of the simpler phase-integral techniques. By using the phase integral techniques, we have derived a fairly complete description of possible modes excited in bell shaped ducts. For each mode a theoretical whistler spectrogram is obtained. Extensions of the theory to apply to nonuniform magnetic fields may lead to predictions of a hyper-fine whistler structure.