For a nonmonotonic Alfv¿n velocity profile, the global model of the magnetosphere can be shown to couple to multiple resonant field lines. The mode structure itself is greatly altered by introduction of nonmonotonicity. Because of the rapid decrease of density at the plasmapause, the variation of the Alfv¿n velocity naturally separates the magnetosphere into inner and outer parts. ULF wave perturbations are found to be trapped in the Alfv¿n velocity valleys and, under certain circumstances, UFL wave amplitudes in the inner magnetosphere are greatly enhanced. The results obtained may explain some ULF perturbations observed in the low-latitude regions of the magnetosphere. ¿ 1989 Americal Geophysical Union