We investigate the properties of a bispherical shell distribution of pickup ions when dispersion of the resonant waves is taken into account. We also extend the method of Johnstone et al. [1991> and Huddleston and Johnstone [1992> for determining the spectra of the quasi-linear self-generated waves to include dispersion. Our specific results assume that all waves propagate along the average magnetic field, and that the waves satisfy the cold electron-proton dispersion relation. The major effect of including dispersion in the analysis is that the particles in a bispherical distribution retain more of their initial energy than in the nondispersive treatment, so the resulting wave spectra are less intense than predicted when dispersion is neglected. The dispersive analysis also encounters the well-known gap in the cyclotron resonance with fast-mode waves which is absent in the nondispersive picture. We present detailed results for pickup protons in the two extreme configurations of magnetic field perpendicular to, and parallel to, the solar wind flow for a range of values of VA/Vsw. We find that when pickup protons are scattered to a bispherical distribution with VA/Vsw=0.1, typical of conditions in the solar wind, the discrepancy in the total energy density of the self-generated waves between the dispersive and nondispersive results amounts to 18% in the perpendicular configuration. This discrepancy falls between 9% and 73% in the parallel configuration, depending on the extent of the particle transport through the resonance gap. These discrepancies are substantially larger for larger values of VA/Vsw. These results indicate that dispersive effects can be important in the correct modeling of the quasi-linear resonant wave-particle interaction between pickup ions and self-generated waves in the solar wind. ¿ American Geophysical Union 1996