Low-frequency electromagnetic turbulence near the proton gyrofrequency observed far upstream of comet P/Halley can be excited by a beam instability driven by relative streaming between cometary protons, solar wind protons, and water group ions. For a given solar wind velocity the growth rates peak at a certain optimum frequency shift (from the exact proton gyrofrequency), and the wavelengths involved can be deduced in a self-consistent way from the dispersion law. Only under ideal conditions when all parameters remain constant would the mode corresponding to the optimum frequency shift grow fastest and might it be possible to observe a nearly constant frequency mode. However, if the solar wind parameters were not constant, then a mode that was in resonance earlier would no longer remain so, and some other mode with a slightly different frequency shift might start to grow fastest, leading to a mixing of many modes. Thus only rarely would one be able to observe a single mode near the proton gyrofrequency, exactly as happens in the observations. Our self-consistent approach yields resonant instabilities with left-handed polarizations in the space frame. ¿ American Geophysical Union 1995