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Statistically determined dispersion relations of magnetic field fluctuations in the terrestrial foreshock

We obtain dispersion relations of magnetic field fluctuations for two crossings of the terrestrial foreshock by Cluster spacecraft. These crossings cover plasma conditions which differ significantly in their plasma $β$ but not in the properties of the encountered ion population, both showing shell-like distribution function. Dispersion relations are reconstructed using two-point instantaneous wave number estimations from pairs of Cluster spacecraft. The accessible range of wave vectors, limited by the available spacecraft separations, extends to $\approx \!2 \times 10^4$ km. Results show multiple branches of dispersion relation, associated with different power of magnetic field fluctuations. We find that Sunward propagating fast magnetosonic waves and beam resonant modes are dominant for the high plasma $β$ interval, while the dispersions of magnetically dominated interval include Alfvén and fast magnetosonic modes propagating Sunward and unit-Sunward.