A prominent yet largely unrecognized feature of the inner magnetosphere associated with particle injections, and more generally geomagnetic storms, is the occurrence of broadband electromagnetic field fluctuations over spacecraft frame frequencies (fsc) extending from effectively zero to fsc ≳ 100 Hz. Using observations from the Van Allen Probes we show that these waves most commonly occur premidnight but are observed over a range of local times extending into the dayside magnetosphere. We find that the variation of magnetic spectral energy density with fsc obeys ɛB=Afsc-α over several decades with a spectral breakpoint at fb ≈ 1 Hz. The values for α are lognormally distributed with α = 1.9 ± 0.6 for fsc < fb and α = 2.9 ± 0.6 for fsc > fb. A is a function of geomagnetic activity with the largest values observed over intervals of decreasing Dst index during the main phase of geomagnetic storms. At these times these waves are nearly always present in the nightside inner magnetosphere and are commonly observed from L = 3 outward. The observed variation of the electric to magnetic field amplitude with fsc is well described by a dispersive Alfvén wave model under the assumption that fsc is primarily a consequence of the Doppler shift of plasma frame structures moving over the spacecraft. The robust anticorrelation between the time rate change of the Dst index and wave spectral energy density coupled with the ability of dispersive Alfvén waves to drive transverse ion acceleration suggests that these waves may boost ion energy density in the inner magnetosphere and intensify the ring current during storm times.