The Cluster STAFF Spectral Analyser measures the magnetic and electric power spectral densities (PSD) δB2 and δE2 in the magnetosheath between 8 Hz and 4 kHz, i.e. between about the lower hybrid frequency and 10 times the proton plasma frequency. We study about 23 h of data on four different days. We do not consider the whistler waves and the electrostatic pulses (which are not always observed) but the underlying permanent fluctuations. Paper 1 (Mangeney et al., 2006) shows why the permanent PSD at a given frequency f depends strongly on the angle ΘBV between the magnetic field B and the flow velocity V: this is observed for the electromagnetic (e.m.) fluctuations, δB2 and δEem2, below the electron cyclotron frequency fce, and for the electrostatic (e.s.) fluctuations δEes2 at and above fce. This dependence is due to the Doppler shift of fluctuations which have a highly anisotropic distribution of the intensity of the wave vector k spectrum, and have a power law intensity ∝k−ν with ν≃3 to 4. In the present paper, we look for parameters, other than ΘBV, which control the intensity of the fluctuations. At f≃10 Hz, δB2 and δE2em increase when the solar wind dynamic pressure PDYNSW increases. When PDYNSW increases, the magnetosheath PDYNMS∝N V2 also increases, so that the local Doppler shift (k.V) increases for a given k. If V increases, a given frequency f will be reached by fluctuations with a smaller k, which are more intense: the variations of δB2 (10 Hz) with PDYNSW are only due to the Doppler shift in the spacecraft frame. We show that the e.m. spectrum in the plasma frame has an invariant shape I1D∝Aem (kc/ωpe)−ν related to the electron inertial length c/ωpe: the intensity Aem does not depend on PDYN, nor on the electron to proton temperature ratio Te/Tp, nor on the upstream bow shock angle θBN. Then, we show results of 3-D MHD numerical simulations of the magnetosheath plasma, which map the regions where the angle ΘBV is ≃90°. The e.m. fluctuations are more intense in these magnetosheath regions, in the spacecraft frame where they are observed in the "whistler" range; and the e.s. fluctuations are less intense in these same regions, in the spacecraft frame where they are observed in the "ion acoustic" range. We conclude that the intensity of the permanent fluctuations in the e.m. range only depends on the Doppler shift, so that from day to day and from place to place in the magnetosheath, the k spectrum in the plasma frame has an invariant shape and a constant intensity. This is observed on scales ranging from kc/ωpe≃0.3 (50 km) to kc/ωpe≃30 (500 m), i.e. at electron scales smaller than the Cluster separation.