A quasilinear theory is presented for the relaxation of the proton and helium distribution functions and the associated excitation of ion cyclotron waves, downstream of the quasi-perpendicular Earth's bow shock. The theory predicts the wave polarization, power and peak frequency, and the proton bulk velocity and temperature anisotropy, sufficiently far downstream of the shock that the ions and waves have relaxed to a quasi-equilibrium. The results are compared with the AMPTE/IRM crossings of the marginally supercritical bow shock documented by Sckopke et al. (1990), for which the number of reflected protons is small and the quasilinear approximation is expected to be valid. The theory starts with the trajectories of the specularly reflected protons and transmitted helium in the laminar electromagnetic fields of the shock transition. In a simplified treatment of the downstream relaxation process, neglecting wave dispersion and assuming that the downstream reflected proton speed v0 is large compared with the Alfven speed VA, the wave and proton quasi-equilibrium predictions are qualitatively in agreement with the observations. In a more precise treatment limited to a perpendicular configuration, in which dispersion is included and v0 may be comparable with VA, the agreement with the observations is generally very good if the contribution of the transmitted core protons is included. One interesting feature is that the minor ion helium may contribute to the wave power, and in return change the proton and helium distribution functions. The helium contribution is estimated but requires further investigation.