On the basis of MHD theory we develop a scheme for distinguishing among the four low-frequency modes which may propagate in a high-&bgr; anisotropic plasma such as the magnetosheath: the fast and slow magnetosonic, the Alfv¿n, and mirror modes. We use four parameters: the ratio of transverse to compressional powers in the magnetic field, the ratio of the wave powers in the thermal pressure and in the magnetic field, the ratio of the perturbations in the thermal and magnetic pressures, and the ratio of the wave powers in the velocity and in the magnetic field. In the test case of an Active Magnetospheric Particle Tracer Explorers/Ion Release Module (AMPTE/IRM) magnetosheath pass near the Sun-Earth line downstream of a quasi-perpendicular shock, the four modes can be clearly distinguished both spatially and spectrally. Near the bow shock, the waves are Alfv¿nic in a large frequency range, 1 to 100 mHz. In the middle and inner magnetosheath, the waves below 10 mHz are Alfv¿nic. The fast mode waves occur in the higher-frequency end of the enhanced spectrum, 80 mHz for the middle magnetosheath and 55 mHz for the inner sheath. The wave enhancement in the intermediate frequencies is slow modes in the inner sheath and mirror modes in the middle sheath. This confirms the earlier report of the existence of the slow mode waves near the magnetopause. These slow waves provide evidence that the magnetopause is an active source of the waves in the sheath. We also show that the measured frequency of a wave is close to an invariant if the magnetosheath flow is in a steady state. Therefore changes in the frequencies of enhanced waves indicate emergence, or damping, or mode conversion of the waves. ¿ American Geophysical Union 1994