New classes of inertial oscillation mode in the form of equatorially trapped boundary waves are discovered. The structure and frequency of the waves are hardly affected by the presence of an inner sphere if the azimuthal wavenumber m>&pgr;(1+&eegr;)/2(1-&eegr;) where &eegr;=ri/ro, the radius ratio of a spherical shell. Also discovered is that equatorially symmetric inertial waves belong to particular subclasses of thermal instabilities. The problem of the inertial waves can therefore be understood within the framework of instability theory. The findings provide valuable insights into many geophysical phenomena, not only for the Earth's fluid core but atmospheres and oceans as well. ¿ American Geophysical Union 1992