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Characteristic curves of partial differential equations (PDEs) in general differ from short wave energy rays. We give conditions for linear, two dimensional second order PDEs that guarantee an exact correspondence between characteristics and energy rays. The findings are applied to time-harmonic, zonally-symmetric small-amplitude equatorial internal waves. It is shown that for a fairly general model of inertia-gravity waves, an exact characteristic-ray correspondence holds. When characteristics of internal waves, trapped in a meridional plane, are followed over several boundary reflections, a convergence towards a limit cycle (called equatorial wave attractor) can generally be found. The results on the characteristic-ray correspondence help to interpret physically equatorial wave attractors. Recent ideas on energy accumulation by near inertial waves, trapped on wave attractors at deep ocean sites are confirmed by our results, at least in the short wave (WKB) sense.