We examine equatorial wave structure in temperature measurements from the High Resolution Dynamics Limb Sounder (HIRDLS) on the Aura satellite. Waves with periods longer than 1 day and zonal wave numbers up to 8 (depending on frequency) are derived from an asynoptic Fourier transform analysis. HIRDLS measurement sampling and resolution afford unprecedented views of the latitude-height structure of equatorial Rossby wave, mixed Rossby-gravity wave, and Kelvin wave modes at altitudes above cloud tops in the tropical upper troposphere and stratosphere. Wave modes with vertical wavelength as short as 4 km can be clearly seen in the data. Kelvin waves comprise a dominant signal throughout the 3 years of HIRDLS measurements, and we further examine time, height, and longitude variations observed in the Kelvin waves. An annual cycle of Kelvin wave temperature amplitudes near the tropopause is observed that may have implications for annual variations in wave-cirrus formation in the tropics. This annual variation can be largely explained by effects of the background wind and stability on Kelvin wave propagation and potential energy in the tropical tropopause layer. At altitudes 20 km and above, the annual cycle gives way to an interannual cycle in Kelvin wave amplitudes that is related to the quasi-biennial oscillation in stratospheric winds. This interannual variation is a signature of Kelvin wave forcing of the descent of westerly winds in the oscillation, and we compute the Kelvin wave fractional contribution to the forcing.