Based on the observed equatorial ocean dynamic characteristics

the effects of a sloping thermocline and Rayleigh friction on the equatorially trapped free Kelvin waves were theoretically studied with a linear one-and-one-half layer reduced gravity model

the multiple-scale method and a small parameter expansion technique. Assuming that main thermocline depth (MTD) variations are slow

i. e. the changes of MTD over one wavelength are smaller than that of the wave amplitude and that wave reflections are negligible

the authors showed by their analytical results that the wavelengths and amplitudes of Kelvin waves are significantly modified by the MTD variations and Rayleigh friction. The results also showed that for an eastward shallowing thermocline

the zonal velocity of the Kelvin waves varies with thermocline depth to the power −7/8. The eastward shallowing of the thermocline depth strengthens Kelvin wave entrapment at the equator. Rayleigh friction reduces the Kelvin wave's eastward velocity while the thermocline acts in the opposite way. The friction causes dispersion of the Kelvin wave

whose dissipation factor does not depend on its wavelength. The friction increases the lateral decay length and causes phase lines of Kelvin waves to slant westward in parabolic arcs.