Tidal response of the solid Earth

In this note, the boundary-value problem for Earth tides is investigated. The tidal motion of the solid Earth is treated as an infinitesimal perturbation superimposed on the hydrostatic equilibrium of a rotating and self-gravitating Earth. Both Lagrangian (material-fixed) and Eulerian (space-fixed) incrementals are defined for describing the tidal perturbations. The linearized differential equations of motion and boundary conditions are derived and given in three different forms, which differ from each other in whether the pure Lagrangian incrementals, or the pure Eulerian incrementals, or a mixed combination of the two are chosen for describing variations in the potential and stress field. Analytical solutions for simple Earth models are discussed. In case of a rotating, elliptical, incompressible and homogeneous Earth, we have found inconsistency in Love's equations of motion and several calculation errors in his analytical expressions. Semi-analytical methods which are mostly used nowadays to determine the Earth tide parameters are presented and the results of different authors are discussed.

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