<p>The location of the Touch Down Point (TDP) of a slack mooring line, closely linked to its residual resistance, requires accurate modeling. We introduce an analytical formulation of seabed contact dynamics as a rigorous boundary condition for the system. Building on this, a modal analysis is developed to fully characterize the natural oscillation modes of a slack mooring line without assumptions on tension perturbations. The framework captures the dynamics of the moving boundary at the seabed and establishes a direct link between the maximum oscillation amplitudes and the onset of shock formation at the contact point. A complementary formulation describes, in static or quasi-static regimes, the relationship between fairlead displacement and seabed contact point motion, providing a solid basis for the study of seabed interaction mechanics. Finally, a numerical investigation contrasts the proposed exact seabed boundary condition with a widely used empirical approximation. The comparison is carried out against two exact benchmarks: (i) natural frequencies from modal analysis and (ii) analytically derived seabed contact point displacement under steady fairlead tension. Results confirm that the new boundary condition improves predictive accuracy without added computational cost. These findings advance the understanding of slack-line dynamics and support future analytical and numerical studies.</p>