Oscillating sessile droplets on a plane, crucial in industrial applications like inkjet printing, spray cooling, and antifogging, present a long-standing challenge in determining viscous damping. Existing solutions are limited to first-order asymptotic approximations of hemispherical droplets, accounting only for viscous dissipation near the wall. Considering further the bulk dissipation that is usually neglected due to its higher-order effect, we obtain a two-term approximation of decay rates for pinned sessile droplets with different contact angles. We demonstrate that the two-term approximation effectively resolves discrepancies between predicted and observed dampings for viscosities in the range of practical interest, supported by experiments and simulations. The two-term approximation also explains why hydrophobic droplets possess much lower damping than hydrophilic ones, and offers a template for damping prediction formulas with potential viscosity measurement applications.

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• Capillary waves
• Drop & bubble phenomena
• Surface tension effects