The paper centres around fundamental solutions of the linearised problem on fluid sloshing in a vertical circular cylindrical tank having a thin rigid-ring horizontal baffle. It develops an analytically oriented approach, which provides accurate approximations of natural frequencies and modes. The singular asymptotic behaviour of the velocity potential at the sharp baffle edge is also captured. A numerical analysis quantifies the natural frequencies and modes versus vertical position and width of the annular baffle. Forthcoming parts will use these approximate fundamental solutions in both nonlinear modal modelling and estimating the damping due to vorticity stress near the baffle.