The early free-surface flow due to an impulsively started point sink in a semi-infinite fluid is analytically studied by employing a power series expansion in time. The full surface elevation to second order is calculated, as well as the third-order elevation at the surface center just above the sink. Gravity enters into the problem only in the third-order surface elevation. A Froude number is defined here with respect to the sink strength and the initial submergence depth. A critical value for the Froude number is analytically determined which is related to dip formation at the free surface. Its value is 15-1/2=0.2582. This critical value is shown to be in good agreement with experimental data and earlier works on dip formation in a container that is drained by a hole in the bottom.