An axisymmetric flow due to a submerged sink in water of infinite depth is considered, with a stagnation point on the free surface above the sink. Forbes & Hocking (1990) calculated numerically a flow for each value of the Froude number F smaller than a critical value Fc. For F close to Fc there is a ring-shaped bump on the free surface. At F=Fc, the crest of the bump becomes a ring of stagnation points. We use the numerical procedure of Hocking & Forbes to show that the bump is the first crest of a train of axisymmetric waves. The wave amplitude decreases with increasing distance from the source. Then we give a local analysis of axisymmetric free-surface flows with a circular ring of stagnation points. We find flows in which the surface has a discontinuity in slope with an enclosed angle of 120° all along the ring. This behaviour is consistent with the numerical solution for F=Fc near the crest of the bump.