In 2-d, one ripple would mean the velocity of water particles move out radially forming a circular wavefront. The Navier Stokes equations say the divergence of velocity has to be zero, but this circular radiation pattern has nonzero divergence. What am I missing here, since water ripples clearly exist?
Waves in fluids carry momentum not mass. When we see ripples diverging from a point, there is no radial (horizontal) movement of mass. Rather the mass moves in the vertical direction, some what sinusoidally.
If you have observed birds floating on water surface, you see them moving up and down but not along the waves.