Inviscid incompressible fluid occupies the region $y>0$, which is bounded by a rigid barrier along $y=0$. At time $t=0$, a line vortex of strength $\kappa$ is placed at position $(a, b)$. By considering the flow due to an image vortex at $(a,-b)$, or otherwise, determine the velocity potential in the fluid.

Derive the position of the original vortex at time $t>0$.