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Take positive vortex strength $\kappa$ to mean anticlockwise circulation. A line vortex at $z_0$ has local velocity potential $\kappa \arg (z-z_0)/(2\pi )$. The image vortex at a plane wall must have strength $-\kappa$ to enforce zero normal velocity at $y=0$. At the initial instant, a potential is thereforeThis is a local, multivalued velocity potential: its gradient is single-valued away from the vortex, but the nonzero circulation precludes a globally single-valued potential around it. For a vortex centred at $(X,Y)$, the velocity isThe real vortex and its opposite image give cancelling $u_y$ at the wall. A point vortex moves with the regular velocity induced by the other vortex, not its singular self-field. Evaluating the image field at $(X,b)$ gives $\dot{X}=\kappa /(4\pi b)$ and $\dot{b}=0$. HenceThe time-dependent velocity potential is obtained by replacing $a$ with $X(t)$ above; the image remains at $(X(t),-b)$. Reversing the convention for positive circulation reverses the translation direction.