= Reflection image of a planar hydrodynamic dipole
{title2=$m'=e^{2i\theta}\bar m$}
For a <complex potential of a two-dimensional potential flow>[complex potential] $w(z)=m/(z-a)$, reflection in the straight wall through zero at angle $\theta$ is $Rz=e^{2i\theta}\bar z$. The image construction $W(z)=w(z)+\overline{w(Rz)}$ gives
$$
W(z)=\frac{m}{z-a}+\frac{e^{2i\theta}\bar m}{z-e^{2i\theta}\bar a}.
$$
On the wall $Rz=z$, this is real, so its <stream function> is constant there. This proves the <impermeability condition>. Thus the image position is $a'=e^{2i\theta}\bar a$ and its moment is $m'=e^{2i\theta}\bar m$. If the point dipole is advected by the regular image flow, the <complex velocity> convention $W'=u-iv$ gives
$$
\frac{d\bar a}{dt}=-\frac{m'}{(a-a')^2}=\frac{\bar m}{4d^2},
$$
where $d=\operatorname{Im}(e^{-i\theta}a)$ is its signed distance from the wall. The last equality follows from $a-a'=2ie^{i\theta}d$. It applies when $d\ne0$.
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