= Line source in a closed-end semi-infinite strip
{title2=$w=\frac Q{2\pi}\log[(f(z)-f(c))(f(z)-\overline{f(c)})]$}
In the impermeable strip $x>0$, $0<y<a$, a <line source> of strength $Q$ at an interior point $c$ has the displayed <complex potential of a two-dimensional potential flow>, with locally consistent <complex logarithm> and $f(z)=\cosh(\pi z/a)$. The <hyperbolic-cosine map of a half-strip> sends the walls to the <real axis>. An equal image <line source> at $\overline{f(c)}$ makes that axis a <streamline>. Since $f'(c)\ne0$, the local singular part remains $Q\log(z-c)/(2\pi)$, proving preservation of strength. With no extra imposed flow, the <complex velocity> tends to $Q/a$ as $x\to\infty$, so the outgoing <volume flux> is $Q$. The source position $c$ has positive imaginary part; assuming it real would place it on a wall.
Back to article page
