Dirichlet Green function for an infinite strip

On the strip $\mathbb{R}\times (0,a)$, with $a>0$, choose the convention $\Delta G=\delta$ and zero Dirichlet boundary conditions on both horizontal sides. The Dirichlet Green function isTo derive the coefficient, use the Fourier sine series completeness relation $\delta (y-c)=(2/a){\sum }_{n\ge 1}\sin (n\pi y/a)\sin (n\pi c/a)$. Each horizontal mode solves $({\partial }_x^2-{\kappa }_n^2)g_n=\delta (x-b)$ with ${\kappa }_n=n\pi /a$. Its decaying Green function is $-e^{-{\kappa }_n|x-b|}/(2{\kappa }_n)$, since the first derivative has jump one. Multiplying by $2/a$ gives the displayed coefficient.

The series converges absolutely when $x\ne b$, vanishes on $y=0,a$, and decays as $|x-b|\to \infty$. At the point source $(b,c)$ it has the expected logarithmic singularity; the source equation is interpreted in the distribution sense. The strip width sets the longest horizontal decay scale $a/\pi$.