Commun. Comput. Phys., 1 (2006), pp. 945-973. Exact Boundary Conditions for Periodic Waveguides Containing a Local Perturbation Patrick Joly 1*, Jing-Rebecca Li 1, Sonia Fliss 11 Domaine de Voluceau - Rocquencourt - B.P. 105, 78153 Le Chesnay Cedex, France. Received 8 November 2005; Accepted (in revised version) 10 April 2006 Abstract We consider the solution of the Helmholtz equation $-\Delta u({\bf x}) - n({\bf x})^2\omega^2 u({\bf x}) = f({\bf x})$, ${\bf x}=(x,y)$, in a domain $\Omega$ which is infinite in $x$ and bounded in $y$. We assume that $f({\bf x})$ is supported in \$\Omega^0:={{\bf x}\in {\Omega} \; | a^-