  Experiments with diblock co-polymer melts display undulated bilayers that
emanate from defects such as triple junctions and endcaps,
\cite{batesjain_2004}. Undulated bilayers are characterized by oscillatory
perturbations of the bilayer width, which decay on a spatial length scale that
is long compared to the bilayer width. We mimic defects within the
functionalized Cahn-Hillard free energy by introducing spatially localized
inhomogeneities within its parameters. For length parameter $\varepsilon\ll1$,
we show that this induces undulated bilayer solutions whose width perturbations
decay on an $O\!\left(\varepsilon^{-1/2}\right)$ inner length scale that is
long in comparison to the $O(1)$ scale that characterizes the bilayer width.
