  We study the modulational dynamics of striped patterns formed in the wake of
a planar directional quench. Such quenches, which move across a medium and
nucleate pattern-forming instabilities in their wake, have been shown in
numerous applications to control and select the wavenumber and orientation of
striped phases. In the context of the prototypical complex Ginzburg-Landau and
Swift-Hohenberg equations, we use a multiple-scale analysis to derive a
one-dimensional viscous Burgers' equation which describes the long-wavelength
modulational and defect dynamics in the direction transverse to the quenching
motion, that is along the quenching line. We show that the wavenumber selecting
properties of the quench determines the nonlinear flux parameter in the
Burgers' modulation equation, while the viscosity parameter of the Burgers'
equation is naturally determined by the transverse diffusivity of the pure
stripe state. We use this approximation to accurately characterize the
transverse dynamics of several types of defects formed in the wake, including
grain boundaries and phase-slips.
