  We construct and justify leading order weakly nonlinear geometric optics
expansions for nonlinear hyperbolic initial value problems, including the
compressible Euler equations. The technique of simultaneous Picard iteration is
employed to show approximate solutions tend to the exact solutions in the small
wavelength limit. Recent work [2] by Coulombel, Gues, and Williams studied the
case of reflecting wave trains whose expansions involve only real phases. We
treat generic boundary frequencies by incorporating into our expansions both
real and nonreal phases. Nonreal phases introduce difficulties such as
approximately solving complex transport equations and result in the addition of
boundary layers with exponential decay. This also prevents us from doing an
error analysis based on almost-periodic profiles as in [2].
