  We rigorously establish the validity of the equations describing the
evolution of one-dimensional long wavelength modulations of counterpropagating
wavetrains for a hyperbolic model equation, namely the sine-Gordon equation. We
consider both periodic amplitude functions and localized wavepackets. For the
localized case, the wavetrains are completely decoupled at leading order, while
in the periodic case the amplitude equations take the form of mean-field
(nonlocal) Schr\"odinger equations rather than locally coupled partial
differential equations. The origin of this weakened coupling is traced to a
hidden translation symmetry in the linear problem, which is related to the
existence of a characteristic frame traveling at the group velocity of each
wavetrain. It is proved that solutions to the amplitude equations dominate the
dynamics of the governing equations on asymptotically long time scales. While
the details of the discussion are restricted to the class of model equations
having a leading cubic nonlinearity, the results strongly indicate that
mean-field evolution equations are generic for bimodal disturbances in
dispersive systems with \O(1) group velocity.
