  High-frequency wave propagation is often modelled by nonlinear Friedrichs
systems where both the differential equation and the initial data contain the
inverse of a small parameter $\varepsilon$, which causes oscillations with
wavelengths proportional to $\varepsilon$ in time and space. A prominent
example is the Maxwell--Lorentz system, which is a well-established model for
the propagation of light in nonlinear media. In diffractive optics, such
problems have to be solved on long time intervals with length proportional to
$1/\varepsilon$. Approximating the solution of such a problem numerically with
a standard method is hopeless, because traditional methods require an extremely
fine resolution in time and space, which entails unacceptable computational
costs. A possible alternative is to replace the original problem by a new
system of PDEs which is more suitable for numerical computations but still
yields a sufficiently accurate approximation. Such models are often based on
the \emph{slowly varying envelope approximation} or generalizations thereof.
Results in the literature state that the error of the slowly varying envelope
approximation is of $\mathcal{O}(\varepsilon)$. In this work, however, we prove
that the error is even proportional to $\varepsilon^2$, which is a substantial
improvement, and which explains the error behavior observed in numerical
experiments. For a higher-order generalization of the slowly varying envelope
approximation we improve the error bound from $\mathcal{O}(\varepsilon^2)$ to
$\mathcal{O}(\varepsilon^3)$. Both proofs are based on a careful analysis of
the nonlinear interaction between oscillatory and non-oscillatory error terms,
and on \textit{a priori} bounds for certain ``parts'' of the approximations
which are defined by suitable projections.
