  We present a new approach to analyze the validation of weakly nonlinear
geometric optics for entropy solutions of nonlinear hyperbolic systems of
conservation laws whose eigenvalues are allowed to have constant multiplicity
and corresponding characteristic fields to be linearly degenerate. The approach
is based on our careful construction of more accurate auxiliary approximation
to weakly nonlinear geometric optics, the properties of wave front-tracking
approximate solutions, the behavior of solutions to the approximate asymptotic
equations, and the standard semigroup estimates. To illustrate this approach
more clearly, we focus first on the Cauchy problem for the hyperbolic systems
with compact support initial data of small bounded variation and establish that
the $L^1-$estimate between the entropy solution and the geometric optics
expansion function is bounded by $O(\varepsilon^2)$, {\it independent of} the
time variable. This implies that the simpler geometric optics expansion
functions can be employed to study the behavior of general entropy solutions to
hyperbolic systems of conservation laws. Finally, we extend the results to the
case with non-compact support initial data of bounded variation.
