  We consider a generalization of the projecting operators method for the case
of Cauchy problem for systems of 1D evolution differential equations of first
order with variable coefficients. It is supposed that the coefficients
dependence on the only variable x is weak, that is described by a small
parameter introduction. Such problem corresponds, for example, to the case of
wave propagation in a weakly inhomogeneous medium. As an example, we specify
the problem to adiabatic acoustics. For the Cauchy problem, to fix
unidirectional modes, the projection operators are constructed. The method of
successive approximations (perturbation theory) is developed and based on
pseudodifferential operators theory. The application of these projection
operators allows to obtain approximate evolution equations corresponding to the
separated directed waves.
