The effect of a small background inhomogeneity on the asymptotic properties of linear perturbations

General regularities in the evolution of one-dimensional unstable linear perturbations on a weakly inhomogeneous background are studied when, at the initial instant, the perturbations are concentrated in the δ-neighbourhood of a certain point. Times are considered when these perturbations do not fall outside the limits of a certain domain of size l such that δ ≪ l ≪ L, where L is the large characteristic size of the background inhomogeneity. With contain assumptions, the effect of the background inhomogeneity on the asymptotic behaviour of the perturbations at long times is taken into account in a general form. The first corrections to the well known asymptotic relation for the evolution of perturbations on a homogeneous background, that arise because of background inhomogeneity, are obtained using Hamilton's method. An example of the use of the proposed approximate method is considered and the error in the approximation is estimated.