This is the first of a two-part study on the propagation of stress gradient in composite materials. Both the plane-strain and axisymmetric motions are considered. In this paper, the reflection and transmission coefficients are derived for the stress, the stress gradient, and the higher-order derivatives of stress at an interface between two elastic media. It is shown that while the reflection and transmission of the stress do not depend on the geometries of the incident wave front and the interface boundary, the reflection and transmission of the stress gradient, and the higher-order derivatives of stress do depend on these geometries. Explicit expressions are derived for the reflection and transmission of the stress gradient in terms of the radii of curvatures of the incident wave front and the interface boundary.