Rocks, soils, and oil and tar sands are complex materials containing pores, cracks, and other defects. If this is the case, their constitutive behaviour can be nonlinear and stress-dependent, which implies that loading changes the properties of the material. If materials react differently to compression and tension, this can have a strong influence on the propagation of seismic waves. A possible approach to describe such materials is heteromodular elastic theory: a piece-wise linear theory with different elastic moduli depending on the stress state. The formulation of such a theory for the 2D and 3D cases is a difficult task. Even for the 1D case, there are only a few dynamical problems solved. One needs a simple model problem to imagine how the signal behaves when passing through such a medium. We consider a 1D problem with a small heteromodularity, obtain its analytical (asymptotical) solution for the case of a suddenly applied harmonic load, and compare it with numerical results. This gives us a general idea of the character of the wave propagation we may expect in such media, and the technique to apply in more complex cases.


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