We propose in this work a model of a continuum with a structure described by an infinite-order equation of motion. The resulting equation of motion provides dispersion for P and S waves, and there are parameter intervals where the wave velocities can decrease and increase with the average size of the microstructure. There are parameter intervals for which the velocity ratio of VSVP is high enough to have a negative Poisson ratio. In the case of long wavelength, we reduce the equations of motion to a fourth-order where they include non-linear and dispersion terms, and we show that solutions exist in the form of progressive waves. In the case of very small deformations, the solution gives an exponential attenuation of the signal. In the case of small deformation, the attenuation depends on the amplitude. The non-linear and dispersion terms cause the generation of multiple combination frequencies from a monochromatic input signal. Our results are that the non-linear effect of multiple frequencies is high because of the increase of non-linearity due to the structure parameter. We also show the existence of solutions for P and S waves with different frequencies and coincident wavelengths. The fact of different contents of frequencies for the same wavelength occurs in seismology and seismic exploration and still needs a scientific explanation.