Backward waves and spatial dispersion

Using a Lagrange chain as an example, it is shown that a backward wave can exist in standard crystals in the presence of the spatial dispersion as well as in high-pass filter chains in the absence of the aforementioned dispersion. As an example, a cell of a 3D backward-wave system is presented. It is noted that such waves can be observed in chiral media, in particular, in cholesteric liquid crystals. An example of a structure where longitudinal waves can propagate is presented. The existence of these waves is due to the spatial dispersion.