Equations arising from modeling elastic media in physics are typically generalisations of wave equations in which different components of the system may have different speeds of propagation; furthermore, the dispersion relation may not be isotropic, and thus the speed of propagation may vary with the direction of propagation.
A particularly simple model for elasticity arises from a two-speed wave equation system of two fields and , with propagating slower than , e.g.
where and for some . This case occurs physically when propagates at the speed of light and v propagates at some slower speed. In this case the null forms are not as useful, however the estimates tend to be more favourable (if the non-linearities are "off-diagonal") since the light cone for is always transverse to the light cone for . One can of course generalize this to consider multiple speed (nonrelativistic) wave equations.
Examples of two-speed models include