# Elasticity

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.

## Two-speed model

A particularly simple model for elasticity arises from a two-speed wave equation system of two fields ${\displaystyle u}$ and ${\displaystyle v}$, with ${\displaystyle v}$ propagating slower than ${\displaystyle u}$, e.g.

${\displaystyle \Box u=F(U,DU),~\Box _{s}v=G(U,DU)}$

where ${\displaystyle U=(u,v)}$ and ${\displaystyle \Box _{s}=s^{2}\Delta -\partial _{t}^{2}}$ for some ${\displaystyle 0. This case occurs physically when ${\displaystyle u}$ 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 ${\displaystyle F,G}$ are "off-diagonal") since the light cone for ${\displaystyle u}$ is always transverse to the light cone for ${\displaystyle v}$. One can of course generalize this to consider multiple speed (nonrelativistic) wave equations.

Examples of two-speed models include