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 $u$ and $v$, with $v$ propagating slower than $u$, e.g.

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

where $U=(u,v)$ and $\Box _{s}=s^{2}\Delta -\partial _{t}^{2}$ for some $0<s<1$. This case occurs physically when $u$ 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 $F,G$ are "off-diagonal") since the light cone for $u$ is always transverse to the light cone for $v$. One can of course generalize this to consider multiple speed (nonrelativistic) wave equations.

Examples of two-speed models include

• Two-speed quadratic-derivative nonlinear wave equations
• Two-speed quadratic NLW