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.

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \Box u = F(U, DU), ~\Box_s v = G(U, DU)}

where $U=(u,v)$ and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \Box_s = s^2 \Delta - \partial_t^2} for some Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 0 < s < 1} . This case occurs physically when Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle F, G} are "off-diagonal") since the light cone for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle u} is always transverse to the light cone for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 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