Past exam of the mathematics course of the University of Cambridge / 2005 / iii / Paper 82 / 1 / a / Solution

Put $Z=\rho \beta =\sqrt{\rho \mu }>0$, the seismic impedance. In a uniform medium, define $w_{+}=v-\sigma /Z$ and $w_{-}=v+\sigma /Z$. The field equations giveThus $w_{+}=2F(x-\beta t)$ and $w_{-}=2G(x+\beta t)$, with arbitrary independent profiles $F,G$. Reconstruction givesThe two characteristic components therefore have $v_{+}=F$, ${\sigma }_{+}=-Z{v}_{+}$ and $v_{-}=G$, ${\sigma }_{-}=Z{v}_{-}$.

The kinetic plus elastic energy mass density is $e=\rho v^2/2+{\sigma }^2/(2\mu )$. Multiplying the equations by $v$ and $\sigma /\mu$ yields $e_t=(v\sigma {)}_x$. Hence the elastic-wave energy flux is $S=-v\sigma$. For a pure forward wave $S_{+}=Z{v}_{+}^2\ge 0$; for a pure backward wave $S_{-}=-Z{v}_{-}^2\le 0$. In the mixture the cross terms cancel, giving $S=Z(v_{+}^2-v_{-}^2)$. Strict sign holds for nonzero components.