Past exam of the mathematics course of the University of Cambridge / 2003 / iii / Paper 70 / 4 / a / Solution

For antiplane shear, the only nonzero infinitesimal strains are $e_{13}=e_{31}=w_{,1}/2$ and $e_{23}=e_{32}=w_{,2}/2$. Their trace is zero. The isotropic constitutive law ${\sigma }_{ij}={\lambda }_L{e}_{kk}{\delta }_{ij}+2\mu e_{ij}$ consequently giveswith every other component zero. The first two equations of motion vanish because the fields are independent of $x_3$. The third is $\rho w_{,tt}={\sigma }_{31,1}+{\sigma }_{32,2}$, henceThe PDF omits the final equals-zero; this is the equation supplied by momentum balance.

For a steadily moving field put $x=x_1-Vt$ and $\beta =\sqrt{1-V^2/c^2}$. In the subsonic case $|V|<c$, the equation becomes ${\beta }^2{w}_{xx}+w_{x_2{x}_2}=0$. With $y=\beta x_2$ it reduces to $w_{xx}+w_{yy}=0$. Thus $w$ is a harmonic function and, on a simply connected region, has a harmonic conjugate; there is a holomorphic function $H(z)$ such thatThis is steadily moving antiplane shear. The representation is local on more general domains unless the conjugate's periods vanish. The subsonic condition is essential: at or above the shear-wave speed the coordinate rescaling is degenerate or the equation is hyperbolic rather than this elliptic harmonic problem.