= 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 gives
$$
\boxed{\sigma_{13}=\sigma_{31}=\mu w_{,1},\qquad
\sigma_{23}=\sigma_{32}=\mu w_{,2},}
$$
with every other component zero. The first two equations of motion vanish because the fields are independent of $x_3$. The third is $\rho w_{,tt}=\sigma_{31,1}+\sigma_{32,2}$, hence
$$
\boxed{\nabla^2w-\frac1{c^2}w_{,tt}=0,\qquad c^2=\mu/\rho.}
$$
The 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 $\beta^2w_{xx}+w_{x_2x_2}=0$. 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 that
$$
\boxed{w=\operatorname{Re}H(z),\qquad z=x+iy.}
$$
This 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.
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