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For the moving semi-infinite crack, take the branch cut of $\sqrt{z}$ on the negative real axis and $\sqrt{x}>0$ ahead of the tip. With steadily moving antiplane shear, face tractions ${\sigma }_{23}^{\pm }=-f(x)$ imply $\mathrm{Im}{F}^{\prime \pm }=f/\mu$. The antisymmetric displacement response gives $F^{\prime -}=-\overline{F^{\prime +}}$. Hence $G=\sqrt{z}{F}^{\prime }$ has jump $G^{+}-G^{-}=-2\sqrt{-x}f(x)/\mu$. The Sokhotski–Plemelj theorem gives the displayed Cauchy integral, assuming convergence and no extra remote homogeneous loading. For $x>0$ it is purely imaginary, so ${\sigma }_{23}={\int }_{-\infty }^0\sqrt{-s}f(s)/(x-s)ds/(\pi \sqrt{x})$. At fixed co-moving loading this ahead-of-tip stress is independent of speed; the displacement still has the factor $1/\beta$.

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.