= Steadily moving antiplane shear
{title2=$y=\beta x_2,\quad\beta=\sqrt{1-V^2/c^2}$}
For subsonic motion $|V|<c$, an <antiplane shear> <displacement> $w(x_1-Vt,x_2)$ satisfies $\beta^2w_{xx}+w_{x_2x_2}=0$. The displayed rescaling makes it a <harmonic function> of $(x,y)$, so locally $w=\operatorname{Re}H(x+iy)$ for a <holomorphic function> $H$. Set $F=\beta H$. The <Cauchy-Riemann equations> then give $\mu F'=\beta\sigma_{13}-i\sigma_{23}$. Keeping this potential normalization explicit avoids losing a factor of $\beta$ in the dynamic <stress> field. At sonic or supersonic speeds the elliptic harmonic reduction does not apply.
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