= Maxwell start-up shear layer
{c}
{title2=$\widehat u(y,p)=\frac Up\exp\left[-y\sqrt{\frac{\rho p(1+p\tau)}{G_0\tau}}\right]$}
For a half-space initially at rest and a wall given a step velocity $U$, <momentum conservation> and the <Maxwell fluid> stress law give $\tau u_{tt}+u_t=(G_0\tau/\rho)u_{yy}$. The displayed decaying <Laplace transform> solution obeys the wall condition. Inverting on a right-half-plane <Bromwich contour> yields a shear signal of speed $\sqrt{G_0/\rho}$. The limit $\tau\to0$ at fixed $G_0\tau=\eta$ gives viscous error-function penetration; $\tau\to\infty$ at fixed $G_0$ gives a sharp elastic front.
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