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For a fixed $\mathrm{Re}\gg 1$, optimize the gain over the initial wave tilt and a later observation time. From the energy profile,Its stationary points satisfyThere are two negative roots in the high-Reynolds-number regime: an early energy minimum and a later maximum. Write them as $T_{-}$ and $T_{+}$. Balancing the large-$|T|$ terms for the early root and the small-$|T|$ terms for the late root givesThus the optimum disturbance starts as a sufficiently tightly wound leading wave and reaches its peak slightly before $k_x=0$; finite viscosity shifts the peak away from the inviscid swing.

Normalize its initial energy at $T_{-}$. The exact maximum ratio isThe prefactor is asymptotic to $(2\mathrm{Re}{)}^{2/3}$, while $g(T_{-})/\mathrm{Re}\to -2/3$ and $g(T_{+})/\mathrm{Re}\to 0$. Therefore optimal transient amplification of a viscous shearing wave givesThe time between the optimum initial tilt and the peak is approximately $(2\mathrm{Re}{)}^{1/3}/(2A)$. Without viscosity, ever larger initial tilts would allow unbounded gain; viscosity damps them before they can swing, selecting the finite optimum. The coefficient above uses the paper's Reynolds number based on $A$, not one based on the full shear rate $2A$.