 Inviscid Couette initial-value Green function

For constant shear $U=\lambda y$, the wall-normal Laplacian of velocity is advected with phase $e^{-i\alpha \lambda yt}$. Inverting $D^2-k^2$, $k^2={\alpha }^2+{\beta }^2$, with its Dirichlet Green function gives the displayed initial-value solution. Impermeable inviscid walls impose zero normal velocity, not zero tangential velocity or no-slip wall vorticity.