Inviscid Couette initial-value Green function ($\tilde{v}(y,t)={\int }_{-1}^1{G}_k(y,s)e^{-i\alpha \lambda st}(D_s^2-k^2){\tilde{v}}_0(s)ds$)

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.