This is a hyperbolic partial differential equation, like the original wave equation, but now of first-order instead of second-order. It is valid for coherent forward-propagating waves in directions near the k 0 -direction. The space and time scales over which E 0 varies are generally much longer than the spatial wavelength and temporal period of the carrier wave. A numerical solution of the envelope equation thus can use much larger space and time steps, resulting in significantly less computational effort.