  We study statistical properties after a sudden episode of wind for water
waves propagating in one direction. A wave with random initial conditions is
propagated using a forced-damped higher order Nonlinear Schr\"odinger equation
(NLS). During the wind episode, the wave action increases, the spectrum
broadens, the spectral mean shifts up and the Benjamin-Feir index (BFI) and the
kurtosis increase. Conversely, after the wind episode, the opposite occurs for
each quantity. The kurtosis of the wave height distribution is considered the
main parameter that can indicate whether rogue waves are likely to occur in a
sea state, and the BFI is often mentioned as a means to predict the kurtosis.
However, we find that while there is indeed a quadratic relation between these
two, this relationship is dependent on the details of the forcing and damping.
Instead, a simple and robust quadratic relation does exist between the kurtosis
and the bandwidth. This could allow for a single-spectrum assessment of the
likelihood of rogue waves in a given sea state. In addition, as the kurtosis
depends strongly on the damping and forcing coefficients, by combining the
bandwidth measurement with the damping coefficient, the evolution of the
kurtosis after the wind episode can be predicted.
