In ocean spectra such as the Pierson Moskowitz or the JONSWAP models, the units of $S(\omega)$ are $m^2/(rad/s)$ (or whatever unit of measurement you are working in). Where does the $m^2$ come from and what does it mean physically?
I understand that the integral of the spectrum over all frequencies, i.e. $\int S(\omega) \,d\omega $, is the variance which means that the integral should have $m^2$ units? Please correct me if I am wrong.