# Units of wave spectrum

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.

• Not sure without more context, but does this table of units help?
– gerrit
Jan 25 '17 at 21:32
• $m^2$ is "square meters" or "meters squared". You can't just "integrate" an equation, you have to state what the integration is in respect to. So is the integration over time, over area, or what?
– MaxW
Jan 25 '17 at 23:19
• By "integral of the spectrum over all frequencies", I meant integrate w.r.t. $\omega$ (angular frequency). So $\int S(\omega) \,d \omega$. Jan 26 '17 at 9:58
• Please edit that into your question Jan 27 '17 at 10:46

I understand that the integral of the spectrum over all frequencies is the variance which means that the integral should have $m^2$ units? Please correct me if I am wrong.
You are correct. If elevation $\eta(t)$ is the measured quantity (units of $m$), the Fourier transform of wave variance $\eta^2$ yields spectrum $S(f)$ with the units of $m^2/Hz$, or if you are working with angular frequency $\omega = 2\pi f$, it yields $S(\omega)$ with the units of $m^2/rad/Hz$.