I trying to find a computing algorithm for the Significant wave height $H_{1/3}$, or $H_\text{s}$ or $H_\text{sig}$ in the time domain. This is defined as the average height of the one-third part of the measured waves—which are $N$ in number—having the largest wave heights:
$$ H_{1/3} = \frac{1}{\frac13 N} \sum_{m=1}^{\frac13 N} H_m $$
For an example set consider $\{100,300,400,400, ..\}$ of height values $H_m$.
Significant wave height, $H_{1/3}$, defined as the average height of the highest one-third of all waves, is historical. In the early days, the wave state was observed visually, which resulted in this definition of $H_{1/3}$. This quantity is not necessarily equal to the contemporary notion of significant wave height, $H_{s}$, which is defined as:
$$ H_{s} = 4 \sqrt{m_0}, $$ where $m_{0}$ is the zero-order moment of the wave variance spectrum:
$$ m_0 = \int{F(f)}\ df = <\!\eta^2\!\!>. $$
$\eta$ is the surface elevation, and $<\!\eta^2\!\!>$ is the time average of the surface elevation variance.
Though $H_{s} \ne H_{1/3}$, Phillips (1977) shows that $H_{s} \approx H_{1/3}$ for a narrow wave variance spectrum $F(f)$. See Dynamics of The Upper Ocean by Phillips, 1977, Cambridge University Press.
If you have time series of $\eta$, you can then calculate $H_{s}$ from $m_0$ in a straightforward fashion. Keep in mind that $H_{s}$ as an integrated quantity only makes sense for linear (small slope) waves that are quasi-uniform in space and time, i.e. $F(f)$ and $H_s$ are changing over a significantly longer time scale than that of $\eta$.
