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 $H_m$.

  • $\begingroup$ Hi, welcome on SciComp. If I understand your question, you just have to evaluate your formula, provided that the set $\{H_m\}$ is ordered for decreasing $H_m$, i.e $H_1 \geq H_2 \geq H_3 \geq \ldots \geq H_N$. $\endgroup$
    – Stefano M
    Commented Sep 6, 2013 at 22:14

1 Answer 1


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 $\eta$.

  • $\begingroup$ Hi @IRO-bot, I'm getting Sea Level values every 10 secs and I want get Hs, H/3 and H/10 every one hour, I was thinking calculate m0 on this way: vitutor.co.uk/estadistica/descriptiva/images/26.gif $\endgroup$
    – Goku
    Commented Sep 9, 2013 at 20:52
  • $\begingroup$ Yes, you can calculate $m_0$ that way. The obvious question is over how many samples to average? This depends on your application. To capture the longest swell (i.e. Pacific ocean), 1 minute should suffice. If your data is from the field and not from the lab/tank, I would average over 10 minutes even, as $H_{s}$ typically changes on much longer time scales. $\endgroup$ Commented Sep 9, 2013 at 21:37

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