Root-Mean-Square Wave Height from the Significant Wave Height

The wave heights in a real sea are all different, so one number must stand for the lot. The significant wave height $H_s$ is the one a buoy reports; it is spectral, defined as four times the standard deviation of the surface. The root-mean-square height $H_{rms}$ is a different summary: the square root of the average of the squared wave heights. This page converts the first into the second, and it matters because a regular wave of the root-mean-square height carries the same energy as the whole irregular sea. That is why wave-energy formulas written for a single wave height are fed this one.

This is a derived relation, not a fit, but it rests on an assumption: that the wave heights follow the Rayleigh distribution, the statistical description in Longuet-Higgins (1952), On the statistical distribution of the heights of sea waves. Under it, with $\sigma$ the standard deviation of the surface, $H_s = 4\sigma$ and $H_{rms} = \sqrt{8}\,\sigma$. Divide the second by the first and $\sigma$ cancels, leaving $H_{rms} = \frac{H_s}{\sqrt{2}}$. Only the ratio of the two summaries is fixed here; the size of the sea comes from $H_s$.

For a surface with $\sigma = 0.5$ m, $H_s$ is 2 m and $H_{rms} = \sqrt{8} \times 0.5 = 1.4142$ m, which is what the page returns for $H_s = 2$ m. Run backward, $H_{rms} = 1.41421$ m gives $H_s = 2$ m again. A quick rule says the root-mean-square height is about $H_s / 1.4$, the same relation with the square root of two rounded; for $H_s = 2.8$ m it gives about 2.0 m.

Because the relation depends on the Rayleigh assumption, it is only as good as that description of the sea, and a sea whose heights depart from it will not obey the ratio exactly. The other height a Rayleigh sea implies from the same significant height is the most probable largest wave, and the power the sea delivers is written for the significant height on the wave power page.