Sea state

Winter, North Atlantic – Water over deck and hatches, storm with huge waves (1958)

The World Meteorological Organization (WMO) sea state code largely adopts the 'wind sea' definition of the Douglas Sea Scale.

The direction from which the swell is coming should be recorded.

In engineering applications, sea states are often characterized by the following two parameters:

• The significant wave height H_1/3 — the mean wave height of the one third highest waves.
• The mean wave period, T₁.

The sea state is in addition to these two parameters (or variation of the two) also described by the wave spectrum ${\displaystyle S(\omega ,\Theta )}$ which is a function of a wave height spectrum ${\displaystyle S(\omega )}$ and a wave direction spectrum ${\displaystyle f(\Theta )}$. Some wave height spectra are listed below. The dimension of the wave spectrum is ${\displaystyle \{S(\omega )\}=\{{\text{length}}^{2}\cdot {\text{time}}\}}$, and many interesting properties about the sea state can be found from the spectrum.

The relationship between the spectrum ${\displaystyle S(\omega _{j})}$ and the wave amplitude ${\displaystyle A_{j}}$ for a wave component ${\displaystyle j}$ is:

${\displaystyle {\frac {1}{2}}A_{j}^{2}=S(\omega _{j})\,\Delta \omega }$

• The International Towing Tank Conference (ITTC) [1] recommended spectrum model for fully developed sea (ISSC[2] spectrum/modified Pierson-Moskowitz spectrum):[3]

${\displaystyle {\frac {S(\omega )}{H_{1/3}^{2}T_{1}}}={\frac {0.11}{2\pi }}\left({\frac {\omega T_{1}}{2\pi }}\right)^{-5}\mathrm {exp} \left[-0.44\left({\frac {\omega T_{1}}{2\pi }}\right)^{-4}\right]}$

• ITTC recommended spectrum model for limited fetch (JONSWAP spectrum)

${\displaystyle S(\omega )=155{\frac {H_{1/3}^{2}}{T_{1}^{4}\omega ^{5}}}\mathrm {exp} \left({\frac {-944}{T_{1}^{4}\omega ^{4}}}\right)(3.3)^{Y},}$

where

${\displaystyle Y=\exp \left[-\left({\frac {0.191\omega T_{1}-1}{2^{1/2}\sigma }}\right)^{2}\right]}$

and

${\displaystyle \sigma ={\begin{cases}0.07&{\text{if }}\omega \leq 5.24/T_{1},\\0.09&{\text{if }}\omega >5.24/T_{1}.\end{cases}}}$

(The latter model has since its creation improved based on the work of Phillips and Kitaigorodskii to better model the wave height spectrum for high wavenumbers.[4])

An example function ${\displaystyle f(\Theta )}$ might be:

${\displaystyle f(\Theta )={\frac {2}{\pi }}\cos ^{2}\Theta ,\qquad -\pi /2\leq \Theta \leq \pi /2}$

Thus the sea state is fully determined and can be recreated by the following function where ${\displaystyle \zeta }$ is the wave elevation, ${\displaystyle \epsilon _{j}}$ is uniformly distributed between 0 and ${\displaystyle 2\pi }$, and ${\displaystyle \Theta _{j}}$ is randomly drawn from the directional distribution function ${\displaystyle {\sqrt {f(\Theta )}}:}$[5]

${\displaystyle \zeta =\sum _{j=1}^{N}{\sqrt {2S(\omega _{j})\Delta \omega _{j}}}\;\sin(\omega _{j}t-k_{j}x\cos \Theta _{j}-k_{j}y\sin \Theta _{j}+\epsilon _{j}).}$

In addition to the short term wave statistics presented above, long term sea state statistics are often given as a joint frequency table of the significant wave height and the mean wave period. From the long and short term statistical distributions it is possible to find the extreme values expected in the operating life of a ship. A ship designer can find the most extreme sea states (extreme values of H_1/3 and T₁) from the joint frequency table, and from the wave spectrum the designer can find the most likely highest wave elevation in the most extreme sea states and predict the most likely highest loads on individual parts of the ship from the response amplitude operators of the ship. Surviving the once in 100 years or once in 1000 years sea state is a normal demand for design of ships and offshore structures.

• Beaufort scale
• Cross sea
• Douglas sea scale

• Bowditch, Nathaniel (1938), American Practical Navigator, H.O. pub No. 9 (revised ed.), United States Hydrographic Office, OCLC 31033357
• Faltinsen, O. M. (1990), Sea Loads on Ships and Offshore Structures, [Cambridge University Press], ISBN 0-521-45870-6