5.7: The 100-Year Wave - Estimate from Short-Term Statistics

Last updated
Save as PDF

Page ID: 47250

Franz S. Hover & Michael S. Triantafyllou
Massachusetts Institute of Technology via MIT OpenCourseWare

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

For long-term installations, it is important to characterize the largest wave to be expected in an extremely large number of cycles. We will make such a calculation here, although the Rayleigh distribution does not adequately capture extreme events over such time scales. Spectra and the consequent Rayleigh height distribution are short-term properties only.

The idea here is to equate \( p(h > h_o) \) from the distribution with the definition that in fact \(h > h_o\) once in 100 years. Namely, we have

\[ p (h > h_{100 yr}) \, = \, \dfrac{1}{100 \textrm{ years} / \bar{T}} \, = \, e^{-2 h_{100 yr}^2 / \bar{h}^{1/3}}, \]

where \(\bar{T}\) is the average period. As we will see, uncertainty about what the proper \(\bar{T}\) is has little effect in the answer. Looking at the first equality, and setting \(\bar{T} = 8\) seconds and \(\bar{h}^{1/3} = 2\) meters as example values, leads to

\begin{align*} 2.5 \times 10^{-9} \, &= \, e^{-2 h_{100 yr}^2/ \bar{h}^{1/3}}; \\[4pt] \log (2.5 \times 10^{-9}) \, &= \, -2 h_{100 yr}^2 / 4; \\[4pt] h_{100 yr} \, &= \, 6.3 \textrm{ meters, or } 3.1 \bar{h}^{1/3}. \end{align*}

According to this calculation, the 100-year wave height is approximately three times the significant wave height. Because \(\bar{T}\) appears inside the logarithm, a twofold error in \(\bar{T}\) changes the extreme height estimate only by a few percent.