5.6: 1/N'th Average Value

We can similarly define a statistic that is the average of the \(1/N\)'th highest peaks. In this case, we are after the average of this collection of \(1/N\) peaks:

\begin{align} \bar{a}^{1/N} \, &= \, E(a | a > a^{1/N}) \\[4pt] &= \int\limits_{a^{1/N}}^{\infty} a \, p(a = a_m | a_m > a^{1/N}) \, da. \end{align}

Note that we use the dummy variable \(a_m\). We have then the conditional probability

\[ p(a = a_m | a_m > a^{1/N}) = \dfrac{p \left[ (a=a_m) \cap (a_m > a^{1/N}) \right]}{p(a_m > a^{1/N})}. \]

Working in nondimensional form, we have

\begin{align} \bar{\eta}^{1/N} \, &= \, \int\limits_{\eta^{1/N}}^{\infty} \dfrac{1}{1/N} \eta p(\eta = \eta_m) \, d\eta \\[4pt] &= \, \dfrac{2qN}{1+q} \int\limits_{\eta^{1/N}}^{\infty} \eta^2 e^{{\eta^2}/2} \, d\eta. \end{align}

Here are a few explicit results for amplitude and height:

\begin{align} \bar{a}^{1/3} \, &= \, 1.1 \sqrt{M_0} \text{ to } 2 \sqrt{M_0} \\[4pt] \bar{a}^{1/10} \, &= \, 1.8 \sqrt{M_0} \text{ to } 2.5 \sqrt{M_0} \end{align}

The amplitudes here vary depending on the spectral broadness parameter \(\epsilon\) - this point is discussed in the Principles of Naval Architecture, volume III, page 20 (E.V., Lewis, ed. SNAME, 1989; see here for a link to this text). Here are some \(1/N\)'th average heights:

\begin{align} \bar{h}^{1/3} \, &= \, 4.0 \sqrt{M_0} \\[4pt] \bar{h}^{1/10} \, &= \, 5.1 \sqrt{M_0}. \end{align}

The value \(\bar{h}^{1/3}\) is the significant wave height, the most common description of the size of waves. It turns out to be very close to the wave size reported by experienced mariners.

Finally, here are the expected highest heights in \(N\) observations - which is not quite either of the \(1/N\)'th maximum or the \(1/N\)'th average statistics given above:

\begin{align} \bar{h}(100) \, &= \, 6.5 \sqrt{M_0} \\[4pt] \bar{h}(1000) \, &= \, 7.7 \sqrt{M_0} \\[4pt] \bar{h}(10000) \, &= \, 8.9 \sqrt{M_0}. \end{align}