# zbMATH — the first resource for mathematics

Statistical properties of wave groups in a random sea state. (English) Zbl 0557.76041
Waves on a random sea have a peak frequency $$\sigma_ p$$. A wave is defined to be one cycle of the sea-surface elevation $$\zeta$$ (t) with frequency $$\sigma$$ in the range $$0.5\sigma_ p\leq \sigma \leq 1.5\sigma_ p$$; this accords with an observer’s intuition, as shown by typical graphs. The Gaussian model is that for $$-T/2\leq t\leq T/2$$, $$\zeta (t)=\sum^{\infty}_{n=0}c_ n\cos (\sigma_ nt+\epsilon_ n)$$, where $$\sigma_ n=2n\pi /T$$ and the phases $$\epsilon_ n$$ are distributed uniformly over (0,2$$\pi)$$. Re-writing this as $\zeta (t)=Re\{A(t)e^{i{\bar \sigma}t}\},\quad where\quad A(t)=\sum^{\infty}_{n=0}c_ ne^{i[(\sigma_ n-{\bar \sigma})t+\epsilon_ n]}=\rho (t)e^{i\Phi (t)}$ defines the real envelope function or wave-amplitude $$\rho(t).$$ Defining $$\eta (t)\equiv Im\{A(T)e^{i{\bar \sigma}t}\}$$ allows $$\rho$$ to be found from $$\rho =(\zeta^ 2+\eta^ 2)^{1/2}$$; $$\eta$$ may be computed by a discretized Hilbert transform $$\eta (t)=\frac{1}{\pi}\oint^{\infty}_{- \infty}(\zeta (s)/(t-s))ds$$ for very long records, or directly from the Fourier coefficients of $$\zeta(t)$$ for shorter records.
It has been shown (Rice 1944-1945, the author 1957) that the probability densities of $$\zeta$$ and $$\rho$$ are Gaussian and Rayleigh, respectively. The length of a wave group is defined as that between two successive upcrossings by the wave-amplitude of some specified level $$\rho$$, and a formula for the mean number $$\bar G(\rho)$$ of waves in a group is found. Similarly a formula for the mean number $$\bar H(\rho)$$ of successive waves exceeding some specified level $$\rho$$ is found, that determines the average length of a run of high waves. $$\bar H(\rho)<\bar G(\rho)$$ always, since $$\bar G(\rho)$$ is the mean repeat period of high runs of mean length $$\bar H(\rho)$$.
Typical wave spectra, and the effect of different filterings, are investigated; agreement between theory and experiment is reasonably good. An argument is given that the distributions of group and high run lengths will both be approximately exponential. This agrees reasonably with the numerically simulated data of Kimura (1980). The correlation between successive wave heights is studied following Uhlenbeck (1943), Rice (1944, 1958), Kimura (1980) by employing various narrow spectrum approximations.
Finally, a simple Markov theory based on that of Kimura (1980) is developed for the distribution of group length and high runs, and compared with the Gaussian theory.
Reviewer: F.J.Wright

##### MSC:
 76M35 Stochastic analysis applied to problems in fluid mechanics 76D33 Waves for incompressible viscous fluids 60G35 Signal detection and filtering (aspects of stochastic processes)
Full Text: