Non‐Gaussian Properties of Second‐Order Random Waves

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Abstract
Under the assumption that the second‐order random wave theory is valid, theoretical solutions for the probabilistic cumulants (or moments), in particular the third and fourth cumulants, of wave elevation associated with a deep‐water unidirectional random wave of arbitrary wave bandwidth are derived. In general, knowing the probability density function is not sufficient to obtain the corresponding power spectral density function, and vice versa. However, through the use of the second‐order random wave theory, the study shows that the probabilistic cumulants and spectral parameters of stationary random wave processes become closely related. In the present paper, numerical attention is given to random waves described by either Pierson‐Moskowitz, JONSWAP, or Wallops spectra. The paper also numerically verifies that the use of the second‐order random wave theory is appropriate only when the significant wave slope is less than about 0.02.