Statistical Distribution of Maximum Surface Displacement in Weakly Nonlinear Random Waves

Abstract

Based on the second-order random wave theory, a joint statistical distribution of the sea surface displacement at a spatially fixed point and its time derivatives of order one and two are derived by using the characteristic function expansion method. From the joint distribution derived, a statistical distribution of the maximum sea surface displacement then is determined. The distribution is found to depend on the frequency spectrum of sea waves associated with the linear approximation and the second-order wave-wave interaction. It is noted that the distribution derived from the present work includes that derived from the linear wave theory as a special case. To demonstrate the nonlinear effect and the significance of the theoretical results developed, the maxima distribution of surface displacement of deepwater unidirectional waves is computed and compared with that obtained by the linear wave model.