Abstract
A run-up of irregular long sea waves on a beach with a constant slope is studied within the framework of the nonlinear shallow-water theory. This problem was solved earlier for deterministic waves, both periodic and pulse ones, using the approach based on the Legendre transform. Within this approach, it is possible to get an exact solution for the displacement of a moving shoreline in the case of irregular-wave run-up as well. It is used to determine statistical moments of run-up characteristics. It is shown that nonlinearity in a run-up wave does not affect the velocity moments of the shoreline motion but influences the moments of mobile shoreline displacement. In particular, the randomness of a wave field yields an increase in the average water level on the shore and decrease in standard deviation. The asymmetry calculated through the third moment is positive and increases with the amplitude growth. The kurtosis calculated through the fourth moment turns out to be positive at small amplitudes and negative at large ones. All this points to the advantage of the wave run-up on the shore as compared to a backwash at least for small-amplitude waves, even if an incident wave is a Gaussian stationary process with a zero mean. The probability of wave breaking during run-up and the applicability limits for the derived equations are discussed.
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