Abstract

In this paper, we study the statistical characteristics of the Riemann wave and the integral of it. Such problems arise in the physics of fronts’ propagation, in particular, during the wave run-up on the shore, a flame front, and phase surfaces. Using the relation of the Lagrangian and Eulerian statistical descriptions, we obtained general expressions for the probability distributions of the front velocity and its displacement. We show that the joint probabilistic distribution of displacement, velocity, and acceleration at the input of a nonlinear medium are necessary to find the probability distribution of displacement. Based on the idea of probability as the time the signal spent in a certain interval, the characteristics of the waves after their breaking were calculated.

中文翻译：

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黎曼波的概率分布及其积分

摘要

在本文中，我们研究了黎曼波的统计特性及其积分。这些问题出现在前沿传播的物理学上，特别是在海岸，火焰前沿和相表面的波浪上升期间。利用拉格朗日和欧拉统计描述的关系，我们获得了前速度及其位移概率分布的一般表达式。我们表明，非线性介质输入端的位移，速度和加速度的联合概率分布对于找到位移的概率分布是必要的。基于概率的概念，即信号在特定时间间隔内停留的时间，计算了波浪破裂后的特征。