Long-wave propagation on a uniformly sloping beach is formulated as a transient-response problem, with initially stationary water subjected to an incident wave. Both the water surface elevation and the horizontal flow velocity on the slope can be represented as convolutions of the rate of displacement of the water surface at the toe of the slope with a singular kernel function of time and space. The kernel, which is typically expressed in the form of an infinite series, accommodates the dynamic processes of long waves, such as shoaling, reflection, and multiple reflections over the slope and yields exact solutions of the linear shallow water equations for any smooth incident wave. The kernel convolution can be implemented numerically by using double exponential formulas to avoid the kernel singularity. The kernel formulation can be extended readily to nonlinear dynamics via the hodograph transform, which in turn enables the instantaneous prediction of nonlinear wave properties and of the occurrence of wave breaking in the near-shore area. This general description of long-wave dynamics provides new insights into the long-studied problem.

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均匀斜率上长波动力学的核表示

在均匀倾斜的海滩上的长波传播被表述为瞬态响应问题，最初静止的水受到入射波的影响。坡面上的水面高程和水平流速都可以表示为坡脚处水面位移率与时间和空间的奇异核函数的卷积。核，通常以无限级数的形式表示，适应长波的动态过程，如浅滩、反射和斜坡上的多次反射，并为任何平滑的入射波产生线性浅水方程的精确解. 核卷积可以通过使用双指数公式在数值上实现，以避免核奇异。核公式可以通过hodograph变换很容易地扩展到非线性动力学，这反过来又可以对非线性波浪特性和近岸区域波浪破碎的发生进行瞬时预测。这种对长波动力学的一般描述提供了对长期研究问题的新见解。