Abstract Wavemaker problem is a fundamental and important issue in the study of marine and coastal engineering. An accurate incident wave field provided by a wavemaker is crucial to explore the problems of wave and wave-structure interaction. Generally, wavemakers start working from a rest state both in physical and numerical tanks. Excessive acceleration induced by sudden motions of wavemaker will result in generating wave instability even wave breaking unexpectedly and has potential harm to wavemaker facilities in physical tanks. It might cause numerical instability in numerical tanks as well. Although different ramp functions can be constructed and introduced to mitigate this issue, theoretical analysis on the performance of different ramp functions is still lacking. In this study, based on the new elementary solution on transient waves given by Chen and Li (Journal of Engineering Mathematics, 2019, 115(1): 121-140), a time-domain analytical model on transient waves with a general ramp function is developed for a vertical flexible wavemaker plate. Two special ramp functions, namely Eatock Taylor's and Lighthill's ramp function, are used to investigate their effects on the spatial-temporal evolution of transient waves. Moreover, based on the open source two-phase flow model Basilisk, nonlinear effect on ramp functions’ performance is investigated by increasing wave steepness of incident waves. Under weakly nonlinear conditions, the numerical solution is consistent with the present analytical solution, which further confirms the validity of the present analytical solution. And it is found that computational cost for the Lighthill's ramp function is less than that of the Eatock Taylor's ramp function in numerical simulations. Finally, it should be pointed out that the present theoretical solution provides an effective way to obtain new insights on the performance of those potential ramp functions which might be useful and applied in future studies.

中文翻译：

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由具有一般斜坡函数的垂直柔性造波板产生的瞬态波

摘要 造波器问题是海洋与海岸工程研究中的一个基础性和重要问题。造波器提供的准确入射波场对于探索波浪和波浪结构相互作用问题至关重要。通常，造波器在物理和数字水箱中从静止状态开始工作。造波器突然运动引起的过大加速度将导致产生波浪不稳定甚至意外破浪，并对物理罐中的造波器设施造成潜在危害。它也可能导致数值坦克中的数值不稳定。尽管可以构建和引入不同的斜坡函数来缓解这个问题，但仍然缺乏对不同斜坡函数性能的理论分析。在这项研究中，基于 Chen 和 Li 给出的瞬态波新基本解（Journal of Engineering Mathematics, 2019, 115(1): 121-140），开发了具有一般斜坡函数的瞬态波时域分析模型，用于垂直柔性造波板。两个特殊的斜坡函数，即伊托克泰勒和莱特希尔的斜坡函数，用于研究它们对瞬态波时空演化的影响。此外，基于开源的两相流模型 Basilisk，通过增加入射波的波陡度来研究对斜坡函数性能的非线性影响。在弱非线性条件下，数值解与本解析解一致，进一步证实了本解析解的有效性。并且在数值模拟中发现，Lighthill 斜坡函数的计算成本小于 Eatock Taylor 斜坡函数的计算成本。最后，应该指出的是，目前的理论解决方案提供了一种有效的方法来获得对那些可能在未来研究中有用和应用的潜在斜坡函数的性能的新见解。