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工程科学与技术:2016,48(3):12-25
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水波问题中线性长波方程和缓坡方程解析解研究进展
(约稿)
Advance in research of analytical solutions to the linear long-wave equation and the mild-slope equation
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投稿时间:2016-04-26    修订日期:2016-04-26
中文摘要: 近20多年来,有关两类主要的水波深度平均方程(即线性长波方程和缓坡类方程)的解析解研究取得了一系列进展。关于线性长波方程,对理想地形(即水深函数为幂函数情形)和拟理想地形(即水深函数为幂函数与一个常数之和的情形),已经构造了一系列准确解析解。其中,针对理想地形所构造的解析解一般为封闭解,而针对拟理想地形所构造的解析解一般只能写成Taylor级数或Frobenius级数的形式。关于缓坡类方程,则于最近构造了一系列Taylor级数形式的准确解析解,解决了国际水波界40多年来的开问题,其中,针对分段单调和分段二阶光滑的二维地形以及分片单调和分片二阶光滑的轴对称三维地形,隐式的修正缓坡方程被成功转化为显式方程。本文拟对20多年来这两类深度平均水波方程解析求解的主要研究进展给予一个较全面系统的综述,并对该方面的研究前景做一些展望。
Abstract:In the past two decades, a series of exact analytical solutions to the two kinds of depth-averaged equations in water waves, i.e., the linear long-wave equation and the mild-slope type equations, have been constructed. For the linear long-wave equation, if the bottom topography is idealized with the water depth being a power function, then the related analytical solution can be written in a closed form. If the bottom topography is quasi-idealized with the water depth being a power function plus a constant, then the related analytical solution can be expanded into a Taylor series or a Frobenius series. For the mild-slope type equations, a number of exact analytical solutions in the form of Taylor series are constructed recently, where the implicit modified mild-slope equation is successfully transformed into an explicit equation for both two-dimensional bathymetries and three-dimensional axisymmetric bathymetries with piecewise monotonicity and piecewise second-order smoothness. In this paper, these advances are summarized and reviewed. And some prospects of the research in the future are made
文章编号:201600420     中图分类号:    文献标志码:
基金项目:国家自然科学基金:请在下栏中列出明细(含项目号和具体课题名)
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刘焕文 约稿 jsu@scu.edu.cn 
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  jsu@scu.edu.cn 
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引用文本:
刘焕文.水波问题中线性长波方程和缓坡方程解析解研究进展[J].工程科学与技术,2016,48(3):12-25.
.Advance in research of analytical solutions to the linear long-wave equation and the mild-slope equation[J].Advanced Engineering Sciences,2016,48(3):12-25.