【作者】 吴岱芸； 黄建亮； 陈树辉；

【Author】 WU Daiyun1,HUANG Jianliang2,CHEN Shuhui2(School of Engineering,Sun Yat-sen University,Guangzhou 510275,China)

【机构】 中山大学工学院；

【摘要】 本文采用双曲函数摄动方法[1-3]来研究明渠水流孤立波问题。明渠水流孤立波问题可以采用以下的行波约化方程来研究[4]d2h/dz2-2/h{dh/dz}-R/F(U-1)h(h3-F(U-1) {dh/dz}+R(h-1(h2+h(1-U2)+(U-1)/(U-1)h=0当弗劳德数F>4时,存在随着分叉参数U改变的周期滚波,其充要条件是无量纲波速1<U<3/2。当R>>1,F=4+σ,σ<<1时,Needham&Merkin将方程简化为含有非线性回复力微小项的范德波方程,采用小参数展开法求得孤立波的波速临界值[5]。本文应用双曲函数摄动法对简化方程进行求解,其结果,参数的分岔值与Needham&Merkin所得结果一致[4],同宿轨线与Maple数值结果非常接近。双曲函数摄动法在求解孤立波问题上具有简单、直观的特点,它能给出数值方法无法给出的解析表达式。对于R<<1的情况将另文讨论。更多还原

【Abstract】 The hyperbolic perturbation method[1-3] is used to study the solitary wave problem of an open-inclined channel.According to Needham & Merkin’s paper[4],the problem can be studied by the travelling wave reduced equation as following d2h/dz2-2/h{dh/dz}-R/F(U-1)h(h3-F(U-1) {dh/dz}+R(h-1(h2+h(1-U2)+(U-1)/(U-1)h=0 When the Froude number F 4,there exists a period roll wave depending on the bifurcation parameter U.The necessary and sufficient condition is the non-dimensional velocity1 U 3 /2.Needham & Merkin simplified the equation to a kind of Van der Pol equation with a small restoring force when R >>1 and F=4+σ,σ<<1.They obtained the critical value of the bifurcation parameter of U by using a small parameter expansion method.In this paper the hyperbolic perturbation method is used to solve the simplified equation.We not only obtain the value of the bifurcation parameter,but also get the solitary wave solution directly.Comparing with Maple numerical method,the hyperbolic perturbation method is more visual and simple in solving solitary wave problem.It can give an analytical expression of the solutions which other numerical method can’t do.We will discuss another case when R<< 1 in another paper.更多还原