Helmholtz方程的指数函数法多波解  

Exp-function method for multiwave solutions of Helmholtz equation

作　　者：纪娟娟[1,2] 郭业才[3] 张兰芳[2] 

机构地区：[1]南京信息工程大学大气科学学院,南京210044 [2]安庆师范学院物理与电气工程学院,安庆246133 [3]南京信息工程大学电子与信息工程学院,南京210044

出　　处：《四川大学学报（自然科学版）》2015年第5期1069-1076,共8页Journal of Sichuan University(Natural Science Edition)

基　　金：江苏省高校自然科学研究重大项目(13KJA510001);高校科研成果产业化推进工程项目(JHB2012-9)

摘　　要：在线性近似下,Helmholtz方程能够描述小尺度大气声波波动.因此分析小规模大气声波波动特性,必须对Helmholtz方程进行求解.大多数求解Helmholtz方程的方法是在某些边界条件和初始条件下进行的,而边界条件和初始条件本身就是在假设、近似的基础上得到的,且会使得计算过程复杂和计算量变大.指数函数法具有多个自由参数是求解非线性问题精确解的一种十分有效、直接的工具,且在求解时不要考虑边界条件和初始条件,对函数进行指数和双曲正切变换的共同作用可近似为线性变换.基于上述思想,本文首先对Helmholtz方程进行反正切变换,将其转化为非线性方程,其次进行单波解、双波解和三波解的假设并对其进行了求解,最后,利用次声波和可听声波对单波,双波和三波解进行了3维空间和2维平面的数值模拟.结果表明,求解方法简单、直观,且在线性近似下,不同频率的声波具有不同的传播特性,双波、三波沿着各自的方向传播,互不干扰.Small-scale atmospheric acoustic wave fluctuations of linear approximations can be described with Helmholtz equation. The Helmholtz equation must be solved in order to analyze the characterization of atmospheric acoustic wave fluctuations. Most of the methods to solve the Helmholtz equation are realized under certain boundary conditions and initial conditions, while boundary conditions and initial conditions are obtained on the basis of assumptions, approximation, and make calculation process complicated and computation amount bigger. Exp-function method with many free parameters is a very effective, direct tool to solve precise solutions of the nonlinear problem and the boundary and initial conditions can be neglected, the interaction of exp-function and hyperbolic tangent transformation can approximate to linear transformation. In this paper, firstly the arctangent transformation is taken toHelmholtz equation and the equation is transformed into nonlinear equation. Then the single-wave so- lution, double-wave solution and three-wave solution are assumed and solved. Finally, the Single- wave solution, double-wave solution and three-wave solution are simulated in 3-dimensional spatial and 2-dimensional planar with infrasound and audible acoustic waves. The results show that different frequency acoustic waves have different propagation characteristics; the double-wave solution and three-wave propagate along the respective direction with non-interference each other under linear ap- proximations.

关 键 词：HELMHOLTZ方程 指数函数法 多波解 数值模拟 大气声学 

分 类 号：O241.82[理学—计算数学]