柱坐标系下亥姆霍兹方程解的分类及典型应用  

Solving Helmholtz equations in cylindrical coordinate system and its typical application in electromagnetism

作　　者：马子寅 陈文琼 梅中磊[1] MA Zi-yin;CHEN Wen-qiong;MEI Zhong-lei(Institute of Optoelectronics and Electromagnetic Information,School of Information Science and Engineering,Lanzhou University,Lanzhou,Ganshu 730000,China)

机构地区：[1]兰州大学信息科学与工程学院光电子与电磁信息研究所,甘肃兰州730000

出　　处：《大学物理》2023年第10期15-19,37,共6页College Physics

基　　金：教育部产学合作协同育人项目(220901663145015;220906517154736)资助。

摘　　要：亥姆霍兹方程是一类用以表征电磁波传播规律的椭圆偏微分方程.对于圆柱坐标系下亥姆霍兹方程的求解问题,大部分教材都使用了分离变量的方法,但是多数内容并没有对分离变量结果做详细的分类和讨论,也没有结合其实际应用加以解释,使得初学者难以理解和掌握.本文对亥姆霍兹方程在柱坐标系下采用了分离变量法进行求解,系统讨论了其解的形式,并通过数值计算和电磁仿真分析了各类解的形式所对应的典型的电磁应用,对电磁场理论教学与研究具有一定的参考意义.Helmholtz equation is a kind of elliptic partial differential equation,which is used to characterize the propagation law of electromagnetic wave.When solving Helmholtz equation in cylindrical coordinate system,most textbooks use the method of separating variables to introduce,but most textbooks do not have a detailed classification and discussion of the results of separating variables,nor do they explain its practical application,which makes it difficult for beginners to understand and master physical meanings.In this paper,the Helmholtz equation is solved by the method of separating variables,the form of solution is discussed systematically,and the typical applications of various results are analyzed by means of electromagnetic simulation and numerical calculation.It has a certain reference significance for the teaching and research of electromagnetic field theory.

关 键 词：亥姆霍兹方程 柱坐标系 波导 谐振腔 

分 类 号：O451[理学—无线电物理]