電磁場的本徵函式
正文
在一定的邊界條件下,分布形式不因激勵方式而定的電磁場模式,是這種邊界條件下的本徵模式。本徵函式和本徵值是數學物理方法的基本概念之一,是表達本徵模式的數學工具,在電磁場模式分析中十分重要,而模式分析則是解給定源的場的有效方法之一。近代數學已把本徵函式和本徵值的研究推進到了新的深度和廣度。對於線性運算元 L,如果其定義域為某類函式(例如在邊界上為零或邊界上法嚮導數為零,而在場區域內二階導數連續且平方可積的函式),則此類中的函式u和常數λ,如能滿足方程Lu=λu,就分別稱為運算元 L在這函式類中的本徵函式和本徵值。在靜態場和簡諧場中常遇到的運算元是
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例如,在均勻填充介質的波導中,簡諧電磁場除可能有橫電磁(TEM)模式外,還有E模(縱向磁場為零)和H 模(縱向電場為零)兩類模式(見電磁波模式)。取管軸為z坐標軸,這兩類模式的一般表達式為
![電磁場的本徵函式](/img/9/22b/ml2ZuM3X5UDM1ETM2IzMxgDM5ETMwADMwADMwADMwADMxAzLzEzL5UzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
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在波導管內介質有縱向間斷面的情形中,除
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在已知的各種正交柱坐標系中,只在直線、圓柱、橢圓柱、拋物柱坐標系中,能用分離變數法求
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在場區橫向延至無限遠的情形(如介質柱內外或導體柱以外)中,除某些特殊情形,如介質柱有相當粗(或導體柱面上敷有相當厚的介質層),可能存在有限個本徵函式(表達表面波的橫向分布)外,一般不存在本徵函式。儘管在這些情況中可能找到既滿足方程
![電磁場的本徵函式](/img/5/15e/ml2ZuM3X0gTM1MTM2IzMxgDM5ETMwADMwADMwADMwADMxAzLzEzL0gzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
在均勻填充介質的柱形諧振腔內,場需同時滿足側壁和端面上的邊界條件。其中Z分量的邊界條件為(en指向側壁法向)
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在其他形式的導體腔內,需研究矢量方程
Hr=0(E模)和 Er=0(H模)
而分為兩種模式,分別用標量![電磁場的本徵函式](/img/d/9f6/ml2ZuM3X2YDO0QTM2IzMxgDM5ETMwADMwADMwADMwADMxAzLzEzL2YzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
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在球形導體表面,德拜勢應滿足邊界條件
![電磁場的本徵函式](/img/9/9b8/ml2ZuM3X1EDN1UTM2IzMxgDM5ETMwADMwADMwADMwADMxAzLzEzL1EzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
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參考書目
林為乾:《微波理論與技術》,科學出版社,北京,1979。
M.博恩和E.沃爾夫著,楊葭蓀等譯《光學原理》,科學出版社,北京,1978。(M.Born and E.Wolf,Principles of OPtics,Pergamon Press,Oxford,1975.)
D.S. Jones,Methods in Electromagnetic Wave Propagation,Clarendon Press,Oxford,1979.