有耗媒质中Maxwell方程组瞬态解的惟一性研究

On the Uniqueness of the Transient Solutions of Maxwell’s Equations in Lossy Medium

在相同的初边值条件下 ,首先给出了有耗媒质中Maxwell方程组的拉普拉斯变换法解和分离变量法解 .通过比较两种解析解的数值曲线得出 :两者的差异在于求解Harmuth解的数值曲线时 ,采取无穷积分的高频截断近似产生了吉布斯现象 ;在计算Harmuth解的数值曲线时不断增大积分上限 ,其数值曲线的振荡不断向不连续点处收缩 ,而且衰减加快 ,因此在求解Harmuth解的数值曲线时 ,若积分区间取为理想无限大 ,则可以预见相同计算精度下两种方法的电场解的数值曲线是一致的 .由此从数值计算的角度 ,验证了有耗媒质中波方程解的惟一性 .计算结果表明 ,在求解由分离变量法所得的无穷积分解析解的数值曲线时 ,吉布斯现象的存在具有普遍意义 .

Laplace transform and variable separation (the so called Harmuth’s technique) are two common methods in deriving the transient solutions of Maxwell’s equations in lossy medium. The computing plots of the solutions from the two methods were inconsistent under the specified boundary initial conditions, which stirred up the debate of the effectiveness of Maxwell’s equations in lossy medium. It is found that the differences between the two numerical curves can be viewed as the Gibbs phenomenon caused by cutting off the high frequency components in computing the infinite integral of the numerical curve of Harmuth’s solution. It can be perceived that the numerical curves of the two transient solutions of electric field strength are consistent with more high frequency components accounted in numerical computation of the infinite integral of Harmuth’s solution. The Gibbs phenomenon should be considered in interpreting the numerical curve of the transient solution with infinite integral derived by the method of separation of variables.