Marengo Edwin A, Devaney Anthony J
Department of Electrical and Computer Engineering, Northeastern University, Boston, Massachusetts 02115, USA.
Phys Rev E Stat Nonlin Soft Matter Phys. 2004 Sep;70(3 Pt 2):037601. doi: 10.1103/PhysRevE.70.037601. Epub 2004 Sep 13.
A general description of localized nonradiating (NR) sources whose generated fields are confined (nonzero only) within the source's support is developed that is applicable to any linear partial differential equation (PDE) including the usual PDEs of wave theory (e.g., the Helmholtz equation and the vector wave equation) as well as other PDEs arising in other disciplines. This description, which holds for both formally self-adjoint and non-self-adjoint linear partial differential operators (PDOs), is derived in the context of both the governing PDE and the corresponding adjoint PDE of the associated adjoint problem. It is shown that a necessary and sufficient condition for a source to be NR is that it obeys an orthogonality relation with respect to any solution in the source's support of the corresponding homogeneous adjoint PDE. For real linear PDOs, this description takes on a more relaxed form where, in addition to the previous necessary and sufficient condition, the obeying of a complementary orthogonality relation with respect to any solution in the source's support of the homogeneous form of the same governing PDE is also both necessary and sufficient for the source to be NR.
本文给出了局域非辐射（NR）源的一般描述，其产生的场被限制在（仅在）源的支撑区域内非零，该描述适用于任何线性偏微分方程（PDE），包括波动理论中的常见PDE（如亥姆霍兹方程和矢量波动方程）以及其他学科中出现的其他PDE。这种描述适用于形式上自伴和非自伴的线性偏微分算子（PDO），它是在控制PDE和相关伴随问题的相应伴随PDE的背景下推导出来的。结果表明，源为NR的充要条件是它相对于相应齐次伴随PDE在源支撑区域内的任何解都满足正交关系。对于实线性PDO，这种描述采用了更宽松的形式，除了上述充要条件外，源相对于同一控制PDE齐次形式在源支撑区域内的任何解满足互补正交关系也是源为NR的充要条件。
