By Malcolm M. Bibby, Andrew F. Peterson
This lecture offers a contemporary procedure for the computation of Mathieu features. those capabilities locate software in boundary worth research corresponding to electromagnetic scattering from elliptic cylinders and flat strips, in addition to the analogous acoustic and optical difficulties, and plenty of different functions in technological know-how and engineering. The authors overview the normal procedure used for those capabilities, convey its barriers, and supply another "tuned" procedure allowing superior accuracy and convergence. The functionality of this technique is investigated for quite a lot of parameters and computer precision. Examples from electromagnetic scattering are supplied for representation and to teach the convergence of the common sequence that hire Mathieu capabilities for boundary price analysis.
Table of Contents: creation / Mathieu capabilities / saw Accuracy utilizing conventional and Tuned equipment / suggested set of rules for Mathieu functionality Computation / Electromagnetic Scattering from undertaking Elliptic Cylinders / Electromagnetic Scattering from an unlimited carrying out Strip / References / Authors' Biographies
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Additional resources for Accurate Computation of Mathieu Functions
2) where Cm and Sm are constants to be determined. 3) 44 5. ELECTROMAGNETIC SCATTERING FROM CONDUCTING ELLIPTIC CYLINDERS on the cylindrical surface at u0 . 7) e magnetic ﬁeld may be obtained as the curl of the electric ﬁeld. For a general vector EN the curl operation is deﬁned r EN D 1 2 cosh u Á cos2 v ˇ Á1=2 Á1=2 ˇ 2 2 ˇ uO cosh2 u cos2 v v O cosh u cos v zO =F ˇ ˇ @ @ @ ˇ ˇ ˇ @u @v @z ˇ Á1=2 Á1=2 ˇ 2 Ev cosh2 u cos2 v Ez =F ˇ Eu cosh u cos2 v For the TM polarization, this simpliﬁes to r EN D D 1 cosh2 u j!
To provide reliable data for comparison purposes, we recompute some of these cases in the following sections. Numerical values for current density and scattering cross section are presented in tabular format to facilitate comparisons with other computations. We explore both the TM z case and the TEz case. 1 THE TM z CASE e speciﬁc cylinder surface is deﬁned by the values of F (hence q) and the coordinate u = u0 . 2) where Cm and Sm are constants to be determined. 3) 44 5. ELECTROMAGNETIC SCATTERING FROM CONDUCTING ELLIPTIC CYLINDERS on the cylindrical surface at u0 .
However, knowledge of its existence means that special attention should be paid to the summations involved so that the consequences are minimized. If subtraction error is determined to be a problem, it can be overcome with the use of higher precision. 7) to estimate the error. v; q// C . 2kv/ . 2a) 22 3. v; q// C . 2k C 1/ v/ 2kC1 Error_ce2rC1 D kD0 h . v; q// C . 2k C 1/ v/ D B2kC1 Error_se2rC1 D kD0 h . v; q// C . 2k C 2/ v/ D B2kC2 Error_se2rC2 D kD0 h . 2k C 2/2 : When either term was less than a certain threshold, in their case 10 10 , Van den Berg and Van Schaik  terminated the summation for that speciﬁc eigenvector.