A scattering from cylindrical objects 9 vector wave

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: e following general theorem � Assume that �(r) is a solution of the ‘scalar’ wave equation solved in any coordinate system � Also consider a known vector field a(r), such that �� a (r ) � 0 Scattering from cylindrical objects 8 Vector wave equation � Now, let us introduce the new vector fields 1 M (r ) � �� �� (r )a (r )� k � 1 N (r ) � �� M (r ) k It can then be shown that the 1st vector field is a solution of the vector wave equation if � � � �� � a � 2 � � � � � � a � � 0 � � � For the derivation use � � � � �� a � � � � � �� � a � � ��� � a � a� 2� � � a �� � �� � � �� �� � a � ��� � a � a� 2� � � � a ��� � � 2 � �� � � � a Scattering from cylindrical objects 9 Vector wave equation � The 2nd vector field is then automatically a solution as well � These two vector fields are linearly independent from each other � They both have zero divergence � Every solution of the vector wave equation in a homogeneous medium can be written as a combination of these vectors, generated by different solutions of the scalar wave equation � One cannot generate more functions by taking more curls. Note that 1 � � N ( r ) � � � � � M ( r ) � kM ( r ) k Scattering from cylindrical objects 10 Vector wave equation � Solution 1: 1 E (r ) � M ( r ) � � � �� ( r )a( r )� k 1 1 j H (r ) � � � � E (r ) � � � � M (r ) � N (r) j��0 j��0 � � Solution 2: 1 E (r ) � N (r ) � � � M (r ) k 1 j H (r ) � � � � N (r ) � M (r ) j��0 � Scattering from cylindrical objects 11 Vector wave equation � Example: choose the scalar function to be a solution of the wave equation (Helmholtz equation) in the Cartesian system � This would be a simple scalar plane wave � 2 � k 2 �� � 0 � � � exp � � jk � r � � � ˆ Take, as an example, a constant a(r), i.e., a (r ) � x ˆˆ M (r ) � � jk � x exp � � jk � r � � � ˆ ˆˆ N (r ) � �k � k � x exp � � jk � r � � M ˆ k N x These correspond to two independent polarizations Scattering from cylindrical objects 12 Vector waves in cylindrical coordinates � For cylindrical objects, a proper choice is to use the constant vector field ˆ a (r ) � z � This field satisfies all the requirements and results in 1 1 ˆ � � �� � z ˆ M � �� �� z k k 1 1 ˆ N � 2 �� � �� � z � � 2 k k 1 �2 k � �2 � ˆ � �� � z � �z �� � � � �2 ˆ � � k � z � �� � � �z � � Scattering from cylindrical objects 13 Vector waves in cylindrical coordinates � We next use cylindrical coordinates �� m,k z � ˆ �� m,kz �� �ˆ �� m,kz ��� ˆ �z �� m,kz �z 1 ˆ �� m, kz � ˆ �� m,kz ˆ M m,kz � ��...
View Full Document

This note was uploaded on 02/26/2013 for the course EE 25227 taught by Professor Akbabi during the Spring '13 term at Sharif University of Technology.

Ask a homework question - tutors are online