# A scattering from cylindrical objects 9 vector wave

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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 � ��...
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## 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.

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