A short way of solving advanced problems in electromagnetic fields and other linear systems

A linear system is characterized in the abstract by a source vector
a
, a response vector
\alpha
and a system matrix
R
, connected by
\alpha = Ra
. Specific interpretations for
a , \alpha
and
R
are given for electromagnetic fields, the scalar product
a \dot Rb
being the reaction. The advantages of this method are illustrated by application to various problems. Mode expansions are defined by the eigenvector equation
Rm = C_{m}m
,
C_{m}
being the eigenvalue and
m
the mode source. This means the mode is generated by a combination of electric and magnetic sources which, apart from the constant multiplier
C_{m}
, are everywhere equal to the electric and magnetic fields, respectively. Such mode expansions are applied to typical waveguide problems. Waveguide theory is set up in terms of unit voltage and unit current mode sources,
v
and
i
, where
v \dot v = 1 = i \dot i
. Then the definitions of waveguide current and voltage coincide with those for circuit theory, e.g., the mode current at cross section
P
is the reaction on a unit voltage source placed at
P
. The method also greatly simplifies scattering and antenna problems, such as the pattern of a monopole antenna which is immersed in a layer of gyrotropic plasma.