Realation between a class of two-dimensional and three dimensional diffraction problems

By means of a certain transformation, a relationship is demonstrated between a class of two-dimensional and three-dimensional scalar or electromagnetic diffraction problems. The basic three-dimensional configuration consists of a perfectly reflecting half plane excited by a ring source centered about the edge and having a variation exp (
\pm i\phi/2
), where
\phi
, is the azimuthal variable; in addition, a perfectly reflecting rotationally, symmetric obstacle whose surface is defined by
f(\rho, z) = 0
(
\rho, z
are cylindrical coordinates) may be superposed about the edge (
z
axis). This problem is shown to be simply related to the two-dimensional problem for the line source excited configuration
f(y, z)= 0
, where
y
and
z
are Cartesian coordinates. Various special obstacle configurations are treated in detail. For the general case of arbitrary electromagnetic excitation, the above-mentioned transformation is used to construct the solution for the diffraction by a perfectly conducting half plane from the knowledge of appropriate scalar solutions, namely those which obey the same equations and boundary conditions, and have the same excitations, as the Cartesian components of the electromagnetic field.