One of the most common integral equation deficiencies is failure to have a unique solution at certain discrete frequencies. At these frequencies, there exist nontrivial solutions of the source-free (homogeneous) form of the integral equation. It has been shown theoretically that neither the H-field equation nor the E-field equation has unique solutions for the current on a conducting body at frequencies corresponding to resonant frequencies of the region enclosed by the conducting surface, but the combined-field equation does have a unique solution [53],[52].
In this section, the scatterer is an arbitrarily shaped perfectly conducting body with a
closed surface .
denotes the electric surface current
induced on
by the incident field (
).
This current satisfies (
) and (
).
The question is whether
(
)
alone is sufficient to determine
, whether (
) alone is
sufficient, or whether both are necessary. The answer
is given by Mautz [53]. In [53], Mautz has proven that the
solution
to (
) is not unique for values of k (wave number) at
which the equations
which are valid when there is no incident field, admit a nontrivial solution.
This solution is called a magnetic cavity mode. Applying the
duality theory, one concludes that the solution
to (
) is not unique
for the same value of k.
The combined field formulation, which is a linear combination of () and
(
), is given by
The solution of () is unique and satisfies both (
) and (
) whenever
is a
positive real number (see [53]).
Since () is the linear combination of (
) and
(
) with a relative weight
, the method of moments formulation obtained from (
) is the same
linear combination of (
) and (
). Hence,
where all matrices and column vectors have the same meaning as in Section
2.2 and 2.3, and is a constant. From experience,
should be between 0.2 and 1.0.