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TitleResonance Scattering: Spectrum of Quasi-Eigen Oscillations of Two-Dimensional Arbitrary Open Cavities
PresenterElena Vinogradova (Macquarie University)
Author(s)Elena Vinogradova
SessionMathematical Physics and Industrial Mathematics
Time11:30:00 2017-09-25

A widely used formulation of electromagnetic scattering from open scatterers is
based on the electric field integral equation. This first kind equation has the
generic difficulty of ill-posedness associated with all first kind equations,
with the consequence that the solution to the discretised equation does not
converge to the true solution as the underlying mesh is refined.

The effective mathematical approach [1], free from this difficulty, is entirely
based on the rigorous Method of Regularization (MoR). By this technique the
problem of the wave diffraction by open cavities is transformed to solving the
infinite second kind system (systems) of linear
algebraic equations $\left( I+H\right) X=B$ for the set of Fourier's
coefficients $X=\left\{ x_{n}\right\} _{0}^{\infty }$ in expansion of the
two-sided surface current density. Here $I$ is identity operator, $H$ is
completely continuous operator in the functional space $l_{2}$, 
$B=\left\{b_{n}\right\} _{0}^{\infty }$ is known right-side of the equation,
to the same class $l_{2}$. The system is solved numerically by the truncation
method. The proven fast convergence rate of the truncated systems along with
possibility to provide any desired (predetermined) accuracy of calculations
makes this method the ideal instrument for calculation of the spectrum of
quasi-eigen values of quasi-eigen oscillations. The dispersion equation arises
as a result of setting $B\equiv 0$ to obtain the homogeneous equation. To find
the non-trivial solutions of the homogeneous equation it is necessary to require
$\det \left( A\right) =0$, where $A=\left\{
A_{nm}\right\} _{n,m=0}^{N}$ is truncated matrix bounded by truncation number

In this talk, the algorithm based on the rigorous MoR is used to examine the
spectrum of quasi-eigen values of quasi-eigen oscillations attributed to the 2D
open perfectly conducting cavities of arbitrary shape. The TM-polarized
quasi-eigen oscillations (modes) are studied. The rigorous analysis carried out
became possible only after developing the proper mathematical approach.
In fact, even for open cavities of the classical shape (circular, elliptic and
rectangular cylinder) the comprehensive studies on this issue are practically
absent. The partial study [1], related to circular slotted cylinder, does not
cover all aspects, arising under computation of the
complex quasi-eigen values. 

The talk is subdivided into two parts. First part is devoted to the computation
of the spectrum of the quasi-eigen values, which all are the complex-valued: the
imaginary part appears as a result of radiation losses through the slot. We
start from the slotted circular cylinder, then continue
study for elliptic and rectangular cylinders with a slot, which may vary in
position along the surface, including non-symmetrically located slots. The
location of the slot on the surface essentially impacts the spectrum of
quasi-eigen values, except for the circular cylinder. Then we analyse the
spectra of an open rectangular cavity with attached flanges, and the cavities,
which may model the inlet ducts: bent structures and few others. We can study
the sensitivity of the spectrum as a function of slot position or angle of
incidence or of other parameters of the problem.

In computing the spectrum of classically shaped open cavities derived from the
closed circular, elliptic and rectangular cylinders, initial approximations for
the real part of the eigen values of the structure come from the corresponding
closed structure. This approximation is not available
for the cavities of general form. To compute the spectrum of slotted cylinders
with cross sections of non-canonical shape we first examine the spectral
characteristics of the condition number $cond\left( A\right) $. This value is
very sensitive to the singular points, lying on the axis $%
k=2\pi /\lambda $, giving the initial approximate real values from which the
procedure of the finding the quasi-eigen values may be started. 

In the second part of the talk we use the data of resonance spectrum obtained
to study the resonance response of the cavities excited by the obliquely
incident E-polarized plane wave. The study includes the computation of the
surface current density, far-field and frequency
dependence of the radar cross-section. In practise the obtained results on
resonance backscattering may be used as additional source for recognition the
targets with negligibly small non-resonance backscattering. In particular, the
developed theory may serve as a robust mathematical
apparatus for accurate evaluation of wave scattering from the jet engine inlets
without limitations on their geometry.

[1] Vinogradova, Elena. 2017. Electromagnetic plane wave scattering by
arbitrary two-Dimensional cavities: Rigorous approach. Wave Motion, 70, 47-64.

[2] Vinogradov, S. S., P. D. Smith, \& E. D. Vinogradova. 2002. Canonical
problems in scattering and potential theory. Part 2: Acoustic and
Electromagnetic Diffraction by Canonical Structures, Boca Raton, FL/New
York/Washington, DC: Chapman \& Hall/CRC.

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