# Details of talk

Title | Resonance Scattering: Spectrum of Quasi-Eigen Oscillations of Two-Dimensional Arbitrary Open Cavities |
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Presenter | Elena Vinogradova (Macquarie University) |

Author(s) | Elena Vinogradova |

Session | Mathematical Physics and Industrial Mathematics |

Time | 11:30:00 2017-09-25 |

Abstract | 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, belonging 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 $N$. 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. \newline [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. |