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A solution of the Helmholtz equation for a circular loop

Keywords:

R.M. Gadelshin – Ph.D.(Eng.), Associate Professor, Kazan National Research Technical University named after A.N. Tupolev
E-mail: garadman@mail.ru


Many tasks of the electrodynamics need to solve the Helmholtz equation. The traditional solving of this equation for limited area con-siders their as more quotient in contrast with tasks for unlimited space. In this case a solution is selecting as the sought spatial distri-bution from ensemble of the possible solutions of strictly Helmholtz equation to satisfy given border conditions, and considered their as more quotient in contrast with solution for unlimited space.
A using a spectral form for the tasks in limited area allows to take into account the particularities of this area, which allow existence of the more full solutions of strictly Helmholtz equation. In task for vibrator, for instance, ensemble of the solutions of the border task for limited area turns out to be more common, than ensemble of the solutions, allowed strictly by Helmholtz equation for unlimited space. As a result more common solutions of this task can be obtained by change of sequence of the solving, i.e. by selection of the sought solution from ensemble of the possible solutions of the border task as satisfying Helmholtz equation.
In tasks for circular loop spectral presentation of solutions of the border tasks is allowed strictly in the form of the typesetting of δ functions only. Herewith it appears that a characteristic of these functions allow the possibility for more common presentation of own function of border task at solving strictly Helmholtz equation. So, for values of the spatial frequency, exceeding undermost «resonance» frequency of the observed area occurs, a «disguise» of separate spectral components when they are presented in Helmholtz equation. This may be explained as manifestation of certain «aperture» effect, appearing for greater importance of the spatial frequency and caused by ambiguity in determination of the phase center of own function of border task at solving strictly Helmholtz equation. «Aperture» effect brings about presence of separate own functions of border task at solving of the Helmholtz equation in the manner of trivial zero components.
The presented solving is available for free values of the spatial frequency of the source task. For illustration a obtained solution is used for description of the field of the inverse dissipation of the circular loop.

References:
  1. Gadel’shin R.M. Obobshhennoe reshenie odnomernogo uravneniya Gel’mgol’cza // E’lektromagnitny’e volny’ i e’lektronny’e sistemy’. 2017. № 2. S. 33−37.
  2. Gadel’shin R.M. Adaptirovanny’e sistemy’ sobstvenny’x funkczij dlya odnomerny’x uravnenij Gel’mgol’cza // E’lektromagnitny’e volny’ i e’lektronny’e sistemy’. 2013. № 12. S. 4−9.
  3. Spravochnik po radiolokaczii / Pod red. M. Skolnika. Per. s angl. V 4-x tomax. Pod obshhej red. K.N. Trofimova. T. 1. Osnovy’ radiolokaczii / Pod red. Ya.S. Iczxoki. M.: Sov. Radio. 1976.

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