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A technique for solving the wave equation and prospects for physical applications arising therefrom


Alexeev Nizhnii Novgorod State Technical University, Minina str. 24, Nizhnii Novgorod, 603600, Russian Federation

A new technique is proposed for solving the Helmholtz equation, based (in contrast to the classical method of separation of variables) on combining the variables. As a result, the wave equation (a partial differential equation) is converted into a single second-order ordinary differential equation. One solution of the latter is the classical exponential with imaginary argument (a simple self-similar solution). The second (the compound self-similar solution) consists of two factors: an exponential of imaginary argument (the first solution) and a tabulated special function. In the two-dimensional case this function is a complementary error function, or, in a special case, the Fresnel integral in complex form. In three dimensions the second factor is the exponential integral function. The physical part of the work is concerned with utilizing the compound self-similar solutions in physical applications involving external problems of electrodynamics and acoustics. These include (in the two-dimensional case) the theory of open waveguide structures.

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Fulltext is also available at DOI: 10.1070/PU1993v036n02ABEH002131
PACS: 02.30.Jr, 02.30.Gp, 41.20.Jb, 02.30.Rz (all)
DOI: 10.1070/PU1993v036n02ABEH002131
URL: https://ufn.ru/en/articles/1993/2/c/
Citation: Kukushkin A V "A technique for solving the wave equation and prospects for physical applications arising therefrom" Phys. Usp. 36 (2) 81–93 (1993)
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Оригинал: Кукушкин А В «Об одном способе решения волнового уравнения и возникающих при этом новых возможностях в некоторых физических приложениях» УФН 163 (2) 81–95 (1993); DOI: 10.3367/UFNr.0163.199302d.0081

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