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Methodological derivation of the eikonal equation
- Authors: Fedorov A.V.1, Stepa C.A.1, Korolkova A.V.1, Gevorkyan M.N.1, Kulyabov D.S.1,2
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Affiliations:
- RUDN University
- Joint Institute for Nuclear Research
- Issue: Vol 31, No 4 (2023)
- Pages: 399-418
- Section: Articles
- URL: https://journals.rcsi.science/2658-4670/article/view/315338
- DOI: https://doi.org/10.22363/2658-4670-2023-31-4-399-418
- EDN: https://elibrary.ru/GCUXWK
- ID: 315338
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Abstract
Usually, when working with the eikonal equation, reference is made to its derivation in the monograph by Born and Wolf. The derivation of this equation was done rather carelessly. Understanding this derivation requires a certain number of implicit assumptions. For a better understanding of the eikonal approximation and for methodological purposes, the authors decided to repeat the derivation of the eikonal equation, explicating all possible assumptions. Methodically, the following algorithm for deriving the eikonal equation is proposed. The wave equation is derived from Maxwell’s equation. In this case, all conditions are explicitly introduced under which it is possible to do this. Further, from the wave equation, the transition to the Helmholtz equation is carried out. From the Helmholtz equation, with the application of certain assumptions, a transition is made to the eikonal equation. After analyzing all the assumptions and steps, the transition from the Maxwell’s equations to the eikonal equation is actually implemented. When deriving the eikonal equation, several formalisms are used. The standard formalism of vector analysis is used as the first formalism. Maxwell’s equations and the eikonal equation are written as three-dimensional vectors. After that, both the Maxwell’s equations and the eikonal equation use the covariant 4-dimensional formalism. The result of the work is a methodically consistent description of the eikonal equation.
About the authors
Arseny V. Fedorov
RUDN University
Author for correspondence.
Email: 1042210107@rudn.ru
ORCID iD: 0000-0002-3036-0117
PhD student of Probability Theory and Cyber Security
6 Miklukho-Maklaya St., Moscow, 117198, Russian FederationChristina A. Stepa
RUDN University
Email: 1042210111@pfur.ru
ORCID iD: 0000-0002-4092-4326
PhD student of Probability Theory and Cyber Security
6 Miklukho-Maklaya St., Moscow, 117198, Russian FederationAnna V. Korolkova
RUDN University
Email: korolkova_av@rudn.ru
ORCID iD: 0000-0001-7141-7610
Candidate of Sciences in Physics and Mathematics, Associate Professor of Department of Probability Theory and Cyber Security
6 Miklukho-Maklaya St., Moscow, 117198, Russian FederationMigran N. Gevorkyan
RUDN University
Email: gevorkyan_mn@rudn.ru
ORCID iD: 0000-0002-4834-4895
Candidate of Sciences in Physics and Mathematics, Associate Professor of Department of Probability Theory and Cyber Security
6 Miklukho-Maklaya St., Moscow, 117198, Russian FederationDmitry S. Kulyabov
RUDN University; Joint Institute for Nuclear Research
Email: kulyabov_ds@rudn.ru
ORCID iD: 0000-0002-0877-7063
Scopus Author ID: 35194130800
ResearcherId: I-3183-2013
Doctor of Sciences in Physics and Mathematics, Professor of Department of Probability Theory and Cyber Security of Peoples’ Friendship University of Russia named after Patrice Lumumba (RUDN University); Senior Researcher of Laboratory of Information Technologies, Joint Institute for Nuclear Research
6 Miklukho-Maklaya St., Moscow, 117198, Russian Federation; 6 Joliot-Curie St., Dubna, 141980, Russian FederationReferences
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