The Solution of the Wave Equation in the Class of Complex Numbers by Introducing Hyperbolic Coordinates

Khoroshavtsev Y. E.

doi:doi:10.11648/j.ijamtp.20180402.15

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The Solution of the Wave Equation in the Class of Complex Numbers by Introducing Hyperbolic Coordinates

Khoroshavtsev Y. E.

Published in International Journal of Applied Mathematics and Theoretical Physics (Volume 4, Issue 2)

Received: 13 May 2018 Accepted: 19 August 2018 Published: 17 September 2018

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Abstract

The finding of the solution of the wave equation, formulated as the Cauchy problem, does not exhaust all possibilities of the theory. The attempt to examine that one by admitting that the time is an imaginary value is made. So the new curvilinear coordinates, named hyperbolic, are introduced in consideration. They allow for hyperbolic equations to extend a field of searching of solutions to the complex plan and give the possibility to apply powerful Fourier’s method. Due to that, the wave equation takes a form of Laplace’s one in polar coordinates. However, the boundary condition differs from well known Dirichlet problem that in this case looses the sence. The new condition is admitted and it is physically formulated as the description of wave from various inertial systems of coordinates. So the result is obtaining proceeding either of the momentum picture of a wave, made from the moving system of coordinates, or on the oscillogram, developed in time The analytic solution that differs from Poisson integral is deduced and gives the formulas of relativistic addition of velocities for points of wave, observing from different inertial systems. That integral was also formally yielded by using the conform translation. Additionally, in the frequencies field those formulas describe the relativistic Doppler’s effect and the red shift in the wave spectrum. For oscillatory boundary condition the solution of the obtained integral gives a description of the shock waves. The fact, that some formulas of Relativity may be deduced by new way, gives the possibility to explain the relativistic theory proceeding from supposition of waving nature of quantum objects.

Published in	International Journal of Applied Mathematics and Theoretical Physics (Volume 4, Issue 2)
DOI	10.11648/j.ijamtp.20180402.15
Page(s)	61-66
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Wave, Equation, Fourier Transformation, Boundary Condition

References

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[2]	Richard P. Feynman, Robert B. Leighton, Matthew Sands. Feynman lectures on physics. Volume 1. London. 1963, p.196.
[3]	Morse P. M. Feshbach H. Methods of theoretical physics. Part 1. New York. McGraw-Hill, 1953.
[4]	Tikhonov A. N., Samarskii A. A. Equations of mathematical physics. Moscow, Pergamon, 1963. (Translated from Russian).
[5]	Sobolev S. L. Partial differential equations of mathematical physics. Pergamon. 1964. (Translated from Russian).
[6]	Yu. E. Khoroshavtsev. Hyperbolic equations in hyperbolic coordinates. Bulletin of Civil Engineers. ISSN 1999-5571. St. Petersburg. April, 2014/2 (43), pp. 152-157. (In Russian).
[7]	N. Gonzalez. An example of pure stability for the wave equation with moving boundary. Journal of mathematical analysis and applications. ISSN 0022-247x. vol. 228, Issue 1, December 1998, pp. 51-59.
[8]	N. Balazs. On solution of the wave equation with moving boundaries. Journal of mathematical analysis and applications. ISSN 0022-247x. Vol. 3, 1961, pp. 472-484.
[9]	Smirnoff V. I. Superior mathematics course. vol. 2. Moscow, Science, Publ., 1974, 656p. (In Russian).
[10]	Yu. E. Khoroshavtsev. The wave interpretation of equations of relativistic dynamics. University Press. St. Petersburg. Mathematical Modeling, Digital Methods and Complexes of Programs. Vol. 10, 2004, pp. 164 – 172. (In Russian).
[11]	B. Forneberg, G. B. Whitham. A numerical and theoretical study of certain nonlinear wave phenomena. Proc. R. Soc. Lond. 1978, pp 373-403.
[12]	Dongbing Zha. Some remarks on quasilinear wave equation in 3-D. Mathematical methods in the applied sciences. Vol. 39, Issue 15, 2016, pp. 4484-4495.
[13]	Ricardo Heras. Lorentz transformations and the wave equation. European journal of physics. Vol. 37, №2, 2016.
[14]	G. M. Muslu, H. Borluk. Numerical solution for a general class nonlocal nonlinear wave equations arising in elasticity. ZAMM. Journal of applied mathematics and mechanics. Vol. 97, Issue 12, 2017, pp. 1600-1610.
[15]	A. Y. Burtscher, R. Donninger. Hyperboloidal evolution and global dynamics for the focusing cubic wave equation. Communications in mathematical physics. Vol. 353, Issue 2, 2017, pp. 549-596.
[16]	W. A. Ahmed, Kh. Saleh. Invariant solutions for a class of perturbed nonlinear wave equations. MDPI, Mathematics. №5 (4), 59, 2017.

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Khoroshavtsev Y. E. (2018). The Solution of the Wave Equation in the Class of Complex Numbers by Introducing Hyperbolic Coordinates. International Journal of Applied Mathematics and Theoretical Physics, 4(2), 61-66. https://doi.org/10.11648/j.ijamtp.20180402.15

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ACS Style

Khoroshavtsev Y. E. The Solution of the Wave Equation in the Class of Complex Numbers by Introducing Hyperbolic Coordinates. Int. J. Appl. Math. Theor. Phys. 2018, 4(2), 61-66. doi: 10.11648/j.ijamtp.20180402.15

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AMA Style

Khoroshavtsev Y. E. The Solution of the Wave Equation in the Class of Complex Numbers by Introducing Hyperbolic Coordinates. Int J Appl Math Theor Phys. 2018;4(2):61-66. doi: 10.11648/j.ijamtp.20180402.15

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@article{10.11648/j.ijamtp.20180402.15,
  author = {Khoroshavtsev Y. E.},
  title = {The Solution of the Wave Equation in the Class of Complex Numbers by Introducing Hyperbolic Coordinates},
  journal = {International Journal of Applied Mathematics and Theoretical Physics},
  volume = {4},
  number = {2},
  pages = {61-66},
  doi = {10.11648/j.ijamtp.20180402.15},
  url = {https://doi.org/10.11648/j.ijamtp.20180402.15},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijamtp.20180402.15},
  abstract = {The finding of the solution of the wave equation, formulated as the Cauchy problem, does not exhaust all possibilities of the theory. The attempt to examine that one by admitting that the time is an imaginary value is made.    So the new curvilinear coordinates, named hyperbolic, are introduced in consideration. They allow for hyperbolic equations to extend a field of searching of solutions to the complex plan and give the possibility to apply powerful Fourier’s method. Due to that, the wave equation takes a form of Laplace’s one in polar coordinates. However, the boundary condition differs from well known Dirichlet problem that in this case looses the sence. The new condition is admitted and it is physically formulated as the description of wave from various inertial systems of coordinates. So the result is obtaining proceeding either of the momentum picture of a wave, made from the moving system of coordinates, or on the oscillogram, developed in time The analytic solution that differs from Poisson integral is deduced and gives the formulas of relativistic addition of velocities for points of wave, observing from different inertial systems. That integral was also formally yielded by using the conform translation. Additionally, in the frequencies field those formulas describe the relativistic Doppler’s effect and the red shift in the wave spectrum. For oscillatory boundary condition the solution of the obtained integral gives a description of the shock waves. The fact, that some formulas of Relativity may be deduced by new way, gives the possibility to explain the relativistic theory proceeding from supposition of waving nature of quantum objects.},
 year = {2018}
}

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TY - JOUR
T1 - The Solution of the Wave Equation in the Class of Complex Numbers by Introducing Hyperbolic Coordinates
AU - Khoroshavtsev Y. E.
Y1 - 2018/09/17
PY - 2018
N1 - https://doi.org/10.11648/j.ijamtp.20180402.15
DO - 10.11648/j.ijamtp.20180402.15
T2 - International Journal of Applied Mathematics and Theoretical Physics
JF - International Journal of Applied Mathematics and Theoretical Physics
JO - International Journal of Applied Mathematics and Theoretical Physics
SP - 61
EP - 66
PB - Science Publishing Group
SN - 2575-5927
UR - https://doi.org/10.11648/j.ijamtp.20180402.15
AB - The finding of the solution of the wave equation, formulated as the Cauchy problem, does not exhaust all possibilities of the theory. The attempt to examine that one by admitting that the time is an imaginary value is made. So the new curvilinear coordinates, named hyperbolic, are introduced in consideration. They allow for hyperbolic equations to extend a field of searching of solutions to the complex plan and give the possibility to apply powerful Fourier’s method. Due to that, the wave equation takes a form of Laplace’s one in polar coordinates. However, the boundary condition differs from well known Dirichlet problem that in this case looses the sence. The new condition is admitted and it is physically formulated as the description of wave from various inertial systems of coordinates. So the result is obtaining proceeding either of the momentum picture of a wave, made from the moving system of coordinates, or on the oscillogram, developed in time The analytic solution that differs from Poisson integral is deduced and gives the formulas of relativistic addition of velocities for points of wave, observing from different inertial systems. That integral was also formally yielded by using the conform translation. Additionally, in the frequencies field those formulas describe the relativistic Doppler’s effect and the red shift in the wave spectrum. For oscillatory boundary condition the solution of the obtained integral gives a description of the shock waves. The fact, that some formulas of Relativity may be deduced by new way, gives the possibility to explain the relativistic theory proceeding from supposition of waving nature of quantum objects.
VL - 4
IS - 2
ER -

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Khoroshavtsev Y. E.

Faculty of Automatic Control Systems, State University of Civil Aviation, St. Petersburg, Russia

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APA Style

Khoroshavtsev Y. E. (2018). The Solution of the Wave Equation in the Class of Complex Numbers by Introducing Hyperbolic Coordinates. International Journal of Applied Mathematics and Theoretical Physics, 4(2), 61-66. https://doi.org/10.11648/j.ijamtp.20180402.15

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ACS Style

Khoroshavtsev Y. E. The Solution of the Wave Equation in the Class of Complex Numbers by Introducing Hyperbolic Coordinates. Int. J. Appl. Math. Theor. Phys. 2018, 4(2), 61-66. doi: 10.11648/j.ijamtp.20180402.15

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AMA Style

Khoroshavtsev Y. E. The Solution of the Wave Equation in the Class of Complex Numbers by Introducing Hyperbolic Coordinates. Int J Appl Math Theor Phys. 2018;4(2):61-66. doi: 10.11648/j.ijamtp.20180402.15

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@article{10.11648/j.ijamtp.20180402.15,
  author = {Khoroshavtsev Y. E.},
  title = {The Solution of the Wave Equation in the Class of Complex Numbers by Introducing Hyperbolic Coordinates},
  journal = {International Journal of Applied Mathematics and Theoretical Physics},
  volume = {4},
  number = {2},
  pages = {61-66},
  doi = {10.11648/j.ijamtp.20180402.15},
  url = {https://doi.org/10.11648/j.ijamtp.20180402.15},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijamtp.20180402.15},
  abstract = {The finding of the solution of the wave equation, formulated as the Cauchy problem, does not exhaust all possibilities of the theory. The attempt to examine that one by admitting that the time is an imaginary value is made.    So the new curvilinear coordinates, named hyperbolic, are introduced in consideration. They allow for hyperbolic equations to extend a field of searching of solutions to the complex plan and give the possibility to apply powerful Fourier’s method. Due to that, the wave equation takes a form of Laplace’s one in polar coordinates. However, the boundary condition differs from well known Dirichlet problem that in this case looses the sence. The new condition is admitted and it is physically formulated as the description of wave from various inertial systems of coordinates. So the result is obtaining proceeding either of the momentum picture of a wave, made from the moving system of coordinates, or on the oscillogram, developed in time The analytic solution that differs from Poisson integral is deduced and gives the formulas of relativistic addition of velocities for points of wave, observing from different inertial systems. That integral was also formally yielded by using the conform translation. Additionally, in the frequencies field those formulas describe the relativistic Doppler’s effect and the red shift in the wave spectrum. For oscillatory boundary condition the solution of the obtained integral gives a description of the shock waves. The fact, that some formulas of Relativity may be deduced by new way, gives the possibility to explain the relativistic theory proceeding from supposition of waving nature of quantum objects.},
 year = {2018}
}

Copy | Download